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H-vtf[e»'H-v}(l-ecosu)
=
S
cos
T — cos
fVr^/J6»'}cosr / J } X {6»'M)}rf M Jo 1-ecosw
J
and if the first side of the equation be generally expansible in a series of multiple cosines of m, instead of being so in particular cases only, its expanded value will always be the one given by the second side of the preceding equation. 3—2
20
ON CERTAIN EXPANSIONS,
[4
Now, between the limits 0 and tr, the function f{eu^~v}cosrx\eul'{-1)} will always be expansible in a series of multiple cosines of u; and if by any algebraical process the function fp cos ryjp can be expanded in the form fp cos rXp= 2-Z«gp8,
(ar« = O
(5);
we have, in a convergent series, y[6«V(-i)jCOsr%{ew'1"-1i} = a0H-2S"ascossM
(6).
i
Again, putting we have
- {1 — Jl - e2} = X
(7),
66
, X ~ e" - = 1 + 2S°° V cos mi 1 r 1 — e cos u
(8).
Multiplying these two series, and effecting the integration, we obtain {^^}du= l
-
i
v. *
yj
\
/
and the second side of this equation being obviously derived from the expansion of fX cos r^X by rejecting negative powers of X and dividing by 2, the term independent of X may conveniently be represented by the notation 2/X cos r^X
(10);
where in general, if FX can be expanded in the form T\ = %_Z(A8\*), we have
[A_ = A,]
f\ = £ 4 0 + 2 " 4,V
(11), (12).
(By what has preceded, the expansion of FX in the above form is always possible in a certain sense; however, in the remainder of the present papier, FX will always be of a form to satisfy the equation F (- ) = FX, except in cases which will afterwards \xy be considered, where the condition A_s = As is unnecessary.) Hence, observing the equations (4), (9), (10), J{ • / '=t I(
°° cos rm 2 cos rvX/X
(13);
from which, assuming a system of equations analogous to (1), and representing by II the product O ^ . . . , it is easy to deduce n I \J-1
s/l-e? 1/•f e « l V(-D e«2V(-D \ e^'"1' X' {eu>li-1}} (1 - e cos «)j J l ' ''-) = 2 _ £ S _ " ...ncosrmf[(2cosr x X)/(X 1 ( X,...)
(14),
4]
IN SERIES OF MULTIPLE SINES AND COSINES.
21
where F (Xu \ j . . . ) being expansible in the form
;
^
o
,
(16),
N being the number of exponents which vanish. The equations (13) and (14) may also be written in the forms
'} = 2 . : cosrm2 c o s r x X ^ * X ^~}^X J
= %_Z2_Z...n(coSrm)ILl2 cosrx^—
*
X
.~^K
+X
^/x
(17),
6
^
^[fCK, X 2 ...)... (18).
y
As examples of these formulas, we may assume (19). Hence, putting xV^-^
+ X-e^-^A,
(20),
and observing the equation > /3i € «V(-i) x '{e«^-i)}
= i_ecoSM
(21),
the equation (17) becomes
e""-1'} = Z-Z cosrmAr{1 'j^Jf^A Thus, if
= cos-\C0SU~e 1 — e cos u
6-v
(22). (23), (24)
'
cos (0 - W) = 2_ £ -y-^— cos rm'{l - | e (X + X"1)} (i (X + X"1) - e} A, ...(25), the term corresponding to r = 0 being ^ { 2 X - 2e - e (Xa +1) + 2e!!X}, — e Again, assuming f f6«V(-i>} - — =
7 1
j
~dm
-J1-^
(l-ecos«)s
(26).
22
ON CERTAIN EXPANSIONS,
[4
and integrating the resulting equation with respect to m,
_ „, sin rm^ ^ sin rm *~~~a 0-2 - X2 - /x2;
which are the formulae required, differing only from those in Liouville, by having X, /A, v, instead of \m, \n, \p; and a, a', a"; /S, /3', / 3 " ; y, 7', y", instead of a, b, c ; a', b', c'; a", b", c". I t is to be remarked, that /3', /3", / 3 ; Y"> 7» y'> a r e deduced from a, a', a", by writing fi, v, \; v, X, /x, for X, /u., 'v. Let 1 + a + /3' + 7" = v; then «u = 4, and we have Xu = /3" - 7', X2u = l + a - / 3 ' - 7 " ,
fj,v = y- a", /i2y = l - a + / 3 ' - 7 " ,
PV = a' - y 8 , J/2U = 1 - a - / 3 ' - 7 " ,
30
ON THE MOTION OF ROTATION OF A SOLID BODY.
[6
Suppose that Aw, Ay, Az, are referred to axes Ax, AY, AT,, by the quantities I, m, n, k, analogous to X, fi, v, K, these latter axes being referred to Awt, Ayt, Azlt by the quantities ln m/t nn kr Let a, b, c; a', b', c'; a", b", c"; a,, bn c,; af, bf, cf; af, bf, cf, denote the quantities analogous to a, ft, 7 ; a.', ft', 7'; a.", ft", 7". Then we have, by spherical trigonometry, the formulae a. = a a, + b af + c a", a' = a' a, + V a,' + c' af, a" = a"a, + b"a't + c"af,
ft ft' ft"
= a b, -\-b bf + c b", =a'b, + b' bf + c'bf, = a'% + b"bf + c"bf,
7 = a ct+b c/ + c c," ; 7' = a' c, + V of + c'c," ; 7" = a"c, + b"cf + c"cf.
