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MATHEMATICAL PAPEKS.
aonHon: 0. J. CLAY and SONS,
CAMBBIDGE DNIVERSITY PRESS WABEHOUSE,
AVE MABU LANE.
ffilasgDfa : 263, ARGYLE STREET.
Eeipjtfl : F. A. BROCKHAUS.
^riB gotfe: MACMILLAN AND CO.
THE COLLECTED
MATHEMATICAL PAPERS
OF
AKTHUE CAYLEY, Sc.D., F.E.S.,
LATE 8ADLERIAK PBOFE880K OF PURE MATHEMATICS IN THE UNIVERSITY OP CAMBRIDGE.
VOL. X.
CAMBRIDGE :
AT THE UNIVERSITY PRESS.
1896
[All Rig/Us reserved.]
CAMBRIDGE :
PRIMED BY J. AND C. F. CLAY,
AT THE UNIVEHSirY PRESS.
(5A
3
l/.lO
ADVEKTISEMENT.
T
HE present volume contains 76 papers, numbered 630 to 705, published
for the most part in the years 1876 to 1880.
The Table for the ten volumes is
^ol
. I. Numbers 1 to
100,
»
II.
101 „
158,
»>
III.
159 „
222,
»»
IV.
223 „
299,
»
V.
300 „
383,
>'
VI.
384 „
416,
»
VII. ,
417 „
485,
l>
VIII. ,
486 „
555,
J9
IX.
556 „
629,
»
X.
630 „
705.
A. R. FORSYTH.
11 June, 1896.
vu
CONTENTS.
[An Asterisk means that the paper is not printed in full.]
PAGE
630. On an expression for \±sm{2p+l)u in terms of %\nu . . 1
Messenger of Mathematics, t. v. (1876), pp. 7, 8
631. Synopsis of the theory of equations ...... 3
Messenger of Mathematics, t. v. (1876), pp. 39 — 49
632. On Aronhold's integration-formula . . . . . . 12
Messenger of Mathematics, t. v. (1876), pp. 88 — 90
*633. Note on Mr. Martin's paper " On the integrals of some diffe-
rentials" .......... 15
Messenger of Mathematics, t. v. (1876), p. 163
634. Tlieorems in trigonometry and on partitions . . . . 16
Messenger of Mathematics, t. v. (1876), p. 164, p. 188
635. Note on the demonstration of Clairaut's theorem ... 17
Messenger of Mathematics, t. v. (1876), pp. 166, 167
636. On the theory of the singular solutions of differential equations
of the first order .......-• 19
Messenger of Mathematics, t. vi. (1877), pp. 23 — 27
637. On a differential equation in the theoiy of elliptic functions . 24
Messenger of Mathematics, t. vi. (1877), p. 29
638. On a q-formula leading to an expression for E^ . . .
Messenger of Mathematics, t. vi. (1877), pp. 63 — 66
C. X.
25
TUl
CONTENTS.
639. An elementary construction in optics .....
Messenger of Mathematics, t. vi. (1877), pp. 81, 82
*640. Further note on Mr. Martin's paper
Messenger of Mathematics, t vi. (1877), pp. 82, 83
641. On the Jleam,re of a spherical surface ....
Messenger of Mathematics, t. vi. (1877), pp. 88—90
642. On a differential relation between ilie sides of a quadrangle
Messenger of Mathematics, t. vi. (1877), pp. 99—101
643. On a quartic curve tvith two odd branches
Messenger of Mathematics, t. vi. (1877), pp. 107, 108
644. Note on magic squares .......
Messenger of Mathematics, t. vi. (1877), p. 168
645. A Smith's Prize Paper, 1877
Messenger of Mathematics, t. vi. (1877), pp. 173—182
646. On the general equation of differences of the second order
Quart. Math. Joum., t. xiv. (1877), pp. 23—25
647. On the quartic surfaces represented by the equation, symmetH
cal determinant =0 . . . . .
Quart. Math. Joura., t. xiv. (1877), pp. 46—52
648. Algebraical theorem ........
Quart. Math. Joum., t. xiv. (1877), p. 53
649. Addition to Mr. Glaisher's Note on Sylvester-' s paper "Develop
ment of an idea of Eisenstein ".....
Quart. Math. Joum., t. xiv. (1877), pp. 83, 84
650. On a q%w.rtic surface ivith twelve nodes ....
Quart. Math. Journ., t. xiv. (1877), pp. 103—106
651. On a special surface of minimum area ....
Quart. Math. Joum., t. xiv. (1877), pp. 190—196
652. On a sextic torse
Quart. Math. Journ., t. xiv. (1877), pp. 229—235
PAGE
28
29
30
33
36
38
39
47
50
57
58
60
63
68
CONTENTS.
IX
PAOE
653. On a torse depending on the elliptic functions . . . . 73
Quart. Math. Joum., t. xiv. (1877), pp. 235—241
654. On certain actio surfaces . . . . . . ' , . 79
Quart. Math. Journ., t. xiv. (1877), pp. 249—264
655. A memoir on differential equations ...... 93
Quart. Math. Journ., t. xiv. (1877), pp. 292—339
656. On the theory of partial differential equations. . . . 134
Mathematische Annalen, t. xi. (1877), pp. 194 — 198
657. Note on the theory of elliptic integrals 139
Mathematische Annalen, t. xii. (1877), pp. 143 — 146
658. On some formulcB in elliptic integrals . . . . . 143
Mathematische Annalen, t. xii. (1877), pp. 369 — 374
659. A theorem on groups ........ 149
Mathematische Annalen, t. xiii. (1878), pp. 561 — 565
660. On the correspondence of homographies and rotations . . 153
Mathemati-sche Annalen, t. xv. (1879), pp. 238—240
661. On the double '^-functions . . . . . . . 155
Proc. Lond. Math. Soc, t, ix. (1878), pp. 29, 30
662. On the double %-functions in connexion with a IG-nodal
quartic s^xrface . . . . . . . . . 157
Crelle's Journal der Mathem., t. lxxxiii. (1877), pp. 210—219
663. Further investigations on the double '^-functions . . . 166
Crelle's Journal der Mathem., t. lxxxiii. (1877), pp. 220 — 233
664. On the l6-nodal quartic surface . . . . • • 180
Crelle's Journal der Mathem., t. lxxxiv. (1878), pp. 238—241
665. A memcnr on the double "^-functions 184
Crelle's Journal der Mathem., t. lxxxv. (1878), pp. 214—245
666. Sur un exemple de reduction d'integrales abeliennes aux fonc-
tions elliptiques . . . . . . • • • 214
Comptes Rendus, t. LXXXV. (1877), pi.. 265—268, 373—376, 426—429,
472—475
62
Z CONTENTS.
PAOB
667. On the hicircular quai-tic — Addition to Professor Casey's
memoir : " On a new form of tangential equation" . . 223
PhU. Ti'ans., t. 167 (for 1877), pp. 441—460
668. On compound combinations ....... 243
Proceedings of the Lit. Phil. Soc. Manchester, t xvi. (1877), pp.
113, 114; Memoirs, ib., Ser. in., t, vi. (1879), pp. 99, 100
669. On a problem of arrangements ...... 245
Edin. Roy. Soc. Proc., t. ix. (1878), pp. 338—342
670. [Note on Mr. Muir's solution of a "problem of arrangement "] 249
Edin. Roy. Soc. Proc., t. ix. (1878), pp. 388—391
671. On a sibi-reciprocal surface 252
Berlin, Akad. Monatsber., (1878), pp. 309—313
672. On the gam£ of matisetrap ....... 256
Quart Math. Journ., t. xv. (1878), pp. 8—10
673. Note on the theory of correspondence 259
Quart. Math. Journ., t. xv. (1878), pp. 32, 33
674. Note on the construction of Cartesians . . . . . 261
Quart. Math. Joum., t. xv. (1878), p. 34
675. On the jieflecnodal planes of a surface ..... 262
Quart. Math. Joum., t. xv. (1878), pp. 49—51
676. Note on a theorem in determinants ..... 265
Quart. Math. Joum., t. xv. (1878), pp. 5.5—57
677. [Addition to Mr. Glaisher's paper "Proof of Stirling's theorem "] 267
Quart Math. Joum., t xv. (1878), pp. 63, 64
678. On a system of quadric surfaces 269
Quart. Math. Joum., t xv. (1878), pp. 124, 125
679. On the regular solids ........ 270
Quart. Math. Joum., t xv. (1878), pp. 127—131
680. On the Hessian of a quartic surface 274
Quart. Math. Journ., t xv. (1878), pp. 141—144
CONTENTS. xl
PAGE
681. On the derivatives of three binary quantics . . . , 278
Quart. Math. Journ., t. xv. (1878), pp. 157—168
682. FormulcB relating to the right line ...... 287
Quart. Math. Journ., t xv. (1878), pp. 169—171
683. On the function arc sin (x + iy) 290
Quart Math. Journ., t. xv. (1878), pp. 171—174
684. On a relation between certain products of differences . . 293
Quart. Math. Journ., t. xv. (1878), pp. 174, 175
685. On Mr. CotterilVs goniometrical problem ..... 295
Quart. Math. Journ., t. xv. (1878), pp. 196—198
686. On a functional equation .... ... 298
Quart. Math. Journ., t. xv. (1878), pp. 315—325; Proc. Lend. Math.
8oc., t. IX. (1878), p. 29
687. Note on the function '^{x)= a- {c-x)^[c {c-x)-b''} . . 307
Quart. Math. Joum., t. xv. (1878), pp. 338—340
688. Geometrical considerations on a solar eclipse . . . . 310
Quart. Math. Journ., t. xv. (1878), pp. 340—347
689. On the geometrical 7'epresentation of imaginary variables by
a real coTvespondence of two planes . . . . . 316
Proc. Lond. Math. Soc., t. ix. (1878), pp. 31—39
690. On the theory of groups ........ 324
Proc. Lond. Math. Soc., t. ix. (1878), pp. 126—133
691. Note on Mr. Monro's papier "On flexure of spaces" . . 331
Proc. Lond. Math. Soc., t. ix. (1878), pp. 171, 172
692. Addition to [578] memoir on the transformation of elliptic
functions .......... 333
Phil. Trana, vol. 169, Part ii. (for 1878), pp. 419—424
693. A tenth memoir on quantics ....... 339
Phil. Trans., vol. 169, Part ii. (for 1878), pp. 603—661
Xli CONTENTS.
694. Desiderata and Siiggestions 401
No. 1. The theory of groiips ;
American Jounial of Mathematics, t. i. (1878), pp. 50 — 52
No. 2. The theoi'y of groxqis ; graphical representation ;
American Journal of Mathematics, t. i. (1878), pp. 174—176
No. 3. The Newton-Fourier imaginary problem;
American Journal of Mathematics, t. ii. (1879), p. 97
No. 4. The mechanical construction of conformable fgitres ;
American Journal of Mathematics, t. ii. (1879), p. 186
695. A link-work for of: extract from a letter to Mr. Sylvester . 407
American Journal of Mathematics, t. I. (1878), p. 386
696. Calculation of the minimum N.G.F. of the binary seventhic . 408
American Journal of Mathematics, t. ii. (1879), pp. 71 — 84
697. On the double '^-functions . . . . . . . . 422
Crelle's Journal der Mathem., t. lxxxvii. (1879), pp. 74 — 81
698. On a theorem relating to covariants ..... 430
Crelle's Journal der Mathem., t. lxxxvii. (1879), pp. 82, 83
699. On the triple "it-functions ........ 432
Crelle's Journal der Mathem., t. lxxxvii. (1879), pp. 134 — 138
700. On the tetrahedroid as a particular case of the 16-nodal quartic
surface 437
Crelle's Journal der Mathem., t. lxxxvii. (1879), pp. 161 — 164
701. Algonthm for the characteristics of the triple ^-functions . 441
Crelle's Journal der Mathem., t. lxxxvii. (1879), pp. 165 — 169
702. On the triple ^-functions 446
Crelle's Journal der Mathem., t. lxxxvii. (1879), pp. 190 — 198
703. On the addition of the double "ii -functions . . . . 455
Crelle's Journal der Mathem., t. lxxxviii. (1880), pp. 74 — 81
704. A memoir on the single and double theta-functions . . . 463
PhU. Trans., vol. 171, Part in. (for 1880), pp. 897—1002
705. Problems and Solutions ........ 566
Mathematical Questions with their Solutions from the Educational
Times, vols. xiv. to LXI. (1871 — 1894) ; for contents, see p. 615
XIU
CLASSIFICATION.
Geometry :
Geometrical constructions : (i) in optics, 639.
(ii) for solar eclipses, 688.
Quartic curves, 643, 667.
Construction of Cartesians, 674.
Correspondence, 673, 683, 689, 694.
Quadric surfaces, 678.
Quartic surfaces, 647, 6.50, 662, 664, 680, 700.
Sextic and other torses, 652, 6.53.
Octic surfaces, 654, 671.
line-geometry, 682.
Fleflecnodal planes, 675.
Flexure of surfaces, 641, 691.
Minimal surface, 651.
Regular solids, 679.
AlfALYSIS :
Trigonometry, 630, 634.
Theory of equations, 631, 694.
Groups, 659, 660, 690, 694.
Combinatory analysis, 634, 644, 668, 669, 670, 672.
Inyariants and covariants, 681, 696, 698.
XIV CLASSIFICATION.
Quantics, 693.
Integration, 632, 633, 640.
Differential equations, 636, 637, 642, 655, 656.
Finite differences, 646, 677, 686, 687.
Elliptic functions and elliptic integrals, 637, 638, 653, 657, 658, 692.
Double theta functions, 661, 662, 663, 665, 697, 703, 704.
Triple theta functions, 699, 701, 702.
Reduction of transcendental integrals, 666.
Problems and solutions, 645, 705.
Miscellaneous, 635, 648, 649, 676, 684, 685, 695.
630]
630.
ON AN EXPRESSION FOR 1 ± sin (2p + 1) i^ IN TERMS OF sinu.
[From the Messengei- of Mathematics, vol. v. (1876), pp. 7, 8.]
Write siii u = x, then we have
I
sin u= X, cos M = V(1— ^),
sin 3w = 3aj - 4a:', cos 3m = (1 - 4av') V(l - a^),
8in5M = 5a;-20«» + 16«», cos 5a = (1 - 12ar' + 16a:«) v/(l -«"),
&c. &c.
It is hence clear, that in general
l-8in(2p+l)M = (l ±a;)((l, xfW
\ +sm(2p + \)u = (\T x) {(1, - xY]\
where (1, x)>' denotes a rational and integral function of x of the order p, and
(1, —xy the same function of —a;; for it is only in this manner that we can have
We, in fact, find
cos'(2p + 1) w = (1 - ar*) {[1, a-'Y{K
1 + sin M = \ -\- X,
1 - sin 3« = {l+x){\- 2x)\
1 + sin 5u = (1 + a;) (1 + 2a; - 'ki?)',
1 - sin 7w = (1 + a:) (1 - 4z; - 4ar + 8a:')^
&c.
and it thus appears that the form is
1 + (-y sin (2;) + 1) M = (1 + x) {(1, a:)pj^
v.. X.
2 ON AN EXPRESSION FOR l±sin(2p+l)w IN TERMS OF Binu. [630
To find herein the expression of the factor (1, xy, write u = j7r— 5 and con8e<|uently
a; = cos 5 ; we have therefore
1 + cos (2p + 1) ^ = (1 + cos 6) {(1, a;y>Y>
where in the second factor on the right-hand side x is retained to stand for its value
cos^. This gives
2 cos' (p+^)e=2 cos» ^e{(i, xyY.
or, what is the same thing,
n .p _co8(p + ^)e
^^' ^^~ cosi^ '
viz. this is
. . .amid
= cmp0-smp0^^^,
which is
- . ^ 1 - cos ^
= cos pa — sm pa — : — ^j— .
•^ ^ sin p
We have
cos pff + i smpB = {a; + i V(l — i"^)]^
= X + i nj{\ — a?) F, suppose,
where X, Y are rational and integral functions of x of the orders p and p — 1
respectively; that is,
cospO = X, sin p6 = 8m0 . Y,
and we have therefore
(l,xy = X-Y(l-x),
which is the required expression for (1, x)p. For instance
p = S, X + i^/(l-x'')Y={x + i^(l-af)}';
that is,
X= -3x +4a^
F= - 1 + ^far", and .-. -{l-x)Y=l- x-4,x' + 4«='
so that X-(l-x)Y=l-4^-4ia? + 8x', ={1, xy,
and hence
1 - sinlu = (1 + x) (1 - 4!X - 4:0^ + Sx'Y,
which agrees with a result already obtained.
The foregoing value of (1, xY may also be written
^^' ""^"^ ^d f''° (^ + 1) ^ - si^i'^l.
which however is not practically so convenient.
The formula corresponds to a like formula in elliptic functions, viz. writing sinam u = x,
the numerator of 1 + (— )? sinam (2p + l)u is
= (l+x){{l, a;)*'P+"j»,
which is (1 + «) multiplied by the square of a rational and integral function of x.
631] 8
631.
SYNOPSIS OF THE THEORY OF EQUATIONS.
[From the Messenger of Mathematics, vol. v. (1876), pp. 39 — 49.]
The following was proposed, as one of the subjects of a Dissertation for the
Trinity Fellowships :
Synopsis of the theory of equations; i.e. a statement in a logical order, of the
divisions of the subject and the leading questions and theorems, but without demonstrations.
In the subject "Theory of Equations," the term equation is used to denote an
equation of the form a;"— pia;"~'+ ... +/)„ = 0, where pi, p^,.., Pn are regarded as known,
and a; as a quantity to be determined ; for shortness, the equation is written f{x) = 0.
The equation may be numerical; that is, the coefficients Pi,Pi,.., Pn are then
numbers ; understanding by number, a quantity of the form a + /8i, where a and /S have
any positive or negative real values whatever; or say, each of these is regarded as
susceptible of continuous variation from an indefinitely large negative to an indefinitely
large positive value : and i denotes V(— !)•
Or the equation may be algebraic; viz. the coefficients are then not restricted to
denote, or are not explicitly considered as denoting, numbers.
I. We consider first numerical equations.
A number a (real or imaginary), such that substituted for x it makes the function
«" - /Jia;"-' + ... ±pn to be =0, or say, such that it satisfies the equation, is said to
be a root of the equation ; viz. a being a root, we have
a"-;jia"-i+... ±p„ = 0, or 8ay/(a)=0;
and it is then shown that a; -a is a factor of the function f{x), viz. that we have
f{!>:) = {x-a)f{x), where f{x) is a function «"-' - jis;"- + • • • i^^-i, of the order n-1,
with numerical coefficients qi, qt,.., qn-i-
1—2
4 SYNOPSIS OF THE THEORY OP EQUATIONS. [631
In general, a is not a i*oot of the equation /, (a;) = 0 ; but it may be so, viz.
/■, (x) may contain the factor x — a; when this is so, /(x) mil contain the factor
{x — aY; writing then /{x) = {x — ay/.^ {x), and assuming that a is not a root of the
equation /, (a;) = 0, x = a is then said to be a double root of the equation. Similarly,
f(x) may contain the factor (x — aY and no higher power, and then x = a is said to
be a triple root ; and so on.
Supposing, in general, that f (x) = (x — a)^ F {x), where a is a positive integer which
may be = 1, and Fx is of the order n — a, then if 6 is a root different from a, we
.shall have x — b a factor (in general a simple one, but it may be a multiple one) of
F(x), and f{x) will in this case become = (a; — a)" (a; — 6/ 4> (a;), where yS is a positive
integer which may be =1, and <Px is of the order n — a—^. The original equation
fx = 0 is in this case said to have a roots each = a, yS roots each = b, and so on.
We have the theorem, a numerical equation of the order n has in every case n
roots, viz. there exist n numbers a, b,... (in general, all of them distinct, but they
may arrange themselves in groups of equal values) such that
f(x) = (x—a)(x — b){x — c)... identically.
If an equation has equal roots, these can in general be determined ; the case is at
any rate a special one, which may be here omitted from consideration. It is there-
fore, in general, assumed that the equation f{x) = 0 under consideration has all its
roots unequal. If the coefficients pi, p,,... are all or any one or more of them
imaginary, then the equation /{x) — 0, separating the real and imaginary parts, may
be written F{x)+i^(x) = 0, where F{x), ^(x) are each of them a function with real
coefficients ; and it thus appears that the equation f{x) = 0 with imaginary coefficients
has not in general any real root; supposing it to have a real root a, this must be
at once a root of each of the equations F{x}=0 and 'P{x) = 0.
But an equation with real coefficients may have as well imaginary as real roots;
and we have further the theorem that for such an equation the imaginary roots enter
in pairs, viz. a + ySi being a root, then will also a - /9t be a root.
Considering an equation with real coefficients, the question arises as to the number
and situation of its real roots; this is completely resolved by means of Sturm's
theorem, viz. we form a series of functions f{x), f (x), f^ (x), . . , /„ (x) (a constant) of
the degrees n, n — 1, . . , 2, 1, 0 respectively ; and substituting therein for x any two
real values a and b, we find by means of the resulting signs of these functions how
many real roots of /(x) lie between the limits a, b.
The same thing can frequently be effected with greater facility by other means,
but the only general method is the one just referred to.
In the general case of an equation with imaginary (it may be real) coefficients,
the like question arises as to the situation of the (real or imaginary) roots, viz. if
for facility of conception we regard the constituents a, yS of a root a + /3t as the
coordinates of a point in piano, and accordingly represent the root by such point;
then drawing in the plane any closed curve or "contour," the question is how many
roots lie within such contour.
631] SYNOPSIS OF THE THEORY OF EQUATIONS. 5
This is solved theoretically by means of a theorem of Cauchy's, viz. writing in
the original equation x + iy in place of x, the function f{x + iy) becomes =P+iQ,
where P and Q are each of them a rational and integral function (with real coefficients)
of {<^> y)- Imagining the point {x, y) to travel along the contour, and considering the
number of changes of sign from - to + and from + to - of the fraction ^ corre-
spending to passages of the fraction through zero (that is, to values for which P
becomes =0, disregarding those for which Q becomes =0), the difference of these
numbers determines the number of roots within the contour. The investigation leads
to a proof of the before-mentioned theorem, that a numerical equation of the order
n has precisely n roots.
But, for the actual determination, it is necessary to consider a rectangular contour,
and to apply to each of its sides separately a method such as that of Sturm's
theorem ; and thus the actual determination ultimately depends on a method such as
that of Sturm's theorem.
Recurring to the case of an equation with real coefficients, it is important to
separate the real roots, viz. to detei-mine limits, such that each real root lies alone
by itself between two limits I and m. This can be done (with more or less difficulty
according to the nearness of the real roots) by repeated applications of Sturm's
theorem, or otherwise. I
The same thing would be useful, and can theoretically be effected, in regard to
the roots of an equation generally, viz. we may, by lines parallel to the axes of
X and y respectively, divide the plane into rectangles such that each (real or imaginary)
root lies alone by itself in a given rectangle; but the ulterior theory, even as regards
the imaginary roots of an equation with real coefficients, has not been developed, and
the remarks which immediately follow have reference only to equations with real
coefficients, and to the real roots of such equations.
Supposing the roots separated as above, so that a certain root is known to lie
alone by itself between two given limits, then it is possible by various processes
(Homer's, or Lagrange's method of continued fractions) to obtain to any degree of
approximation the numerical value of the real root in question, and thus to obtain
(approximately as above) the values of the several real roots.
The real roots can also frequently be obtained, without the necessity of a previous
separation of the roots, by other processes of approximation — Newton's, as completed
by Fourier, or by a method given by Encke — and the problem of their determination
to any degree of approximation may be regarded as completely solved. But this is
far from being practically the case even as regards the imaginary roots of such
equations, or as regards the roots of an equation with imaginary coefficients.
A class of numerical equations which need to be considered, are the binomial
equations x» - a = 0, where a, = a + /Si, is a complex number. The foregoing conclusions
apply, viz. there are always n roots, which it may be shown are all unequal. Supposing
6 SYNOPSIS OF THE THEORY OF EQUATIONS. [631
one of these is 0, so that d" = a, theu, assuming x = Oy, we have y" — 1 = 0, which
equation (like the more general one af — a = 0) has precisely n roots ; it is shown
that these are 1, », w', . . , a»"~', where o» is a complex number a + ^i such that
a? + j3' = l, or, what is the same thing, a complex number of the form cca 6 + % aia 0 ;
and it then at once appears that 0 may be taken = — . We have thus the
trigonometrical solution of the equation a;" — 1 = 0. We may also obtain a like
trigonometrical solution of the first-mentioned equation a;" — o = 0. We are thus led
1
to the notion (a numerical) of the radical a", regarded as an »i-valued function, viz.
any one of these being denoted by ^{a), then the series of values is
^{a),to^(a),.., a,''->(/(a).
1
Or we may, if we please, use ^(a), instead of a", as a symbol to denote the ra-valued
function.
It is not necessary, as regards the equation a;" — 1 = 0, to refer here to the
distinctions between the cases n a prime, and a composite, number.
As the coefiScients of an algebraical equation may be numerical, all which follows in
regard to algebraical equations, is (with, it may be, some few modifications) applicable
to numerical equations; and hence, concluding for the present this subject, it will be
convenient to pass on to algebraical equations.
II. We consider, secondly, an algebraical equation
a;»— p,a;»-' + ...=0,
and we here assume the existence of roots, viz. we assume that there are n quantities
a, b, c, ... (in general, all of them different, but in particular cases they may become
equal in sets in any manner), such that
af*—piX''-^+...=(x-a)(x-b)....
Or, looking at the question in a different point of view, and starting with the roots
a, b, c, ... as given, we express the product of the n factors x — a, x — b,... in the
foregoing form, and thus arrive at an equation of the order n having the n roots
a, b, c In either case, we have
Pi-Xa, p^^tah,.., pn-ahc...,
viz. regarding the coefficients pi, Pi,.., pn as given, then we assume the existence of
roots a, b, c, . . . such that pi = Set, &c., or regarding the roots as given, then we write
Pi> Pi> &c., to denote the functions 2a, 2a6, &c.
It is to be noticed that, in virtue of
a;" - pia;"-» + . . , = (a; - a) (a; - 6), &c.,
631] SYNOPSIS OF THE THEORY OF EQUATIONS. 7
or of the equivalent equations p, = Sa, &c., then
a"-^,a"-'+ ... = 0,
6" -pM~' + ...=0,
(viz. it is for this reason that a, b, ... are said to be roots of a;"— j9,a;'^> + ... =0);
and, moreover, that conversely from the last-mentioned equations, assuming that a, b,...
are all different, we deduce
Pi = Xa, jOa = tab, &c.,
and
ai^-PiX'>^^ + ... =(x-a)(x- b) ....
Observe that, if for instance a = b, then the two equations a" — pia"~' + . . . = 0,
6"— 2J,6"~' + ... = 0 would reduce themselves to a single equation, which would not of
itself express that a was a double root, that is, that {x — aY was a factor of
a;" — /)!«""' + &c. ; but by considering b as the limit of a + h, h indefinitely small, we
obtain a second equation
na"-' - (n - l)p,a"-' + . . . = 0,
which, with the first, expresses that a is a double root; and then the whole system
of equations leads, as before, to the equations p^ = 2a, &c. But this in passing : the
general case is when the roots are all unequal.
We have then the theorem that every rational symmetrical function of the roots
is a rational function of the coefficients ; this is an easy consequence from the less
general theorem, every rational and integral symmetrical function of the roots is a
rational and integral function of the coefficients.
In particular, the sums of powers 2a', 2a', &c., are rational and integral functions
of the coefficients.
An. ordinary process, as regards the expression of other functions 2a"6^, &c., in
terms of the coefficients, is to make them depend on the functions 2a", &c., but this
is very objectitmable ; the true theory consists in showing that we have systems of
equations
Pi =ta,
f Pt = tab,
1 j3,' = 2a» + 22a6,
■ Pj = 2a6c,
■P1P2 = 2a'6 + 32a6c,
, pi' = ta' + Sla'b + 6tabc,
&c., &c.
where, in each system, there are precisely as many equations as there are root-functions
on the right-hand side, e.g. 3 equations and 3 functions 2a6c, ta% la'. Hence, in
each system, the root-functions can be determined linearly in terms of the powers and
products of the coefficients.
8 SYNOPSIS OK THE THEORY OF EQUATIONS. [631
It follows that it is possible to determine an equation (of an assignable order)
having for roots any given (unsymmetrical) functions of the roots of a given equation.
For example, in the case of a quartic equation, roots (a, b, c, d), it is possible to find
an equation having the roots ab, ac, ad, be, bd, cd, being therefore a sextic equation ;
viz. in the product (y — ab){y — ac) (y - ad) {y — be) (y — bd) {y - cd), the coefficients of
the several powers of y will be symmetrical functions of a, b, c, d, and therefore
rational and integral functions of the coefficients of the original quartic equation.
In connexion herewith, the question arises as to the number of values (obtained
by permutations of the roots) of given >msymmetrical functions of the roots ; for instance,
with roots (a, b, c, d) as before, how many values are there of the function ab -\-cd;
or, better, how many functions are there of this form ; the answer is 3, viz. ab + cd,
a4i-\-bd, ad-\-bc; or, again, we may ask whether it is possible to obtain functions of a
given number of values, 3-valued, 4-valued functions, &c.
We have, moreover, the very important theorem that, given the value of any
unsymmetrical function, e.g. ab + cd, it is in general possible to determine rationally
the value of any .similar function, e.g. (a + bf + (c + df.
The cb priori ground of this theorem may be illustrated by means of a numerical
equation. Suppose, e.g. that the roots of a quartic equation are 1, 2, 3, 4; then if it
is given that a6 + cd=14, this in effect determines a, b to be 1, 2 (viz. a = 1, 6 = 2,
or else o = 2, 6 = 1) and c, d to be 3, 4 (viz. c = 3, d=4, or else c = 4, d = 3); and
it therefore in effect determines (a + 6)' + (c + d)' to be = 370, and not any other
value. And we can in the same way account for cases of failure as regards particulai-
equations ; thus, the roots being 1, 2, 3, 4, as above, a-b = 2 determines a to be = 1
and b to be = 2 ; but if the roots had been 1, 2, 4, 16, then a'6 = 16 does not
uniquely determine a and b, but only makes them to be 1 and 16, or else 2 and 4,
respectively.
As to the d posteriori proof, assume, for instance, ^, = a6 + cd, yi = (a + 6)* + (c + dy,
and so <, = ac + d6, ya = (a + c)' + (d + 6)', &c. — in the present case there are only the
functions <,, «j, t, and y,, y^, y,— then yi + y^ + y^, tiyi + t^y^ + tiy^, ti'yi + 1,% + ti'^s will
be respectively symmetrical functions of the roots of the quartic, and therefore rational
and integral functions of its coefficients, that is, they will be known.
Imagine, in the first instance, that ti, t^, t, are all known; then the equations
being linear in y,, y^, y,, these can be expressed rationally in terms of known functions
of the coefficients and of tu t^, <,. that is, y^, y,, y, will be known. But observe
further, that y, is obtained as a function of ti, t^, 1% symmetrical as regards t,, <,:
it can consequently be expressed as a rational function of ^ and of <j + <s, tjtj, or,
what is the same thing, of ^i and ti-\-t^+ti, <i<2 + ii<3 + <2<3, titjt,; but these last will
be symmetrical functions of the roots, and as such expressible rationally in terms of
the coefficients : that is, y, will be expressed as a rational function of ^i and of the
coefficients, or, ti being known, y, will be rationally determined.
We may consider now the question of the algebraical solution of equations, or,
more accurately, that of the solution of equations by radicals.