Then expressing a, b, c; a', V, c'; a", b", c"; a/t bn c,; af, bf, cf; terms of I, m, n; I,, mt, nn after some reductions we arrive at khtv kk, {ft" - 7 ) kkt (7 — a ' ) kkt (a' - ft")
af,
bf,
cf,
in
= 4 (1 — Ul — niml — nnff, = 4II 2 suppose, = 4 (I +1, + ^m - nmf) IT, = 4 (m+ ml + Ipn — lin,) II, = 4 (n + n, + ynp. - mnf) II;
and hence II =l — ll/ — mmt — nn/, H/j, = in + ml + ltm — lmn
UX=l+l/ + n/m— nmt, Tlv = n +n, + m,n — tnnl,
which are formulas of considerable elegance for exhibiting the combined effect of successive displacements of the axes. The following analogous ones are readily obtained : P =1 +\l + fim + vn, Pml = fi — m — \n + vl,
Plt = A, — I — vm + fin, Pnt = v — n— fd + \m :
and again, P,m =yu — ml-\- \nt — vl,,
P(n = v-n/
+ fd, — \mr
These formulae will be found useful in the integration of the equations of rotation of a solid body. Next it is required to express the quantities p, q, r, in terms of X, fi, v, where as usual
dft
, dft'
„ dft"
Differentiating the values of ft>c, ft'/c, ft"ic, multiplying by 7, 7', 7", and adding, up = 2X' (7/* - 7'X + 7") + 2 / (7X - y'fi + y"v) + 2v'(-'Y~y'v
+7'»,
6]
ON THE MOTION OF ROTATION OP A SOLID BODY.
31
where A.', //, v', denote - j - , -£, -4-. Reducing, we have dt at ctt icp=2 (X' + vp' - v'n) : from which it is easy to derive the system icp = 2 ( X'+vp- v'fjb), icq = 2 ( - vX' + fi + v'X), KV = 2( fiX'-Xfi+v' ); or, determining X', fx, v', from these equations, the equivalent system 2X,' = (1 + X2) p + (\{JL- v ) q + (v\+ fi) r,
r.
The following equation also is immediately obtained, K = K (\p +fJ-q + vr). The subsequent part of the problem requires the knowledge of the differential coefficients of p, q, r, with respect to X, fi, v; X', //, v. It will be sufficient to write down the six
>w
^^j
n- - ^ - ~r fj/\
—
2f
,
from which the others are immediately obtained. Suppose now a solid body acted on by any forces, and revolving round a fixed point. The equations of motion are dt dX'~dX = ~dX' d_ dT _dT^dV dt dfi dfi dfi' dt dv' where
dv
dv'
T = \ {Atf- + Bq* + Cr*); V = 2 [](Xdx + Ydy + Zdz)] dm;
Xdx + Ydy or if Xdx Ydy ++1Zdz is not an exact differential, -^-', -=—, -=-, are independent
symbols standing for
d*
dg
dz
dX
dX
dX
32
ON THE MOTION OP ROTATION OF A SOLID BODY.
[6
see Mecanique Analytique, Avertissement, t. I. p. V. [Ed. 3, p. VII.] : only in this latter case V stands for the disturbing function, the principal forces vanishing. Now, considering the first of the above equations dT
. dp
~AP-3k , ... whence, writing
A1S
,
dq
n
dr
*-fiL + OriL
9 =
K
2
—
, , . , , dp p, q, r, K, for ff,
J jrp
Jt ^'
D
+B
dq _ ,
dr die _ , ^ , o,.'
9 9 (AP
' "
VB
= pf 2t =
A
T)
-— = qi*
ri
A
T>
ra+ri5
5— 4>,
^— <j),
$
C-B ,\( . A-O~,\r.
B-A
and considering p, q, r as functions of 0, given by these equations, the three latter ones take the form K _ dX dfi dv 4qr d(ji d(j> d' K
_
d\ d4+
d/j, . dv dj>+ dj>'
_
dX
dfi
dv
=
^dtf)"
dij>+
d$]
4^ K
v
of which, as is well known, the equations following, equivalent to two independent equations, are integrals, Kg = Apil+M-ff-
fi)
v2) + 2Bq (X/J,-v)
+2Bq(/j,v+\)
+ 2 O (v\ + /*),
+ Cr
where g, g', g", are arbitrary constants satisfying
To obtain another integral, it is apparently necessary, as in the ordinary theory, to revert to the consideration of the invariable plane. Suppose g' = 0, g" = 0, g" = 0\0 - /*„) k, K0Bq = 2 (fioVf, + Xo) k,
naGr =
(1 + v? - V - Mo2) &;
whence, and from K0 — 1 -f X02 + yu/ + i/oa, «0Cr = (2 + 2J;,,2 — *0) k, we obtain (2 + 2v')k
(1 + ty8) ilj?