631] SYNOPSIS OF THE THEORY OF EQUATIONS. 9
In the case of a quadric equation x'+px + q = 0, we can find for x, by the
assistance of the sign V( ) or ( )* an expression for a: as a two-valued function
of the coeflacients p, q, such that, substituting this value in the equation, the equation
is thereby identically satisfied, viz. we have
giving
a^= hP'-q+P^ap'-q)
+px = -:^p^ ±p^/{\p'-q)
+g = +g
ai'+px + q =0,
and the equation is on this account said to be algebraically solvable, or, more accurately,
to be solvable by radicals. Or we may, by writing u; = — ^p+z, reduce the equation
to z'=^p^ — q, viz. to an equation of the form z^=a, and, in virtue of its being thus
reducible, we may say that the equation is solvable by radicals. And the question for
an equation of any higher order is, say of the order n, can we by means of radicals,
that is, by aid of the sign ^( ) or ( )"•, using as many as we please of such
signs and with any values of m, find an w-valued function (or any function) of the
coefficients, which substituted for x in the equation shall satisfy it identically.
It will be obsei-ved that the coefficients p, q, ... are not explicitly considered aa
numbers, but that even if they do denote numbers, the question whether a numerical
equation admits of solution by radicals is wholly unconnected with the before-mentioned
theorem of the existence of the n roots of such an equation. It does not even follow
that, in the case of a numerical equation solvable by radicals, the algebraical expression
of X gives the numerical solution ; but this requires explanation. Consider, first, a
numerical quadric equation with imaginary coefficients ; in the formula x= — ^p± */(^p^ — q),
substituting for p, q their given numerical values we obtain for x an expression of the
form x = a + ffi ± V(7 + 8t), where a, y3, y, B are real numbers ; this value substituted
in the numerical equation would satisfy it identically and it is thus an algebraical
solution ; but there is no obvious d priori reason why the expression V(7 + Bi) should
have a value = c + di, where c and d are real numbers calculable by the extraction
of a root or roots of real numbers ; it appears upon investigation that \/(7 + Bi) has
such a value calculable by means of the radical expression s/yi'f + S^) ± 7} ; and hence
that the algebraical solution of a quadric equation does in every case give the
numerical solution of a numerical quadric. The case of a numerical cubic will be
considered presently.
A cubic equation can be solved by ladicals, viz. taking for greater simplicity the
cubic in the reduced form a?-qx — r = Q, and writing x = a + b, this will be a solution
if only 3a6 = g, and a' -1- &• = r, or say H»' + ^) = i^ ! whence
ex. 2
10 SYNOPSIS OF THE THEORY OF EQUATIONS. [631
and therefore
a six-valued function of q, r. But then writing h = ^- , we have, as may be shown,
a + h a three-valued function of the coejfficients ; it would have been wrong to com-
plete the solution by writing h = ^{^r ±sf{\r^ — -^(f)], since here {a + h) would be
given as a 9-valued function, having only 3 of its values roots, and the other 6 values
being irrelevant. An interesting variation of the solution is to write x = ah{a + h),
giving 0*6* (a' -H 6*) = r and 3a'6' = 5', or say i(a'-i- J') = f-, a'&' = ^y; whence
and therefore
{i(a'-6')}> = |ar»-5V3'),
« = ^{fJ±^V(ir--^9')}, 6 = ^{t->^^^(ir'-^9»)[,
and here although a, b are each of them a 6- valued function, yet, as may be shown,
ab{a-\-h) is only a 3- valued function.
In the case of a numerical cubic, even when the coefificients are real, substituting
their values in the expression
*• = -e^fi'- ± V(i'-= - ^ff')} + [i? - ^{ir + Var= - ^^))],
this may depend on an expression of the form ^{f + Si), where y and S are real
numbers (viz. it will do so if Jr" — -^^ is a negative number), and here we cannot
by the extraction of any root or roots of real numbers reduce ^(y + Bi) to the form
c + di, c and d real numbers; hence, here the algebraical solution does not give the
numerical solution. It is to be added that the case in question, called the "irreducible
case," is that wherein the three roots of the cubic equation are all real ; if the roots
are one real and two imaginary, then, contrariwise, the quantity under the cube root is
real, and the algebraical solution gives the numerical one.
The irreducible case is solvable by a trigonometrical formula, but this is not a
solution by radicals; it consists, in effect, in reducing the given numerical cubic (not
to a cubic of the form z' = a, solvable by the extraction of a cube root, but) to a
cubic of the form 4a::' — 3a; = a, corresponding to the equation 4 cos* ^ — 3 cos d = cos 3^
which serves to determine cos 0 when cos Sd is known.
A quartic equation is solvable by radicals; and it may be remarked, that the
existence of such a solution depends on the existence of 3-valued functions such a.s
ab + cd, of the four roots (a, b, c, d); by what precedes, ab + cd is the root of a cubic
equation, which equation is solvable by radicals ; hence ab + cd can be found by radicals ;
and since abed is a given value, ab and cd can each be found by radicals. But by
what precedes, if ab be known, then any similar function, say a + b, is obtainable
rationally ; and, consequently, from the values of a + b and ab we may by radicals
631] SYNOPSIS OF THE THEORY OF EQUATIONS. 11
obtain the value of a or h, that is, an expression for a root of the given quartic
expression ; the expression finally obtained is 4-valued, corresponding to the different
values of the several radicals which enter therein, and we have therefore the expression
by radicals of each of the four roots of the quartic equation. But when the quartic
is numerical, the same thing arises as in the cubic: the algebraical expression does
not in every case give the numerical one.
It will be understood from the foregoing explanation as to the quartic, how in
the next following case, that of a quintic equation, the question of the solvability by
radicals depends on the existence or non-existence of i- valued functions of the five
roots (a, h, c, d, e); a fundamental theorem on the subject is that a rational function
of 5 letters, if it has less than 5, cannot have more than 2 values; viz. that there
are no 3-valued, or 4-valued, functions of 5 letters; and by rfeasoning, depending in
part upon this theorem, Abel showed that a general quintic equation is not solvable
by radicals: and a fortiori the general equation of any order higher than 5 is not
solvable by radicals.
The general theory of the solvability of an equation by i-adicals depends very
much on Vandermonde's remark, that supposing an equation is solvable (by radicals)
and that we have therefore an algebraical expression of x in terms of the coefficients,
then substituting for the coefficients their values in terms of the roots, the resulting
value of the expression must reduce itself to any one at pleasure of the roots a, b, c, . . . ;
thus in the case of the quadric equation where the solution is a; = + ^p ± \/(lj^ — q),
writing for p, q their values a + b, ab, this is a;=H('^ + ^) ± V{(* — ^)''j]. =a or 6
according to the value of the radical. But it is not considered necessary in the
present sketch to go further into the theory of the solvability of an equation by
radicals. It may oe proper to remark that, for quintic equations, there are solutions
analogous to the trigonometrical solution of a cubic equation, viz. the quintic equation
is here in effect reduced to some special form of quintic equation ; for instance, to
Jerrard's form af^ + ax + b = 0 or to some form presenting itself in the theory of elliptic
functions; but the solutions in question are not solutions by radicals. And there are
various other interesting parts of the theory which have been excluded from consideration.
2—2
12 [632
632.
ON ARONHOLD'S INTEGRATION-FORMULA.
[From the Messenger of Mathematics, vol. v. (1876), pp. 88 — 90.]
The fundamental theorem in Aronhold's Memoir, " Ueber eine neue algebraische
Behandlungsweise der Integrale...n(a;, y)dx, &c.," Grelle, t. LXI. (1863), pp. 95 — 145, is
a theorem of indefinite integration. The form is
. f dx _, (a^+hv+g)x + (h^+br,+f)y+g^+fr] + c
J(ax + ^lf + 'Y)(hx + by+f) ^ aiv + ^y + y
where y is a certain irrational function of x, determined by a quadric equation, and
the other symbols denote constants connected by certain relations; viz. writing, for
shortness,
U = {a, b, c, f, g, K^x, y, 1)-, ={a,,...\x, y, If for shortness,
that is,
that is.
or
= aaP + 2hxy + hf+ %fy -^Igx-'rc;
Tf=(a, h, c,f, g, K^x, y, V^^, rj, 1), =(a, ...$a;. y, l$f j;, 1),
= (aa + % +g) ^+{]ix + by +/) i; +gx ^fy + c,
(a^ + Aij + 5r) a; + (/if + 61; +/) 2/ + 5r^ +/»/ + c ;
{P , Q, R)='{ax + hy+g, hx + by+f, gx+fy+c),
(P., Qo, -Ro) = (a? +hv+g, Af + bv +/. g^ +fv + c),
n =cuc + ^y+y,
(A, B, C, F, 0, H) = (bc-p, ca-g\ ah-h\ gh-af, h/-bg, fg -ch),
632] ON aronhold's integration-formula. 13
then y is determined as a function of x by the equation U=0, that is,
(a, b, c, f, g, h^x, y, \f = Q;
or, what is the same thing,
hy = -[}uc +/+ V(- Co? + 'lGx-A)];
the constants a, y8, f, 17 are such that
(a, 6, c,/ fir, A$?, ^, 1)^=0,
af + /3'7 + 7 = 0,
that is,
and the value of A is given by
h? = -{A, B, G, F, G, H^a, yS, 7)^
The theorem may therefore be written
where the several symbols have the significations explained above.
The verification is as follows. We ought to have
Adx_ Ppdx + Q^dy adx+ 0dy
~nQ~ W il '
when dx, dy satisfy the relation P dx + Q dy = 0, viz. substituting for dy the value
^r— , the equation becomes
A^PqQ-PQo aQ-^P
ft If ft '
that is, substituting for ft its value,
AW = (PoQ-PQo)ic^ + ^y + y)-{oiQ-0P)W.
On the right-hand side, substituting for W its value,
coefif. a = a: (P„Q - PQo) - Q {P<,<« + QoV + Ro), =QoR- QRo,
coefi-. yS = y {P,Q - PQ,) + P (P,x + Q,y + E„), = R,P - RPo,
(as at once appears by aid of the relation U=Px+ Qy + R = 0),
coefi". 7 =PoQ — RQo-
The equation to be verified thu.s is
AW =
a ,
/3.
1
Po,
Qo,
Ro
p,
Q,
R
14 ON aronhold's integration-formula. [632
which, substituting therein for P, Q, R, P„ Q,, R^, their values, and writing
is in fitct
Air=(^l,...^\, fi, u^a, /9, 7).
We have identically
(o,...$<c, y, iy.(a,...^^, V, ly- W' = (A,...l\, ^, vY,
which, in virtue of (a, ...$f, rj, 1)' = 0, gives
W'=-{A,...'$\,^,,vy■,
and since A»= - (.4, ...Jo, yS, 7)', the equation is thus
Vl-(il,...$a, ^. 7)»}.V{-(^,...][X, ,., ,/)»}= (4,... $\, ^, .,$a, j3, 7).
that is,
(J,...$a, /3, 7^(A...]li>., f^, vy-[{A,...-$\, M, i-Ja, /8, 7)P = 0.
The left-hand side is here identically
= K{a,. . .$7M - 0v, OLv -y\, ffK- afi.y :
substituting for X, n, v their values, we find
viz. in virtue of fl„=0, these are =-fn, -57!!, - ffl, and the quadric function is
= ifn'(a, ...$f, 17, 1)', vanishing in virtue of the relation (a, ...Jf, »/, 1)' = 0.
The equation in question
V{-(4...$a, A 7)»1V{-(^...$X, ^. vy\={A...\\, ij,. v-^a, yS, 7)
is thus verified, and the theorem is proved.
633]
15
633.
NOTE ON MR MARTIN'S PAPER, "ON THE INTEGRALS OF
SOME DIFFERENTIALS."
[From the Messenger of Mathematics, vol. v. (1876), p. 163.]
The Note refers to a detail in a process of integration.
16
[634
634.
THEOREMS IN TRIGONOMETRY AND ON PARTITIONS.
[From the Messenger of Mathematics, vol. v. (1876), p. 164, and p. 188.]
If
then
A+B-{-G+F-\-G + H^O,
BmA+FsmB-k-F ainC + F, coaF, siaF
siaA + GaiaB + O siaC + 0, cos 0 , sin G
sin j4 + fl^sin B + H ain C + H, cos H, sin H
= 0.
Let Mn = number of partitions of n, no part less than 2, the order attended to ; e.g.
if K=7, the partitions are 7, 52, 25, 43, 34, 322, 232, 223, ih = 8; the series is
«2= 1,
"3= 1,
"4= 2,
«,= 3,
it,= 5,
th= 8,
ih=l3,
'(, = 21,
where each
term
is
the
sum
of the
next preceding
two
terms
635]
17
635.
NOTE ON THE DEMONSTRATION OF CLAIRAUT'S THEOREM.
[From the Messenger of Mathematics, vol. v. (1876), pp. 166, 167.]
It seems worth while to indicate what the leading steps of the demonstration are.
The potential of the Earth's mass upon an external or superficial point is taken
to be
where F,, F„ F,, ... are Laplace's functions of the angular coordinates.
The surface is assumed to be a nearly spherical surface »• = «(!+ u), where
M = Mj + M, -I- &c., and zti, Mj,... are Laplace's functions of the angular coordinates. To be
a surface of equilibrium, with an equation F+^wVsin^ ^ = (7, the latter must be
equivalent to the equation r = a(l + w), and it follows that we have
F, = Foai*i,
F, = FoaX - iw'a" (J - cos» 6),
&c.,
which values are to be substituted in the expression for F.
The whole force of gravity (due to the attraction and the centrifugal force) is
taken to be g, = - -7- ( F + ^ mV' sin" ^), and it follows that
g=^{l+u, + 2u,+ ...)-^m'a-^a)%(i-coa'e),
Cb
C. X. 3
18 NOTE ON THE DEMONSTRATION OF CLAIRAUT's THEOREM. [635
which is of the form
Sr= G |l + M,-$^(i-co8'^)+ 2«,+ ...|.
Taking the Earth to be the spheroid of revolution
r = a{l + €(J-co8»^)),
u, = e (^ — cos" ^), Mj = 0, &c.,
then
and the equation is
or say
^=G{l-(f^"-e)(i-co8«^)}.
5f = (?{l-(fm-6)(i-cos>^)},
«u*a
where m, = -^- , is the ratio of the centrifugal force at the equator to the force of
gravity, which is the theorem in question. The expression " it follows " has been twice
used as meaning it follows as a mere analytical consequence, in the proper degree of
approximation, the steps of the deduction being purposely omitted.
636] 19
636.
ON THE THEORY OF THE SINGULAR SOLUTIONS OF DIFFER-
ENTIAL EQUATIONS OF THE FIRST ORDER.
[From the Messenger of Mathematics, vol. vi. (1877), pp. 23 — 27.]
In continuation of the former paper with this title {Messenger, vol. ii., 1873, pp. 6 — 12,
[545]), I propose to discuss various particular examples, chiefly of cases in which the
differential equation is of the form (L, M, N"^p, l)- = 0, where L, M, N are rational and
integral functions of {x, y), and whether it admits or does not admit of an integral
equation (P, Q, R\c, 1)' = 0, where P, Q, R are rational and integral functions of {x, y).
The singular solution of the differential equation
(Z, M, N\p, 1)» = 0.
if there be a singular solution, is »9 = 0, where S is either = LN — M^, or a factor of
LN — M*. But in general LN—M^ is an indecomposable function, such that LN— M^ = 0
is not a solution of the differential equation, and this being so, there is no singular
solution ; viz. a differential equation {L, M, N'^p, \y = 0, where L, M, N are rational
and integral functions of {x, y), has not in general any singular solution.
Consider now a system of algebraical curves U = 0, where U is as regards {x, y)
a rational and integral function of the order m, and depends in any manner on an
arbitrary parameter C*. I say that there is always a proper envelope, which envelope
is the singular solution of the differential equation obtained by the elimination of C
from the equation ^=0, and the derived equation in regard to {x, y). It follows
that the differential equation {L, M, N'^p, 1)^ = 0, which has no singular solution, does
not admit of an integral of the form in question U=0, viz. an integral representing a
system of algebraic curves.
* The expressions in the text may be understood as extending to the case where U is a, function of any
number (o) of constants c, , c, Ca, connected by an (o-l)fold relation, U thus virtuaUy depending on a
single arbitrary parameter.
3—2
20 ON THE THEORY OF THE SINGULAR SOLUTIONS OF [636
The theorem just referred to, that the system of algebraic curves 17=0 has
always an envelope, is an interesting theorem, which I proceed to prove. Assume
that in general, that is, for an arbitrary value of the parameter, the equation U=0
represents a curve of the order m, with B nodes and k cusps (and therefore of the
class n, with i inflexions and t double tangents, the numbers m, S, k, n, r, i being
connected by Pliicker's equations); for particular values of the parameter, the values
of 8 and K may be increased, or the curve may break up, but this is immaterial.
The consecutive curve U+8cdcU=0 is a curve of the same order m, with 8 nodes
and K cusps, consecutive to the nodes and cusps of the original curve U, and the two
curves intersect in m' points; but of these, there are 2 coinciding with each node,
and 3 coinciding with each cusp of the curve U = 0, as at once appears by drawing
a curve with a node or a cusp, and the consecutive curve with a consecutive node
or cusp ; the number of the remaining intersections is = m- — 2S — 3/c, and the envelope
is the locus of these m' — 2S — Sk points. Observe that the two curves have in common
?i' tangents ; but of these, 2 coincide with each double tangent and 3 coincide with
each stationary tangent of the curve U=0, viz. the number of the remaining common
tangents is =n*— 2t— 3i (which is =m^— 28— 3/^): and that these n*— 2t— 3i common
tangents are indefinitely near to the m' - 28 — 3k common points respectively, and are
in fact the tangents of the envelope at the m' — 28 — 3/e points respectively. Now in
an algebraic curve we have m 4- n = m' — 28 — 3k, viz. the number m' — 28 — 3« cannot
be =0, and we have therefore always an envelope the locus of the system of the
to' — 28 — 3/c points. It might be thought that the conclusion extends to transcendental
curves ; if this were so, the result would prove too much, viz. it would follow that a
differential equation (L, M, N\p, 1)^ = 0 without a singular solution had no general
integral ; but it will appear by an example that the theorem as to the envelope does
not extend to transcendental curves.
Ex. 1.
j»» - (1 - 2/-) = 0, that is, dy^ - (\ - y^) da? = 0.
Here there is no algebraical integral, but there is a quasi-algebraical integral of
the form (P, Q, R\c, 1)' = 0; viz. starting with the form y = sin(a;+0) and expressing
sin G and cos G rationally in terms of a new parameter, this is
d^ (y + cos x) — 2c sin x-\-{y — cos x) = 0,
where the coefficients are one-valued functions of (x, y). The discriminant of the
differential equation in regard to p and that of the integral equation in regard to c
are each =y'— 1, and we have a true singulai* solution y^—\=0.
Ex. 2.
(l_a^)p»_(l_y)=0,
that is,
(1 - !t?)dy' -{\-f)da?==0.
We have here an algebraic integral of the proper form, which is at once derived
from the circular form
C = cos"'a; + cos~*y
636] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 21
by changing the constant, viz. this is
& - icayy -{l-a?- y") = 0.
The two discriminants are here each = (a^ — 1) (y^ — 1), and we have
as a true singular solution. The curves are in fact the system of conies (ellipses and
hyperbolas) each touching the four lines a; = 1, x= — 1, y = 1, y = —\.
Ex. 3.
(1 - yOi>' - 1 = 0, that is, {\- y')dy^- -da? = Q.
This is an extremely interesting example : the curve is the orthogonal trajectory
of the system of sinusoids y = sin {x + c), which is the integral of Example 1 ; and we
thus at once see that the real portion of the curve is wholly included between the
lines y= — 1, y= + \, being an infinite continuous curve, having a series of equidistant
cusps alternately at the one and the other line, and obtained by the continued
repetition of the finite portion included between two consecutive cusps on the same
line. The discriminant of the differential equation equated to zero gives 2/^ — 1=0,
the equation of the two lines in question ; but this does not satisfy the differential
equation, and it is consequently pot a singular solution ; by what precedes, it appears
that it is, in fact, a cusp-locus.
We thus see that the curves which represent the integi-al equation have no real
envelope; but it is to be further shown that there is no imaginary envelope, and that
the curve obtained by the elimination of the parameter is, in fact, made up of a
(imaginary) node-locus and of the foregoing cusp-locus.
The curve is properly represented by taking x, y each of them a one- valued
function of the parameter 0, viz. we may write
y = cos 0,
X = c + ^0 - 1 sm20.
In fact, these values give
^ = -sin^, J = i(l-cos2^) = 8in^0,
and therefore
1 -1
P~ sin ^ ~ V(l -y^)'
that is, (1 — y'')j9' — 1 = 0, the differential equation.
It is obvious that to a given value uf the parameter there corresponds a single
point of the curve; and it is to be shown that, conversely, to a given point of the
curve corresponds in general a single value of the parameter.
22 ON THE THEX)RY OF THE SINGULAR SOLUTIONS OF [636
Suppose the coordinates of the given point are y = co8 0, x = c + ^a— ^sm2et, where
a is a determinate quantity; then, to find 0, we have
cos ^ = cos a, 20 — sin 26 = 2a.- sin 2a,
The first equation gives 6 = 2mir ± a, and the second equation then is
imir ± 2a T sin 2a = 2a — sin 2a ;
viz. taking the upper signs, this is imw = 0, giving m = 0 and 0 = a; and, taking the
lower signs, it is nnr = a — sin a, which, a being given, is not in general satisfied ;
hence to the given point there corresponds only the value a of the parameter 0. If,
however, a is such that a — sin a is equal to a multiple of v, say r-n-, then the last-
mentioned equation is satisfied by the value m = r, so that to the given point of the
curve correspond the two values a and 2rir — a of the parameter ; these values are
in general unequal, and the point is then a node; but they may be equal, viz. this
is so if o = r7r (the point on the curve being then y = cosr7r, =+1, x=c + ^rir), and
the point is then a cusp; showing what was known, that there are on each of the
lines y = — \, y = + l, an infinite series of equidistant cusps.
More definitely, suppose a = rir ±0, where /3 is a root of the equation 2/9 — sin 2/8 = 0,
then
sin 2a = + sin 2/3, 2a - sin 2o = 2r'7r + (2/S - sin 2/S) = 2r7r,
and to the given point on the curve correspond the two values a and 2r-jr — a of
the parameter. If /8 = 0, we have, as above, the cusps on the two lines y = -\-\,
y = — \ respectively ; but if yS be an imaginary root of the equation 2/3 — sin 2/8 = 0,
then we have an infinite series of nodes on the imaginary line y = cos r-rr cos /3 ; and
there are an infinite number of such lines corresponding to the different imaginary
roots of the equation 2y3 — sin 2/3 = 0.
From the form in which the equation of the curve is given, we cannot directly
form the equation of the envelope by equating to zero the discriminant in regard to
the constant c; but we may determine the intersections of the curve by the con-
secutive curve (corresponding to a value c + Sc of the constant), and thus determine
the locus of these intersections.
Consider for a moment the curves belonging to the constants c, Ci, and let 0, 0i
be the values of the pai-ameter 6 belonging to the points of intersection; we have
cos ^ = cos ^i, 4c -1- 2^ — sin 2^ = 4ci -(- 2^1 — sin 2^, ; we have 0i = 2rir + 6, but we cannot
thereby satisfy the second equation ; or else ^i = 2r7r — 6, giving
4c + 20 - sin 26 = 4c, + 4>r7r -20+ sin 26,
that is, 20 — sin 20 = 2ci — 2c + 2»-7r ; and we have thus corresponding to any given value
of r a series of values of 0, viz. these are 0 = ttt -f- /8, where /8 is any root of the equa-
tion
2^- sin 2/3= 2c, -2c.
636] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 23
In particular, taking Ci = c, the intersections are given by 0 = r7r + /3, where /8 is
any root of the equation 2/9 — sin 2/8 = 0 ; viz. we have thus an infinite number of
intersections lying on each of the lines y = cos m- cos y8. If /3 = 0, the intersections lie
on the two lines y = 1, y = —l respectively ; if /9 be an imaginary root of the
equation 2/3 — sin 2/3 = 0, then they lie on the imaginary lines y = cos rir cos 0. But by
what precedes, it is clear that in the former case the intersections are nothing else
than the cusps on the lines y=l, y = — 1; and in the latter case nothing else than
the nodes on the lines y = cos rtr cos /3 ; viz. there is no proper envelope, but instead
thereof we have lines of cusps and of nodes.
Ex. 4.
that is,
(1 - f)dy' - (1 - a^)da^=0.
I have not examined this; the curve is the series of orthogonal trajectories of
the conies of Example 2, and the integral equation may be represented by ij = cos 6,
X = cos if), where c = {26 — sin 20) — (2<^ — sin 2^).
Equating to zero the discriminant of the differential equation, we have (l—y'){l—a:^)=0,
viz. the four lines x = l, x = — l, y=l, y = —l; this is not an envelope, but a locus
of cuspe. ^
24 [637
637.
ON A DIFFERENTIAL EQUATION IN THE THEORY OF ELLIPTIC
FUNCTIONS.
[From the Messenger of Mathematics, vol. vi. (1877), p. 29.]
In the differential equation
Q._Q(, + l)_3 = 3(l-^)f.
considered Messenger, t. iv., pp. 69 and 110, [594] and [597], writing Q = x and
k+ T=y, the equation becomes
" Z + xy-a? '
and we have, as a particuleir solution.
To verify this, observe that from the value of y
and the equation becomes
3 j^{(a;'-6a^-3)>-64a^}
|^(«»-1>'= J(a^_i)(-B»_9) '
viz. this is
{a? -iy{as'-Q)={x'-Qa?- 3)«-64ar',
which is right.
638] 25
638.
ON A 9-FORMULA LEADING TO AN EXPRESSION FOR E,.
[From the Messenger of Mathematics, vol. vi. (1877), pp. 63 — 66.]
It is to be shown that we have identically
_ 1 - 9g - 25^^ + 49^ + Slg" - ...
or, what Lb the same thing,
(l-2, + 2^-2,.+ ...)-16(^, + ^-^4^ + ...)
_ 1 + 9g + 2o(f + 499" + Slg'" + . . .
~ 1 +5' + ^=+5«4-?'°+ ...
{A);
(B),
where the form (A) is that intended to be made use of, but the form (B) is rather
more convenient for the demonstration.
We have
(l-2, + 2^-2,'+...)'=l + 8|j=^^ + J^,-j^4-...}.
(Jacobi, Fund. Nova, p. 188, Oes. Werke, t. i., p. 239), taking the formula as there
written down, and changing q into — q.
Also, if for a moment •
X = 1 + 2 + ^' + 5" + 2'" + &c.,
and
^, dX
^'- dq'
C. X.
26 ON A (/-FORMULA LEADING TO AN EXPRESSION FOR E,.
SO that
then
qX' = q + Sf + 6^+ I0q"> + &c.,
Z + 85Z' = 1 + 9? + 259* + 4V + 81g" + &c,
60 that the right-hand side of (B) is
Z + 8gZ' ,^o„X'
But (Fund. Nova, p. 185, Oes. Werke, t. i., p. 237),
so that
l-g'.l-q'.l-q'...
1-q .l-(f.l-(f ...'
Z' _ -2q _ 4^ _ 6g* _
Z~l-9» 1-9* l-g« ■■■
1 3g' 00*
1— Q' 1— grs 1 — 5*
And the equation (5) intended to be proved thus becomes
1+ 8-
-l+8ol-^*--^^2' ^-
39'
og*
1 —q 1—5^ 1—5'
viz. omitting the terms unity, dividing by 8q, and transposing, this is
1-g^ l-5*l-g« ■"
2g I 4(f 6g'
q' i. -q' l—q*
L ^_JS* _ =0
1-q l-f l-g» ■■
The second and third lines unite together, and the equation becomes
_ 1 2q _ S^ ^ _
~ T+^ ■•■ r+ 3= ~ f +^ ■*" TTg* ■■■
l-g' l+5»l-7' H-^
t
■q 1— g** i—q° i—q
1 39' _ j9L_ V _ ^Q,
l-Q 1-'^ 1-Q' 1-9' ■■■ '
[638
638] ON A (^-FORMULA LEADING TO AN EXPRESSION FOR E^. 27
or, collecting and arranging,
1+q 1+5= l+q' 1+q^ l+(f
, 1 3g- 5q*
an identity which it is easy to verify to any number of terms. But to prove it
directly, we have only to add the pairs of terms in the alternate columns; calling the
left-hand side Fq, we thus obtain
^ ^\ l+q' 1+y 1+5* •••
1—q^ 1 — g«
viz. this equation is Fq = 2qF(q^); and thence
Fq=:2Y+"-F{q*) = 2'q'+^+*F(^) = &;c.;
we thus have Fq = 0.
The equation (fi), or, what is .the same thing, the equation (A) is thus proved.
Reverting to the equation (A), we have
(l+2q + 2q*+...y='^,
IT
(Jacobi, Fund. Nova, p. 188, Oes. Werke, t. i., p. 239),
(ib., p. 135; ib., p. 189),
if q = e ^, and K, E^ are the complete functions FJc, EJc.
The left-hand side of the equation is thus
and we have
/_ J 2Ea _ TT^ l-9q'- 25^ + 'i9q' + Big" - ...
V kJ'iK"' l-q'-q^ + q' + q'o-...
which is a new expression for £^, as a ^-function. The expression on the right-hand
side presents itself, Clebsch, Theorie der Elastidtdt (Leipzig, 1862), p. 162, and must
have been obtained by him as a value for ( — 1+ „M; but there is no statement
that this is so, nor anything to show how this form of ^-function was arrived at.
Mr Todhunter called my attention to the passage in Clebsch.
4—2
28
[639
639.
AN ELEMENTARY CONSTRUCTION IN OPTICS.
[From the Messenger- of Mathematics, vol. vi. (1877), pp. 81, 82.]
Consider two lines meeting at a point P, and a point A ; through A, draw at
right angles to AP, a line meeting the two lines in the points U, V respectively ;
and through the same point A draw any other line meeting the two lines in the
points U', V respectively ; also let the points u', v' be the feet of the perpendiculars
let fall from U', V respectively on the line UV: then we have
Au''^ Av''
''AU'^AV
The theorem can be proved at once without any difficulty. It answere to the optical
construction, according to which, if UPV represents the path of a ray through
a convex lens AP, then the thin pencil, axis U'P and centre U', converges after
refraction to the point V, where U'V are in lined with A the centre of the lens;
considering as usual the inclinations to the axis as small, we have approximately
AV = Av, AU' = Au', and the theorem is
1 1
AU''^ AV'
1 J^
AU'^AV
1^
AF'
if AF is the focal length of the lens.
In the original theorem, the line UV need not be at right angles to AP, but
may be any line whatever; the projecting lines U'u' and V'v' must then be parallel
to AP, and the theorem remains true.
640] 29
640.
FURTHER NOTE ON MR MARTIN'S PAPER " ON THE INTEGRALS
OF SOME DIFFERENTIALS."
[From the Messenger of Mathematics, vol. VI. (1877), p. 82.]
See paper, Number 633; this further note relates also to a detail.
30 [641
641.
ON THE FLEXURE OF A SPHERICAL SURFACE.
[From the Messenger of Mathematics, vol. vi. (1877), pp. 88 — 90.]
It is known that an inextensible spherical surface, or to fix the ideas the spherical
quadrilateral included between two arcs of meridian and two arcs of parallel, may be
bent in suchwise as to be part of a surface of revolution, the meridians and parallels
of the spherical surface being meridians and parallels of the new surface, and, more-
over, the radius of each parallel of the spherical surface being in the new surface
altered in the constant ratio k to 1. We have, in fact, on the spherical surface, writing
p for the latitude and q for the longitude, and the radius being unity,
a: = cospcosq,
y = cos^ sin q,
z=smp,
values which give
da? + dy^-\- dz- = dp" + cos= p df.
This last equation is satisfied by the values
9
X
= C0SpC08|;,
y = cos p am ^ ,
z=^E{k, p),
where E {k, «), = I ^J{\—J(^6m"p)dp, is the elliptic function of the second kind; or rather,
Jo
this is so when A; < 1, but the same notation may be used when k > 1. These values
give the deformation in question.
The two cases to be considered are k<l, and k>\\ we take in each case a
spherical quadrilateral A BCD (fig. 1), bounded by AB (an arc of the equator), the
arc of parallel CD, and the two arcs of meridian AD and BG. In the first case,
there is no limit to the latitude AD, = BG, or taking these =90°, we may in place
641]
ON THE FLEXURE OF A SPHEBICAL SURFACE.