(l+
Hence, writing h = Apl* + Bq^ + C?\*, the equation
, ., i , reduces itselfC to
or, integrating,
4 dv0 h + kr -= -TT = ; 1 + v02 d(f> (k + Cr) pqr' 4 t a n - „„ = f ^
^
•
The integral takes rather a simpler form if p, q, be considered functions and becomes
of r,
2tan-^ = f A ± ^ C
/
wdv — vdw
vdu — udv
the integral of which may be written in the form J
^
L +
WV-y — VWX
e-a^
+J^L
UWy — WUX
VUX — VXU
where, on account of (17),
and also in the form
l
(24);
fu + gv + hw = 0
(25),
where / , g, h are connected by
ft^ f
c^a^ 9
9
this last equation is satisfied identically by
f
7,2
«2
°
c
n2
n2
(97)
°?L
Restoring x, y, z, xlt yx, zx for u, v, w, ux, vlt wlt the equations to a line of curvature passing through a given point xlt yx, zx, on the ellipsoid, are the equation (14) and (6s_c2)
a2 {yiz2 - yW)
( c s_ a s)
(ffl._6.)
b2 (zyW - zW)
_
c2 ( ^ y - x2y,2)
^v" ''
or again, under a known form, they are the equation (14) and
(& 2 -c 2 )^
tf-a?
f
tf-K
*_
B * C * a* + C * A * ¥ + A * B ? d>~
[
>'
7]
LINES OP CURVATURE OP AN ELLIPSOID.
39
From the equations (14), (29) it is easy to prove the well-known form
in fact, multiplying (29) by m, and adding to (14), we have the equation (30), if the equations 1
62-c2
1.
c2 - a2 1 _
1
I
a2-62 1
1
—h in
1
^- — =
1
are satisfied. But on reduction, these take the form (B2 - (?) 6 + (b2 - c2) m6 + ma? (62 - c2) = 0, 2
1
2
2
2
(32),
2
(C - A ) 9 + (c - a ) md + mb* (c - a ) = 0, (^42 -B*)e + (a2 - 62) md + m& (a2 - .&2) = 0, and since, by adding, an identical equation is obtained, m and 6 may be determined to satisfy these equations. The values of 6, m are
^kr» m
_ bV (B* - O2) + cV {(? - A2) + a262 (A* - B2) ~ (a 2 - 62) (6 2 - c2) (c2 - *•)
( 3 3 )(
}-
40
[8
8. ON LAGRANGE'S THEOREM. [From the Cambridge Mathematical Journal, vol. in. (1843), pp. 283—286.] THE value given by Lagrange's theorem for the expansion of any function of the quantity x, determined by the equation x = u + hfx
(1),
admits of being expressed in rather a remarkable symbolical form. The a priori deduction of this, independently of any expansion, presents some difficulties; I shall therefore content myself with showing that the form in question satisfies the equations
^.JF'xfadx^.JF'xdx Fx = Fu for h = 0
(2), (3),
deduced from the equation (1), and which are sufficient to determine the expansion of Fx, considered as a function of u and h in powers of h. Consider generally the symbolical expression
Jh) m v °l v i n g m general symbols of operation relative to any of the other variables dh entering into ah. Then, if B,h be expansible in the form
Bh = t(Ahm)
(5),
it is obvious that (^
(6).
8]
ON LAGRANGE'S
THEOREM.
41
For instance, u representing a variable contained in the function
Sh, and taking
a particular form of (f> (h -~\,
From this it is easy to demonstrate d du
where H'A denotes -jy 'Sh, as usual. ah
Hence also
of which a particular case is
^{(^Y^F'ufue^A^iKJ-Y^F'ue^l J du (\duj
Also,
J
dh {\duj
(11). j
(§uf'm~1 (F'u^fu) = Fu for h = 0
^ '
( 12 )-
Hence the form in question for Fx is
Fx=(±y^-iWue>/u) from which, differentiating with respect to u, and writing F instead of F',
i-kTx'Kdu)
^""' '
(14)
-
a well-known form of Lagrange's theorem, almost equally important with the more usual one. I t is easy to deduce (13) from (14). To do this, we have only to form the equation \-hf'x~'~"\du)
^"J " " ' '
(15)>
deduced from (14) by writing Fxf'x for fx, and adding this to (14), Fx = (-j-\
M
(FueVu) -h(j-)^h
(Fuf'u e^™)
du[ eVu) c.
(16). 6
42
ON LAGRANGE'S THEOREM.
[8
In the case of several variables, if x = u + hf(x,
#1...),
x1 = u1+h1f1{x,
xl...),
&c
(17),
writing for shortness F, / , / . . .
for F(u,
«,...),
/ ( u , u,...),
/ i ( « , «,-..). •••
t h e n the formula is
{where / ' ( * ) is written to denote -j- f(x, or the coefficient of AnV>
in the expansion of F(x,
i8
=
T
xx ...), &c.)
Xt
)
(19)
^ V ^
1.2 ... w. 1.2 ... From the formula (18), a formula may be deduced for the expansion of F(x, xx ), in the same way as (13) was deduced from (14), but the result is not expressible in a simple form by this method. An apparently simple form has indeed been given for this expansion by Laplace, Mecanique Celeste, [Ed. 1, 1798] torn. I. p. 176; but the expression there given for the general term, requires first that certain differentiations should be performed, and then that certain changes should be made in the result, quantities z, z , are to be changed into zn, z^ ; in other words, the general term is not really expressed by known symbols of operation only. The formula (18) is probably known, but I have not met with it anywhere.
43
9. DEMONSTRATION OF PASCAL'S THEOREM. [From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 18—20.] LEMMA 1. Let U = Ax + By + Cz = Q be the equation to a plane passing through a given point taken for the origin, and consider the planes TJX = 0,
U2 = 0,
Us = 0,
Ut = 0,
Us = 0,
U6 = 0 ;
the condition which expresses that the intersections of the planes (1) and (2), (3) and (4), (5) and (6) lie in the same plane, may be written down under the form A2,
As,
At
B1,
2? 2 ,
B3,
Bt
Clt
O2,
Cs,
Ct
As,
Ait
As,
Ae
B3,
B A,
Bs,
B6
C3,
Ct,
( 7 6 , Ce
. LEMMA
= 0.
Ax,
.
2. Representing the determinants &c.
by the abbreviated notation 123, &c; the following equation is identically true: 345.126 - 346.125 + 356.124 - 456.123 = 0. 6—2
44
DEMONSTRATION
OP PASCAL'S
[9
THEOREM.
This is an immediate consequence of the equations
Xx,
x3,
xit
xs,
xe
x3,
2/s,
2/4,
2/5,
2/6
2/3, 2/4,
3s,
£4,
Z»,
Z6
*^3,
*^4,
^5,
^6
2/3,
2/4,
2/5,
2/6
2/l,
2/2,
•^I,
^2,
= 0.
xt,
Consider now the points 1, 2, 3, 4, 5, 6, the coordinates of these being respectively x1, y l t z± xe, y6, ze. I represent, for shortness, the equation to the plane passing through the origin and the points 1, 2, which may be called the plane 12, in the form
consequently the symbols 12s, 12,,, 12, denote respectively y^ — y^,
zxx2 — z2xlt
xxy2—x2yx,
and similarly for the planes 13, &c. If now the intersections of 12 and 45, 23 and 56, 34 and 61 lie in the same plane, we must have, by Lemma (1), the equation 12., 45s, 23*, 56,,,
.
12,, 45,, 23,, 56,,
.
12,, 45,, 23,, 56,,
.
.
= 0.
23*, 56^, 34*, 61* 23j, 56Z, 34,, 61, Multiplying the two sides of this equation by the two sides respectively of the equation = 612.345, 2/2,
•
x 4, 2/3,
2/4,
zt, and observing the equations = 612,
112 = 0, &c.
45
DEMONSTBATION OF ]PASCALS TB[EOB
9] this becomes
612
.
= 0,
•
645, 145, 245, 623, 123,
423, 523
.
156, 256, 356, 456, .
. 534
361, 461, 561 reducible to 612 534
= 0;
145, 245, 123, 156,
423
256,
•
356, 456 361, 461
or, omitting the factor 612 . 534 and expanding, 145 . 256 . 423.361 + 245.123. 456.361 - 245.123.356. 461 - 245.156 . 423.361 = 0. Considering for instance xe, ys, ze as variable, this equation expresses evidently that the point 6 lies in a cone of the second order having the origin for its vertex, and the equation is evidently satisfied by writing x6, y6, ze = x1} ylt zlt or x3, y3, z3, or x4, yit e4, or xs,ys, zs, and thus the cone passes through the points 1, 3, 4, 5. For xe,y6, zB=a;2, y2, za, the equation becomes, reducing and dividing by 245.123, 452". 321 - 352. 421 + 152 . 423 = 0, which is deducible from Lemma (2), by writing xe, y6, ze = #2, y2, z2, and is therefore identically true. Hence the cone passes through the point (2), and therefore the points 1, 2, 3, 4, 5, 6 lie in the same cone of the second order, which is Pascal's Theorem. I have demonstrated it in the cone, for the sake of symmetry; but by writing throughout unity instead of z, the above applies directly to the case of the theorem in the plane. The demonstration of Chasles' form of Pascal's Theorem (viz. that the anharmonic relation of the planes 61, 62, 63, 64 is the same with that of 51, 52, 53, 54), is very much simpler; but as it would require some preparatory information with reference to the analytical definition of the similarity of anharmonic relation, I must defer it to another opportunity.
46
[10
10. ON THE THEORY OF ALGEBRAIC CURVES. [From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 102—112.] SUPPOSE a curve defined by the equation U = 0, U being a rational and integral function of the mth order of the coordinates x, y. It may always be assumed, without Ios3 of generality, that the terms involving xm, ym, both of them appear in U; and also that the coefficient of ym is equal to unity: for in any particular curve where this was not the case, by transforming the axes, and dividing the new equation by the coefficient of ym, the conditions in question would become satisfied. Let Hm denote the terms of U, which are of the order m, and let y — ax, y — fix ...y — \x be the factors of Hm. If the quantities a, fi ...X are all of them different, the curve is said to have a number of asymptotic directions equal to the degree of its equation. Such curves only will be considered in the present paper, the consideration of the far more complicated theory of those curves, the number of whose asymptotic directions is less than the degree of their equation, being entirely rejected. Assuming, then, that the factors of Hm are all of them different, we may deduce from the equation C = 0 , by known methods, the series
y = ax + a.'+ — +
y =Xx + X -{
,
(1).