31
of the quadrilateral ABCD consider the birectangular triangle ABE; the new form of
this is A'B'E', where the radius OA', =k.OA, = ^•, is less than the original radius
unity, but OE', =EJe, is greater than the same radius unity. The surface has at E'
a conical point, the semi-aperture of the cone being =tan-'p, if as usual k' = \/{\-lif)\
to verify this, writing for convenience q = 0, we have for the meridian section x = cos^,
z = E{k, p), and thence
dz
V(l
dz
—. , which for « =
smp
dz
p=90° gives ^
= -k'.
Observe also that for p = 0, ^ = xs , viz. the surface of i-evolution cuts the plane of
the equator at right angles.
There is no limit to the arc AB, it may be = 360°, viz. we must in this case
cut the hemisphere along a meridian to allow of the deformation; or it may exceed
360°, the hemisphere spherical surface being in this case conceived of as wrapping
indefinitely over itself, and we may instead of the half lune E'A'B', consider the lune
included between two meridians extending from pole to pole, and therefore the whole
spherical surface, conceived of as wrapping indefinitely over itself; the result is, that
this may be deformed into a surface of revolution, which, in its general form, resembles
that obtained by the revolution of an arc less than a semi-circle round its chord ;
the half-chord being greater, and the versed-sine less than the radius of the original
sphere.
If A; > 1, there is obviously a limit to the latitude AD, = BC, of the spherical
quadrilateral; viz. this is equal to 8in~'T- Supposing that in the quadrilateral ABCD
iC
(fig. 1) the latitude has this limiting value, then (see fig. 2) the new form is
A"B"C"D", where along the bounding arc CB" the tangent plane is horizontal ; viz.
as before ^ = -^!l!^!??^^ , =0 for » = 8in-J. It is to be observed, that the
dx smp k
radii for the parallels A"B" and CD" are k and ^-cos^ respectively; the difference of
32
ON THE FLEXURE OF A SPHERICAL SURFACE.
[641
these is k(l—coap), which, however great k is, must be less than the arc of meridian
^ \ COS t)
A"D", =»; substituting for k the value -. — , the condition is — -. -<p, viz. this
smj* Binp '^
Fig. a.
is tan i^p < p, which is true for every value up to ^ = 90°. But, more than this, we
should have
k'{l- eospY + E^ (k, p) < f,
viz. writing as before k = -. — , this is
° smp
Ei^p'p)<p'-'^-'^P'
this must be true, although (relating as it does to a form of E for which k is greater
than 1) there might be some difficulty in verifying it.
There is, as in the first case, no limit to the value of AB, viz. this may be
= 360°, the spherical zone being then cut along a meridian, or it may be greater
than 360° ; and, moreover, the spherical quadrilateral may extend south of the equator,
but of course so that the limiting south latitude does not extend beyond the foregoing
value sin~' r : viz. we may have a zone between the latitudes ± sin"' v , which may
be a complete zone from longitude 0° to 360° or to any greater value than 360°.
The result is, that the zone is deformed into a surface of revolution, which in its
general form resembles that obtained by the revolution of a half-circle or half-ellipse
about a line parallel to and beyond its bounding diameter, the bounding half-diameter
being less, and the gi-eatest radius of rotation greater, than the radius of the original
sphere.
6421
-■ 33
642.
ON A DIFFERENTIAL RELATION BETWEEN THE SIDES OF A
QUADRANGLE.
[From the Messenger of Mathematics, vol. vi. (1877), pp. 99 — 101.]
Let the sides and diagonals YZ, ZX, XY, OX, OY, OZ of a quadrangle be
/, ff, h, a, b, c, and let the component triangles be denoted as follows :
A=AYZO, ={b, c.f),
B = AZXO, ={c, a, g),
C=AXYO, =(a,b,h),
n = AXYZ. =(f,g,h}.
viz. A, B, C, n are the triangles whose sides are (b, c, f), (c, a, g), (a, b, h), (/, g, h)
respectively, so that il=A + B + C. Then we have between (a, b,c,f,g, h) an equation
giving rise to a differential relation, which may be written
n (Aada + Bbdb + Cede) - {BCfdf-{- CAgdg + ABhdh) = 0.
This may be proved geometrically and analytically. First, for the geometrical proof,
it ia enough to prove that, when a and b alone vary, the relation between the in-
crements is Aada + Bbdb = 0 ; for then a and g alone varying, the relation between
c. X. 5
34
ON A DIFFERENTIAL RELATION BETWEEN
[642
the increments will be Q,ada-Cgdg = 0 (as to the negative sign it is clear from the
figure that a, g will increase or diminish together): and we thence at once infer the
general relation.
We have consequently to prove that, considering a and h as alone vaiiable,
Adda + Bhdh = 0 ;
or, what is the same thing,
iida : -bdb = XOZ : YOZ.
The points XYZ remain fixed ; but 0 moves through the infinitesimal arc 00',
centre Z, which may be considered as situate in the right line OM drawn from 0
at right angles to ZO, and meeting XY produced in the point M. And then, writing
for a moment /.OXY=X, /.OYX=Y, /l0MY=M, we find at once
da = 00' cos{X-\-M),
- db = 00' cos (Y-M);
ada _acos(X + M)
that is,
da _ cos (X + M)
'S~c6i(Y-M)'
or
bdb b cos (Y-M)-
But drawing Xa, Yfi each of them at right angles to ZO, we have a cos (X + M) = Xa,
b COS (Y - M) = Y^, and evidently XOZ : YOZ = Xa : F/3; whence the equation is
ada XOZ . • . • .i • j i j.-
— 7. ji = ttttw . which is the required relation.
OCK* lUZ
For the analytical proof, it is to be observed that the relation between a, b, c,
/, g, h is a. quadric relation in the quantities a^ 6=, c', /^ g", h^ respectively; this
may be written
1 re» a*
by + by + <^h* + c*h' - (b' + (t) g-/r - {g'' + h'} 6V
-((>^-,r){g^--h'}
+ {b'--h')(c^-g^)
^I'^-c^-^-h!'
+ 1
+ 1
= 0;
say for a moment this is .4 + Ba^ + Ca* = 0, where
A= b'g* + by + c'h* + &h^ +fHb^ - A") (c- - f)
- (6» + c=) g^h^ - ig"- + h?) ¥c'
B = -{¥- c') ig' - h') +/'{- ¥ -C-f- h?) +f\
C= r-;
then we have as usual
, , ada + J r, bdb + &c. = 0,
642] THE SIDES OF A QUADRANGLE. • 35-
where
= Ca- + 15.
J du
But in virtue of u = 0, we have
(Ca' + ^By= C{Ca* + Ba' + A) + l(B'- AG),
that is, ^^„ = ^/(B'-4iAC); and here B'-4iAC is a quai-tic function of/'-, which is
easily seen to reduce itself to the form
r--i9 + hy/' -{9- h)\r- -{h + cff- -{b- cy.
The coefficients of Mb, cdc, &c., are given as expressions of the like form; substituting
their values, the differential relation is
V{/' -(9 + hyf' -ig- Kyf' -{b + cyp - (6 - cy] ada + &c. = 0,
which is, in fact, the foregoing result.
It is right to notice that there are in all 16 linear factors,
f+9 + f',
b + c+f.
c + a+g.
a + b + h
say
d ,
f .
9 .
h ,
-f+9 + l>'
-b + c+f,
- c+a+ff.
— a + b + h
d' ,
/'.
9\
h'.
f-9 + k
b-c+f.
c-a+g.
a — b + h
d".
/".
9",
h".
/+9-l'>
b + c-f,
c+a — g,
a + b — h
d'".
/'",
9"'>
W"
and this being so, the coefficients of ada, bdb, cdc, fdf, gdg, hdh, are
^/(dd'd"d"' . f/'f"f"l - '^i99'9"9"' ■ hh'h"h"' ),
^{dd'd"d"' . gg'g"g"' ), - ^{hh'h"h"' . fff'f" ),
^{dd'd"d"' . hh'h"h"' ), - V(//7T" • 99'9"9"').
respectively.
We may imagine the quadrilateral ZOXY composed of the four rods ZO, OX,
XY, YZ (lengths c, a, h, f as before) jointed together at the angles, and kept in
equilibrium by forces B, G acting along the diagonals OY(=b), ZX (= a) respectively.
We have c, a, h, f given constants, and the relation ^ (a, b, c, f, g, h) = 0, which
connects the six quantities is the relation between the two variable diagonals (g, b);
by what precedes, the differential relation <f>'g . dg + <^'b . db = 0 is equivalent to
HBbdb — CAgdg = 0. By virtual velocities we have as the condition of equilibrium
Bdb + Gdg = 0 ; hence, eliminating db, dg we have
B __ G__
nBb CAg '
or, say
B _G I 1
b ' g~AXYO.AZYO- AXYZ.^ZXO'
viz. the forces, divided by the diagonals along which they act, are proportional to the
reciprocals of the products of the two pairs of triangles which stand on these diagonals
respectively. The negative sign shows, what is obvious, that the forces must be, one
of them a pull, the other a push.
5—2
36 [643
643.
ON A QUAKTIC CURVE WITH TWO ODD BRANCHES.
[From the Messenger of Matfiematics, vol. vi. (1877), pp. 107, 108.]
It is a known theorem that the branches of a plane curve are even or odd ; viz.
two even branches, or an even and an odd branch (whether of the same curve or of
different curves) intersect in an even number (it may be 0, and this is to be under-
stood throughout) of real points; but two odd branches (of the same curve or of
different curves) intersect in an odd number of real points*.
In particular, a right line is an odd branch, and hence it meets any even branch
of a curve in an even number of real points, and an odd branch in an odd number
of real points ; or (what is the same thing) an even branch is one which is met by
any right line whatever in an even number of real points; and an odd branch is one
that is met by any right line whatever in an odd number of real points.
It is to be observed, that the simple term branch is used to denote what has
been called a complete branch, viz. the partial branches which touch an asymptote at
its opposite extremities are considered as parts of one and the same branch, and so
in other cases. Thus a quadric curve, whether ellipse, parabola, or hyperbola, is one
even branch; a cubic curve is either one odd branch, or else it is an odd branch
and an even branch ; and generally a curve of an odd order has always an odd number
of odd branches, and a curve of an even order has always an even number of odd
branches.
A curve without nodes has at most one odd branch ; for if there were two, these
would intersect in a real point, which would be a real node on the curve. In parti-
cular, a quartic curve having two odd branches must have a real node ; this however
may be, as in the instance about to be given, a node at infinity.
A simple instance of a quartic curve with two odd branches is that represented
by the equation
{a? - 1) (y» + 1) - 2mxy = 0,
* The two branches must be distinct branches; a branch whether odd or even does not of necessity
intersect itself (have upon it any real node), bat it may intersect itself in an odd, or an even, number of real
points.
643] ON A QUARTIC CURVE WITH TWO ODD BRANCHES,
or, what is the same thing,
37
where
or say
a- = ^ {2 + m- + m x/(4 + 7n^)],
-= J {2 + m= - m V(4 + m%
80 that m being positive a>l, and the curve consists of two real branches included
between the lines a; = a, a;= -, and the lines a;=-a, x = -- respectively; each of these
lines touches the curve in a real point, viz. a; having any one of the last-mentioned
values, the value of y at the point of contact is y=^^; and between each pair of
lines we have the asymptote x = + l or x = -l. Hence the curve has the form shown
in the figure, and it is thereby evident, that each branch of the curve is met by
■<>-
any real right line whatever in one real point, or else in three real points. The
numerical values in the figure are a = f, m = ^, whence also x = a or — , y=l, and
x = — a or - , w = — 1.
The curve has two nodes at infinity, viz. writing the equation in the form
{af-z'')(y + z'')-man/z'' = 0,
that is,
a^y^ + z^(x'-y''-7nay) + z* = 0,
it appears that the points (z = 0, ai = 0), (z = 0, y = 0) are each of them a node. The
first of these (z = 0, x = 0) is the real intersection of the two odd branches: the other
of them is a conjugate point.
38 [644
644.
NOTE ON MAGIC SQUARES.
[From the Messengm- of Mathematics, vol. vi. (1877), p. 168.]
In a magic square of any odd order, formed according to the ordinary process,
there is a tolerably simple analytical expression for the number which occupies any
given compartment ; thus taking the square of 21, let the dexter diagonals (N.W. to S.E.)
commencing from the N.E. corner compartment, be numbered 1, 2, 3,.., 20, 21, 20',
19', . . , 2', 1', the diagonals of course containing these numbers of compartments respect-
ively ; and in any diagonal let the compartments reckoning from the top line be
numbered 1, 2, 3,.., respectively; then if D^^^ (or D'g^^ as the case may be) denotes
the number in the compartment </> of the diagonal 6 or ff, we have
2) ,9+,,^= 20(9+ 10 + ^,
i),fl ,«= 206' + 231 + .^ (-21).
/>W.,« = -22^ + 430 + <^,
Ua ,« = -22^ + 231 + <^(-21),
where in the second and fourth expressions the term —21 is to be retained only if
<{)> 0; if (f> 1^ 6, it is to be omitted. There would be a like fonnulse for a square
of any odd order, and it would be easy to %vrite down the formulae for the general
value 2n + l: but I have preferred to give them for a specific case.
645]
39
645.
A SMITH'S PRIZE PAPER, 1877.
[From the Messenger of Mathematics, vol. vi. (1877), pp. 173—182.]
The paper was as follows:
1. Show (independently of the theory of roots) how, if a; satisfies an equation
of the order n, a given rational function of x can in general be expressed as a
rational and integral function of the oi-der n — 1. State the theorem in a more
precise foi-m, so as to make it true universally.
2. Investigate the form of the factors of 1 + sin(2}i + 1)* considered as a
function of sin a; ; and give the formula in the two cases, 2?t + 1 = 3 and 5 respectively.
3. Write down the substitutions which do not alter the function ab + cd; and
explain the constitution of the group.
4. Find in a form adapted for calculation an approximate value for the sum of
the middle 2o + 1 terms of the expansion of (1 + 1)=^, n being a large number, and
a small in comparison therewith.
Obtain thence a complete and precise statement of the theorem that in a large
number of tosses the numbers of heads and tails will probably be nearly equal.
5. A point in space is represented on a given plane by its projections from
two fixed points. Show how a problem relating to points, lines, and planes, is
thereby reduced to a problem in piano; and apply the method to construct the line
of intersection of two planes each passing through three given points.
6. A weight is supported on a tripod of three unequal legs resting on a smooth
horizontal plane, their feet connected in pairs by strings of given lengths. Show how
to determine the tensions of the several strings.
7. Explain the ordinary configuration of a system of isoparametric lines on a
spherical surface ; for instance, what is the configuration when there are two points
of minimum value, and one point of maximum value, of the parameter ?
40 A smith's prize paper, 1877. [645
8. Find the attraction of an infinite circular cylinder, of uniform density, on a
given exterior or interior point.
9. Determine the number of arbitrary constants contained in the equation of a
surface of the order r which passes through the curve of intersection of two given
surfaces of the orders m and n respectively.
10. Find, for the several values of p, the number of the conies passing through
'p given points and touching h—p given lines ; and, in each case, show how to obtain
(in point-coordinates or line-coordinates, as may be most simple) the equations of the
conies satisfying the conditions in question.
11. Investigate the theory of the linear transformation of a ternary quadric
function into itself
12. Explain the theory of the solution of a partial differential equation, given
function of x, y, z, p, q, »•= arbitrary constant H; where p, q, r are the differential
coefficients of the dependent variable u in regard to the independent variables x, y, z
respectively.
I propose, not (as in former years) to give complete solutions, but only to notice
in more or less detail the leading points in the several questions.
1. The expression is of course required, not only for a given integral function
of X, but for a given fractional function. The case where the given function is
integral presents no difficulty ; when the given function is fractional, the most simple
case is where it is = — — ; supposing the equation to be / {x) = 0, here dividing
SG ~~' Cb
f{x) by x — a, we have a quotient R{x) which is a rational and integi-al function of
an order not exceeding n — 1, and a remainder which is =/(«) ; that is,
=^)=ii(.)4-^:
x—a x—a
or, in virtue of the given equation — - = — ii(a;), viz. we have thus in the
° ^ x—a ^ x—a
requiied fonn. But ify"(a) = 0, then we do not obtain such an expression of .
3u ^~ €b
It has to be shown that the like considerations apply to any fractional function, and
the precise form of the theorem is, that any rational function of x which does not
become infinite for any value of x satisfying the given equation, can be expressed as a
rational and integral function of an order not exceeding n—\.
2. The function 1 — sin(2w-|- l)a; is a rational and integral function of sin a;, of
the order 2n4-l; which if n is even (or 2n + l = 4p+l) contains, as is at once
seen, the factor 1 — sin x, but if n is odd (or 2n+\ = 4ep — 1) the factor (1 + sui x).
Sill oc
Suppose that any other factor is 1 — r— , where sin a not = + 1 ; then this will be
a double factor if only sin a; = sin o satisfies the condition
0 = -j-i — fl - sin (2« + 1) x],
o . sm a; ' ^ '
645j A smith's prize paper, 1877. 41
., , . „ cos(2n+ l)a; ^, , .
that IS, 0= ^^^- ; the value in question gives sm(2w+ l)a;= 1, and therefore
co8(2n+l)a;=0; and it does not give cosa; = 0; hence every such factor 1-^?
... sin a
is a double factor, or we have
1 - sin (2n + 1) a; = (1 + sin a;) n f 1 - ~^]\
\ sm a/
Or the like result might be obtained by considering instead of 1 - sin (2n + 1) x,
the more general function
sin (2n + 1) a + sin (2n + 1) a;,
and finally assuming a = Jtt.
3. Relates to a theory which is not, but ought to be, treated of in the text
books of the University. See Serret's Algehre Superieicre, t. ii.. Sect. IV.
The substitutions which leave ab + cd unaltered are
1 I 1, that is, the letters remain unchanged,
(ab), that is, a and b are interchanged,
(cd), that is, c and d fere interchanged,
(ab) (cd), that is, a and b and also c and d are simultaneously interchanged,
(ac) (bd), same with a and c, b and d,
(ad) (be), same with a and d, b and c,
(acbd), that is, we cyclically change a into c, c into b, b into d, and d into a,
(adbc), that is, we cyclically change a into d, d into b, b into c, and c into a.
a
/8
7
S
e
?
0
viz. we have eight substitutions 1, a, /9, 7, S, e, f, ^ forming a group; that is, the
product of any two of them, in either order, is a substitution of the group (or,
what is the same thing, the effect of the successive performance of the two upon
any arrangement abed is the same as that of the performance thereon of some other
substitution of the group); thus we have 0.'=!, ^ = 1, 7"=!, a/8 = y3a = 7, &c. ; the
system of these equations, which verify that the set of substitutions form a group,
defines the constitution of the group — thus to take a more simple instance, a group
of 4 may be 1, a, a^ o^ («*= 1) or 1, a, /3, a^, (d? = \, ^ = 1, a/3 = /9a).
4. The expression of the general coefficient is
1.2...2n
1.2 ...n-a.1.2... n + a'
which can be transformed by the well-known formula
1.2...«=n"+*V('»-)e-",
C. X.
42 A smith's prize paper, 1877. [645
viz. the coefficient thus becomes
2«" 1
W? a\»+«+* ■
V(»i7r)/j_«\"-+»/j^aV
Now a is supposed small in comparison with »^, and the factors in the denominator
have the logarithms
and
hence the denominator is = e" , and the final approximate value of the coefficient is
2s»
nj{nir)
6 n
Hence, converting as usual the sum into a definite integral, we have the sum of the
2a + 1 coefficients
= -r, — c e « da,
's/(«'r)y_,
or, what is the same thing,
For the chance that the number of tosses lies between n+ a and n — a, this has
merely to be divided by 2*" ; hence writing a = kn, the chance that the number may
be between n{l +A;) and m(l —k) is
1 f*Vn
V(t)J -tV«
where observe that the integral, taken with the limits oo , — x has the value VC"")-
Considering A as a given fraction however small, by increasing n we make
k i\/(n) as large as we please, and therefore the integral, as nearly as we please
= V(t), or the chance as nearly as we please = 1 ; and hence the complete and
precise statement of the theorem, viz. by sufficiently increasing the number of tosses,
the probability that the deviation from equality shall be any given percentage (as
small as we please) of the whole number of tosses, can be made as nearly as we
please equal to certainty.
Further, restoring a in place of kn, the chance of a number between n + a and
jt — a is
1 [•"*
Vn
645] A smith's peize paper, 1877. 43
which when -t—-. is small is =-77 — ;, (more accurately -77 — r-, when a is small);
V(«) V(w7r) V •' V(«7r) /
hence, however large o is, the chance of a deviation from equality not exceeding
± a, continually diminishes with n, and by making n sufficiently large becomes as
small as we please.
5. The point is represented in the given plane by two points which lie in
lined with a fixed point (say 0) of that plane, viz. 0 is the intersection of the
given plane by the line which joins the two projecting points.
A line is represented on the given plane by two lines, viz. these are the
projections of the line from the two given points; each point of the line is represented
by the points of intersection of the two lines by any line through 0.
A plane may be represented on the given plane by means of its trace thereon,
and of the two points {in lined with 0) which represent any point of the plane.
Thus any problem relating to points, lines, and planes, in space is converted into
a problem of plane geometry. For instance, to find the trace on the given plane of
a plane through three given points A, B, C, the three points are represented by
means of the pairs of points A^, A^; Bi, B^; C,, C„ the points of each pair lying
in lined with 0 ; the required trace passes through the intersections with the given
plane of the lines _ BG, CA, AB respectively, and we hence find it as the line
through the three points which are the intersections of Bfii, Bfi^, of G^Ai, G^A^,
and of AiB^, AJi^ respectively; that these points are in a line is a theorem of
plane geometry, which, if not previously known, would have at once been given by
the construction.
6. The solution ought obviously to be obtained from the principle of virtual
velocities ; taking a, b, c for the lengths of the legs, /, g, h for the lengths of the
strings, and z for the height of the summit, ^ is a known function of a, b, c, f, g, h,
(2 is in fact = -jr , where F, the volume of the tetrahedron, is a given function of
a, i", t^ f, 9, h; and A, the area of the base, is a given function of / g, hj.
Writing then F, G, H for the tensions, and W for the weight, and regarding
z, f, g, h aa variable, the principle gives
Wdz + Fdf+Odg+Hdh = 0,
that is,
respectively.
7. The ordinary case is when an isoparametric line has on one side of it
larger values, on the other side of it smaller values of the parameter; the case
where the isoparametric line is a line of maximum, or of minimum, parameter is
excluded.
6—2
44 A smith's prize paper, 1877. [645
The lines in the neighbourhood of a point of maximum, or of minimum, parameter
are ovals surrounding the point in question, each oval being itself surrounded by the
consecutive oval. Supposing that there are two points of minimum parameter, we
have round each of them a series of ovals, until at length an oval belonging to
the one of them comes to unite itself with an oval belonging to the other, the two
ovals altering themselves into a figure of eight. Surrounding this we have a closed
curve (in the first instance a deeply twice-indented oval) which (in the case supposed
of there being, besides the two points of minimum parameter, a single point of
maximum parameter) is in fact an oval surrounding the point of maximum parameter,
and the remaining curves are the series of ovals surrounding that point. If we
project stereographically from the point of maximum parameter (so that this point
is represented by the points at infinity) we have a figure of eight, each loop
containing within it a series of continually diminishing closed curves, and the figure
of eight itself surrounded by a series of continually increasing closed curves.
8. The investigation by means of the Potential presents the difficulty that the
Potential of the infinite cylinder has no determinate value, as at once appeal's from
the limiting case where the cylinder is reduced to a right line; the difficulty is
perhaps rather apparent than real, inasmuch as the partial differential equations
dV d^V dV
contain only differential coefficients -5— , -j-^ , where -j- as representing an attraction,
and therefore also j— , are determinate. But it is safer to work directly with the
Attraction; the Attraction of an infinite line acts in the perpendicular plane through
the attracted point, and is inversely proportional to the distance ; the problem is
thus reduced to the plane problem of a circle of uniform density, force varying as
(distance)"', attracting a point in its own plane. This is precisely similar to the
case of a sphere with the ordinary law of attraction ; dividing the circle into rings,
each ring exerts an attraction = 0 upon an interior point, and an attraction as if
collected at the centre upon an exterior point. Hence, writing a for the radius of
the cylinder, and r for the distance of the attracted point, the attraction is =irr
for an interior point, and = — ;- for an exterior point.
9. The theory is precisely the same as for curves; taking the surfaces to be
U=0 of the order m, and F=0 of the order n, the general form of the equation
of a surface of the order r (r not less than m or n) is LU + MV = 0, where L is
the general function of the order r — m, and M the general function of the order
r—n; and so long as r is less than m + n, we obtain the required number of
arbitrary constants as the sum of the numbers of the coefficients of L and of M,
less unity. But as soon as r is =m. + n a modification arises, viz. we obtain here
an identity by assuming L=V, M = — U, and so for any larger value of r, we have
an identity by assuming L=V<}), M=— U<f>, where <^ is the general function of the
order r — in — n.
10. The numbers are known to be 1, 2, 4, 4, 2, 1, which values are obtained most
easily (though not in the way which is theoretically most interesting) by finding for
645] A smith's prize paper, 1877. 45
the first three cases the equation of the required conic in point-coordinates; and then,
by changing these into line-coordinates, we have the equations for the remaining three
cases.
p = o: 5 points. The equation of the conic is
(a, b, c,f, g, h\x, y, zf^O,
and we have -5 linear equations to determine the ratios of the coefficients; the number
is therefore = 1.
p=4: 4 points and 1 line. Taking 17=0 and F= 0, the equations of any two
conies each passing through the four points, the equation of the required conic will
be U + \V=Q, and the condition of touching a given line gives a quadric equation
for X. ; the number is therefore = 2.
p = 3: 3 points and 2 lines. In the same manner, by taking U = 0, F= 0, W=0,
for the equations of any three conies through the three points ; or if the equations
of the lines containing the three points in pairs are x=0, y=0, z = 0, then the
equations of the three conies are yz = 0, zx = 0, !vy = 0, and the equation of any conic
through these points is fyz + gzx + hxy = 0 ; the conditions of touching two given lines
^x + riy+^z=0 and ^x+v'y + ^'z=^0, are
v/ vf + v^r v*? + VA V? = 0, v/ vr + ^ff vv + va v?' = o ;
we have thus the ratios »Jf : \Jg : njh linearly determined in terms of V?> V'?. &c, ;
there is no loss of generality in taking Vf. Vf each with a determinate sign, the
signs of ^Jf), &c. being then arbitrary, we have 2*, =16 values of >Jf : 'Jg : \/K and
therefore one-fourth of this = 4, for the number of values of f : g : h; that is, the
number is =4.
11. This is a known theory; taking x^, y„ ^, for the linear functions of x, y, z,
which are such that
(o, b, c,f, g, h^x„ yu z,y = {a, h, c, f, g, h'^x, y, z)\
then assuming a;,, y, , 2,= 2f-a;, 2i;— y, 2f— ^ respectively, we have
(a,. ..$2^-0:, 1r,-y, 'i.ii-zy = {a,...\x, y, z)\
or, omitting terms which destroy each other, and throwing out the factor 4, this is
an equation which is satisfied identically by assuming
a^-yh7i-\-g^=ax + hy+gz . -p-rj+fi^,
h^+bv+/^ = hx+by+fz + v^ . -Xf,
g^+fv+c^=ga;+/y + cz-fi^+\v • >
46 A smith's prize paper, 1877. [645
where \, fi, v are arbitrary; viz. multiplying by f, rf, f, and adding we have the
equation in question. The three equations determine f, »;, f as linear functions of
X, y, z; and we have thence a;, , y, , ^, as linear functions of x, y, z; viz. this is a
solution containing three arbitrary constants \, /a, v.
12. The partial differential equation might equally well have been proposed in the
form, given function of x, y, z, p, q, r = 0, viz. the equation then is <f) (x, y, z, p, q, r) = 0,
the general partial differential equation involving the three independent variables x, y, z,
and the derived functions p, q, r of the dependent variable u, but not involving the
dependent variable u. The question is therefore in effect as follows: to find p, q, r
functions of x, y, z connected by the foregoing equation, and, moreover, such that
pd^ + qdy + rdz is an exact differential ; for then writing u = j(pda; + qdy + rdz), we have
the solution of the given partial diffei'ential equation.
Whatever be the method adopted, it comes out that the solution depends on the
integration of the system of ordinary differential equations
dp _ dq _ dr _dx _dy_dz
d<f> d<j> d^ d<f> d<l> d<f> '
dx dy dz dp dq dr
and the answer consists first in showing this, and secondly, in shelving how from an
integral or integrals of the system we pass to the solution of the partial differential
equation.
Considering the partial differential equation in the form actually proposed, we may
instead of <f> write H, where H will stand for that given function of x, y, z, p, q, r which
is the value of the arbitrary constant H; making this change, and putting the fore-
going equal quantities equal to the differential dt of a new variable, the system of
ordinary differential equations is
dp^_dH dq^_dH dr^_dH
dt dx ' dt dy ' dt dz '
dx _ dH dy _ dH dz _ dH
dt~ dp' di~ dq' di~ dr '
where H is & given function of x, y, z, p, q, r. This is, in fact, the Hamiltonian system
of equations ; and it was in view to the connexion that the partial differential equation
was proposed in its actual form.
646]
,47
646.
ON THE GENERAL EQUATION OF DIFFERENCES OF THE SECOND
ORDER.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877),
pp. 23-25.]
CONSIDER the equation of dififerences
Ux ^ ax—\ Ux—i "T Ox — 2 ^^X— 2 )
viz. we have
and thence
Ui = a,Mi + b
oMo,
«j = aji«s + 6iMi,
M4 = as«3+ b-ith,
Us = UiUt + bsUi,
Ug = aiUi + btUi,
&c.,
«3 =
lh +
as
Mo,
Ut = a^a^ai
Ui +
asCla
60 Mo,
+ aA
+62
+ aiba
s = a^asaaOi Mj +
a4a3aa
Mo,
+ a^chbi
+ 0462
+ atajb^
+ 0363
+ aia,b3
+ k
63
48
ON THE GENERAL EQUATION OF DIFFERENCES
[646
+ 050463
+ ajOji,
+ 0,0364
+ 6«6,
w, = atatOiO^ai Ui + 0,040,03 6(,«,,
+ 05040,6,
+ 05040,63
+ 0,030,63
+ 0,030,64
+ 0,6,6,
+ 0,646,
+ 0,6463
&c.
It is now easy to see the law ; viz. writing for instance
M, = 54321 . It, + 5432 . 6oM„,
then 54321 has a leading term 0,040,030, : it has terms derived from this by changing
any pair 03a, into 6,, 0,03 into 63, O4O3 into 6„ 0,04 into 64: it has terms derived by
changing any two pairs 040,, 030, into 636,; OjOi, OaO, into 646,; 0,04, 0,03 into 6463,
and so on ; where observe that the expression a pair denotes the product of two con-
secutive o's.
And, similarly, 5432 has a leading term 05040,03 ; the other terms being derived
from this in the same manner precisely.
The solution of «» = Za; (om^-, — m^^s) is included in, and might be deduced from
the foregoing, but it is convenient to obtain it .separately. Supposing for greater
simplicity that it_, = 0, i(.„=l (or, what is the same thing, Mo=1, Ui-lja), then we find
Mo = l,
M, = ?,0,
u, = kka--l„
M, = IsliliO,^ -
i-hh
a.
«4 = IJJJia* —
+ hi A
a- + hh,
1*5 = lulJ^Llia" —
hhhh
a» +
hhh
+ khhk
+ hhh
+ WA
+ hkk
+ h
kh
h
a,
&c..
646] OF THE SECOND ORDER. 49
viz. we may for example write
M, = /, 4321 . a' - 4321 (•) a» + 4321 (:) a ;
where
4321 denotes IJslili:
in 4321 (•), we omit successively each number, viz. we thus obtain
432 + 431 + 421 + 321 ,
in 4321 (:), we omit successively each two non-consecutive numbers, viz. the omitted
numbers being 1, 3; 1, 4; 2, 4, we obtain
42 + 32 + 31,
and so on, the omissions being each three numbers, each four numbers, &c., no two of
them being consecutive; thus in 654321 (.•.), the omissions are 5, 3, 1, and 6, 4, 2; or
the symbol is
642 + 531 ,
As an application, a solution of the differential equation t- (« j^j + (« — a)y = 0
is y = Uv + UiX + luaf + &c., where n^M„ = au,^i — m„_3, and in particular I^m, = au„ ; the
equation of differences is thus of the form in question, and retaining i„ in place of
its value, =n°, the solution is tia=l, Ui = lia, 11^=1^1^0? — l^, &c. ut suprci. The
differential equation was considered by the Rev. H. J. Sharpe, who mentioned it to
Prof Stokes.
C. X.
50
[647
647.
ON THE QUARTIC SURFACES REPRESENTED BY THE EQUATION,
SYMMETRICAL DETERMINANT = 0.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877),
pp. 46 — 52.]
Consider the equation
V= a, h, g, I =0,
h, h , f, m
ff, f, 0, n
I, m, »i, d
where for the moment (a, b,...) denote linear functions of the coordinates {x, y, z, w).
This is a quartic surface having 10 nodes; viz. if we write {A, B,...) for the first
minors of the determinant, then the cubic surfaces .4=0, 5 = 0, ... have in common 10
points which are nodes of the quartic surface.
Suppose that (a, h, c, f, g, h) are linear functions of the form (x, y) ; then, observing
that every term of V contains as a factor
a.
h.
9
h,
h.
f
9'
/.
c
or one of its first minors, it is clear that the line x = 0, y=0 is a double line on
the surface. But the number of nodes is now less than 10 ; in fact, writing (x = 0,
y = 0), we make each of the fii-st minors of V to vanish; that is, the cubic surfaces,
which by their intersection determine the nodes, have in common the line (x = 0, y = 0),
and there is a diminution in the number of their common intersections. I do not
pursue the enquiry, but pass to a different question.
647] ON QUARTIC SURFACES REPRESENTED BY A PARTICULAR EQUATION. 51
I, in fact, take the terms (a, ...) of the determinant to be homogeneous functions
of {x, y, z, w) of the degrees
0,
1,
1,
2,
1,
2,
0
1
1,
2,
2,
I
0,
1,
1,
0
respectively, viz. a, d, I are constants, g, h, m, n linear functions, and b, c, f quadric
functions of the coordinates ; V = 0 still represents a quartic surface ; and it appears
by a general formula that the number of nodes is = 8. But we can easily show this
directly; and further, that the 8 nodes are the intersections of three quadric surfaces;
or say that the quartic surface is octadic. For denoting as before the first minors by
A, ..., then B, C, F are each of them a quadric function of the coordinates, viz. we
have
B=d{ac —g^ ) — cl'- — an- + 2gln,
C = d(ab -h^)- arri' - bl- + 2hlm,
F=d {gh — af) + Pf + mva - nlh — Img,
and we have identically
BC-F"- = (ad-l'')V,
80 that throwing out the constant factor ad — I', the equation of the surface is
BC-F^ = 0,
and it has 8 nodes, the intersections of the three quadric surfaces B=0, C=0, .^=0.
By equating to zero any other minor of the determinant V, we have a surface passing
through the 8 nodes; we have for instance the quartic surface
a, h, ff ':=0.
h, b, f\
ff' f, c \
Suppose now (and in all that follows) that, the degrees being as already mentioned,
we further assume that b, c, / are quadric functions of the form («, yY, g, h linear
functions of the form {x, y); then since each term of V contains either
I a, h, g
h. h, f
or one of its first minors, it is clear that the line {x = 0, y = 0) is a double line on
the surface. But in the present case there is not any diminution in the number of
the nodes ; in fact, writing a; = 0, y = 0, and therefore b, c, f, g, h each = 0 (but not
a = 0), the minors B, C, F none of them vanish ; that is, the line a; = 0, y = 0 is not
a line on any one of the quadric surfaces, and the quadric surfaces intersect as before
in an octad of points.
7—2
52
ON THE QUARTIC SURFACES REPRESENTED BY
[647
The equation V = 0 thus represents a quartic surface having a double line, and
also 8 nodes forming an octad.
We may without loss of generality write d = 0; in fact, the determinant is unaltered
if we add to the fourth column 0 times the first column, and then to the fourth
line 0 times the first line ; the determinant is thus of the original form, but in place
of d it has d + 201 + 0'a, which by properly determining 0 can be made = 0. And
then changing the original I, m, n, the equation is
V =
a, k, g, I
h, h, f, m
9, f, c, n
I, m, n, 0
= 0.
Or, writing for shortness,
K =
a, h, g
h, h, f
and denoting the minors hereof by (a, b, c, f, g, h), then the equation is
V = (a, b, c, f, g, h\l, m, nf = 0,
where the degree of K is 4, and the degrees of a, b, c, f, g, h are 4, 2, 2, 2, 3, 3
respectively, those of I, m, n being 0, 1, 1 respectively.
The nodes are, as before, the intersections of the quadric surfaces 5 = 0, 0=0,
^=0, viz. (d being now =0) the values are
-B = cP-2gln +an^,
-C=bf-2hlm + am^,
F =fl^ — glm — hln + amn.
But, according to a previous remark, the nodes lie also on the quartic surface ^ = 0 ;
viz. this is a set of four planes intersecting in the line a; = 0, y = 0.
Now, in general, any plane through the line a; = 0, y = 0 meets the surface in this
line twice and in a conic ; if the plane is, y= 0x, we have
a. &, c, / 5^, A = a', b'a?, c'a?, fa?, g'x, Kx,
where a', b', c', /', g', h' are functions of 0 of the degrees (0, 2, 2, 2, 1, 1) respectively ;
and thence also
a, b, c, f, g, h=aV, bV, cV, fir^, g'a?, hV,
where a', b', c', f, g*, h' are functions of 0 of the degrees 4, 2, 2, 2, 3, 3 respectively;
the equation of the surface thus becomes (a', b', c', f, g', h'$^x, m, n'f = 0; viz. this
is a quadric equation which, combined with the equation y — 0x — O, determines the
647] THE EQUATION, SYMMETRICAL DETERMINANT = 0. 53
conic in question. But for each of the planes K=0, we have (a', b', c', f , g', W^lx, m, nf
a perfect square, or the conic a two-fold line ; we have thus the 8 nodes lying in
pairs on four lines, say the four ''rdys," in the four planes ^"=0 respectively; each
of these mys meets the double line a; = 0, y = 0 in a point ; and we have thus on
the double line 4 points, which are in fact pinch-points of the surface (as to this
presently). It has just been stated that for the plane passing through the nodal line
and a ray, the conic is a two-fold line (the ray twice) containing upon it a pair of
nodes ; more properly, the conic is the point-pair composed of the two nodes.
We can find through the nodes four different plane-pairs ; in fact, forming the
equation
this is
P (c + 2\f+ V6) -'2l{g + \h) (n + \m) + a{n+ Xmf = 0 ;
or, as this may also be written,
[a (n + \m) -l{g + \A)p + l^h- 2\f + Vc) = 0,
where b, c, f and therefore also b — 2Xf-l-\'c are of the form (x, y)-; say that we have
b — 2Xf-|-X% =(p, q, r^x, yY, where p, q, r are of coui-se quadric functions of \;
determining X by the quartic equation pr — q^ = 0, we have b — 2Xf-i-X-c a perfect
square, = (ax + /Sy)- suppose ; and we have thus the plane-pair
[a(n + \m)-l(g + \h)y-l"-{ax + 0yy = O
containing the eight nodes; viz. there are four such plane-pairs. The two planes of
a plane-pair intersect in a line called an "axis"; that is, we have four axes each
meeting the nodal line; and we have thus also through the nodal line and the four
axes respectively four planes, which are " pinch-planes " of the quartic surface (as to
this presently).
It has just been seen that the equation B—2\F+O\^ = 0 (where X is arbitrary)
is expressible in the form
[a (n + Xm) -l{g + \li)Y + P (p, q, r^x, yf = 0,
viz. this is the equation of a quadric cone having its vertex on the nodal line at
the point x = 0, y =0, an — Ig + \ (am — lh) = 0; this is, in fact, a cone touching the
sur&ce, as at once appears by writing the equation of the cone in the form
G
that is,
1
^ {BC-F- + (\C-Ff]=Q,
[-l^S! j^(XG-Ff]^0\
we thus see that, taking for vertex any point whatever on the nodal line, there is a
circumscribed quadric cone.
S4
ON THE QUARTIC SURFACES REPRESENTED BY
[647
For each of the above-mentioned four values of X, the quadric cone breaks up
into a plane-pair; each plane of the plane-pair is thus a "trope" or plane touching
the surface along a conic; viz. this is the conic passing through the intersection of
the plane (or say of an axis) with the nodal line and through four nodes of the
surface. We have thus 8 tropes, intei-secting in pairs in the four axes (and the inter-
section of each axis with the nodal line being a pinch-point). Moreover, joining the
nodes in pairs, we have four rays, each meeting the nodal line, the plane through it
and the nodal line being a pinch-plane; this is illustrated in the figure.
As to the pinch-planes and pinch-points, remark first that a plane through the
nodal line is in general a bitangent plane, its two points of contact being the points
where the conic in such plane meets the nodal line. When the two points of contact
come to coincide, the plane is a pinch-plane ; viz. this happens when the plane passes
through a ray, the conic being then the ray twice repeated. And secondly, at a point
on the nodal line there are in general two tangent planes, viz. these are the tangent
planes to the quadric cone belonging to such point ; when the two tangent-planes
come to coincide the point is a pinch-point, and this happens when the point is the
intersection of the nodal line with an axis, for then (the quadric cone breaking up
into the two tropes through the axis) the two tangent planes become the plane
through the axis taken twice.
Each section through the nodal line is a conic, and the polar of the nodal line
in regard to this conic is a point ; the locus of this point (for different sections
through the nodal line) is a right line which may be called simply the "polar." To
prove this, considering the section by the plane y = 6x, we have to find the pole of
the line a; = 0 in regard to the conic
(a', b', c', f, g', \i'\lx, VI, nr=0;
647] THE EQUATION, SYMMETRICAL DETERMINANT = 0. 55
this ia Ix : m : n = a' : h' : g' , viz. \i g=g^x + g^, h = h^x + hiy, this is
Ix : m : n = a : g^ + g^O : /(„ + h^d,
or joining hereto the equation y = 6x, we have
Ix : ly : m : n = a : aO : gn+g^d : ho + hjO,
where I, a, g„, gi, h„, Ai are constants; m, n are linear functions of the coordinates
(x, y, z, w). The equations represent, it is clear, a right line which is the polar in
question ; and they may be written
Ix _ h-im — g, n ly _ hi,m — g^n
a hgo-Kg^' a Kga-h^g^'
When the plane passes through a ray, the conic becomes, as was stated, the point-
paii' composed of the two nodes in such ray ; the harmonic in regard to these two
points of the intersection of the ray with the nodal line is thus a point on the
polar : that is, the polar meets the ray ; and the two nodes are situate harmonically
in regard to the intersections of the ray with the nodal line and the polar respectively.
The polar may be arrived at in a diiferent manner, viz. if instead of a plane
through the nodal line we consider a point on the nodal line, this is the vertex of
a circumscribed quadric cone ; and taking the polar plane of the nodal line in regard
to this cone, then considering the point as variable, the different polar planes all pass
through a line which is the polar in question. And hence, taking for the point the
intersection of the nodal line with an axis, it appears that the axis meets the polar;
and, moreover, that the two tropes through the axis are harmonics in regard to the
planes through the axis, and the polar and nodal line respectively.
Collecting the foregoing results, we have a quartic surface as follows:
We have two lines, a nodal line and a polar; meeting each of these, four lines
called "rays" and foui- other lines called "axes." On each ray, harmonically in regai-d
to its intersections with the nodal line and the polar, two nodes of the surface (in
all 8 nodes): through each axis, harmonically in regard to the planes through it and
the nodal line and the axis respectively, two tropes of the surface (in all 8 tropes).
In each trope (or, what is the same thing, in its conic of contact) are 4 nodes;
through each node (or, what is the same thing, touching its tangential quadricone) are
4 tropes; the relation of the nodes and tropes may be thus represented, viz. taking
the pairs of nodes to be 1, 2 ; 3, 4 ; 5, 6 ; 7, 8 ; and those of tropes to be I, II ;
III, IV ; V, VI ; VII, VIII ; then we have
56 ON QUAETIC SURFACES RKPRE8ENTED BY A PARTICULAR EQUATION. [647
I II lU rv V VI VII VIII
1
2
•
•
•
•
3
4
•
•
•
.
5
6
•
•
•
7
8
•
•
•
viz. reading horizontally or vertically, the dots show the tropes througii each node, or
the nodes in each trope.
The plane through any ray and the nodal line is a pinch-plane of the surface, its
point of contact being the intersection of the ray with the nodal line; and the inter-
section of each axis with the nodal line is a pinch-point of the surface, the tangent
plane being the plane through the axis and the nodal line; the surface has thus
4 pinch-planes and 4 pinch-points.
648]
57
648.
ALGEBRAICAL THEOREM.
[From the Quarterly Journal of Pwe and Applied Mathematics, vol. xiv. (1877), p. 53.]
I WISH to put on record the ifollowing theorem, given by me as a Senate-House
Problem, January, 1851.
If {a + /9 + 7 + ...{P denote the expansion of (a + yS + y-l- ...)'', retaining those terms
Naf'^'''f... only in which
fe + c-l-d + ...>p-l, c + d-l-...>p-2, &c., &c.,
then
a;" = (;b+ o)»-n {a|' (a; + a + /S)"-' + i»i(m - 1) {a+^Y{x + a. + fi + yy-"
- ^n(n -l){n - 2) {a+ 0 + yY {x+ a + 0 + y + S)"-^ + &c.
The theorem, in a somewhat different and imperfectly stated form, is given, Burg,
Crelle, t. i. (1826), p. 368, as a generalisation of Abel's theorem,
{x + o)» = ai» + no (a; + /S)""' + ^n (n - 1) a (a - 2/3) (a; + 2/3)"-"
+ ^{n-l)(n -2){n -S)a(a-3ffy (x + 2^y + &c.
C. X.
58 [649
649.
ADDITION TO MR GLAISHER'S NOTE ON SYLVESTER'S PAPER,
"DEVELOPMENT OF AN IDEA OF EISENSTEIN."
[From the Quarterly Journal of Pure amd Applied Mathematics, vol. xiv. (1877),
pp. 83, 84.]
The formula (11) [in the Note], under a slightly different form, is demonstrated by
me in an addition [263] to Sir J. F. W. Herschel's paper " On the formulae investigated
by Dr Brinkley, &c.," Phil. Trans, t. CL., 1860, pp. 321—323. The demonstration is in
effect as follows : let u denote a series of the form \-\-hx + ca?-¥ da? + ..., and let m*
(where i is positive or negative, integer or fractional) denote the development of the i-th
power of u, continued up to the term which involves a;", the terms involving higher
powers of x being rejected ; u", u', w", . . , and generally ?t* will denote in like manner
the developments of these powers up to the terms involving a;", or, what is the same
thing, they will be the values of u* corresponding to i = 0, 1, 2, . . , s. By the formula
m'=1 + y(m — 1)+ ^ — »— (« — !)''+ ... as far as the term involving (« — 1)", ?t* is a rational
and integral function of i of the degi-ee n, and can therefore be expressed in terms
of the values u", w", u\.., u" which correspond to i = 0, 1, 2, . . , n. Let s have any one
of the last-mentioned values, then the expression
i.i—l.i—2...i — n 1
«. «-l. . .2.1.-1. -2. ..-(n-s)'
which as regards i is a rational and integral function of the degree n (the factor i — s
which occurs in the numerator and denominator being of course omitted), vanishes for
each of the values i=0, 1, 2,.., n, except only for the value i — s, in which case it
becomes equal to unity. The required formula is thus seen to be
. „ (i . I — 1 . t — 2 . . . t — «
M' = 2 -^
i-s s.«-l ...2.1.-1.-2,
-^¥^A'
649] ADDITION TO MR GLAISHER's NOTE ON SYLVESTER'S PAPER. 59
where the summation extends to the several values s=0, 1, 2,.., n; or, what is the
same thing, it is
M» = 2 ■' - ^-
i-s 1.2 ...s. 1.2... (n
or, changing the sign of i, it is
-)"■}■
-i_v(^'-*'+i-*'+2-t+n i-yi ,\
" -^\ i + 's 1.2...s.l.2...»i-s"r
where, as before, s has the values 0, 1, 2, . . , n successively. Or, what is the same
thing, we have
^ _ _ \i.i + l.i + 2...i + n (-)»! ^ |
^-'■""■^1 t + ~s 1.2...«.1.2...n-8 ••»)'
where the term corresponding to s = 0, as containing the factor Co, „ vanishes except
in the case n = 0 (for which it is = 1) ; and omitting this evanescent term, this is in
fact the formula (11).
8—2
60 [650
650.
ON A QUARTIC SURFACE WITH TWELVE NODES.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877),
pp. 103—106.]
Write foi* shortness
a = /3-7, /=o-S, af=p,
b = y-a, g = ^-S, hg = q,
c = a — yS, h= y — S, ch = r;
then, 0 being a variable parameter, the surface in question is the envelope of the
quadric surface
. (o + ey aghX' + (/3 + 0)"- bh/Y' + (y + 0f cfgZ^ + (S + 0y a6c F^ = 0 ;
viz. this is
ta^aghX^ . taghX"- - taaghX^ = 0.
There are no terms in X*, &c. ; the coefiBcient of Y^Z^ is
y'cfg.bfh + ^hfh.cfg-2^bfh.ycfg,
which is
= bcpgh (0 - yf, = a'bcf'-gh, = abcfgh . p.
Hence the whole equation divides by abcfgh, and throwing out this factor, the result is
p{Y'-Z'> + X'W') + q(Z'X'+Y''W') + r(X'Y' + Z''W') = 0,
or, observing that p + q + r = 0, this may also be written
p{YZ + XWy+q{ZX+YWy + r{XY + ZWf = 0,
650] ON A QUARTIC SURFACE WITH TWELVE NODES,
diiid £l1so
p{YZ-xwy + q (zx - Ywy+r (XY- zwy = o.
The more general equation
ip, q, r, I, m, ti^YZ + XW, ZX + YW, XY+ZWy==0
represents a quartic surface (octadic) having the 8 nodes
(1, 0, 0, 0), (T, 1, 1, 1),
. (0, 1. 0, 0), (1, T, 1, 1),
(0, 0, 1, 0), (1, 1, T, 1),
(0, 0, 0, 1), (1, 1, 1, 1).
61
We have
p. XW +YZW
q. FF» + YZW
r. ZW^- + YZW
I. 2XYZ + W{Y^+Z^)
m. 2XYW+Z(W^+ F»)
n. 2XZW+Y(W'' + Z%
p. YZ'
q. YW^
r. YX-'
I
+ XZW
+ XZW
+ XZW
2XYW + Z(W- + X')
m. 2YZX + W(Z^ + X^)
n. 2YZW+X{W' + Z%
dwU =
p. X'W +XYZ
q. Y'W +XYZ
r. Z'W +XYZ
I. 2WYZ + X {Y^ + Z'^)
m. 2WZX +Y{Z- +Z»)
n. 2WXY + Z (X^ + Y').
dzU =
p. Y'Z +XYW
q. X^-Z +XYW
r. W'Z +XYW
I. 2ZXW+Y{W' + X')
VI. 2YZW + X{W'+Y^)
n. 2ZXY + W(X'+Y'),
Hence there will be a node
1, 1,1, I, a p + q + r+ 21- 2m - 2n = 0,
1, 1, T, 1, ... p + q + r - 21 + 2m - 2n = 0,
T, T, 1, 1, ...p + q + r-2l-2m+2n = 0,
1, 1, 1, 1, ...p + q + r-\-2l+2m + 2n = 0;
or say there will be
1 of these nodes if p + q + r + 21 + 2m + 2n =0,
2 p + q + r+2l = 0, m + n = 0,
3 p + q+r=2l = —2m=-2n,
4 p + q + r = 0, 1=0, m = 0, n=0;
62 ON A QUARTIC SURFACE WITH TWELVE NODES,
viz. the surface having the 12 nodes is the original surface
where
p(7Z+XWy + q(ZX+ YWy + riXY+ZWy,
p + q + r=0.
The Jacobian of the quadrics
YZ+XW-=0, ZX+YW=0, XY+ZW = 0.
18
viz. the equations are
W, Z, Y, X
Z, W, X, Y
Y, X, W, Z
= 0;
X^-X (7» ^ Z'- + W^) + 2YZW = 0.
Y^ -Y (Z* + X'-+W^)+2ZXW = 0,
Z^ -Z (X- +Y'-+ W) + 2XYW = 0,
W*-W(X'+Y"-+ Z^) + 2XYW = 0,
each of which is satisfied in virtue of any one of the pairs of equations
(Y-Z^O, X-W = 0)
(Z-X = 0, Y-W = 0)
(X-Y=0, Z- W = 0)
(Y+Z=0, X+W = 0),
(Z+X = 0, Y-\-W = 0).
(X+Y = 0, Z+W = 0).
[650
80 that the Jacobian curve is, in fact, the six lines represented by these equations.
Any two of the three tetrads form an octad, the 8 points of intereection of
three quadric surfaces: a figure representing the relation of the 12 points to each
other may be constructed without difficulty.
Each tetrad is a sibi-conjugate tetrad quoad the quadric X'' + Y- + Z' + W-=:0.
The three tetrads are not on the same quadric surface.
651] 63
651.
ON A SPECIAL SURFACE OF MINIMUM AREA.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877),
pp. 190—196.]
A VERY remarkable form of the surface of minimum area was obtained by Prof.
Schwarz in his memoir "Bestimmung einer speciellen Minimal-flache," Berlin, 1871,
[Ges. Werke, t. I., pp. 6 — 125], crowned by the Academy of Sciences at Berlin. The
equation of the surface is
1 + fiv + v\ + \fi = 0,
where X, fi, v are functions of x, y, z respectively, viz.
_ _ r de
and y, z are the same functions of /*, v respectively. A direct verification of the
theorem that this is a surface of minimum area, satisfying, that is, the differential
equation
r (1 + <f) - 2pq8 + t{l+f) = 0,
is given in the memoir; but the investigation may be conducted in quite a different
manner, so as to be at once symmetrical and somewhat more general, viz. we may
enquire whether there exists a surface of minimum area
1 +/*!/+ i/X + \/t = 0,
where the determining equations are
X"" = a\* + 6V + c,
(i^ = a/i* + hfi? + c,
v'^ = av* + bv^ + c,
64 ON A SPECIAL SURFACE OF MINIMUM AREA. [651
fX' = -5-, &c.j. I find that the coefficients a, b, c must satisfy four homogeneous
quadric equations, which, in fact, admit of simultaneous solution, and that in three
distinct ways ; viz. assuming a = 1, the solutions are
a = l, b= Y. c= 1.
a = l, 6 = -2,c=l,
a = l, 6 = -|, c = -i;
that is,
X'= = \^ + J^X= + 1 1= f (f V + f V + f )},
which gives Schwarz's surface:
X'» = \* - 2V + 1 or X' = ± (X» - 1),
which, it is easy to see, gives only x + y + z=: const. ; and
\'= = V-f\=-^, =(\=-l)(X=+J).
which is a surface similar in its nature to Schwarz's surface.
The investigation is as follows: the condition to be satisfied by a surface of
minimum area U = 0 is
(a + b + c)(Z'+ F^ + Z=)-(a, b, c, f, g, h^X, Y, Z)^ = 0,
where (X, Y, Z) are the first derived coefficients and (a, b, c, f, g, h) the second
derived coefficients of U in regard to the coordinates. Considering U &s a, function
of X, /*, V, which are functions of x, y, z respectively, and writing {L, M, N) and
(o, h, c, f, g, h) for the first and second derived functions of U in regard to X, fi, v,
also X', X" for the first and second derived functions of X in regard to x, and so
for ix, fi." and v, v" : we have
(X, Y, Z) = (L\', M,,',Nv'),
(a, b, c, f, g, h) = (aX'» + ZX", b^'' + M,i,", cv'^ + Nv", f^'v, gv'\', h\y),
and for the particular surface U = 1 + fiv + v\ + \fi = 0, the values are
(Z, M, N, a, b, c,f, g, h) = {fi + v, p + \, \ + fi, 0, 0, 0, 1, 1, 1).
Hence the condition is found to be
2^'V'» (X + /i) (X + v)
+ 2v'V^'' (fi + v)(fi + \)
+ 2XVH''+X)(«'+m)
- X" (fi + v) ((X + 1/)' /» + (X + fiy !/'»)
-im"{v +x) [(fi + \y I/'' +(n + vy x'»}
- p" (X + m) {{v + fif ^'' +{"+ X)' fi''] = 0,
651] ON A SPECIAL SURFACE OF MINIMUM AREA. 65
or saj' this is
- 2\" (fj. + v) {(X + vy fji'- + (\ + fif v'^] = 0.
We have to write in this equation \'- = ok* + bX' + c, and therefore \" = 2a\' + b\,
&c. ; the left-hand side, call it il, is a symmetrical function of X, fj,, v, and is con-
sequently expressible as a rational function of
p, =\ + ii + v,
q, =iJ,v + v\ + \/i,
r, = X/ii/.
We ought to have fl = 0, not identically, but in virtue of the equation 1 -|- g' = 0,
that is, n should divide by I -\- q; or, what is the same thing, H should vanish on
writing therein g = — 1.
To effect the reduction as easily as possible, observe that we have (X +/*) {\ + v) = X' + q;
and therefore
I.H'V (X + n)(\ + v) = IXy'V^ + qlfi''v'^.
Similarly, in the second term,
(jjL + v){\ + vy= {v + \)(vf + q) and (fi + v) (\ + fif = (li + \) (fi' + q).
The complete value of Q, thus is
il = 2(Aq + B)-[(C + I))q + E + F],
where
A = Sxy»».'», B = S/[t'=I/'^
C = SXX" (vy + ^iPv'% D = 2X" (!/>'= -I- ^V=),
Er=l.XK"{fJi'-'+v'% F = %\"{v/ji" + fiv'').
We find without difficulty
A = u^ ( q*- ^q''pr + ^r"- + 2p''r=)
+ ab{-2(f+ qY + ^W - 3r= - 2^r)
-\-aci 4^= - Sg^j" -I- 8pr -I- 2p^)
+ ¥ ( q'-2pr)
+ be (- 49- -I- 2p')
+ c'^ ( 3),
iJ = a' ( gV + 2pr»)
+ a6(- 4gr=' + 2jjV^)
-I- ac (- 2g^ + q-p- + iqpr - 3r' - 2p'r)
-I- b' ( 3r=)
-t- 6c ( 2(7^ — 4ipr)
+ c' (-2q +2^'),
C, X. ®
66
[651
(- 69* + 39V + 1 2qpr - 9r= - 6j9V)
8q^-l6qp^ +16pr + 4p«)
2q' - ipr)
iq + 2p%
Iq'pr - 2qr^ - 4pV=)
(— iqpr + 2^r)
(- Aq^ + 2qp^ - 2pr)
2pr)
2q).
49V -Hpj-^)
(- 12qT^ + 6j9V)
49' + 2^^ + Sgrjor - 6?^ - 4pV)
29- — 4pr),
4.f -I2qpr + 12i^)
qpr - Sr")
2q^ + 9p» -pr),
where in each line the terras are arranged according to their oi-der in p, r.
Substituting, we find
c
= a' (
+ a6(-
■\r ac{
+ 6' (
+ he (
D
= a" (
+ a6(-
+ oc (•
+ 6» (
+ he (
E
= + a= (
+ a6(
+ ac{-
+ 6» (
+ 6c (
F
= a- (
+ ai (
+ oc (
+ 6= (
+ fec(-
a= a" (
- 29» + 69»j9r - 8gV )
+ rt6(
2q*- ff- <fpr+ *qr^)
+ ac
[ - 2fp' + Uqpr - 12i^ )
+ ¥ {
- 3qpr+ 3r^ )
+ bc (
- 29= + qp^ - Spr )
+ c» (
2</+2p= );
iz. writing q — —
1, this is
ft= aM
2 - 6jt>r- 8?-»)
+ a6(
2+ ^«- pr- 4r')
+ ac (
- 2p^ - 14pr - 12r»)
+ 6'-' (
3pr+ 30
+ 6c<
-2- p'- 3pr )
+ c»
[-2-2p> );
651]
ON A SPECIAL SURFACE OF MINIMUM AREA.
67
or, what is the same thing, it is
( 2«= + 2ab - 26c - 2c=)
+ p^ ( ab- 2ac — be + 2c')
+ |)r (- 6a^ - ab- 14ac + 36^ - 36c )
+ r= (- 8a= - 4a6 - 12ac + 36=^ ) ;
so that, writing for convenience a = 1, the equations to be satisfied are
2 -2c= -1-2(1- c) 6 = 0,
- 2c + 2c='+ (1- c) 6 = 0,
-6 -14c + 36^- (l+3c) 6 = 0,
- 8 - 12c + 36-° - 46 = 0.
The first and second are (1 -c)(2 + 2c + 26)= 0 and (1 - c)(-2c+ 6) = 0; viz.
they give c=l, or else 6 = — |, c = ^. In the former case, the third and fourth
equations each become 36-' — 46 — 20 = 0, that is (36 — 10) (6 — 2) = 0; in the latter case,
they are satisfied identically ; hence we have for a, 6, c the three systems of values
mentioned at the beginning.
This completes the investigation ; but it is interesting to find the values assumed
by the other factor of fl on substituting therein for a, b, c the foregoing several
systems of values. We have in general
n = - 2aY + 2abq* -• 2bcq^ + 2d'q
+ jir {- abcf' - iacq" + bcq +2c'' )
+pr{ dd^q' — aiq-^liacq — 36-g — 36c)
-f r* (- 8a Y + 4tabq -\2ac + 36= )
= - 2a»(^ -fl) + 2a6 (y" - 1) - 26c (g= - 1) -f 2ci'{q-\-l)
+2)= [-ab ((/=' + l)-2ac(5'=-l)-t-6c(f/-|-l))
+ pr [ Qa-{<f+l)- a6(g''-l) + (14ac-36=)(g'-t-l)}
-t-r= { -8o» (9=-l)-l-4a6(5-|-l)}
= (9+ 1) /-2a''(g'-^»-f-(/--^-|-l)-F2a6(5'-g'-' + 5'-l)--26c(g'-l)-l-2c=
+f [- ab{q'-q^-l)-2ac(<i-l) + bc]
+pr [ Qa^ {f + q + \)- ab{q-l) + (\^ac-W)\
-fr" { -8a'(g-l)-|- 4a6}
Hence writing, first, a=c=l, 6 = -^, we obtain, after some reductions,
il = {q-¥l){-2q{q-\){q'-iiq + \)+f{q-l){-^q-2)+pr{Gq^-^q-\Q) + r^-^ + ^];
secondly, writing a= c = 1, 6 = — 2, we obtain
n=^{q + \)[-2{q + \Y{q'-\-\)+P'.2{q-\f+2pr{'&f-2q + Q)-Sr^q];
and, thirdly, writing a=l, 6 = — J, c = — f, we obtain
n = (5 + 1) {(- 2?* + ty' - Is' + f g) +P' (- ig' -h S? - J,?) -hi>»- (6?= - -^2 - Y) + »•'(- 8? -^ ^)}.
9—2
68 [652
652.
ON A SEXTIC TORSE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877),
pp. 229—235.]
The torse having for its edge of regression or cuspidal edge the curve defined by
the equations as = cos <^, y = sin ^, z = cos 2<j), is an interesting and convenient one for
the construction of a model, and it is here considered partly from that point of view.