1-
and these being obtained, we have, identically, U = (y - ax -a' - ...) (y - fix- fi' - . . . ) . . . (y -Xx
-X'
- ...)
(2),
10]
ON THE THEORY OP ALGEBRAIC CURVES.
47
the negative powers of x on the second side, in point of fact, destroying each other. Supposing in general that fx containing positive and negative powers of x, Efx denotes the function which is obtained by the rejection of the negative powers, we may write (8),
«x...
the symbol E being necessary in the present case, because, when the series are continued only to the power x~m+1, the negative powers no longer destroy each other. We may henceforward consider U as originally given by the equation (3), the m ( m + l ) quantities a, a'...a<m>, /3, fi'... /3<m), ...X, X'...X(m> satisfying the equations obtained from the supposition that it is possible to determine the following terms a(m+i)) ygc»+i) . . . \ C H - D , ... so that the terms containing negative powers of x, on the second side of equation (2), vanish. I t is easily seen that a, /3 ... X, a', / 3 ' . . . X' are entirely arbitrary, a", fi"...X" satisfy a single equation involving only the preceding quantities, a.'", j3'"... X'" two equations involving the quantities which precede them, and so on, until a(m), yS(m) ... X(m), which satisfy (m—1) relations involving the preceding quantities. Thus the m ( m + l ) quantities in question satisfy \m(m — 1) equations, or they may be considered as functions of m (m + 1) — \m (m — 1) = \m (m + 3) arbitrary constants. Hence the value of U, given by the equation (3), is the most general expression for a function of the m th order. I t is to be remarked also that the quantities a(m+1), /S(m+1) ... X(m+1), are all of them completely determinable as functions of a, /3 ... X,... a<m>, /3<m> ... X « . The advantage of the above mode of expressing the function U, is the facility obtained by means of it for the elimination of the variable y from the equation U—0, and any other analogous one V = 0. In fact, suppose V expressed in the same manner as U, or by the equation -Bx...- — ) ...(y-Kx...-^-)
(4),
n being the degree of the function V. I t is almost unnecessary to remark, that A, B ...K,...A are to be considered as functions of £n(re + 3) arbitrary constants, and that the subsequent Ain+1), B(n+1> ...Kin+1) ... can be completely determined as functions of these. Determining the values of y from the equation (3), viz. the values given by the equations (2); substituting these successively in the equation (5), analogous to (2), and taking the product of the quantities so obtained, also observing that this product must be independent of negative powers of x, the result of the elimination may be written down under the form E
f | ( Z ) +
... (6),
48
ON THE THEORY OF ALGEBRAIC CURVES.
[10
the series in { } being continued only to x~mn+1, because the terms after this point produce in the whole product only terms involving negative powers of x. It is for the same reason that the series in ( ), in the equations (3) and (4), are only continued to the terms involving x~~m+1, #~"+1 respectively. The first side of the equation (6) is of the order mn, in x, as it ought to be. But it is easy to see, from the form of the expression, in what case the order of the first side reduces itself to a number less than mn. Thus, if n be not greater than m, and the following equations be satisfied, A=a,
A®=a® ...A«-v = **-»,
r$>n
(7),
1
B = ft B® =/3< » . K=K,
R® = «w ... KP-v = KC-1*,
V > u,
the degree of the equation (6) is evidently mn- r — s... - v, or the curves U= 0, F = 0 intersect in this number only of points. If mn — r — s... — v = 0, the curves JJ=O and V=0 do not intersect at all, and if mn — r—s — v be negative, = —w suppose, the equation (6) is satisfied identically; or the functions U, V have a common factor, the number a> expressing the degree of this factor in x, y. Supposing the function V given arbitrarily, it may be required to determine U, so that the curves U=0, V= 0 intersect in a number mn—k points. This may in general be done, and done in a variety of ways, for any value of k from unity to \m (m + 3). I shall not discuss the question generally at present, nor examine into the meaning of the quantity mn — | m ( m + 3) {— \m (2m — m — 3)} becoming negative, but confine myself to the simple case of U and V, both of them functions of the second order. It is required, then, to find the equations of all those curves of the second order which intersect a given curve of the second order in a number of points less than four. Assume in general
then A", B" satisfy A" + B" = 0, and putting B" = .KD, and therefore A" = - - 7 ^ - 5 , A —H -A — li and reducing, we obtain
V=(y - Ax - A')(y -BxSimilarly assume
U= E (y — ax — a'
B') + K. ) (y — fix — /3' — — ) ,
then a", /3" satisfy a" + /3" = 0, and putting fi" = ~~3, reducing, we obtain
V = (y - ax - a') (ff-fix- ff) + k.
and therefore a" = - ——=, and
10]
ON THE THEORY OF ALGEBEAIC CURVES.
49
Suppose (1) U=0, V= 0 intersect in three points, we must have a = A, or the curve U=0 must have one of its asymptotes parallel to one of the asymptotes of V= 0. (2) The curves intersect in two points. We must have a = A, a' = A', or else a = A, fi = B; i.e. JJ=O must have one of its asymptotes coincident with one of the asymptotes of the curve F = 0, or else it must have its two asymptotes parallel to those of V = 0. The latter case is that of similar and similarly situated curves. (3) Suppose the curves intersect in a single point only. a' = A', <x" = A", which it is easy to see gives
Then either a = A,
or else a = A, a! = A', /3 = B, which is the case of one of the asymptotes of the curve U=0, coinciding with one of those of the curve V=0, and the remaining asymptotes parallel. As for the first case, if a, ax are the transverse axes, 6, 6X the inclinations of the two asymptotes to each other, these four quantities are connected by the equation a? _ tang cos2 \Q ~a? ~ tan 6X cos2 \Q^'
and besides, one of the asymptotes of the first curve is coincident with one of the asymptotes of the second. This is a remarkable case; it may be as well to verify that £7~=0, V=0 do intersect in a single point only. Multiplying the first equation by y — Bx — B', the second by y —fix—fi',and subtracting, the result is (A-P)(y-Bx-B')-(A-B)(y-0x-/3') = O, reducible to
Combining this with V = 0, we have an equation of the form y — Bx — D = 0. from this and y — Ax — C = 0, x, y are determined by means of a simple equation.
And
(4) Lastly, when the curves do not intersect at all. Here a = A, a! = A', /3 = B, /3' = £', or the asymptotes of U=0 coincide with those ef F = O ; i.e. the curves are similar, similarly situated, and concentric: or else a = A, a' = A', a" = A", /3 = B; here U ={y - Ax - A'){y -Bx- $') +K, or the required curve has one of its asymptotes coincident with one of those of the proposed curve; the remaining two asymptotes are parallel, and the magnitudes of the curves are equal. In general, if two curves of the orders m and n, respectively, are such that r asymptotes of the first are parallel to as many of the second, s out of these asymptotes
c.
7
50
ON THE THEORY OF ALGEBRAIC CURVES.
[10
being coincident in the two curves, the number of points of intersection is ran — r—s; but the converse of this theorem is not true. In a former paper, On the Intersection of Curves, [5], I investigated the number of arbitrary constants in the equation of a curve of a given order p subjected to pass through the mn points of intersection of two curves of the orders m and n respectively. The reasoning there employed is not applicable to the case where the two curves intersect in a number of points less than mn. In fact, it was assumed that, W = 0 being the equation of the required curve, W was of the form uU+vV; u, v being polynomials of the degrees p — m, p — n respectively. This is, in point of fact, true in the case there considered, viz. that in which the two curves intersect in mn points; but where the number of points of intersection is less than this, u, v may be assumed polynomials of an order higher than p — m, p — n, and yet uTJ+vV reduce itself to the order p. The preceding investigations enable us to resolve the question for every possible case. Considering then the functions U, V determined as before by the equations (3), (4), suppose, in the first place, we have a system of equations a = A, Assume
$ =B
0 = H (t equations)
(8).
P = (y — ax — ...){y — fix — ...) ... (y — 6x — ...), Q ={y -Ax-.
..){y-Bx-...)...(y-Hx
T =(y -ix-...)
whence
... {y-KX
U = PT,
-...);
-...),
V=QV.
Suppose . U-ET.
V= = EV.P(ET + AT) -ET.Q (EV + = ET. EV . (P - Q) + EV . P. AT - # T . Q. = E [ET. EV. (P - Q) + EV. PAT - E? . Q. = II suppose.
In this expression ET, EW are of the degrees m — t, n — t, AT, A ^ of the degree - 1, and P, Q, P — Q of the degrees t, t, t - \ respectively. The terms of II are therefore of the degrees m + n — t — 1, m — 1, n — 1 respectively, and the largest of these is in general m + n — t — 1. Suppose, however, that m + n — t — 1 is equal to m — 1 (it cannot be inferior to it), then t = n; *P becomes equal to unity, or A'*' vanishes. The remaining two terms of II are ET (P — Q), PAT, which are of the degrees m - 1, n—1 respectively. II is still of the degree m—1, supposing m >n. If m = n, the term PAT vanishes. II is still of the degree m — 1. Hence in every case the degree of II is m + n — t — 1: assuming always that P — Q does not reduce itself to a degree lower than t — 1, (which is always the case as long as the equations
10]
ON THE THEORY OF ALGEBRAIC CURVES.
51
a' = A', ft'= B' 6' = H' are not all of them satisfied simultaneously). It will be seen presently that we shall gain in symmetry by wording the theorem thus: the degree of II is equal to the greatest of the two quantities m + n — t — 1, m — 1. Suppose next, in addition to the equations (8), we have
a' = A',
0' = B'
t,'=F,
t' equations, t$>t... (8').
Then, taking T', V, P', Q' the analogous quantities to T, M*, P, Q, and putting
EW.U-ET.