The edge is a quadriquadric curve, the intersection of the cylinder ar' + y^=l with
the parabolic hyperboloid z = a? — y'^\ the cylinder regarded as a cone having its vertex
at infinity on the line a; = 0, 2/ = 0, viz. the vertex is on the hyperboloid, or the curve
is a nodal quadriquadric (the node being thus an isolated point at infinity on the line
in question), and the torse is consequently of the order 8 — 2, =6, viz. it is a sextic
torse.
The edge is a bent oval situate on the cylinder 3?-Vy-=\, such that, regarding <^
as the azimuth (or angle measured along the circular base from its intereection with
the axis of x), the altitude z is given by the equation z = cos 2<^ ; viz. there are in
the plane xz, or, say in the planes xz, x'z, two maxima altitudes z=\, and in the
plane yz, or, say in the planes yz and i/z, two minima altitudes z = — \. The sections
by these principal planes are, as is seen at once, nodal curves on the surface ; they
2
are, in fact, the cubic curves z=i — r , viz. here as x increases fi-om ±1 to ± oo ,
Or
2
z increases from the before-mentioned value 1 to 3, and z = — Z + —„, viz. as y increases
y
from +1 to ±<xi , z decreases from the before-mentioned value - 1 to — 3. The two
half-sheets (which meet in the cuspidal edge) intersect each other along these nodal
lines, in such^vi8e that the section of the surface by any axial plane (plane through
the line a; = 0, y = 0) is a curve having a cusp on the cuspidal edge, and such that
when the axial plane coincides with either of the principal planes x = 0, y = 0, the
(552] ON A SEXTIC TOESE. 69
two half-branches of the curve coincide together with the portions which lie outside
the cylinder a? + y'^= 1, in fact, the portions referred to above, of the nodal curve in
the plane in question ; the portions which lie inside the cylinder are acnodal or isolated
curves without any real sheet through them. It may be added, in the way of
general description, that the section of the surface by any cylinder af + y^ — c- (c>l)
is a curve of the form z = C cos (20 ± B), 0 the angle along the base of the cylinder
from the intersection with the axis of a;; C, B are functions of c; viz. we have for
the two half sheets respectively
z=Ccos(20 + B) and z = C cos {20 - B),
each curve having thus the two maxima + C, and the two minima —C; and the two
carves intersect each other at the four points in the two principal planes respectively;
viz. the points for which ^ = 0, 90°, 180°, 270°, and z=C cos B, -CcosB, CcosB, -CcosB
accordingly.
Proceeding to discuss the surface analytically, we have for the equations of a
generating line
X — cos d> M — sin rf) z — cos 2(i
^~Z= x^= o • ol ' =p suppose,
— sin ^ cos 9 — 2 sm 2<f) '^ ^'^
or say
X = cos (f) — p sin <^,
y = sin </) + p cos (f),
z = cos 2</) — 2/3 sin 2<)>,
which equations, considering therein p, <}> as arbitrary parameters, determine the surfece.
Writing a; = 0, we find y = —. — — , and then z= — S + 2 sin^ <f), viz. we have
2
x=0, .2 = — 3 + — , for section in plane yz ;
if
and, similarly, writing y = 0, we find x = -r , and then z=S — 2 cos' ^, viz.
2
y = 0, z = S— — ioT section by plane xz.
By what precedes, these are nodal curves, crunodal for the portions
(y = ±l to +oo,« = -l to -3) and (a; = + 1 to ±co, z=l to 3)
respectively, acnodal for the remaining portions y<±l, x< ±1 respectively.
Writing x = rcos0, y = r sin 0, so that the coordinates of a point on the surface
are r, 0, z, where r='^(x' + y') is the projected distance, 0 is the azimuth from the axis
of X, and z is the altitude, we have
r cos 0 = cos <f>— psin <f>,
r sin 5 = sin <f>+ p cos <^,
z = cos 2^ — 2/3 sin 2<^.
70 ON A 8EXTIC TORSE. [652
We have ?•'= !+/»'; and thence also, if tana = 2/>, =±2V('^ — 1). that is,
1 ^ 2 V(»^ - 1)
then
z = V('i»-» - 3) COS (2^ + &),
showing that for a given value of r (or section by the cylinder aF + y = »•») the
maximum and minimum values of z are z= ± ^/(ir* — 3).
But proceeding to eliminate ^, we find
?•= cos 2^ = (1 - p-) cos 2<p - 1p sin 2^,
r» sin 2^ = 2p cos 2<^ + (1 - p") sin 2<^ ;
or multiplying these by 1 + ^p^ and 2p' and adding
»•= {(1 + 3p-) cos 26 + 2/3» sin 2^} = (1 + p-f (cos 2^ - 2/j sin 2<^),
that is,
■)'' {(3»-= - 2) cos 2^ ± 2 (r' - 1)3 sin W] = i-*z ;
or, finally,
r»z = (3r= - 2) cos 2^ + 2 (?•= - 1)J sin 2^,
which is the equation of the surface in terms of the coordinates r, 6, z.
Observing that (3r^ — 2V + 4 (»•=-!)' = >•* (4?^— 3), we may write
^ V'(4r- - 3) cos |8 = 3r» - 2,
r= ^(4?^ - 3) sin /3 = 2 (r» - 1)»,
and therefore also
and the equation thus becomes
^ _ 2(r=-l)«
tan^ = -^_/,
^ = V(4»-=-3)cos(2^ + /3),
where z is the altitude belonging to the azimuth 6 in the cylindrical section, radius r.
The maxima and minima altitudes are + \f{4tr^ - 3), and these correspond to the values
^ = ±i/3. i^ + ^/S, TT + J/3, fTT + ^/S; it is to be further noticed that when ?• = !, we
have /3 = 0, but as r increases and becomes ultimately infinite, /3 increases to \ir,
that is, ^/8 increases from 0 to Jtt.
It may be noticed that the surface is a peculiar kind of deformation, obtained by
giving proper rotations to the several cylindrical sections of the surface z = v'(4?*' — 3) cos 26 ;
viz. in rectangular coordinates this is r^z = \/(4?"^ — 3) («° — y% that is,
{a? + 3/5)» z= - (4 {x" + y^)-Z]{a?- y'Y = 0.
To obtain the equation in rectangular coordinates, we have
16(r»-l)»'5[° = 0.
i^z-^(a^-f)
viz. this is
r^z' - 2z (3r-"- 2) (a? - y') + (3r= - 2)» (l - ^^"^ - 16 (r» - 1)'^' = 0,
652] ON A SEXTIC TORSE. 71
or, what is the same thing, it is
r*2^ - 2z (3j^ -2)(af-f) + (3r= - 2)-^ - *^- {4 (»-^ - 1 )» + (3r« - 2)'} = 0,
viz. the term in { } being »•* (4>-^ — 3), this is
r^z" -2z(S7^- 2) (of - y") + (Sr^ - 2)'' - 4«^- (4?-= - 3) = 0,
or say
a» (a? + iff - 2z {Za? + ^f-2)(a?- y") + {Zx" + Zy- - 2)= - ^a?y^ (4a? + 4y - 3) = 0.
This may also be written
\z{a?-f)-^sc'-^'-+2Y+ iaihf (z^ - 4>af - 4.y'' + 3) = 0,
a form which puts in evidence the nodal curves
a; = 0, ay = - 3y» + 2, and y = 0, za? = ^x" - 2.
It shows also that the quadric cone z* — \oi? — 4y^ + 3 = 0 touches the surface along
the curve of intersection with the surface z{a? — y^) — Z {a? + y'^) + 2=0. This is, in
fact, the curve of maxima and minima of the cylindrical sections, viz. reverting to the
form z = iJi^tr* — 3) cos {26 + yS), or, if for greater clearness, attending only to one sheet
of the sui-face, we write it 2 = V(*^^ — 3) cos {26 — /8), we have a maximum, z = ^(4?^ — 3),
for 26 = ^ (or 29r + /S), giving
„- ' 3r'-2 3r»-2
cos2^=co8;(3, = „-,7-ri — 5\. = — s — :
»•* \/(4» — 3) ■ r-z
and a minimum, z = — 'J{4^ — 3), for 26= 7r+ 0 (or Stt + /3), giving
Sr^-2 3r^-2
cos 26 = — cos /3 = —
r»V(4r='-3)' r^z '
viz. the locus is «» = 4(r'-3), z {afl - y'') = Sr' - 2 ; aud for «= V(4»--'- 3)cos(2^ + /3) we
find the same locus, viz. the equations of the locus are
^» _ 4a? _ 4y2 + 3 = 0, z{x'-f-)-S!ifi-Sy^ + 2 = 0,
as above.
To put in evidence the cuspidal edge, write for a moment ^=z — a? + y', the
equation becomes
{?(a? -y*) + {r^-l){r'-2)- ia^y^}- + 4a;y (f-- + 2f (a? - y') + {r"- -l){r'- 3) - 4^y-} = 0 ;
viz. this is
C'r' + 2^{x'-y''){r'-l){r^-2) + {r'-iy{r'-2f- 4^afy' {r^-iy = 0,
or writing the last term thereof in the form
-[r'-(a^-y'yi{r'-iy,
and then putting r* = 1 + U, the equation is
f{l + 2U+U') + 2^U{U-l){a^-f) + U'{U-lf-U'{{U+iy-{«^-fy} = 0;
viz. this is ^-r ^
{^-U{a^-f)Y + 2U{^' + ^U{aP-f)-2U'] + ^U' = 0,
72 ON A SEXTIC TORSE. [652
showing the cuspidal edge f=0, U=0, viz. z=x'-y\ a!' + y» = l. Moreover, along the
cuspidal edge the surface is touched by ^ - U (ai' - f) = 0, that is, by z-{it^-y*) = 0;
and at the points where this tangent surface again meets the surface we have
(ar* - yy {af + y"+S) — 4-=0; viz. the surface contains upon itself the curve represented
by this last equation, and z — (x*-y*) = 0.
As a verification, in the form
{« (a? - y») - 3a? - 3y' + 2)= + 4a?/ (^^ - 4a? - 4y» + 3) = 0
of the equation of the surface, write z-ar' — y*. If for a moment a? + ?/= = \, af-y^ = /i,
then the value of ir is z= Xfi, and the equation becomes
(\m' - 3\ + 2)' + (V - /t») (XV - 4\ + 3) = 0,
that is,
/*= (V - 6V + 8X - 3) - 4\' + 12X= - 12\ + 4 = 0 ;
or, what is the same thing,
(\-1V{m^(X + 3)-41 = 0,
so that we have (X- 1)^ = 0, or else /i-(X + 3)- 4 = 0; viz. (a? 4-y''-l>' = 0, or else
(a? — y^y (a? + 3/* + 3) — 4 = 0, agreeing with the former result.
In polar coordinates, the surface is touched along the cuspidal edge by the surface
z = r* cos 20, and where this again meets the surface we have r* (r* + 3) cos' 2^ — 4 = 0.
For the model, taking the unit to be 1 inch, I suppose that for the edge of
regression we have
a; = 2 cos <f>, y = 2 sin 0, z = 5 + ('45) cos 2^ ;
viz. the curve is situate on a cylinder radius 2 inches. And I construct in zinc-plate
the cylindric sections, or say the templets, for one sheet of the surface, for the several
radii 2, 3, . . , 8 inches ; taking the radius as k inches, the circumference of the cylinder,
or entire base of the flattened templet, is = 2k-7r ; and the altitude, writing 20 in place
of 20 — ^ as above, is given by the formula z= 5 + (-45) V(^''' — 3)cos 2^, so that the
half altitude of the wave is = ("45) \/{l<? — 3) ; having this value, the curve is at once
3A? — 8
constructed geometrically. We have, moreover, cos /8 = .^ .^ — . ; the numerical values
then are
3A?-8
k
2kir
(-45)V(A?-
-3)
-3)
i/3
2
12-57
0-45
1-00
0°
3
18-85
110
•86
15
4
2513
1-62
•69
23
5
31-42
2-11
•57
27i
6
37-70
2-59
•48
30J
7
43-98
305
•42
32i
8
50-27
351
•36
34
the altitudes in the successive templets being thus included between the limits 5 ± 0*45,
5 + 1-10,.., 5+3-51.
653]
73
653.
ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877),
pp. 235—241.]
K
On attempting to cover with paper one half-sheet of the foregoing sextic torse,
[652], I found that the paper assumed approximately the form of a circular annulus of
an angle exceeding 360°, and this led me to consider the general theory of the
construction of a torse in paper, and, in particular, to consider the torses such that
when developed into a plane the edge of regression becomes a circular arc. It is
scarcely necessary to remark that, to construct in paper a circular annulus of an
angle exceeding 360", we have only to take a complete annulus, cut it along a radius,
and then insert (gumming it on to the two teiminal radii) a portion of an equal
circular annulus ; drawing from each point of the inner circular boundary a half-
tangent, and considering these half-tangents as rigid lines, the paper will bend round
them 80 a.s to form the half-sheet of a torse having for its edge of regression this
inner boundary, which will assume the form of a closed curve with two equal and
opposite maxima and two equal and opposite minima, described on a cylinder, and
being approsdmately such as the curve given by the equations
X = cos 6, y = &in6, z = m, cos 26.
Considering, in general, an arc PQ (without inflexions) of any curve, and drawing
at the consecutive points P, F, P", &c. the several half- tangents PT, FT, P"T",...,
then, considering these as rigid lines and bending the paper round them, we have
the half-sheet of a torse, having for its edge of regression the curve in question
now bent into a curve of double curvature. It is, moreover, clear that the edge
of regression has at each point thereof the same radius of absolute curvature as the
original plane curve; in fact, if in the plane curve PF =ds, and the angle T'PT
between the consecutive half-tangents PT and FT' be =d4>, these quantities ds and
C. X. 10
74 ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS, [653
d^ remain unaltered in the curve of double curvature; and the radius of absolute
curvature is given by the equation pd<f> = ds. In particular when, as above, the arc
is a circular one, say of radius =o, then, however the paper is bent, the edge of
regression has at each point thereof the radius of absolute curvature = a.
Consider on any given surface, at a given point P thereof, and in a given
•direction, an element of length PP", then (under the restrictions presently mentioned)
we can determine the consecutive element P'P", such that the curve PP'P"... shall
have at P a radius of absolute curvature = a ; in fact, r being the radius of
curvature of the normal section of the surface through the element PP", the radius
of curvature of the section inclined at an angle 6 to the normal section is =rcos^;
so that we have only to take the section at the inclination 6, = cos"' - to the
J r
normal section, and we have the consecutive element P'P" such that the radius of
absolute curvature of the curve PP'P" is =a. The necessary restriction, of course, is
that r > a ; thus, if at the given point P the two principal radii of curvature are
of the same sign (to fix the ideas, let the two principal radii and also a be each
of them positive), then we may on the surface determine a direction PQ, for which
the radius of curvature of the normal section is = a ; and then the direction of the
element PP" may be any direction between PQ and the direction PR, corresponding
to the greatest of the two principal radii.
Having obtained the element P'P", we may, if the radius of absolute curvature
at P" be given, construct the next element P'P", and so on ; that is to say, on a
given surfiice starting from a given point P and given initial direction PP", we can
(under a restriction, as above, as to the curvature at the different points of the
surface) construct a cui've having at the successive points thereof given values of the
i-adius of absolute curvature ; viz., the value may be given either as a function of
the coordinates of the point on the surface, or as a function of the length of the
curve measured say from the initial point P; it is in this last manner that in what
follows the value of the radius of absolute curvature is assumed to be given.
We may thus, taking on paper an arc PQ with its half-tangents, apply it to a
given surface, the point P to a given point, and the infinitesimal arc PP" to an
element PP" in a given direction from the given point ; and we thus obtain the
half-sheet of a torse having for its edge of regression a determinate curve upon the
surface. In particular, the arc PQ may be circular of the radius a, and the surface
be a circular cylinder of radius a \ and we thus obtain the toi-se having for edge of
regression a curve on the cylinder radius a, and such that the radius of absolute
curvature is at each point = a. There are three cases according as a > a, a = a,
or a < a ; it is to be remarked that if a> a, then the curve must at each point
cut the generating line of the cylinder at an angle not exceeding co8~'-, but that
in the other two cases the angle may have any value whatever; and, further, that
in every case when the angle is = 0, viz. when the curve touches a generating line
of the cylinder, then the osculating plane of the curve coincides with the tangent
plane of the cylinder.
653] ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS. 75
The analytical theory is very simple. Taking x, y, z ss, functions of the length
8, we have
(S)'+(i:r-(S'-^
the condition, which expresses that the radius of absolute curvature is = a, then is
By what precedes, the point {x, y, z) may be taken to be upon a given surface, say
upon the cylinder a?-\- y-=a?; and we may then write x = a cos 6, y = a. sin 6. Taking-
instead of s any independent variable u whatever, and using accents to denote the
derived functions in regard to u, the equations become
x'- + y'- +/2 =s'=,
x"^ + y""- + z"^-s"'- = \s'\
x = a cos 0, y = a sin 0.
From the last two equations we obtain
x"- + y'' = O.'0'-, x""- + y"- = a= (^ = + 6'^),
and the first two equations thus become
a'ff" + z'^ = s'"-,
and from the first of these we find
0/
„ a^0'd"+z'z"
* ~ (a'0'^ + z")i '
whence the second equation is
a^(^0'. + 0')+z - („.^.. + y-y-- „. .
or reducing, this is
iar0" + z'-) (0"^ + ^*) + (^V^ - 2(9'(?'Vz" - a^0'^0"^) = J^, {oe0'- + z'J.
Taking here 0 as the independent variable, we have 0'=l, 0" = O, and the equation
becomes
or, what is the same thing,
z"^ = ^ (af + z'^y - (a' + zf'y.
ao.
Write here
a- + z'- = n-,
10—2
76
then
ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS.
[653
/' =
nn'
and the equation becomes
or say
V(n»-a»)'
acdil _,„
V(n»-rf'.n'-aV)~ '
viz. this equation determines Q as a function of 0, and we then have
(ds==nd6,
X = a cos d,
, y = a sin ^,
equations which determine x, y, z, s as functions of the parameter 6, and give thus
the edge of regression of the toi-se in question.
It is clear that the formulae are very much simplified in the case a=a, where
the radius of absolute curvature a is equal to the radius a of the cylinder; but it
is worth while to develope the general case somewhat further.
Considering the elliptic functions sn m, en it, dn m, to the modulus A; (= A;') = . a. ,
v(2)
Assume
then
fl = -
dn = -
V(aa) dn u
ic sn w '
\l{aa) en u du
k sn'' M '
D.-'-a? =,"° ( dn' M - ° A:° sn^ M 1
Ar sn' M V a J
aoL
1^ sn' u
1-1 +
.?)i'sn'«}.
Q.* - a?a* = ,i^V- (dn* u - h" sn« u\
k* sn* u ^ '
cC'd
and hence
i*sn'
— (1 -2i'sn'M), =
g'a'
i*sn*M
cn'M,
d^ =
A" sn « dw
y{l-(l+?)^sn'«}'
A \/(<M[) dn M du
^{l-(l-H^)A.sn'«}'
da^
653] ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS. 77
We have thus z=^k \/{cm) u, no constant of integration being required, viz. m is a
mere constant multiple of z : and the first and second equations then give s and 6
as functions of u, that is, of z; but it is obviously convenient to retain u instead
of expressing it in terms of z. As regards the form of these integrals observe that,
writing sn m = X, we have
dX
mi =
and thence
de=
k?\dX
d8 =
k VCaa) dX
each of which is in fact reducible to elliptic integrals, but I do not further pursue
this general case.
In the particular case a = a, we have
1 - (l + - j A^ sn» u = cn^ u,
and the equations become
,rt Id'snudu , kadnudu
dff = , ds = ,
en M en M
which admit of immediate integration ; viz. we have
. , k^. dn u+k'
^=h'^''^d^^k"
or determining the constant so that 0 may vanish for u=0, say
Ar* , /dn u + k' l-k'\
and
. , A^ , /dn u + k l-k\
.=P« log (5-^-^3;
viz. to verify these results we have
d0 ,P
du
^ hf [dnu+kf dn w - A
_ ^ sn M en tt _ Tji sn w
~ dn» u — k"' ' ~ en M '
and
du (1 + sn M 1 — sn u)
A;a en u dn u _ ka dn u
1 — sn" w ' en M
78 ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS. [653
Hence, recurring to the original equations, and writing for convenience a = a = l,
we see that a solution of the simultaneous equations
IS
a; = cos ^, y = sin 6, z = ku,
^ , /dn u+k' l-k'\ , , , /I + sn M\
- , &* , /dn u+k l-k\ , , , /I + sn M\
where, as before, k = k' = -tt^ .
Restoring the radius a, and writing the system in the form
x—a cos 6, y = a sin 0, 2 = kau,
„ .k?, /duM + i;' \-k'\ 1, , /l+snM\
we see that, as u passes from m = 0 to m = ^, and therefore 2 from 2 = 0 to
z = kaK (K the complete function -^i JT^r) . then 6 and s each pass from 0 to oo ;
and, similarly, as u passes from m = 0 to u = — K, that is, as ^ passes from 0 to
— kaK, then 6 passes from 0 to oo , and s from s = 0 to s= — oo; viz. the curve
makes in each direction an infinity of revolutions about the cylinder. Developing
the cylinder, a6 becomes an ^-coordinate ; viz. we have thus the plane cui've
z = koM,
It^a, fdnu + k' 1 - k'\
_ &*a , /dn M + k l-k\
'-i Y ^"^^ \dn u-k' -l+kT
which is a curve extending from the origin in the direction x positive, to touch at
infinity the two parallel asymptotes z=± kaK ; and conversely, when such a plane
curve is wound about the cylinder, there will be in each direction an infinity of
revolutions round the cylinder.
654]
79
654
ON CERTAIN OCTIC SURFACES.
[From the Qmrterly Journal of Pure and Applied Mathematics, vol. xiv. (1877),
pp. 249—264.]
I. CONSIDEE the torse generated by the tangents of the quadriquadric curve, the
intersection of the two quadric surfaces
aa? + by" + cz- +dvf =0,
aV + by + c'2- + d V = 0 ;
then, writing
be' - b'c = a', ad' - a'd =/',
ca'-c'a = 6', bd'-b'd = ^',
ab'-a^ = c', cd'-c'd=A',
and therefore
a'f' + b'g' + c'h' = 0,
the equation of the torse, writing for gi'eater convenience (a, b, c, f, g, h) in place of
(a', h', c', /', g', h'), but understanding these letters as signifying the accented letters
W, b', c', /'. g', h'), is
a*/Y2* + byz'x' + &h?(&u(^
+ a^f^a^vf + b^y^w^ + d'h^z^w*
+ 2b'd'ghx'y^z^ - 2cY-ahx*fw'' + Wfaga^z^w'
+ l&a-hffz'^a? - ^(ihfbfifz^jfi + 2c-g-bhfoi?w-
+ 2a^y^fgzi^!i?f - 2b'^h?cgz*oshu^ + Id'h-cfz^yHif
- 2bcg''hhHhf^z^ - Icahfhifz-'d? - 2ahfY-w*a?y''
+ 2{bg- ch) (cA - a/) (af- bg) a^f-zV = 0.
80 ON CERTAIN OCTIC SURFACES.
If in this equation we write
c''h'=c, c'h''=h;
[654
and therefore
be' - b'c = —
ca — c a =
^{hg) '
ab'-a^ =
and consequently
then the equation becomes
^{ch) '
ad'-a'd =
bd' - b'd =
cd' - c'd =
/
9
^(bg) '
h
(a/)^ + {bg)^ + (ch)^ = 0;
+ Ihcod^yH- - Icfa^yhip- + 1hfod'z''vi?
+ 2cay*z'^a? — tag-fzHo"^ + 2cgy*xW
+ 2ahz*3?y^ - 2bhz*xhv- + 2ahz*y'w''
- 2ghit}*y^z- - 2hfw*z^a? - ^fgvfxY
+ 2 {Q)gf - (chf] {{chf - (ffl/)*} {(a/)* - (c/t)*} a^^^'^w' = 0.
This same equation, without the relation
and with an arbitrary coefficient for x^y'^z'^w'^ ; or say, the equation
+ pxhu* + gyx^ + ^''■2*«^
+ 2bcx*y''z^ - 2cf«i^y-'w^ + 2hfaifzhtP
+ 2cafz''a? - 2agy*z'W + 2cgy^a?w-
+ 2ahz^ahj* - 2bh2'a?'uf + 2ahz*yhu'
- 2ghw*y'z^ - 2hfw*z^a^ - 2fgu/*x'y^
+ 2kafyz^if = 0,
where a, b, c, f, g, h, k are arbitrary coefficients, is the general equation of an octic
surface having the four nodal curves
X =0, . hz^vj^ — gvt^y^ + ay V = 0,
J, = 0, - hzhifi . +/wW + 6zW = 0,
2=0, gfv? -fw'^a? . + ca?y^ = 0,
w = 0, - ayV - bz^si? - ca?y^ . = 0.
654] ON CERTAIN OCTIC SURFACES. 81
In feet, the equation of the surface may be written in the form
V)* f/V + gy + h-z* - 2ghy^z* - 2A/iV - 2fga?y^}
+ 2vP (- cfx^- - agy*z* - bhz*a^ + 2ka^yV^
[+ bfx'z- + cgy*a? + aha^y^ j
+ [ay-z' + h2:'a? + c*^^) " = 0,
which puts in evidence the nodal curve
w = 0, — ay-z^ — bz'^oc^ — cxy = 0 :
there are three similar forms which put in evidence the other three nodal curves.
The four curves are so related to each other that every line which meets three
of them meets also the fourth curve; there is consequently a singly infinite series of
lines meeting each of the four curves; these break up into four series of lines each
forming an octic scroll, and each scroll has the four curves for nodal curves respectively;
that is, each scroll is a surface included under the foregoing general equation, and
derived from it by assigning a proper value to the constant k. To determine these
values, write
\ + IJL + V =0,
af bg cli .
r^ + -^ + - = 0,
equations which give four systems of values for the ratios (\ : /^ : v). We have then
k = af--^ + bg + cA^ — ,
X /x \
viz. k has four values corresponding to the several values oi (k : fi : v).
The scroll in question is M. De La Goumerie's scroll Si ; the equation of the
scroll 2j is consequently obtained from the octic equation by writing therein the last-
mentioned value of k.
It is to be noticed that k is, in effect, determined by a quartic equation; and,
that, for a certain relation between the coefficients, this equation will have a twofold
root. Assuming that this relation is satisfied, and assigning to k its twofold value,
the resulting scroll becomes a torse ; that is, two of the four scrolls coincide together
and degenerate into a toree; corresponding to the remaining two values of k we have
two scrolls, companions of the torse. In order to a twofold value of k, we must have
af_ bg _ch
and thence
(a/)* + (65-)4 + (cA)* = 0;
or, what is the same thing,
(«/+ bg + clif - 27 abcfgh = 0.
C. X. 11
82 ON CERTAIN OCTIC SURFACES. [654
If for a moment we write af=CL*, bg = 0', ch = 'f, and, therefore, o + /3 + 7 = 0; then
for the twofold root, we have X : /i : i/ = a : yS : 7, and consequently
A; = a'(7-/3) + y8'(a-7) + 7'(/3-a)
= (a-/9)r;8-7)(7-a),
that is,
* = {(«/)* - (f>g)^\ {(bgf - (cA)*l ((c/0* - (a/)*),
which agrees with the result in regard to the octic torse.
If in the octic equation we write {x, y, z, w) in place of (ar", y^, z-, w'), then we
have the quartic equation
+ fx^ + 5ry w» + h^zhi}'
+ ^hcxHjz — 2cfa?yw + 2bfa?zw
+ Icay^zx — lagy'^zw + tcg^fxw
+ 2abz-xy — 2bhz'xw + iahz^yw
— ighv^yz — ihfu/'zx — 2fguPxy
+ 2kxyzw = 0,
which is the equation of a quartic surface touched by the planes x = 0, y=0, z=0,
w = 0, in the four conies
X =0, . hzw — gwy + ayz = 0,
y = 0, — hzw . -'rfwx + bzx = 0,
z =0, gyw —fwx . + cxy = 0,
w = 0, — ayz — bzx — cxy . = 0,
respectively.
II. The octic surface
U = b^c'/'a? + c'ayf + aWtV +fYhHt^
- 2a'cg (bg - ch) fz^ - 2b'ah (ch - of) !fia? - 2c%f (of- bg) a^f
+ 2a''6A( „ )y^z»+2b^cf{ „ )z^3fi+2c^ag{ „ )^
- 2f'bc ( „ ) afw'' - 2g^ca ( „ ) i/^w" - 2h''ab ( „ ) zhti'
+ 2fghi „ )x!W+2g"-h/( „ )yV+2hYg( „ )z'-uf
+ /' (by + c'/i' - 'ibgch) Wa^ + g^ (d'h* + a'/^ - 4>chaf) w*y* + h' {a^p + I/'g" - iahfg) vr'z*
+ a'( „ )y*z*+bH „ )z*a^+(f( „ )a^
- 2gh (bcgh - a'/' - 2afbg - 2afch) w'y''z'
- 2bh ( „ ) z*a^-
+ 2cg ( „ ) y*aiW
+ 26c ( „ ) ai'y^z^
654] ON CERTAIN OCTIC SURFACES. 83
- 2V {cahf- by - 2bgaf - 2bgch) ■w^z'^ofi
- 2c/ ( „. ) a^yi^
+ 2ah{ „ )2*y^vfi
+ 2ca( „ )y*x^z^
- 2fg (abfg - (?h? - 2chaf - 2chbg) vfah/'^
-^ag{ „ )y*z^w''
+ 26/( „ )a*zhi^
+ 2a6 ( „ ) 2*0^^2
+ ^nofiy'^zhv^ = 0,
where the values of the coefficients indicated by ( „ ) are at once obtained by the
proper interchanges of the letters, and where fl is an arbitrary coefficient, is a surface
having the four nodal conies
X =Q, . cf - bz' +fw- = 0,
y =0, -caf' . +az' + gw- = 0,
z =0, baS'-ay^ . hw' = 0,
w = 0, -far' - gy^ -hz' . = 0.
In fact, writing the equation under the form
w»e + {far'+gf + hz'Y X (6V«* + d'ay + a^b^z* - 2a^bcy''z'' - ib^caz'a? - 2c»a6«»y») = 0,
we put in evidence the nodal conic w = 0, /ar" + gy'' + A^= = 0 : and similarly for the other
nodal conies.
It is to be observed, that the complete section by the plane w = 0 is the conic
fa? + gy-\-hs^ = 0, twice repeated, and the quartic
ft'c'a;* + (?aY + a*'^ - 2a''bcy'z^ - 2ab'cz''sc?' - 2ahc^a?y'' = 0 :
the latter being the system of four lines
"'j.^j.'^— ft * y ^ ft
The plane in question, w = 0, meets the other nodal conies in the six points
(a; = 0, -V-c2' = 0), (y = 0, c2'-aa? = 0), {z = 0, cw^-6y>=0),
which six points are the angles of the quadrilateral formed by the above-mentioned
four lines.
The four conies are such, that every line meeting three of these conies meets
also the fourth conic. The lines in question form a double system : each of these
11—2
S4 ON CERTAIN OCTIC SURFACES. [654
systems has, in reference to any pair of nodal conies, a homographic property as
follows ; viz. considering for example the two conies in the planes z = 0 and w = 0
respectively, if a line meets these conies in the points P and Q respectively, and
through these points respectively and the line x=0, y = 0 we draw planes, then the
system of the P planes and the system of the Q planes correspond homographically
to each other, the coincident planes of the two systems being the planes x = 0 and
y = 0 respectively.