V=W,
we have, as before,
IT = E {EV. EW. (P' - Q') + EW. P'AV - ET. The degree of which case the m + n — if — 2, greatest of the the case where
P'- Q' is t'-2 degree may be n — 1, m — 1. quantities m+n t' = n — 1 would
(unless simultaneously a" = A", /3" = B" £' = F\ in lower). The degrees, therefore, of the terms of II' are Or we may say that the degree of II' is equal to the — t' — 2, m — 1 ; though to establish this proposition in require some additional considerations.
Continuing in this manner until we come to the quantity II1*"11, the degree of this quantity is the greatest of the two numbers m + n — t{k~1] — k, m — 1. And we are satisfied, so that the series may suppose that none of the equations a{k) = A[k) II, II' II i*"1* is not to be continued beyond this point. Considering now the equation of the curve passing through the mn — t — t'... — t{k~1} points of intersection of 27=0, V— 0, we may write
W = uU+vV+pU+p'W
+p*-« n*-1>=0
for the required equation; the dimensions of u, v, p, p' ...pik"1) p — m, p—n;
p-m —n + t+1 or p — m+1; p — m—n + t' + 2 or p—m + 1;
(9),
being respectively
p — m — n + t{k~l) +k or p — m + 1,
the lowest of the two numbers being taken for the dimensions of p, p' ...p{k~1}. Also, if any of these numbers become negative, the corresponding term is to be rejected. In saying that the degrees of p, p' p(k-v have these actual values, it is supposed 1 that the degrees of II, II' II*" actually ascend to the greatest of the values p — m — n + t+1,
or m — 1 ;
m + n — t' — 2, or m — 1 ;
— m + n — = i/3 0 + 3 ) + i (P ~ m - n +l) (fi - m - n + 2 ) - mn + v - A where
iip + k — m — n+1
A = 0, if p + k—ra — n + 1 be negative or zero
be positive; and y is given by the equation (9).
(12),
10]
ON THE THEORY OF ALGEBRAIC CURVES.
53
Also, if 0 be the number of points through which the curve W = 0 can be drawn, including the points of intersection of the curves U=0, F = 0 , then 0=(f> + (mn-t-t'
-i*- 1 )) or
0=hp(p + 3) + ?(p-™>-n + l)(p-'m-n + 2)+V-A~t-t'...-t*-v
(13).
Any particular cases may be deduced with the greatest facility from these general formulae. Thus, supposing the curves to intersect in the complete number of points mn, we have 8 being zero or unity according as p < m + n — 1 or p > m + n — 1. Reducing, we have, for p $> m + n — 3,
0 = \p (p + 3) + J (p — m — n +1) (p — m — n + 2) — mn, and for p > m + n — 3,
4> = ip (P + 3 ) ~ mn> Suppose, in the next place, the curves have t parallel pairs of asymptotes, none of these pairs being coincident. Then p$> m + n — t— 2, $ = |JO (p + 3) + \ (p — m — n + 1) (p — m — n + 2)— mn, p>m + n — t — 2, p 1f> m +n — 2, 0 = \p (p + 3) + £ (p - m - n + 2) (p - m - n + 3), p >m + n — 2, <j) = \p (p + 3) — mn +1, In these formulae, if t be equal to 2 or greater than 2, the limiting conditions are more conveniently written p^m
+ n — t—2;
p $> m + n— t— 2>m + n — 4 ;
p>m + n — 4>.
Similarly may the solution of the question be explicitly obtained when the curves have t asymptotes parallel, and t' out of these coincident, but the number of separate formulae will be greater. In conclusion, I may add the following references to two memoirs on the present subject: the conclusions in one point of view are considerably less general even than those of my former paper, though much more so in another. Jacobi, Theoremata de punctis intersectionis duarum curvarum algebraicarum; Crelle's Journal, vol. xv. [1836, pp. 285—308]; Pliicker, Theoremes gendraux concernant les equations a plusieurs variables, d'un degre1 quelconque entre un nombre quelconque d'inconnues. D° vol. XVI. [1837, pp. 47—57].
54
ON THE THEORY OF ALGEBRAIC CURVES.
[10
Addition. As an exemplification of the preceding formulae, and besides as a question interesting in itself, it may be proposed to determine the asymptotic curves of the rth order of a given curve having all its asymptotic directions distinct,—r being any number less than the degree of the equation of the given curve. A curve of the rth order, which intersects a given curve of the mth o r c j e r i n a number of points, = mr — -|r (r + 3), is said to be an asymptotic curve of the rtb order to the curve in question. Suppose, as before, U = 0 being the equation to the given curve, DEFINITION.
U=E [y -ax-
} ... [y -\x
...
...
and let 6, $ ... to denote any combination of r terms out of the series a... X, and &', '... &/, &c. ... the corresponding terms out of a'... X', &c. Then, writing -*»...-_)(y-^...-^-—) \J/(m)
(where the quantities (m», ^i™-1), ? » , n'...Wm> are entirely determinate, since, by what has preceded, &, '...£l' satisfy a certain equation, 6", ",...il" two equations Q\m)t <j><m); ... il(™)(m-l) equations), we have V=0 for the required equation of the asymptotic curve. It is obvious that the whole number of asymptotic curves of the order r, is n (n — 1) ... (n — r + 1), viz. 1 . 2 ... r curves for each combination of —^
1 2
asymptotes. Some particular instances of asymptotic curves will
be found in a memoir by M. Plucker, Liouvilles Journal, vol. I. [1836, pp. 229—252], Enumeration des courbes du quatrieme ordre, &c. The general theory does not seem to be one to which much attention has been paid.
55
11]
11. CHAPTERS IN THE ANALYTICAL GEOMETRY OF (n) DIMENSIONS. [From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 119—127.] CHAP.
1. On some preliminary formulce.
I TAKE for granted all the ordinary formulae relating to • determinants. It will be convenient, however, to write down a few, relating to a certain system of determinants, which are of considerable importance in that which follows: they are all of them either known, or immediately deducible from known formulae. Consider the series of terms
•"-lj
the number of the quantities A
**-2
-"-n
K being equal to q (q < n).
Suppose q + 1
vertical rows selected, and the quantities contained in them formed into a determinant, this may be done in
-=—~
—-—~- different ways. The system of determinants
so obtained will be represented by the notation (2),
and the system of equations, obtained by equating each of these determinants to zero, by the notation ! =0.
(3).
5G
ANALYTICAL GEOMETRY OF (n) DIMENSIONS.
[11
The -^—= ", ^ — ~ equations represented by this formula reduce themselves to r J 1.2 (n-q + 1) H (n — q) independent equations. Imagine these expressed by (l) = 0, (2) = 0 (n-q)=0 (4), any one of the determinants of (2) is reducible to the form (5), where ®1( ©2...@»_3 are coefficients independent of xlt xt...xn. be replaced by
The equations (3) may =0
\x,
•(6),
and conversely from (6) we may deduce (3), unless = 0
(7).
T 2 , . . . Tn
(The number of the quantities \ , may also be expressed in the form
is of course equal to n.) The equations (3)
/J,...T
(8), ... o)9^T2,... \qAn ... + wqKn the number of the quantities X, ft ...a> being q. And conversely (3) is deducible from (8), unless = 0.
\ , Xq,
(9).
...
CHAP. 2. On the determination of linear equations in xlt x2, ... xn which are satisfied hy the values of these quantities derived from given systems of linear equations. It is required to find linear equations in xlt...xn which are satisfied by the values of these quantities derived—1. from the equations £(' = 0, W = 0 ... US' = 0; 2. from the equations S " = 0, 23" = 0 ... J i , " = 0 ; 3. from &'"=(), W = 0...W = 0, &c. &c, where npn,
...+
(1),
Bnxn,
and similarly ^ " , 23",..., ST", 23'",..., &c. are linear functions of the coordinatesxlt x2,... xn.
57
ANALYTICAL GEOMETRY OF (ft) DIMENSIONS.
11]
Also r', r"... representing the number of equations in the systems (1), (2).. . and k the number of these given systems, « - 1 or
(k-
r ' + r" +• ...
Assume
••• =
... = &c
X
(2),
the latter equations denoting the equations obtained by equating to zero the terms involving x1, those involving x2, &c.... separately. Suppose, in addition to these, a set of linear equations in V, /jf... X", fi"... so that, with the preceding ones, there is a sufficient number of equations for the elimination of these quantities. Then, performing the elimination, we thus obtain equations M* = 0, where ? is a function of a-'u # 2 ... which vanishes for the values of these quantities derived from the equations (1) or (2) ...&c. The series of equations M* = 0 may be expressed in the form
jj 8 ' , 23', ...
BX +
k
J
[
pkMPNP ...u
(B, 1),
instead of (B, 2), that form being principally useful in showing the relation of the function u to the theory of the transformation of functions. It may immediately be seen, that in the equations (B), (C) we may, if we please, omit any number of the marks of variation (•), omitting at the same time the corresponding signs 2, and the corresponding factors of the series L, M, If ... Also, if u be such as only to satisfy some of the equations (A); then, if in the same formulte we omit the corresponding marks ('), summatory signs, and terms of the series L, M, N... , the resulting equations are still true. From the formulae (A) we may obtain the partial differential equations 2 2 ... last... -r-: V 2 2 ...(rat \
I u = 0, or pu,
(D),
dpst...) ...
d ™ - } u = 0, or pu, r drfit..../ 1
according as a. is not equal, or is equal, to /3; and so on : the summatory signs referring in every case to those of the series r, s, t,..., which are left variable, and extending from 1 toTOinclusively. To demonstrate this, consider the general form of u, as given by the first of the equations (A). This is evidently composed of a series of terms, each of the form cPQR...(p Is/*/...,
in which
factors),
l s / ' i , " . . . . . . . (m terms)
Q, R, &c. being of the same form; and we have d .,-=—— , u = cQR... 2 2 ...last... ast... dfist...) \ and
22...
d
''' dfist...
Is/*/ ...,
d J 5 - T — P + &C.+ &C, dfist...
I s / ' * / ' . . . . . . . (m terms)