Conversely, if through the line (a; = 0, y = 0) we draw the two homographically
related planes meeting the two conies in the points P and Q respectively, then, for
a proper value (determined by a quadratic equation) of the constant k(=^-i-0) which
determines the homographic relation, the line PQ will be a line meeting each of the
four conies, and will belong to one or other of the above-mentioned two systems,
as k is equal to one or the other of the two roots of the quadratic equation. The
scroll generated by the lines meeting each of the four conies, or what is the same
thing, any three of these conies, is primd facie a scroll of the order 16 ; but by
what precedes, it appears that this scroll breaks up into two scrolls, which will be
each of the order 8. Moreover, each scroll has the four conies for nodal curves; and
since the equation U=0 is the general equation of an octic surface having the
four conies for nodal curves, it follows, that the equation of the scroll is derived
from that of the octic surface U=0, by assigning a proper value to the indeterminate
coefficient fl; so fthat there are in fact two values of il, for each of which the
surface 17 = 0 becomes a scroll.
To sustain the foregoing conclusions, take x=6'y, x=0y for the equations of the
two planes through the line (a; = 0, y = 0), which meet the 2:-conic and w-conic in the
points P and Q respectively ; then the equations of the line PQ are
V(/i9= + 9){x- ffy) + V(- h) (6' -e)z=0,
- 'Jibe'"- -a)(x- dy y + V(- h) id' -6)w= 0,
or, writing therein d' = kd, the equations are
V(/^ +9)(,x- key) + V(- h) {k -l)ez =0,
-V(6*»^ -»)(«- ey)+s/{-h)(k-l)ew = 0.
To find where the line in question meets the plane y = 0, we have
V(/(9-^ +g)x + ^/{-h)ik-l)ez=^0,
- sj{hk^e' -a)x + V(- A) (^• - 1) (9w = 0,
and thence
(/(?» +g)a? + h (k - If e^z'' = 0,
(6/c»^ -a)i^ + h{k-lf e^w- = 0,
or multiplying a, g and adding
{af+hgk?)!c'-\-h{k-\)'{a^ + gvf') = Q,
or assuming
af^-hgk' + ch(]e-\f = <i,
654] ON CERTAIN OCTIC SURFACES. 35
the equation is
— cx^ 4- az^ + gvfl = 0.
That is, k being determined by the quadric equation af->thgk^+ch{k-\y = 0, the line
PQ meets the y-conic y = 0, -ca^+ az' + gw- = 0; and, in a' similar manner, it appears
that the line PQ also meets the a;-conic x = 0, cy" — hz^ +fvf = 0.
Writing for greater symmetry \ : -k : k-\=\ : ^ : v, vie have
X + /i + V =0,
af\- + hgfi? + chv" = 0,
so that there are two systems of values of (\, fi, v) corresponding to, and which may
be used in place of, the two values of k respectively.
Starting now from the equations
(/(9» + g) (key - xY + h(k- ly d'z-^ = 0,
(bk'e"--a){0y -a:y- + h(k-l)''&hv'' = 0,
the elimination of 0 from these equations leads to an equation f7=0, of the above
mentioned form but with a determinate value of the coefficient.
The process, although a long one, is interesting and I give it in some detail.
Elimination of 0 from the foregoing equations.
We have
U=MU [(fe-- +g) (k0y -xy + h(k-iy e'z'l
where 11 denotes the product of the expressions corresponding to the four roots of
the equation
(bh'0'-a)(0y-xy + h(k-iy0Hi/' = O.
Observing that this equation does not contain z, and that the expression under the
sign n does not contain w, it is at once seen that the product 11 is in regard to
{z, w) a i-ational and integral function of the form (z'', w^)^; and since, in regard to
{z, w), U is also a rational and integral function of the same form (z", vp)*, it is clear
that the factor M does not contain z or w, but is a function of only (x, y). To
determine it we may write z = 0, w = 0: this gives
c» (6«" - ay^y (fa? + gy-J = MW (f0' + g) (k0y - xy,
where
(blt?0'-u)(0y-xy = O,
and the values of 0 are therefore +7^-^,, -r-ivk, -,-■ Hence substituting and
k>J{b)' k^(b) y y
observing that
eh'{k-iy = {af+bgk-y,
86 ON CERTAIN OCTIC SURFACES. [654
it is easy to find
a;* A'(A:-l)»•
that is, we have
{bli?&'-a){ey-xf + h(k-lf9W = 0.
cc
If for greater convenience we write 0= --<f>, then this formula becomes
if
where
(b^ai'ifi'' - ay') (^-l)- + h(k-iy vfi^- = 0,
or, what is the same thing,
Suppose that the terms in U which contain z'' are = @z- ; then we have
or, what is the same thing,
® = A(l=ly^ ^S.^^'n'c/*^,^^ +^3/=) {k<i> - \r,
where 11' refers to the remaining three roots <f>«, (^3, ^^ ; this may also be written
Hence, observing that we have identically
and writing <^ = + ^^^^, 4> = t> h=\/(— 1) as usual}, we find
n {<f,w V(/) ± iy V(5')l = -|^ [c {^ V(/) ± iy VC^^'/s'w^] y^
n(^<^-l) JA^ih^-af + hv^);
whence, writing for shortness
A = [c {a; V(/) + ty V(5')1 -/^r^^] [c {a; V(/) - iy ^/igf] - fgv^l
= cy'V + cyy* +f'g^v/* + icfg'y^vf - icfgic^' + 'i&fga^f,
654] ON CERTAIN OCTIC SCTRPACES. 87
we find
and thence
n (/a;2</)= + gy^) (k<f> - 1 )» = ^' ^^^~ ^^" A (6^ - ay' + hw^f ^ ,
and consequently
Hence, writing
we may calculate separately the terms
A B
-j A
and
2 ^ ^ + ^
[x^>J{f) + iysJ{g) x^-s/(f)-iy'J(g),
The first of these is
^{k-iy{fai' + kfgyy (bx" - ay' + hvfly ^'^' ^' '^^''
if for shortness
{x, y, wY = {fa? + %2/') [4 {(2 - A) 6«= - aj/^ + A (1 - k) vPf
-2 (ha?- ay' + hvfi) {(6 - 6;fc + A;^) 6«= -ay' + h(\- kf it?]]
+ ik'(k-l)gy'-(ba?-ay'+hu?){(2-k)ba?-ay' + h(l-k) v?}:
the second is
2
" (k-lf{fa?+h'gy'y^ ^'^' ^' '^^'''
if for shortness
(x, y, wf = {{fa? - k?gy') {cfa? - cgy' -fgv?) - ^ckfga?f]
X [fgbl(?a? + [2cA {k-lf-y a/] gf +fgh (k - If v?}
+ 2 [k'bg -ch{k- iy\fga?f [c {k+\) {fa? - kgf) - kfgv?\ ;
and hence
e = ^f^^j^ ,y \^^ [x, y, wY + 2 {bx' - af + hiiff {X, y, w)«],
which must be a rational and integral function of {x, y, w).
In partial verification of this, observe that, because U contains the terms
H?cf{ch - af) a?z' + 2Q,a?y"-z'w\
88 ON CERTAIN OCTIC SURFACES. [654
0 should contain the terms
2h^cf(ch -a/)af + 2£la?y^i}',
viz. in 0 the term in a? should be =2Ifcf{ch — af)af^.
Now writing y = 0, w — 0, we have
A = c*/ V,
{x, y, w]' = by {4 (2 - kf -2(6-6* + 1<P)} of,
= by{4,-4>k+2t)af,
{x, y, wf = hcpg¥af ;
and hence the requii-ed term of 0 is a;* multiplied by
¥&fh. (4 - 4it + 'i.k') + 2hl>cfg1<? :
viz. the coefficient is
= 26V [c'l (2 -2k + *=) + bgk'l
= 2b'cf[cli + ch (1 - kf + bgk^],
which in virtue of the relation af+ bgk? + ch (1 — k)- becomes, as it should do,
= 2b-cf{ch - af).
The actual division by (fa^ + k^gy^y would, however, be a very tedious process, and
it is to be observed, that we only require to know the term 2na^?/V of 0. We may
therefore adopt a more simple course as follows : the terms of 0 which contain w^
are = (Aa;^ + 2naf^^ + By*) vf, hence writing for a moment
{x, y, w}' = P + Quj\ {«>, y, wy = R + Sw%
and obseiTing that we have
A = c= (/ar^ + gf-y- - Ic'fg {fa? -gy') w' + &c.,
(baf - ay^ + hw^y = {bod^ - ay'^y + 2h (ba? - ay*) v/' + Sic.,
we have
( fa? + kfgyj (Aai' + 2na?y"- + By*) = (?h {fa? + gyj Q - 2(?fgh (fa? -gf) P
+ (ba?-ay^y8+ 2/t {ba?-ay-)E.
But in this identical equation we may write a?=a, y^=b, which gives
(a/ + tbgy {Aa? + 2fia6 + B¥) = c?h (af + bgy Q - 2(?fgh (af- bg) P ;
and from the equation
{x, y, w]^ = P + Qn?,
we have
r, r^ , / ^ , .^^T* i(l - k) oh + (1 - k) hv?}"']
P+Qn? = (af+ bgl?) [ |^ ^J^^ _ ^\ _^ ^J ^^ ' J
+ 4>k?(k- 1) bghw- (1 - k) ab.
= - ch (k-iy {4 (k-iy (a^b- + 2w^abh) -2(o-6k + k?) hw^\
-4,k'(k-iyab'ghv?,
654] ON CERTAIN OCTIC SURFACES,
that is,
Q= {k-\y ahh [ch (- 6fc= + 4* + 2) - Wh\,
whence
{k-\y'{Aa? + 2nah + B¥) = ah(af+ hgf [ch (- Q¥ + 4/1- + 2) - 4A;%]
+ 8/5r(a/-65r)(i-l)»(a6)=.
But we have
Aa? +B¥ = - 2ab (af- hg) (- afbg + (?h? + 2cyta/+ 2c%),
and thence
2 (^ - 1)'' n = (a/+ 6£f)=' [c/t (- m-' + ik + 2)- 4'k'bg]
+ (k- ly (of- hg) /- 2afbg + 2cW + 4cAa/+ 4c%\
or
(A: - ly ft = (a/+ 65r)= [cA (- 'ik" + 2^ + 1) - 2k?hg-]
+ {k-lf {af- bg) [Safbg + c-h' + 2chaf+ 2chbg].
Writing - 3A;= + 2^ + 1 = - 3 (A - 1 )^- 4 (k - 1), this is
89
n = (af+bgr[ch{-S-^^)-^/^\^,bg
+ (a/- 65-) [3a/i'5^ + Ch- + 2chaf+ 2chbgl
or since 1 : —k : k—l = X : fj. : v; and writing for shortness (a/, bg, ch) = (a, ^, y),
this is
n = (a + ^ylY(-3-^^-^d + (a-0){Sa0 + y- + 2ya+2y0},
which is the value of il : viz. the conclusion arrived at is that, eliminating 0 from
the equations
(/&-- +g){key-xf + h{k-lfe'-z" =0,
{h¥0'-a){ dy-xy + hik-iye^w'-^O:
where k denotes a determinate function of af, bg, ch, viz. writing af, bg, ch=a, ^, y
and 1 : —k : k—\ —\ : fi : V, viQ have
\ + /t + V =Q,
aX'+^fi'+yv^^O,
equations which serve to determine k: the result of the elimination is the octic
equation
b'df'af + . . . + 2nxyzhv^ = 0,
where il has the last-mentioned value.
C. X. 12
90
ON CERTAIN OCTIC SURFACES.
[654
The value of fl is unsymmetrical in its forni, and there are apparently six values;
viz. writing
- 3 - ^) - ?^° /SJ + (a -^)(-S+ a/9 + 7'),
-S-'^^)-^^ly\-(y-a)(S + ya+n
_ 3 - ^) - ?->;i 4 - (« - ^3) (-S + a/8 + y),
V J v )
A =(/3 + 7y ^
B =(7+a)'(^
C =(a + /3)»
£, = (7 + a)»|/3
where for shortness S—2{^y + ya + a^), the six values would be A, B, C, A^, B^, C,.
But we have really
A=B = G = -A, = -B,=^-C,\
so that 11 has really only two values, equal and of opposite signs, or, what is the
same thing, fl^ has a unique value. In fact, writing for shortness
\ + li + v = P, oX= + /3/*= + 7«/== JT,
we find at once the identity
\^ {A + A,) = (y3 + yf{- 2X - 4XaP),
80 that A = -A„ in value of P = 0, Z = 0. And similarly B = -Bu G = -G,.
But the demonstration of the equation A=B is more complicated. We have
A-B=-Sai0 + yy-ioc{fi+yy^-2y(^ + yy^^^ + (fi-y)iS + l3y + c^)
+ 3yS(7 + a)^ + 4/3(7+ay- + 2a(7 + a)=-'-(7-a)(£f + 7a + /8»),
that is,
\W{A-B) = {-3a(/3+ yf + S^(y + ay+{^ -y){S + ^y + a')-(y -a)(S+ya + ^-)} \Y
-4a(/3 + 7)-X./i"
-2y(B + yyv'(i''
+ 4)8(7 + °-y v^y
+ 2a{y + ay\\
654]
ON CERTAIN OCTIC SURFACES.
91
or, denoting for a moment the coefficient of X^* by K, and writing also yv^ = X - a\- — fifj?,
v = P —\ — /I, this is
= ifxy
- 4a (/3 + 7)^ Xfj."
- 2 (/3 + 7)- fi^ (X - aX- - /3/i-)
+ 4/3 (7 + a)= \XP - \ - /i)
= - 2 0 + 7)= fi'X + 4/8 (7 + a)-^ \>P
+ 2a(7+a)2X*
- 4/9(7+ a)»\>
+ j- 4^ (7 + a)^ + ^ + 2a (0 + 7)=} W
- 4a (^ + 7)= V»
+ 2/3(/9 + 7)>S
and here the coefficient of V/** is found to be
= 2 {ayS (a + ^) + 7 (a - 0)"- - 87= (a + /8)|.
Hence, the terms without X ot P are =2V, where
V = o (7 + a)^«
- 2/9 (7 + a)»\>
+ (0/8 (a + /9) + 7 (a - /9)' - 3r (« + yS)} X>=
-2a(/9+7>'\/ti'
+ /3(/3+7)V,
and this is identically
(a+7)V'
= + 27\/i V X -{
+ (/8 + 7)m;
a\= + /3/a= + 7(P-\-/i/ = Z,
we have the first factor
(a + 7) X' + (/3 + 7) At' + 27 V = -X - 7-?^ + 27-P (>. + /I*),
and consequently
W (4 - 5) = - 2 (;9 + 7f /t'Z + 4/S (7 + afX'^P
+ 2 {Z -7^^ + 27P (\ + /a)l {a(7 + a)\^ - 2 (/37 +70 + a/3) \/t + ^ (/S + 7) /*'! ;
viz. in virtue of P = 0, Z = 0, we have A=B. And thus
A^B = G=-A, = -B, = - (\:
so that the only values of fl are, say, A and - yl.
12—2
where observing that
a(7 + a)\=
- 2 (/87 + 7a + a/3) V
92
ON CERTAIN OCTIC SURFACES.
[654
Reverting to the original equations
iP" +9)(key-a:y + h(k-iye'z' =0,
(6if ^ - a) ( 0ij-icy + h {k - \)- GW = 0,
say these are
(a, b. c. d, eK iy = 0,
(a', b', c'. d'. e'K iy = 0.
then the coefiicients in the two equations have the values
f^y\ bkY,
- 2kfxy, - 2blt^xy,
fa? + gky- +h{k- If z\ hk'a? - ay^ + h {k - 1)» <
- 2gkxy, 2axy,
go?, - aa?,
where observe that only c contains z'^, and only c' contains lif. The result of the
«limination is
a, b, c, d, e =0;
a , b , c , d , 6,
a, b, c, d, e,
a , b , c , d , e,
a', b', c', d', e'
a', b', c', d', e',
a', b', c', d', e',
a', b', c', d', e',
■viz. here the only terms which contain s? and vf are
c W= V c'«aV,
and hence the terms in «* and vfi are
h^ {k -Vf!^. a^'b'k'x'y* + A« (A - 1)« iif" .pg^k'x'y*,
= h^k" {k-\Yoc*y* (a'b-h^^ +fYh:W),
viz. these are
or assuming that the determinant contains as a factor the function b''(ff'ii?+ ... + 2ila?y''z°w\
with a properly determined value of fl, we see that the other factor is =h^k*(k — iyx*y*,
which agrees with a preceding result.
655]
93
655.
A MEMOIR ON DIFFERENTIAL EQUATIONS.
[From the Quarterly Jottrnal of Pure and Applied Mathematics, vol. xiv. (1877),
pp. 292—339.]
We have to do with a set bf variables, which is either unipartite (x, y, z,...),
or else bipartite (x, y, z, ...\ p, q, r, ...), the variables in the latter case corresponding
in paii-s x and p, y and q, &c.
A letter not otherwise explained denotes a function of the variables. Any such
letter may be put = const., viz. we thereby establish a relation between the variables ;
and when this is so, we use the same letter to denote the constant value of the
function. Thus the set being {x, y, z; p, q, r), H may denote a given function
pqr — xyz; and then, if £r= const., we have pqr — xyz = H (a constant). This notation,
when once clearly understood, is I think a very convenient one.
The present memoir relates chiefly to the following subjects :
A. Unipartite set (a;, y, z,...). The diiferential system
dx _dy _dz _
and connected therewith the linear partial differential equation
xf+Yf + zf+... = 0:
dx dy dz
also the lineo-differential
Xdx+Ydy-\-Zdz+ ....
B. Bipartite set {x, y, z,...; p, q, r,...). The Hamiltonian system
dx _dy _ dz _ _ dp _ dq ^ dr ^
dB~dB~dB~'~ _dH~ _dH'' dJJ ""
dp dq dr dx dy dz
94 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
and connected therewith the linear partial differential equation
otherwise written
dHd£_dHde dHde_dHde
dp dx dx dp dq dy dy dq '" '
_d{H, 6) d{H,6) _
d(p, w) d {q, y)
where H denotes a given function of the variables: also the Hamiltonian system as
augmented by an equality =dt, and as augmented by this and another equality
dH . dH dH
_ jTT ( dH dH dH \
X dp ^ dq dr '"/'
C. Bipartite set {x, y, z,...; p, q, r, ...). The partial differential equation
H = const., where, as before, H is a given function of the variables, but p, q, r, ...
are now the differential coefficients in regard to x, y, z,... respectively of a function
F of these variables, or, what is the same thing, there exists a function
V = I (pdx + qdy + rdz +...),
of the variables x, y, z,
In what precedes, I have written {x, y, z, ...) to denote a set of any number n
of variables, and (x, y, z, ...; p, q, r,...) to denote a set of any even number 2n of
variables, and the investigations are for the most part applicable to these general
cases. But for greater clearness and facility of expression, I usually consider the case
of a set {x, y, z, w), or {x, y, z; p, q, r), &c., as the case may be, consisting of
a definite number of variables.
The greater part of the theory is not new, but I think that I have presented
it in a more compact and intelligible form than has hitherto been done, and I have
added some new results.
Introductory Remarks. Art. Nos. 1 to 3.
1. As ah'eady noticed, a letter not otherwise explained is considered as denoting
a function of the VEiriables of the set ; but when necessary we indicate the variables
by a notation such as z =z{x, y)\ z is here a function (known or unknown as the
CEise may be) of the variables x, y, the z on the right-hand side being in fact a
functional symbol. And thus also z = z{x, y), = const, denotes that the function z (x, y)
of the variables x, y has a constant value, which constant value is =z, viz. we thus
indicate a relation between the variables x, y.
2. The variables x, y, &c., may have infinitesimal increments dx, dy, &c. ; and
the equations of connexion between the variables then give rise to linear relations
between these increments, the coefficients therein being differential coefficients and.
655]
A MEMOIR ON DIFFERENTIAL EQUATIONS.
95
as such, represented in the usual notation; thus if z = z (x, y), we have dz = -^ dw -\- ^- dy,
cujc ^y
dz dz
where -r- , t- are the so-called partial differential coefficients of z in regard to x, y
respectively. If we have y = y («), then also dy = ~^ dx, and the foregoing equation
becomes
^^^^^_^dzdjX
\dx ay dxj
but considering the two equations z — z{x, y) and y = y (x) as determining z as a
function of x, say z = z (x), we have dz = ^ dx ; whence comparing the two formulae
d {z) _ dz dz dy
dx dx dy dx'
d(z)
The
where , is the so-called total differential coefficient of z in regard to x.
. d(z) dz
distinction is best made, not by any difference of notation ) -, ,~ , but by appending
in any case of doubt the equations or equation used in the differentiation. Thus we
have , where z = z (a;, y): or, as the case may be, ^ where z = z(x, y) and y = y {x).
3. A relation between increments is always really a relation between differential
coefficients : but we use the increments for symmetry and conciseness, as in the case
dj2c du dz
of a differential system ^ = -^ = ^ , or in a question relating to the lineo-differential
Xdx + Ydy + Zdz, for instance in the question whether this can be put = du.
Notations. Art. Nos. 4 to 6.
4. Functional determinants. If a, b, c,
then the determinants
da
da
da
da
da
dx'
dy
y
dx'
dy'
dz
db
db
db
db
db
dx'
dy
dx'
dy'
dz
dc
dc
dc
dx '
dy'
dz
are for shortness represented by
are functions of the variables x,y,z,w,...,
&c.,
d{a, b) d(a, b, c) „
d(x, y)' d(x, y, z)'
96
A MEMOIR ON DIFFERENTIAL EQUATIONS.
[655
the notation being especially used in the first-mentioned case where the symbol is
; ' — (. It is sometimes convenient to extend this notation, and for instance
d{x, y)
use ,-~^ — —. to denote the series of determinants
d{x, y, z)
da
da
da
cUc'
dy'
Tz
db
db
db
dx'
dy'
dz
which can be formed by selecting in every way two columns to form thereout a
determinant; the equation
d(a, b) ^Q
d{x, y, z)
will then denote that each of these determinants is = 0.
The analogous notation
d (a, b, c)
d{x, y)
would denote non-existent determinants, viz. there are here not columns enough to
form with them a determinant : and the notation is not required.
5. In the case of a bipartite set {x, y, z,...; p, q, r,...), if a, b are any
functions of these variables, we consider the derivative
(n h\ ^ ^^' ^) .d{a, b) d (g, 6)
^"' "^-dip, x)^d(q, y^dir, z)^-'
viz. (a, b) is used to denote the sum of the functional determinants on the right
hand.
6. Taking again {x, y, z, w, ...) as the variables, then in the theory of the
lineo-dififerential Xdx -^Ydy ■{- Zdz +Wdw ■'r ..., we use certain derivative functions analogous
to PfafRans. They may be thus defined; viz. considering the numbers 1, 2, 3, 4, ... as
corresponding to the variables x, y, z, w, ... respectively, we have
1 = Z, 2 = F, 3 = Z, 4 = TT, &c.,
dy dx' dz dx' "'
123 = 1.23 + 2.31-1-3.12
\dz dyj \dx dz J \dy dx I
1234 = 12 . 34 -I- 13 . 42 -t- 14 . 23
^fdX _dY\ (d^_dW\ /dX _dZ\ idW _dY\ /dX _dW\ (dT _dZ\
~\dy dx)\dw dz J \dz dx) \dy dw) \dw dx ) \dz dy)'
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 97
and, adding for greater distinctness the next following cases,
12345= 1.2345+ 2.3451+ 3.4512+ 4.5123+ 5.1234,
123456 = 12 . 3456 + 13 . 4561 + 14 . 5612 + 15 . 6123 + 16 . 2345,
where of course 2345, &c., have the significations mentioned above.
Dependency of Functmis. Art. Nos. 7 and 8.
7. Two or more functions of the same variables may be independent, or else
dependent or connected ; viz. in the latter case any one of the functions is a function
of the others a = a (x), b = b (x), the functions a, b are dependent, but if
a=a(x, y), b = b(x, y),
then the condition of dependency is
d{a, b)
d{x, y) "'
and, similarly, if a = a {x, y, z), b = b (x, y, z), then the conditions of dependency are
i d{a, b)
d{x, y, z)
= 0,
viz. if the equations thus represented are all of them satisfied, the functions are
dependent, but if not, then they are independent.
Observe that, when a=a(x, y, z), b=b (x, y, z) as above, if we choose to attend
only to the variables x, y, treating ^; as a mere constant, there is then a single condition
of dependency -,' x = 0, and so if we attend only to the variable x, treating y, z as,
d y^, y)
mere constants, then a and b are dependent. Thus when a = x, b=x' + y, the functions
a, b are independent if we attend to both the variables x, y; dependent if y be
regarded as a constant.
8. Further when a = a{x, y), b = h (x, y), c = c (x, y), the functions a, b, c are
dependent ; but when a=a{x, y, z), b = b (x, y, z), c=c (x, y, z), the condition of depen-
dency is
d (g, b, c) _ Q .
d{x, y, z) '
and so when a = a{x, y, z, w), b = b{x, y, z, w), c = c{x, y, z, w), the conditions of
dependency are
d{a, b, c) ^^
d(x, y, z, w)
viz. if all the equations thus represented are satisfied, the functions are dependent;
but if not, then they are independent. And so in other cases.
13
C. X.
98 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
The General Differential System. Art. Nos. 9 to 22.
9. Taking the set of variables to be (x, y, z, w), the system is
dx _dy _dz _ dm
'X~Y~~Z~W'
and we associate with this the linear partial differential equation
xf^Yf^zf^wf = 0.
dx ay dz dw
10. It is tolerably evident that the differential equations establish between
X, y, z, w a threefold relation depending upon three arbitrary constants; in fact,
regarding (a?, y, z, w) as the coordinates of a point in four-dimensional space, and
starting from any given point, the differential equations determine the ratios of the
increments dx, dy, dz, dw, that is, the direction of passage to a consecutive point ;
and then again taking for x, y, z, w the coordinates of this point, the same equations
give the direction of passage to the next consecutive point, and so on. The locus
of the point is therefore a curve, or we have between the coordinates a threefold
relation, and (the initial point being arbitrary) we have a curve of the system
thi-ough each point of the four-dimensional space, viz. the relation must involve three
arbitrary constants. But this being so, the constants will be expressible as fimctions
of the coordinates, viz. the threefold relation involving the three constants will be
expressible in the form a = const., b = const., c = const., where a, h, c denote respectively
functions of the coordinates (x, y, z, w).
11. Supposing that one of the relations is a = const., it is clear that the increment
, da , da , da -, da ,
da, = J- dx + -j-dy+ -^ dz + J— dw,
must become = 0, on substituting therein for dx, dy, dz, dw, the values X, Y, Z, W
to which by virtue of the differential equations they are proportional, viz. that we
must have identically
Xf + Yp + Z^+W^^=0.
dx dy dz dtu
Conversely, when this is so, we have da = 0, by virtue of the differential equation.
We say that a is a solution of the partial differential equation, and an integral
of the differential equations, viz. any solution of the partial differential equation is
an integral of the differential equations, and any integral of the differential equations
is a solution of the partial differential equation, or, this being so, we may in general
without risk of ambiguity, say simply a is an integral*; similarly b and c are
integrals, and, by what precedes, there are three integrals a, b, c.
* Viz. we use indifferently, in regard to the differential equations and to the partial differential equation,
the term integral, which is appropriate to the differential equations ; the appropriate term in regard to the
partial differential equation would be solution.
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 99
Observe that, in speaking of an integral a, we mean a function of the variables-
the differential equations give between the variables the relation a = const., and when
this is so, we use the same letter a to denote the constant value of this function.
12. In speaking of the three integrals a, b, c we mean independent integrals;
any function whatever ^a of an integral a, or any function whatever <f> (a, b) of two
integrals a, b, is an integral (viz. it is an integral of the differential equations, and
also a solution of the partial differential equation), but such dependent integrals give
nothing new, and we require a third independent integral c, viz. we need this to
express the threefold relation between the variables, given by the differential equations,
and also to express the general solution ^ (a, b, c) of the partial differential equation.
13. By what precedes the analytical condition, in order that the integrals a, b, c
may be independent, is that they are such as not to satisfy the relations
d(a, b, c) ^Q
d{x, y, z, w)
14. We moreover see d posteriori, that there cannot be more than three inde-
pendent integrals; in fact, if a, b, c, d are integrals, then, considering them as
solutions of the partial differential equation, we have four equations which by the
elimination therefrom of X, Y, Z, W, give
d(o, h, c, d) _
d{x, y, z, w)
and this is the very equation which expresses that a, b, c, d are not independent.
15. The notion of the integrals may be arrived at somewhat differently thus:
take a, b, c, d any functions of the variables, and write
A=X^+Yp+zf+Wp,
da: dy dz dw
and the like for B, C, D: then replacing the original variables w, y, z, w by the
new variables a, b, c, d, the differential equations become
da _db _dc _dd
A~B~'G~B'
where A, B, C, D are to be (by means of the given values of a, b, c, d a& functions
of X, y, z, w) expressed as functions of a, b, c, d. If then A=0, 5=0, C=0, the
differential equations become
da _db _dc _dd
viz. we have da = 0, db = 0, dc = 0, and therefore a = const., b = const., c = const., that
is, we have the integrals a, 6, c as before.
13—2
100 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
16. There is no general process for obtaining an integral a of the differential
equations. Supposing such integral known, we can introduce it as a variable, in
place of one of the original variables, say w, viz. we thus reduce the system to
dx _dy _dz _da
X~Y'"Z~'0'
where X, Y, Z now denote the values assumed by these functions upon expressing
therein w as a function of x, y, z, a, viz. they are now functions of x, y, z, a. The
system thus breaks up into do = 0 and the system
dx _dy _dz
in which last (by virtue of the first equation, or a = const.) a is to be regarded as
a constant ; the original system of three equations between four variables is thus
reduced to a system of two equations between three variables. Supposing h to be
an integral of this reduced system, h is given as a function of x, y, z, a, but upon
substituting herein for a its value as a function of x, y, z, w, we have b a function
of the original variables x, y, z, iv, and b is then a second integral of the original
system.
17. In like manner supposing a and b to be known, we reduce the system to
the single equation
dx _dy
X~T'
where X, Y are now functions of x, y, a, b; supposing an integral hereof to be c,
we have c a function of x, y, a, b; but upon substituting herein for a, b their values
as functions of x, y, z, w, we have c a function of x, y, z, w, and as such it is the
third integral of the original system.
18. It may be remarked that if, to the original system, we join on an equality
=• dt, viz. if we consider the system
^ _dy _dz _d'w ._ J .
where X, Y, Z, W are as before functions of the variables {x, y, z, w), then the
integrals a, b, c of the original system being known, we can by means of them
express for instance X as a function of x, a, b, c, and we have then, const. =t— l ^ ,
where the integration is to be performed regarding a, b, c as constants ; writing f -y = t,
but after the integration replacing a, b, c by their values as functions of x, y, z, w,
we have t a function of x, y, z, w; and we say that t—r is an integral ; putting
it = const, we use also t to denote the constant value of the integral 1—1-^ in
question. Observe that here, the integrals o, b, c being known, the last integral t-r
is obtained by a quadrature.
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 101
19. The result would have been similar, if the adjoined equality had been = -s,
{T a function of x, y, z, iv), but in reference to subsequent matter, I retain the
dV
equality = dt, and adjoin a second equality = _. (fi a function of x, y, z, w) ; we
have then the integral t — r as before, and another integral V — I -^ , where n, X
are first expressed as functions of x, a, h, c, but after the integration a, b, c are
replaced by their values as functions of {x, y, z, w), say this is the integral V —\;
this, when the integrals a, b, c are known, is (like t — r) obtained by a quadrature.
20. Attending only to the adjoined equality = dt, we can by means of the four
integrals express each of the variables x, y, z, w as a, function of a, b, c, t — r; viz.
these four equations, regarding therein t — r as a variable parameter, are in fact
equivalent to the equations a = const., b = const., c = const., which connect the variables
X, y, z, w with the integrals a, b, c regarded as constants.
21. All that precedes is of course applicable to a system of n — 1 equations
between n variables, the number of independent integials being = n — 1.
22. I take an example with the three variables x, y, z; the differential equations
being i
dx _ dy _ dz
tio (.y - z)~ y {z - «!)~ z {x - y)'
and therefore the partial differential equation
, .de , , .d6 ^ , .d0 -
The integrals are a = x + y +z, b— xyz ; and it will be shown how either of these
integrals being known, the system is reduced to a single equation between two
variables, say x, y.
First, a being known, =x + y + z as before, we have
x{y-z) = x{x-\-'2.y-n), y(z-x) = y(a-2x- y),
and the system is
dx dy
x(x + 2y-a) ~ y(a-2x-y)'
which has the integral b = xy(a-x-y); observe that this is a solution of the partial
differential equation
x(x + 2y-a)^ + y(a-2x-y)^=0.
For a putting its value we find b = xyz.
102 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
Secondly, b being known, = xyz as before, we have
a'(y-«) = a'3'-|. y{z-x) = --xy,
and the system is
dx _ dy
h~b •
which has the integral a = x + y -\ ; observe that this is a solution of the partial
differential equation
/ h\de (h \de ^
['^-y)di+[x-''y)Ty-^-
For b putting its value, we find a = x + y + z.
The Multiplie): Art. Nos. 23 to 29.
23. First, if there are only two variables (a,-, y), the system consists of the
single equation
dx _ dy
Z~T'
which may be written
Ydx-Xdy = 0.
Hence, if a be an integral, we have
-dx + ^^dy^O;
the two will agree if there exists a function M such that
dx dy
and thence, in virtue of the identity
d da _ d da
dy dx dxdy'
we find
dMX dMY ^
dx dy '
or, as this may also be written,
ydM , ^dM , „/dZ , dY\ -
dy \dx dy .
as the condition to determine the multiplier M. Supposing M known, we have
M {Ydx — Xdy) = da, or say a = \M (Ydx — Xdy), viz. the integral a is determined by a
quadrature.
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 103
24. In the case of three variables (x, y, z), the system is
dx _dy _dz
or, writing these in the form
Ydz-Zdy = 0, Zdx-Xdz = 0, Xdy~Ydx = 0,
the couree which immediately suggests itself is to seek for factors L, M, N, such
that, a being an integral, we may have
L{Ydz- Zdy) + M {Zdx - Xdz) + N{Xdy - Ydx) = da,
but this does not lead to any result. The course taken by Jacobi is quite a different
one : he, in fact, determines a multiplier M connected with two integrals a, b.
25. Supposing that a, b are independent integrals, we have
X^+Yp + Zp^O,
dx dy dz
Xf+Yf + Zf=0;
dx dy dz
and determining from these equations the ratio of the quantities X, Y, Z, we may,
it is clear, write
MX,MY,MZ = i^„ i^\, ip^.
d (y, z) d (z, x) d (x, y)
It may be shown that we have identically
d d{a, b) d d (a, b) d d (a, b) _
dx d (y, z) dy d (z, x) dz d (x, y) '
and we thence deduce
djMX) d(MY) d^Z) ^ ^ .
dx dy dz '
or, what is the same thing,
da; dy dz \dx dy dz) '
as the condition for determining the multiplier M.
26. The use is as follows: supposing that M is known, and supposing also that
one integral a of the system is known, we can then by a quadrature determine the
other integral b. Thus, supposing that we know the integral a, =a(x, y, z), we can
by means of this integral express z in terms of x, y, a; and hence we may regard
the unknown integral b as expressed in the like form, b = b(x, y, a). The original
values of T- , -j- , -r- become on this supposition
db db da db db da dh^da
dx da dx' dy da dy ' da dz '
104 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
aud we thence find
d (a, b) d (a, b) d{a, b) _ da db da dh d (a, b)
d (y, z) ' d(z, x)' d (x, y) dz dy' dz dx' d {x, y) '
We have therefore
ifv MV— da db da db
' dz dy' dz dx'
and, consequently,
^^^fx^^fy^y- =^^^^-^^yy^
dz
viz. M, -r , Y, X being all of them expressed as functions of x, y, a, the expression
on the right-hand is a complete differential, and we have
b=i^^{Ydx-Xdy);
dz
that is, the integral b is determined by a quadrature.
27. Thus, in the example No. 22,
dX dY dZ^^
dx dy dz '
and a value of the multiplier is = 1. Supposing that the given integral is a—x + y+z,
da
dz
then J- = 1, and we have accordingly 1 as the multiplier of the equation
y{a — 2x — y)dx + x{a — x — 2y) dy = 0,
viz. this equation is integrable per se. Supposing the given integral to be b = xyz,
then -j-=xy, viz. we have — as the multiplier of the equation
az xy
[i - *y) ^ + (y - ^y) dy = 0,
and we thus in each case obtain the other integral as before.
28. The foregoing result may be presented in a more symmetrical form by taking
in place of x, y any two variables u=u(x, y, z), v = v (x, y, z).
Supposing the integral a known as before, the system then is
du _dv _da
U~V~~0'
where U, V=X y-+ Y , +Z -,- , X ^-+ Y-j- +Z ^-, these being expressed as functions
dx dy dz dx dy dz o j-
of u, V, a; or, what is the same thing, we have Vdu — Udv = 0, a being in this
equation regarded as a constant.
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 105
From the foregoing values of MX, MY, MZ, we deduce
MU, MV-'^^'^' "' ^^ <^(^. «. b)
d{x, y, z)' d{x, y, z)'
But forming the values of du, dv, da, db, we have an equation, determinant = 0, which
equation may be written
d{x, y, z) d{x, y, z) d{x, y, z) d{x, y, z) '
or, writing herein da = 0, this is
viz. this is
or say
d(x, y, z) d{x,y,z) d {x, y, z)
M ( Vdu - Udv) = db t^""' "• '^l ,
' d{x, y, z)'
where, on the right-hand side, everything must be expressed in terms of u, v, a. It
thus appears that on expressing the final equation as a relation Vdu — Udv = 0 between
the variables u and v, the multiplier hereof is ilf -=- -77^ — ' — { ■ li u, v=x, y, this agrees
with a foregoing result.
29. The theory is precisely the same for any number of variables. Thus, if there
are four variables x, y, z, w, we have
MX,MY,MZ,MW = ^'^, .ji^J^%, d(a^ _d^^
d(y, z, w) d(z, w, x) d{w, x, y) d{x, y, z)'
and, we have between the functions on the right-hand an identical relation, in virtue
of which
djMX) d(MY) ^ djMZ) ^ d(MW)^^^_
dx dy dz dtv '
then, supposing that a value of M is known, and also any two integi-als a, b, and
that by means of these the equation to be finally integrated is expressed as a relation
Vdu — Udv = 0 between any two variables u and v, the multiplier of this is
_ ^ d(u, V, a, b)
~ ^ d(x, y, z, w)'
where U, V and this multiplier are to be expressed in terms of u, v, a, b.
The general result is that, given a value of the multiplier, and also all but one
of the integrals, the final integral is expressible by a quadrature.
C. X. 14
106 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
P/affian
Theorem. Art. No. 30.
30.
According as
the variables
are
we have
«,
Xdx
= du,
!^, y.
Xda:+ Ydy
= \du,
X, y, z,
Xdx+Ydy + Zdz
= \du + dv.
X, y, z, w,
Xdx+Ydy + Zdz +
Wdw
= Xdu + fjdv.
and 80 on ; viz. the theorem is that, taking for instance two variables, a given lineo-
dififerential Xdx + Ydy is = Xdu, that is, there exist X, u functions of x, y, which verify
this identity, or, what is the same thing, such that we have
Y rr ~. du du
^' ^ = ^Tx' ^d^'
and so, in the case of three variables, there exist \, u, v functions of x, y, z, such that
y „ „ _ du dv du dv du dv
' ' dx dx' dy dy' dz dz'
The problem of determining the functions on the right-hand side is known as the
Pfaffian Problem; this I do not at present consider, but only assume that there exist
such functions.
The Hamiltonian System, its derivation from, the general System. Art. Nos. 31 to 34.
31. Considering a bipartite set (x, y, z: p, q, r), the general system of differential
equations may be written
dx dy _dz _ dp _ dq _ dr
T~'Q~R~-X~-Y~-Z'
But by the Pfaffian theorem we may write
Xdx + Ydy + Zdz + Pdp + Qdq + Rdr = ^dp + ijdo- + fdr,
viz. there exist f, 17, f, p, a, t functions of the variables x, y, z, p, q, r, such that we
have
V t^P ^ da ^dr „ fc<^P ■ da dr
^^^da^ + ^dx+^di'-' ^^^dp-^^'dTp^^Tp'--
and we have the foregoing general system expressed by means of these given functions
f . Vt K> p) ffy T of the variables.
32. But the lineo-differential
Xdx + Ydy + Zdz + Pdp + Qjdq + Hdr
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 107
may be of a more special form ; for instance, it may be a sum of two terms = fdp + rjdtr :
or, finally, it may be a single term = ^dp, and in this case we have the Hamiltonian,
system, viz. writing H in place of p, if we have
Xdx + Ydy + Zdz + Pdp + Qdq + Rdr = ^dH,
where H is a. given function of the variables, then the system is
dx _ dy _ dz _ dp _ dq dr
dB~ dS~ dB~ ~dH~ J^dH'^^dB'
dp dq dr dx dy dz
which is the system in question.
33. Any integral « of the system is a solution of
dHd6,dHd0, dH dd dH dd dH d0 dH dO
dp a
viz. writing, as above
dp dx dq dy dr dz dx dp dy dq dz dr '
(H .X rf(^. 0) d(H, 6) d{H, 6)
^"'""^ d{p,x) + d{q.y)^d{r,z)
=» Jast-mentioned expression,
the partial differential equation is {H, d) = 0 ; and, conversely, any solution of this
equation is an integral of the differential equations.
34. It is obvious that a solution of {B, 6) = 0 is II; hence the entire system of
independent solutions may be taken to be H, a, b, c, d; or, if we choose to consider
a set of five independent solutions a, b, c, d, e, then we have H = H{a, b, c, d, e) a
function of these solutions.
An Identity in regard to the Fimctions {H, 0). Art. Nos. 35 and 36.
35. Taking the variables to be (x, y, z, p, q, r), and H, a, b to be any functions
of these variables, we have the identity
(H, (a, b)) + (a, (b, ID) + (b> {H, a))=0,
which is now to be proved. For this purpose we write it in the slightly different form
((a, b), H)^(a, (6, H))-{b, {a, H)).
The first term on the right-hand side is
(da d da d da d da d da d _da d\
dp dx dq dy dr dz dx dp dy dq dz dr)
operating upon
'dhdH dbdH dbdH_dbdH_dbdH_dbdH\_
[dp dx dq dy dr dz dx dp dy dq dz dr)'
14—2
c
108 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
and if we herein attend to the terms which contain the second differential coefficients
of H, these are symmetrical functions of a, h. For instance,
d-H m ■ . • da db
dm
da db da db
dxdy
" dp dq dq dp '
d'H
da db da db
da; dp
" dp dx dx dp'
d'H
da db da db
dxdq
dp dy dy dp '
Hence, forming the like terms of the second terms (6, (a, H)) and subtracting, the
terms in question all vanish : and we thus see that (a, (6, H)) — {b, (a, H)) is a linear
function of the differential coefficients
dH_ dH dH dH dH d^
dx ' dy' dz ' dp ' dq' dr '
36. Attending to any one of these, suppose -j- , the coefficient of this
in (a, (6, H)) is = (a, -j-\
i„ (6, (a, i.)) „ (.,|), .-(|,.).
wherefore, in the difference of these, it is
{'■fHt'")' =l<"-'>-
Hence, for the several terms
d^ dM dH dH dH dH
dx' dy' dz ' dp' dq ' dr '
the coefficients are
/d^ d^ d^ _d d ^\{ h\
\dp' dq' di-' ~dx' ~d'y' ~dz)^'^' ^^''
or, what is the same thing, we have
(a,{b,H))-(b,(a,H)) = i(a,b),H),
the identity in question.
The Poisson-Jacobi Theorem. Art. Nos. 37 to 39.
37. The foregoing identity shows that if {H, a) = 0, and {H, 6) = 0, then also
{H, (a, b)) = 0; or, what is the same thing, if a and b are solutions of the partial
differential equation (H, 0) = O, then also (a, 6) is a solution; or, say, if o, 6 are
integrals, then also (a, b) is an integral.
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 109
Supposing that the set is (x, y, z, p, q, r), so that there are in all five integrals
d, h, c, d, e, then the theorem may be otl^erwise stated, we have (a, b) a function of
the integrals a, b, c, d, e.
Observe that, knowing only the integrals a and b, we find (a, 6) as a function of
X, y, z, p, q, r, this may be =0, or a determinate constant, or it may be such a
function that by virtue of the given values of a and b it reduces itself to a function
of a and b; in any of these cases the theorem does not determine a new integral. But
if "contrariwise the value of (a, b), obtained as above as a function of the variables, is
not a function of a, b, then it is a new integi-al which may be called c.
38. To obtain in this way a new integral, we require two integrals a, b other
than H; for knowing only the integrals a, H, the theorem gives only (a, H) an
integral, and we have of course (a, H) = 0, viz. we do not obtain a new integral.
But starting from two integrals a, b other than H, we may obtain as above a
new integral c; and then again {a, c) and (6, c) will be integrals, one or both of
which may be new. And it may therefore happen that in this way we obtain all the
independent integrals a, b, c, d, e; or the process may on the other hand terminate,
without giving all the independent integrals.
The theory is obviously applicable throughout to the case of a bipartite set
(x, y, z,..., p, q, r, ...) of 2n variables.
39. It may be remarked here that, in the Hamiltonian system, a value of the
multiplier is if = 1 ; and consequently, if in any way all but one of the integi-als,
that is, 2?! — 2 integrals, be known, the remaining integral can be found by a
quadrature.
It is further to be noticed that, if we adjoin a new variable t and a term =dt
to the system of equations; then the 2m - 1 integrals of the original system being
known, all the original variables can be expressed in terms of the 2m -1 integrals
regarded as constants and of one of the variables say x : we then have
dt = dx -T- -J— ,
dp
or
dH
t — e= Idx -h
dp'
or say
'=^-r- dp'
viz. if after the integration we suppose the 2n-l integrals replaced each of them
by its value, we have
e=t-^(x, y, z,..., p, q, r,...),
which is the remaining or 2nth integral of the original system as augmented by
the term =dt.
110 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
The Poissoti-Jacobi theorem peculiar to the HamilUmian Form. Art. Nos. 40 to 45.
40. Taking for greater simplicity the set (x, y, p, q), and writing
Xdx + Ydy + Pdp + Qdq = ^p+ vd<T,
then the general system
dx _dy _ dp _ dq
P~^~-X~-Y'
becomes
da; _ dy _ dp dq
^ dp dp dq dfy V dx dxl V dy dy'
and the corresponding partial differential equation {6 the independent variable) is
f(p, 6) + 71 {a, 0) = O,
f„ ^. rfO>. 0) , dip, 0) rf(<7. 0) dja. 0)
^P' ^^-d(p, x)^d{q, yy ^'"' ''^ d{p, xV~d{q, yY
It is to be shown that if a, b are solutions, viz. if we have
implying of course
{p, a) {a, b)-{p, b)(a; «) = 0,
then it is not in general true that we have («, b) a solution ; that is, not in general
true that
f(p, (a, 6)) + i?(<7, (a, 6))=0;
the condition for the truth of this equation is in fact ^ = a function of p, <r, but
when this is so, ^dp + rjda is \dH, viz. there exist \, H functions of p, o- (and
therefore ultimately of x, y, p, q) satisfying this equation, and the system is really
Hamiltonian.
41. We consider whether it is true that
^(p,ia,b)) + v{'r,{a,b)) = 0.
We have identically
((a, b), p) + ((6, p), a) + ((/>, a), 6) = 0,
((a, b), <r) + ((6, a), a) + ((o-, a), 6) = 0,
so that multiplying by f, r), and adding, the equation in question is
f[((&. P), «) + ((/>. «)> i)]+[((6. «r), a)+(ia, a). 6)]=0.
But in virtue of the equations satisfied by a, b, we may write
{p, a) = Iv, {b, p) = - (p, b) = - mt),
{a, a) = - l^, (b, <r) = - (a, b) = m^
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. Ill
where I, m are indeterminate functions of x, y, p, q; and the equation in question
now becomes
^[-(mv, a) + {lv, b)]+v[(m^, a)-(l^, b)] = 0-
that is,
? [- m (v, a) - V {m, a) + l(f], b)+r) {I, b)]
+ V[ m(la) + ^(m,a)-l(^,b)-^{l,b)] = 0;
viz. omitting the terms which destroy each other, this is
- m^ (v, a) + l^ (v, b) + mv (?, a) - h, H b) = 0.
Substituting for wif, &c., their values, we have
{a, b) (v, a) - (<r, a) (r,, b) + (p, b) (f , a) - (p, a) (?, b) = 0;
and the question is whether this is implied in the equations
f (p, a) + v {<T, a) = 0,
^{p,h)+^{a,b) = 0.
42. Write 17 = k^, the equation in question is
(<r, b){Kl a)-(<7, a)(«f, b) + (p, b)(l a)-(j>, a)(f, b) = 0;
viz.
(f. a)[(/>. i) + *(«•, i)]-(f i)[(p. a) + «(<^. «)]+r[(<^. «•)(«, a) -{a, a)(«, 6)] = 0;
and we wish to see whether this is implied in
{p, a)-\- K (a, a) = 0,
{p, b)+ic (a, b)=0,
(a, b)(p, a) -(a, a)(p, b) = 0;
or, what is the same thing, whether these last equations imply
{a, b) (k, a) - (a, a) (k, b) = 0.
Suppose « is a function of p, a, then, as is at once seen,
. diK , . dtK , ,
(AC, a) = ^(p, a) + ^(«7, a).
that is,
which give
and thence
(o-, h){K, a) -(a, a)(K, b) = p^[{<r, b){p. a) - (a, a)(p, b)];
viz. K being a function of p and a, the two equations imply the third.
112 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
43. But we wish to prove the converse, viz. that, if the two equations imply
the third, then « is a function of p, <t.
Now the equations
(«r, b){K, a) -{a, a)(«, 6) = 0, (o-, h)ip, a) -{a, a)(p, b) = 0,
we transformable into
d (a-, k) d(b, a) _ . d(<r, p) d(b, a) _ .
d(p, x) d,(p, x)~ ' d (p, x)d(p,x)~
+ <!' V'
+ 7. V'
+ P. 9.
+p. 9.
+?. V'
+F. y.
+ «, q,
+ «, 5-
+*', y,
+ a;, y.
the lines after the first being the corresponding terms with q, y, &a instead of p, x.
And if independently of the values of a, b, one of these equations implies the other,
we must have
d{v, k) d{<T, k) dja-, k) d(a, k) d (a, k) d(a; k)
d{p, x)' d{q, y)' d{p, qV d(p, y)' d{x, q)' d(a;, y)'
proportional to the like expressions with a; p instead of a, k; say these are
d{a^ ic) ^ ^d{cr^ ^^
d{p, x) d{p, x)'
44. Assume k a function of p, a, x, y: we have
. / ^^ ^^ -^^ ^p ^^ ^g. ^x ^ j^^ /^ ^ , ^ ^^\ _dicd(<r, p) dicda „
the condiu.-^ = dp[Tpd^'^ da- dx "*■ di) dx [dp dp ^ da dp) ~ dp d\p, x) '^dxdp' *^''- '
when this is 8. ,jg become
therefore ultimately
Hamiltonian. dKdJ^^p) ^d^da^^dic p)
dp d ip, x) dx dp d (p, x) '
41. We consider wL
dx d{<T, p) dx dff _ . d{<r, p)
„, ^ •, . n dpd{q7y) dydq~ d(q, y)'
We have identically
((a, Uk djff, p) ^ ^ ^ ^ d{<7, p)
((a,b)y(P'9^ rf(i>>9)'
so that multiplying by f, ,,, and al^ + ^ ^ = A ^j^'-^J .
1, y) dydp d(p, y)
p) dx da- _ , d {a, p)
But in virtue of the equations satisfiet^ dx dq~ d (q, x) '
(p, a)= It), dja-, k) ^ ^d (a, p)
(o-, a) = - l^, (I d Wy) d (x, y) '
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 113
Hence, unless i/ i — 0, we have A = j- . The remaining five equations then are
dK da- _ dx da- _ .
dx dp ' dy dq~ '
dK dcr _ . die da _ da dK da dK _.
dy dp ' dx dq ' dx dy dy dx '
die die
which give, and are satisfied by t- = 0, j- = 0, viz. we then have k a function of
p, a without X, y which is the theorem in question.
45. The proof fails if ^ , { = 0. But here, unless also ,, ' '^{ = 0, we can,
^ d(p, q) d{x, y)
by assuming in the first instance k a function of p, <t, p, q, prove in like manner
that K is a function of only p and a: if however we have as well ,) :;=0 and
^ '^ dip, q)
,4—'— < = 0, the last-mentioned process would also fail, but it can be shown the
d{x, y) ^
conclusion holds good in this case also; hence the conclusion that the Poisson-
Jacobi theorem holds good only for a Hamiltonian system.
\
Conjugate Integrals of the Hamiltonian System. Art. Nos. 46 to 51.
46. For greater clearness, let n = 4, or let the variables be x, y, z, w, p, q, r, s;
the system of differential equations therefore is
dx _dy _ dz _dAV _ dp _ dq _ dr _ ds
M~dH~dB~dH~^dS~^^~^^~ _dH'
dp dq dr ds dx dy dz dw
and any integral hereof is as before a solution of {H, 0) = O. Assume that the
integrals are H, a, b, c, d, e, f, so that
{H,a) = 0, (H.b) = 0, (ZT, c) = 0, (H, d) = 0, (H, e) = 0, (H,f) = 0.
Considering here a as denoting any integral whatever, that is, any solution
whatever of the partial differential equation (H, 6) = 0, it is to be shown that it is
possible to determine 0 so as to satisfy a.s well this equation (H, d) = 0, as also
the new equation (a, 0) = 0.
47. We, in fact, satisfy the first equation by taking
0, =0(H, a,b, c, d. e,f),
any function whatever of the seven integrals. But, 0 having this value, we find
K «.(«, Hy§H^. «)f +(«, i/^Ha. «)S.(., ^)g-(«. 4'+<"'/>r
c. X. 15
114 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
or, since the first two terms on the right-hand vanish, the equation (o, 6) = 0 thus
becomes
(a, b)^+(a, c)^^+ia, d)^ + (a, e)^ + (a./)^=0.
But by the Poisson-Jacobi theorem (a, b), &c., are each of them a solution of
{H, 0) = 0, viz. they are each of them a function of H, a, b, c, d, e, f. This is
then a linear partial differential equation wherein the variables are H, a, b, c, d, e, f;
or, since there are no terms in j-jv , t- , we may regard a, H as, constants, and
treat it as a linear partial differential equation in b, c, d, e, f, the solutions of the
equation being in fact the integrals, or any functions of the integrals, of
db dc _ dd _ de _ df
(a, b)~{a, c)~(a, d)~(a, e)~(a, /)'
48. Suppose any four integrals are b', c', d', e', so that a general integral is
<^ (H, a, b', c', d', e'), then 6', c', d', e' qua functions of H, a, b, c, d, e, f are integrals
of the original equation (H, 5) = 0 ; hence changing the notation and writing b, c, d, e
in place of these accented letters we have (H, a, b, c, d, e) as solutions of the two
equations {H, 6) = 0, (a, ^) = 0 ; viz. a being any integral of the first of these equations,
we see how to find four other integrals (6, c, d, e) which are such that
(H,a) = 0, (H,b) = 0. {H,c) = 0, (H, d) = 0, {H, e) = 0,
(a , b) = 0, (a , c) = 0, (a , d) = 0, (a , e) = 0.
49. We proceed in the same course and endeavour to find 0, so that not only
{H, 6) = 0, (a, 6) = 0, but also (6, 6) = 0. Assuming here 0 = d (H, a, b, c, d, e) an
arbitrary function of the integrals, the first and second equations are satisfied ; for the
third equation, we have
,, />> ,. T.\ d0 ,, .d0 ,, j^d0 . ,, , OP . ,, ^\d0 . ,, , (tp
(6, 0)^Q>, H)^+ib, a)^ + (6. 6)^+(6, c)^+(b, d)^ + ib, e)^;
viz. the first three terms here vanish, and the equation (6, ^) = 0 becomes
,, , d0 ,, ,, d0 ,, , d0 .
where, b, c, d, e being solutions as well of (H, 0)= 0 as of (a, 0) = 0, we have (b, c) a
solution of these two equations, and as such a function of H, a, b, c, d, e; and so
{b, d) and (b, e) are each of them a function of the same variables. The above is
therefore a linear partial differential equation wherein the variables are H, a, b, c, d, e,
■ J . , . d0 d0 dd J IT J.
but as the equation does not contam -m , -j- , or jT , we may regard It, a, b as con-
stants ; and the solutions of the equation are, in fact, the integrals of
dc dd de
(6, c) (6, d) (b, e)'
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 115
50. Supposing that any two integrals are c', d', so that a general integral is
<i>{H, a, h, c', d'), then c', d' qua functions of H, a, b, c, d, e are integrals of the
former equations (ff, 6) = 0, (a, 6) = 0, so that again changing the notation, and writing
c, d instead of the accented letters, we have (H, a, b, c, d) as solutions of the three
equations {H, d)=0, (a, e) = 0, {b, (9) = 0, viz. a being any solution of the first equation,
and b any solution of the first and second equations, we see how to find two others
c, d, of the same two equations, which are such that
{H,a) = 0, (H,b) = 0, (H,c) = 0, {H, d) = 0,
(a, b) = 0, (a , c) = 0, (a , d) = 0,
(b, c)=0, (6, d) = 0;
or, attending only to the integrals H, a, b, c, these are integrals of the equations
(H, 0) = O, (a, 0) = O. (b, e) = 0, such that
(H, o) = 0, (H. b) = 0, (H, c) = 0, (a, b) = 0, (a, c) = 0, {b, c) = 0.
We here say that H, a, b, c are a system of conjugate solutions. Attempting to
continue the process, it would appear that there is not any new independent integral d,
such that (H, d) = 0, (a, d) = 0, (b, d) = 0, (c, d) = 0 (the first three of these are
satisfied by the integral d found above, but the last of them is not) ; we may,
however, taking d an arbitrary function of H, a, b, c, replace H hy d; viz. we thus
have the four integrals a, b, c, d, sich that
(a, b) = 0, (a, c) = 0, (a, d) = 0, (b, c) = 0, (6, d) = 0, (c, d) = 0,
and which are consequently said to form a conjugate system.
51. The process is of course general, and it shows how, in the case of a
Hamiltonian system of 2n variables, it is possible to find a system H, a, b, ... , f con-
sisting of H and n — 1 other integrals, or, if we please, a system of n integrals
a, h,...,f,g, such that the derivative of any two integrals whatever of the system is
= 0; any such system is termed a conjugate system.
Hamiltonian System — the function V. Art. Nos. 52 to 58.
52. Taking a Hamiltonian system with the original variables x, y, z, p, q, r, we
adjoin the two new variables t, V, forming the extended system
dx _ dy _ dz _ dp _ dq _ dr , dV
dH~dH~dH~ 'dH~ dE dH "'" dH ^ dH ^ dH'
dp dq dr dx dy dz ^ dp ^ dq dr
Supposing the. integrals of the original system to be a, b, c, d, e, we have
H = H{a, b, c, d, e) a determinate function of these integrals; also an integral
T =«-</>(», y, z, p, q, r) and an integral \=V--^{x, y, z, p, q, r); these integrals,
exclusive of the last of them, serve to express x, y, z, p, q, r as functions of
a, b, c, d, e, t-r; and the last integral then gives F=\+a function of the last-
mentioned quantities.
15—2
116 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
63. We consider the differential expression
dV—pdx — qdy — rdz,
which, treating the integrals as constants, that is, in the expressions of V, x, y, z,
regarding t as the only variable, is at once seen to be =0; hence, if we regard all
the integrals as variables, the value is
= dX, + ilda + Bdh + Cdc + Bdd + Ede,
without any term in dr, since this enters originally in the form dt — dr, and there-
fore disappears with d<.
The coefficients A, B, G, D, E are of course functions of a, b, c, d, e, t — r;
it is to be shown that they contain t — r linearly, viz. that in these coefficients
respectively the coefficients of f — t are
dH dH dH dH dH
da' db ' dc ' dd ' de
where H is expressed as above in the form H (a, b, c, d, e) ; this being so, the entire
term in t — r will be (t — r)dH; each coefficient, for instance A, has besides a part
A', which is a function of a, h, c, d, e without t — r, or chxinging the notation and
writing the unaccented letters to denote these parts of the original coefficients, the
final result is
dV - p dx - qdy - r dz = (t - t) dH + dK + Ada + Bdh + Cdc + Ddd + Ede,
where H stands for its value H {a, b, c, d, e), and A, B, C, D, E are functions of
a, b, c, d, e without t—r.
54. To prove the theorem, we have
and thence
. _ dV dx dy dz
da " da ^ da da '
dA _ dT _dpdx_dqdy_drdz_ d'x _ d^y _ d^z
dt dadt dt da dt da dt da " dadt da, dt da dt
_ d \dV dx dy dz
~ da\lS,~Pjt~'^Tt~'''dt
dp dx dq dy dr dz dp dx dq dy dr dz
da dt da dt da dt dt da dt da dt da '
dV
and then substituting for , , &c., their values from the system of differential equations,
the first line vanishes, and the second line becomes
dH dp dH dq dH dr dH dx dH dy dH dz
dp da dq da dr da dx da dy da dz da'
_dH
~ da '
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 117
and hence A=(t-r)^ +A', and the like for the other coeflScients B, C, D, E,
which is the theorem in question.
55. We may have between two coefficients of the formula, for instance, D and E, a
relation 3— = ^ , and I will for the present assume, without proving it, the theorem
that if a, b, c are conjugate integrals, then this relation ^ -v-v = 0, holds good,
merely mentioning that the proof depends on the consideration of certain symbols
[a, b], which are the converses, so to speak, of the symbols (a, b), viz. considering
the variables a;, y, z, p, q, r as given functions of a, b, c, d, e, t— r, then we have
d(y, x) dig, y) d{r, z)
■• ' -■" d(a, b)'^d{a, byd{a, b)'
The assumption is used only in the two following Nos. 56 and 57.
56. Supposing then that a, b, c are conjugate integrals, we have ^ -j j = 0,
and there exists therefore ^, a function of a, b, c, d, e, such that
d(j} = A' da + F db+ C dc + B dd + E de,
{A', B', C functions of the same quantities a, b, c, d, e), we have therefore
dV-pdx-qdy-rdz = dX + (t-T)dH + d^-ir{A-A')da + {B-B')db + {C-G')dc.
Taking as above a, b conjugate integrals (a, b) = 0, and c any function whatever
of a, b, H, then a, b, c are conjugate integrals, and the formula holds good. Suppose
further that a, b, H are absolute constants, then dH =0, da = 0, dh = 0, dc= 0, and
the formula becomes
dV—pdx — qdy—rdz = dX + d^;
or, writing this under the form,
pda; + qdy + rdz=dV—dX — d<}>,
it follows that pdx + qdy + rdz is an exact differential, a theorem which may be
stated as follows: viz. if a, b are conjugate integrals of the Hamiltonian system, and
if from the equations Zr=con.st., a = const., 6 = const., we express p, q, r as functions
of X, y, z, then pdx+qdy+rdz is an exact differential ; or, what is the same thing,
p, q, r are the differential coefficients j~ > j > j oi U a. function of x, y, z.
This is, in fact, a fundamental theorem in regard to the partial differential equation
jff^= const., and it will presently be proved in a different manner.
57. If, as before, a and b are conjugate integrals, then, writing as we may do
X in place of \ + 0, and finding F as a function of x, y, z, a, b, H from the
equation
V = \+ Up dx + qdy + r dz),
118 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
and again treating a, b, H as variable, we have
dV-pdx-qdi/-rdz = dK + (t-T)dH + Ada+Bdb,
where A, B are functions of the integrals a, b, c, d, e, that is, they are themselves
integrals, which may be taken for the integrals d, e, or we have
dV—pdx — qdy — rd£ — d\ + {t— t) dH + dda + e db ;
we have therefore
dV_, dV_
dS,-'^' db~^'
equations which, on substituting therein for a, b, H their values as functions of
f!t y> ^) P> 1> *"> determine the integrals d, e, which with a, b, H or a, b, c, are the
remaining integrals of the Hamiltonian system ; and further
dV ^
which, when in like manner, we substitute therein for a, b, H, their values as
functions of x, y, z, p, q, r, determines t, the remaining integral of the system as
increased by the equality = dt.
58. Reverting to the general theorem No. 52, let Xs,, y^, Zo, Pa, ?o. ^o. t„ be cor-
responding values of the variables x, y, z, p, q, r, t; and let a^, &c., ..., Vo be the
same functions of x^, yo, z„, p„, q^, r^, t^ that a, &c., ..., V are of the variables; we
have a = ao,..., c = eo, and corresponding to the equation
dV —pdx - qdy -rdz =d\ + (t -r)dH + Ada + ... + Ede,
the like equation
dVu — podxa — qody„ - r^dza = d\ + {to — r) dH + Ada + ... + Ede.
Hence, subtracting
dV- dF„ = (< — Q dH +p dx + qdy + r dz — ptdx^ — q^dya — r„ dzo,
or, considering only IT as an absolute constant,
dV— dVo = pdx + qdy + rdz—podxo — qadyo — n dzo]
viz. if from the equations £r = const, a = ao, b = b^, c = Co, d = do. 6 = 6^, we express
p, q, r, po, 5i), n as functions of x, y, z, x,,, y,, z^, H, then
pdx + qdy + rdz-podxo-qo dy^ — r^ dz„
will be an exact differential. And in particular regarding x^, yo, z„ as constants, then
pdx + qdy + rdz is an exact differential, viz. there exists a function
V=\ + l(pdx + qdy + r dz).
We have thus again arrived at a solution of tlie partial differential equation ir = const.
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 119
The Partial Differential Equation H = const. Art. Nos. 59 to 70.
59. In what just precedes we have, in fact, brought the theory of the Hamiltonian
system into connexion with a partial differential equation, viz. we have determined
the variables p, q, r sts functions of x, y, z such that pdx + qdy + rdz is an exact
differential =dV; but we now consider the subject in a more regular manner.
The partial differential equation is 5" = const, viz. here H denotes, in the first
instance, a given function of p, q, r, x, y, z, where p, q, r are the differential
coeflScients of a function V of x, y, z, or, what is the same thing, there exists a
function V of x, y, z such that pdx + qdy + rdz = dV; and then, this function H
being constant, we use the same letter H to denote the constant value of the
function. The equation H = const, is the most general form of a partial differential
equation of the first order which contains the independent variable only through its
differential coefiicients p, q, r, and it is for convenience put in a form containing
the arbitrary constant H, which constant might without loss of generality be put = 0
or =any other determinate value.
60. We seek to determine p, q, r as functions of x, y, z, satisfying the given
equation H = const., and such that we have pdx+ qdy + rdz an exact differential
= dV; this would be done if we can find two other equations K = const, and
L = const., such that the values of p, q, r obtained from the three equations give p, q, r
functions having the property in question. Attending to only two of the equations,
say H = const, and K = const., we have here p, q, r functions of x, y, z, such that
pds + qdy + rdz is an exact differential, and two of the equations which serve to
determine p, q, r as functions of x, y, z are .£r = const., ^1' = const. We have to
prove the following fundamental theorem, viz. that (H, K) = 0.
61. In fact, from the equations £r= const., ^= const., treating x, y, z as inde-
pendent variables, we have
dH dHdp dHdq dHd^^^
dx dp dx dq dx dr dx '
dK dK dp dK dq dKdr^Q^
dx dp dx dq dx dr dx '
,. . dp 111 dK dH ,
and if from these equations in order to elmnnate ^ we multiply "Y ^ • ~ ^p > ^^^
add, we find
d(K. H) d(K, H)dq djK, E)dr_^_
d{p, x) ^ ' dip, q) dx d (p, r) dx
and, in precisely the same way,
d{K, H) djK, H) dp d(K, H) dr _^
diq, y) d(q,p) dy^ ' d(q, r) dy
d{K, H) d(K, H) dp d(K, H) dg ^ ^q
d (r, z) d (r, p) dz d (r, q) dz '
120 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
Adding these together, we have
^A, xi;+ d{q,r) \dy dz)^ d(r, p) \dz dx)^ 'd{p, q) \dx dyl '
viz. ifpdx + qdy + rdz be an exact differential, then (H, K) = 0, which is the theorem
in question.
62. In the case where the variables are (x, y, p, q), we have simply
viz. pdx + qdy being a complete differential, (K, H) = 0. Conversely, if {K, H) = 0,
then T^ — ^^ = 0, and pdx + qdy is an exact differential ; viz. this is so unless
, ' — ~ = 0 ; this equation would imply that K, H considered as functions of p, q,
are functions one of the other: and, supposing it to hold good, we could not from
the equations H==0, K = 0 determine p, q as functions of x, y, for, eliminating one
of the variables p, q, the other would disappear of itself. We hence obtain the
complete statement of the converse theorem, viz. the functions H, K being such that
it is possible from the equations H—0, K = 0 to express p, q as functions of x, y,
then, if {H, K) = 0, we have pdx + qdy an exact differential.
63. Returning to the case of the variables (x, y, z, p, q, r), if p, q, r are
determined as functions of x, y, z by the three equations H = 0, K =0, L = 0, then,
by what precedes, in order that pdx+qdy + rdz may be a complete differential, we
must have {H, K) = 0, {H, L) = 0, {K, X) = 0 ; and it further appears that if these
equations are satisfied, then we have, conversely,
dr dq _- dp dr _ dq dp _.
dy dz ' dz dx ' dx dy '
that \a, pdx + qdy + rdz is an exact differential; viz. this is the case unless we have
between H, K, L the relation
d{H, K) d(H, K) d{H,K)
d{q, r) '
d{r,p) '
d {p, q)
H,L ,
H,L ,
H, L
K, L ,
K, L ,
K, L
= 0.
where in the determinant the second and third lines are the same functions of
H, L and K, L respectively that the first line is of H, L.
The determinant is, in fact, equal to the square of
d{H, K, L)
d{p, q, r) '
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 121
and, if it vanish, it is impossible, by means of the equations H = 0, K = 0, L = 0,
to determine p, q, r as functions of x, y, z. Hence, if the last-mentioned equations
are such that by means of them it is possible to effect the determination, and if,
moreover, (H, K) = 0, {H, L) = 0, {K, L) = 0, then pdx + qdy+rdz will be an exact
differential.
64. Considering H as given, we have, by what precedes, K, L solutions of the
linear partial differential equation (H, d) = 0; and since also K, L must be such
that {K, L) = 0, they are conjugate solutions; or in conformity with what precedes,
using the small lettere a, b instead of K, L, we have the following theorem for the
integration of the partial differential equation H = const., where as before H is a,
given function of x, y, z, p, q, r.
Find a and b, such that H, a, b are a system of conjugate solutions of the linear
partial differential equation {H, 6) = 0: then from the equations H = const., a = const.,
b = const., determining p, q, r as functions of a, b, H, and in the result treating
these quantities as constants, we have pdx + qdy + rdz an exact differential =dV,
and thence
F=\+ I {pdx + qdy + r dz),
an expression for V containing thd three arbitrary constants \, a, b, and therefore a
complete solution of the given partial differential equation H = const.
The theorem applies to the case where n has any value whatever, viz. if there
are n variables x, y, z, ... , then we have to find the n — 1 integrals a, b, c, ... ,
constituting with H a system of conjugate integrals ; and the theorem holds good.
In particular, if n = 2, or the independent variables are x and y, then we find any
solution a of the partial differential equation {H, d) = 0; the values p, q derived from
the equations //^ = const., a = const., give V=\ + j{pdx + qdy), a complete solution.
65. But there is a different solution depending on the consideration of corre-
sponding values; viz. if the independent variables be as before x, y, z, p, q, r, and it
«o. yo, Zo, Po, ?o, n are corresponding values of x. y, z, p, q, r, then, taking a, b, c, d, e
to be integrals of (H, d)=0: so that U is here a given function of a, b, c, d, e,
since the number of independent variables is = 5 : and representing by a^, b^, c, d^, e„
the like functions of x^, yo, Zo, P<» q<» n, we form the equations
/r= const., a = Oo, b = bo, c = Co, d = do, e = eo-
We have the theorem that, expressing by means of these equations p, q, r, as
functions of x, y, z, x„, y„, z„, H, and regarding therein x„ y„, z„ H as constants,
we have pdx + qdy + rdz an exact differential, and therefore
V=\+\{pdx-^qdy + rdz),
C. X. ^6
122
A MEMOIR ON DIFFERENTIAL EQUATIONS.
[655
a solution of the equation i/^ = const, involving the arbitrary constants X, x^, y„ Zj
-(one more than required for a complete solution).
The theorem is here stated in the form proper for the solution of the partial
dififerential equation H = const. ; a more general statement will be given further on.
66. I take first n = 2, or the independent variables to be x, y\ here p, q are
determined by the equations a = a^, b = ba, c = Co, H = const., and it is to be shown
that pdx + qdy = dV.
Considering p, q, p^, q^ as functions of the independent variables x, y, then
dq^
differentiating in regard to x, and eliminating ^, -^ ,
da da dq da doo da^
dx dq dx' dp' dp^ ' dq^
dx
= 0,
, we find
db db dq dh dbo db^
dx dq dx' dp' dp„ ' dq„
dc dc dq dc dc^ dco
dx dq dx' dp' dp„ ' dq„
dH dHdq dH ^
dx dq dx' dp'
0
viz. this is
But in the same way
ditto, bo) (d (H, c) ^ d(H, c) dq) ^^ ^ ^
dipo. qi,)\d(p, a;) d(p,q)dx)
ditto, 6o) {diH, c) ^ djH, c)dp{ ^ ^^ ^Q.
dipo, qo)\diq, y) diq, p)dy]
+ &c. = 0,
adding these two equations we have
ditto, bo) \,jj ._^d iH.jc) /dq_dp\
dipo, qo)\ ' dip, q)\dx dy)
the terms denoted by the &c. being the like tei-ms with b, c, a and c, a, b in
place of a, b, c. We have iH, a) = 0, iH, b) — 0, iH, c) = 0, and the equation, in
fact, is
L^dia^,bo)diH,c)'\/dq dp\^^_
\ dipo, qo) dip, q)}\dx dy) '
VIZ. we
have T^ — r^ = 0, the condition for an exact differential.
dx dy
67. Coming now to the case where the independent variables are x, y, z, we
proceed in the same way with the equations 5^ = const., a = ao, 6 = 6o. c = Co, d = do,
« = Co- Differentiating in regard to x, and eliminating
dp dq dpo dqo dro
dx' dx' dx' dx' dx'
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 123.
ttJ*
we find for 3- the equation
djcp, dp, e,) (dr d(a, b, H) ^ d{a, b, H))
d (po, 9o. n) (da; d (r, p, q) d (x, p, q)] ^'~
We have in the same way for -^ the equation
d(co, dp, Co) (dp d(a, b, H) d{a, h, H)\
d(po. qo. n) \dz dip, r, q) ^ d {z, r,q)]^ '^- ' "'
whence, adding, we obtain
dr _ dp\ d {a,J,^ ) _^ d(a, b, H) ^ dja, b, H)) ^^ ^ ^
\.dx dz) d{r, p, q) d{x, p, q) d{r, z, q)]
where the terms denoted by the &c. are the like terms corresponding to the different
permutations of the letters a, b, c, d, e.
The equation may be simplified; we have identically
— j-{o, H)- J- {H, a)- J (a, b) = -^. r^ + -^^ f ;
dq^ ' dq^ ' dq^ ' d{x, p, q) d (z, r, q)
JJT
or, since {H, a) = 0, (H, b) = 0, the left-hand side is simply - -5— (a, b), and the
equation becomes
d(c,, rf„ Co) r /*• _ d/p\ d(a, b,^ _dH . J , „ _ ^
d(p„ qo. r,)\[dx ~dz) d(r;p, q) dq ^''' "^l + ^c.-U.
68. This ought to give t -^ = 0 ; it will, if only
^{dp^Jo^ ]
{d(po, qo, ro)' ']
which is thus the condition which has to be proved. By the Poisson-Jacobi theorem,
(a, b) is a function of a, b, c, d, e: if we write
^"'" ° d(po, Xo) d(q„, yo) d(r„, ^o)'
then (tto, bo) is the same function of a^, h, Co, do, 60 ; but these are equal to a, b, c, d, e
respectively, and we then have (a, b)={ao, bo), and the theorem to be proved is
l\i^''^^-^^iao,bo)\ = 0.
d (po, qo, »•.) J
But, substituting for (a,, h) its value, the function on the left-hand side is, it
is easy to see, the sum of the three functional determinants
d{uo, bo, Co, d,. Bo) djuo, bp, Cp, do, Bq) djap, h, Cp, dp, Bq)
d~(po, q„ To, po, Xo)' d(po, qo, n, qo, Vo)' d(po, qo, n, r„, z,)'
16—2
124 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
and each of these, as containing the same letter twice in the denominator, that is,
as having two identical columns, is =0; the theorem is thus proved. And in the
same way J^ — ^ , j^~j ^.re each =0; that is, pdx+qdy-k-rdz = dV.
dv dt)
69. The proof would fail if the factors multiplying 3 ^^ , &c., or any one of
these factors, were =0. I have not particularly examined this, but the meaning must
be that here the equations a = Oo, &c., H = const., fail to give for p, q, r expressions
as functions of x, y, z, x„, y„, z„, H; whenever such expressions are obtainable, we have
p dx + qdy + r dz = dV.
The proof in the case of a greater number of variables, say in the next case
where the independent variables are x, y, z, w, would probably present greater difficulty,
but I have not examined this.
70. Taking the independent vaiiables to be x and y, we may from the equations
a = ao, b = ha, c-=c„, ff= const, (which last equation may also be written H = 11^= const.)
find p, q. Pa, 5o as functions of x, y, x^, y^, H; and we have then the theorem that,
considering only H a& a. constant,
pdx + qdy —p„ dx^ — q„ dy^ = dV.
To show this, we have to prove the further equations -^ 4- ^ = 0, &c. ; we find
dp
dxo
^ jrf(6o, Co) d{a, H)) dE d (oo, 6., c„)^ ^
\d{po, qo) d{p, q)\ dq d(x„, p„, q,)
dpo „ (d{b, c) d (gp, H^)) _ dHo d (a, b, c) ^ ^
dx \d{p, q) d{pa, go)J dqo d{x, p, q)
and it is to be shown that the coefficients of -t^ , -p are equal and of opposite
signs, and that the other two terms are equal ; viz. this being so, subtracting the
two equations, we have the required relation -r^ + -^ = 0. Now H, Hi, are the same
functions of a, b, c and of «„, K, c^', and there is no real loss of generality in assuming
c = H, C(i = J?o; but this being so, the first coefficient is
djbo, Ho) d(^,_H) ^ d (H„ a,) d (6, H)
d{po. 9o) d(p, q) d{p„ qo) d{p, q)'
and the second is
d(p, H) d(ao, H,) ^ d{H. a) d{bo^ H^)
d{p, q) d(po, qo) d {p, q) d(p„, </„) '
which only differ by their signs. As regards the other two terms, we have identically
^(6. H) + ^{H. a)+ ^^-(a, 6) = ^^-—^.
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 125
which, in virtue of (a, H) = 0, (6, H) = 0, becomes
J- (a , 6 ) = T-) ! — ^ ;
dq ^ d(x, p, q)'
similarly,
dS, ,^ _ d (gp, bo, Hq)
Hence the terms in question are
dH dH,, ,. dHdH,, ,,
- d^ d?„ ^«- ^«)' -~dq d^<"'^>'
which are equal in virtue of (a, 6) = (aj, 6o); and, similarly, the other conditions might
be proved. But the proof would be more difficult in the case of a greater number
of variables.
Examples. Art. Nos. 71 to 79.
71. The variables are taken to be x, y, z, p, q, r. As a first example, which will
serve as an illustration of most of the preceding theorems, suppose pqr —1 = H; the
Hamiltonian system, with the adjoined equalities, is here
dx _dy az _dp _dq _dr _ ^ dV
qr~rppq 0 0 0 Spqr '
The integrals of the original system may be taken to be
a=p,
b =7,
c =r,
d = qy-px,
e =rz — px,
and there is of course the integral H = pqr-\, which is connected with the foregoing
five integrals by the relation H=abc—\.
We form at once the equations
(a, 6)=0, (a, c)=0, {a, d) = -a, (a, e) = -a,
(b, c) = 0, (6, d)= b, (b, e)= 0,
(c, d) = 0, (c , e) = c,
(d.e)= 0;
hence it happens that no two of these integrals a, b, c, d, e give by the Poissoa-
Jacobi theorem a new integral. To show how the theorem might have given a new
integral, suppose that the known integrals had been a = p + q, and e = rz-px, then
(a, e) = —p: or the theorem gives the new integral a = p.
126 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
We have as a conjugate system a, b, c; also the conjugate systems H, a, b;
H, a, c; H, b, c; H,b, e; H, c, d; H, d, e; but the first three of these, considering
therein H as standing for its value abc—1, are substantially equivalent to the first-
mentioned system (a, b, c).
72. Postponing the consideration of the augmented system, we now consider the
partial differential equation pqr =1 + H, where if is a given constant and p, q, r
denote the differential coefficients of a function V. The most simple solution is that
given by the conjugate system H, a, b, viz. here p, q, r are determined by the
1 4- //
equations p = a, q = b, pqr = 1 + H, that is, r = — j — ; or, introducing for symmetry the
constant c, where abc = 1 + H as before, then ?• = c, and we have
V= \ + I {adx + bdy + cdz), =\ + ax-\-by-\-cz,
where a, b, c are connected by the just-mentioned equation abc = 1 + H. This is there-
fore a solution containing say the arbitrary constants X, a, b, and, as such, is a
complete solution.
But any other conjugate system gives a complete solution, and a very elegant one
is obtained from the system H, d, e. Writing for symmetry ;8 — o, 7 — a in place of
d, e, we have here to find p, q, r from the equations
H =pqr—\, qy—px — ^ — a, rz — px = <y — a;
or, if we assume 6 = px— a, then
H=pqr—\; px, qy, rz — O+a, 0 + ^, 0 + y
respectively, whence
{l+H)xyz = (e+a)(e + ^)ie + y).
which equation determines ^ as a function of x, y, z (in fact, it is a function of the
product xyz), and then
e + a e + fi e + y
•* X y z
and we have
/0-t-a, d + ^ , 0 + y
V
= X + j{'ydx + '^-dy + '^/dz).
There is no difficulty in effecting the integration directly by introducing ^ as a new
variable, and we find
F=X-H3^-alog^^-^log^±-^-7log^-+'>'.
Or, starting from this form, we may verify it by differentiation; the value of dV is
,fl/„ a ^ 7 \ adx Bdy ydz
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 127
where the term in d6 is
which, from the equation which determines 0, is
\iB y z ) '
and the value of dV is thus
J-±^d.^'-±^dy^'±ydz.
X y '' z
The solution contains apparently the four constants \, a, y8, 7, but there is no loss of
generality in writing, for instance, a = 0, and the number of constants really contained
in the solution may be regarded as 3.
73. To show how the equations Zf = const., a=ao, h — hf,, c = Co, d = d^, e=eo
give a solution; remarking that these equations are pqr — \=H, p=p„, q = qo, r=r^,
qy-px = q^yt, —p<,Xa, rz-px = r^o —po^o, we find
p{x-x„)=q(y-y„) = r(z-z<,),
and consequently p, q, r =
^^'■"^^ (x-x,)i ' ^^'■"'^^ (y-y.)^ ' ^^'^^ (.-.,)* '
respectively : whence
V =\+ \(pdx + qdy + rdz),
= X + S^(l+H)(x- x,)^ (y - y„)* {z - z,)\
which is the solution involving the four constants \, a,„, y„, Zg.
If in the foregoing value of V we consider a-o, yo, z^ as variables, then p, q, r
having the values just mentioned, and po, q„, r„ being equal to these respectively, we
obviously have
dV = pdx+ qdy + rdz —podxo — qodyo — rodzo.
74. Considering now the augmented Hamiltonian system, we join to the foregoing
integrals a, b, c, d, e, the new integrals « - t = and F - \ = Zpx. And then expressing
all the quantities in terms of t — r,
X =bc(t — t),
y =ca(t-T) + ^,
z —ah{t — r)-\--.
0
p =a, q = h, r = c, H = abc—1,
V=\ + Sabc{t-T).
128 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
Forming from these the expression for dV—pdx — qdy — rdz, the term in dt — dr
disappears ; there is a term in t — r, the coefficient of which is
3d . abc — ad.bc — bd.ca — cd.ab,
which is =d.abc, or the term is (t — T)dH; and we have, finally,
d e
dV — pdx — qdy — rdz = d\ + (t — r)dH — bd j- — cd- ;
0 c
viz. t enters only in the combination {t — t) dH, which is the fundamental theorem.
Considering ^ as a determinate constant, this term disappears.
We may show how this formula leads to the solution of the partial differential
equation pqr =1+H; treating i/ as a definite constant, then in order that the
formula may give dV — pdx—qdy ~rdz = d\, or V-\+\{pdx + qdy+rdz), as before,
d e
the last two terms of the formula must disappear; this will be the case if j ^^^ '
are constants, or, say, d = 6/3, e = 07, /3 and 7 being constants. But, this being so, we
have qP = qy — px, fr^ = rz - px, that is, px = q {y — fi) = r (z — 7), pqr = 1 + H, giving the
values of p, q, r; and then
V=X+j{pdx + qdy + r dz), = \ + 3 ^(1 + ^) «* (y - /9)H^ - 7)*,
which is substantially the same solution as is obtained above by a different process.
Or, again, observing that we have
d e
dV — pdx — qdy — r dz — d\ ■\- {t — r) dH — dd — de-j-db — dc,
0 c
then, taking H, b, c constants, we have
dV —pdx — qdy — r dz = d\ — dd— de,
which, changing the value of \, gives the before-mentioned solution
V= \ + ax + by + cz, (abc = 1 + H).
75. As a second example, suppose
the augmented system is
dx _dy _dz _dp _dq _dr _ , dV
p q r X y z p' + ^+r''
corresponding to the dynamical problem of the motion of a particle acted upon by a
repulsive central force equal to the distance.
The integrals of the original system may be expressed in various forms, viz.
the quotient of any two of the expressions x +p, y +q, z + r, or of any two of the
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 129
expressions x—p,y — q,z — r is an integral, or again the product of any expression of the
first set into any expression of the second set is an integral : we may take as integrals
We have then
CM/
dt = ,, . , that is, t~T= log [x + ^{a? - a)j = log {x +p),
giving x-\-p = ef^~^, and thence the other quantities x — p, y + q, &c. For greater
symmetry, I introduce a new set of constants a, b, c, a', b', c, and I write also e'~' = T,
g-t+T^y (where 2^'=!). We then have
x = aT + a'T, p = aT- a'T',
y = bT+b'T', q=bT-b'T',
z = cT+c'T', r=cT-c'T';
also, comparing with the values obtained as above,
a =\ , b =^8 , c =^6 ,
We have, moreover,
IT = - 2 (aa' + 66' + cc')= - Ha + /9 + 7)-
76. We find
jP + q^ + i^ = H+{a'' + b' + c') T' + (a' + b'' + c'') T'\
and thence
V=\+{{p'' + q-' + r^)dt
= \ + H{t-T)-^^{a? + ¥ + c=) r-i(a'^ + 6'^ + c'O T'\
We may from this obtain the expression for
dV — pdx — qdy —rdz,
when everything is variable. The terms in {dt-dr), as is obvious, disappear; omitting
these from the beginning, we have
dV=d\ + {t-T)dH + {ada + bdh + cdc) T"- - {a! da' + b'db' + c'dc') 2"' :
also
pdx = {aT- a'T) {Tda + T'da),
= daiaT^-a') + da'(-aT'' + a):
thence forming the analogous expressions for qdy and rdz, we have
pdx-i-qdy + rdz = (ada + bdb + cdc)T'- (a'da' + b'db' + c'dc') T''
- (a'da + b'db + c'dc) + (a da' + bdb' + c dc),
17
C. X,
130 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
whence
dV-pdx- qdy — rdz = d\. + (t — T) dH + a'da + b'dh + c'dc — ada'-bdh'-cdc';
or, in place of a, b, c, a', b\ c', introducing a, ^, y, B, e, and attending to the value
of -BT,
dV-pdx-qdy-rdz=dX+{t-T)dn+\dH+^^dh-\-\'^df.
77. Suppose H, S, e absolute constants, this becomes
d{V— \) = pdx + qdy + rdz,
. or
F= \+j(pdx + qdy + r dz),
and we have thus a solution of the partial differential equation
p^ + q- + r' = ic' + y'' + z' + 2H;
viz. p, q, r are here to be determined as functions of x, y, z by the equations
p'' + q^ + r"" = x' + y'' + z^ + 2H,
y +q =h{x+p),
z +r = e (a; +p).
We have
2H+ a^ + y^ + z^ = p' + {y-S (x+p)Y+ \z - e (x+p)Y ;
or, on the right-hand side, vfntmgp^ = {x+py — 2x(x+p) + af,
left „ „ a^={x-py-2x(x+p)+p!',
the equation is
{1 + S' + e'){x + py-2(x+ Sy + ez){x + p) - 2H = 0,
which gives p as a function of x, y, z. But the result is a complicated one, except
in the case H.=^\ we then have
_ 2 (a; + gy + e^)
2S (a; + Sw + ez)
y-\-q =
z +r
l + B' + e^ '
2e (x + Sy + ez)
' l + S'+e" '
and thence
v=x-H^^f^z^H^-^^^^l±^,
a complete solution of the partial differential equation
p' + q'+r^ = x'' + y^ + z\
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 181
More symmetrically, we have the solution
(ax +hy + czf
7= X _ ^ (^ + 2/. + ^=) + v__^
+ c'
,2 '
as can be at once verified.
78. In the same particular case H = 0, introducing the corresponding values
Po, qi>> I'd, ^0. Vo, ^0. we find a very simple expression for V— V„, as a function of
X, y, z, x^, y^, Zt. We have, writing 2'o = e'»~'^, T^' = 6-''+'', and therefore T^T^ = 1,
y, = hT, + h'T:, g„ = bT, - b'Z'.
ZQ^ClQ'rCJ.fJy ?'o = CiQ C J. Q y
and thence
x-x, = a{T- T,) + a' (^- ij , =(r- T,) (a- ^j .
Forming the analogous quantities y — ya, &c., we deduce
{x-x,f-^(y- y,y +(z-Zoy = (T- Zy |a= + 6= + c^ + (a" + b" + c") ^j^fj^ ,
{x + x,r + {y + y,y + (z + z,y = {T+ T,y |a» + 6^ + c» + (a'» + 6'= + c'») ^^ .
But we have
F- F, = i {(a» + 6« + c») (2" - r„') - (a'^ + 6'^ + c'O (^, - ^,)}
= Hr" - n') |a' + 6' + C + (a'' + &'• + c") j,y ,
and hence the required formula
F- F. = J V{(a; - !c,y + (y- y,y ^(z- z,y] >J{{x + x,y + {y + y,y + {z + z,y],
or, say, for shortness,
= iV(i2)V('Sr).
79. We ought, therefore, to have
Jd V(-B) VCiS) =pdx + qdy+rdz —po dxo — q<,dyo-ndz„,
where p, q, r, p^, q„, Vo denote as above, and consequently
pi + qt + .,-^ = ai' + y' + z% Po^ + q,^ + r,^ = x,^ + y,^ + z,\
17—2
132 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655
We have in fact
and thence
p'+q* + r' = i{R + S+2(af + y' + z''-x,'-y,'-z,%=a^+y' - z\
p.' + go' + r„» = i (E + -S - 2 {x' + y^ + z' - a;„- - ^o' - 2.')}> = a:.' + y.' + ^o'.
or the last-mentioned iresults are thus verified.
Partial Differential Equation containing the Dependent Variable: Seduction to Standard
Form. Art. Nos. 80, 81.
80. The equation /f = const, is the most general form of a partial differential
equation not containing the dependent variable V; but if a partial differential equation
does contain the independent variable, we can, by regarding this as one of the dependent
variables, and in place of it introducing a new independent variable, exhibit the
equation in the standard form if = const., H being here a homogeneous function of
the order zero in the diflferential coefficients. Thus, if the independent variables are
X, y, the dependent variable z, and its differential coefficients p, q, then the given
partial differential equation may be H, = H (p, q, x, y, z), = const. But we may
determine ^ as a function of x, y by an equation F = const., V being a desired
dV dV
function of x, y, z; and then writing p, q, r for the differential coefficients -j-, ,- ,
J- , we have p = — -, q = — ", and the proposed partial differential equation becomes
H \-^> -l- ^. y, y= const.
viz. this is an equation containing only the differential coefficients p, q, r of the
dependent variable V, a function of x, y, z. And, moreover, H is homogeneous of
the order zero in p, q, r; consequently
dH dH dH ^
Pd^-'^l^+''dr=^'
dV
or, in the augmented Hamiltoninn system, the last equality is = -tx , so that an
integral is F= const. ; as already stated, this is the equation by which z is determined
as a function of x, y.
81. Thus, if the given partial differential equation be pq — 2: = if, we here consider
the equation ^ — z = H. The Hamiltonian system is
i^dx _ r^dy _ — r^dz _dp _dq _dr [ dV\
^ ~ p" ~ ' ipq ~ 0~"0~T \ ~o) '
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 133
having the integrals
a=p,
b=q,
c=px-qy,
r q
-£._!
^~pq r='
(whence H= — abe). We have H, a, b, a system of conjugate integrals and, in terms
of these,
p = a. 9 = 6, r = y/(^^y;
hence, writing \ for the constant value of V, we have
\=ji^ada;+bdy + ^ [^gj dz\^ .
that is,
X = aa; + 6y + 2 i^[ab {z + H)],
or say, *
4Mb(z + H)=((uc + bi/- \y,
a solution containing really the two constants \ and r, and thus a complete solution
of the given equation pq — a = ^. We have, in fact,
2ah p = a {ax + hy — X),
2ab q = 6 {ax + 6y — X) ;
that is,
4o'6»pq = o5(aa; + 6y-X)», =4a'6»(« + fl),
or
pq = ^ + /T,
as it should be.
134 [656
65Q,
ON THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS.
[From the Mathematische Annalen, t. xi. (1877), pp. 194 — 198.]
In what follows, any letter not otherwise explained denotes a function of certain
vai-iables {oc, y, p, q), or (x, y, z, p, q, r), &c., as will be stated in each particular case.
An equation a = const, denotes that the function a of the variables is, in fact, a
constant (viz. by such equation we establish a relation between the variables) : and when
this is so, we use the same letter a to denote the constant value of the function in
question; I find this a very convenient notation.
Thus if the variables are x, y, z, p, q, r and if p, q, r are the differential coefficients
in regard to x, y, z respectively of a function V of x, y, z, then H (as a letter not
otherwise explained) denotes a function of x, y, z, p, q, r and considering it as a given
function,
H = const.
will be a partial differential equation containing the constant H. For instance, if H
denote the function pqr — xyz, H = const, is the partial differential equation, pqr — xyz = H
(a given constant).
The integration of the partial differential equation, H = const., depends upon that of
the linear partial differential equation
(H, 0) = O,
where as usual (H, 0) signifies
a (H, @) ^ a(H. 0) ^ 3(H, 0)
a(p, x) d(q, y) d{r, z) '
It can be effected if we know two conjugate solutions a, h of the equation (H, 0) = O,
viz. a, 6 as solutions are such that (H, a) = 0, (H, 6) = 0, and (as conjugate solutions)
are also such that (a, 6) = 0 ; in this case if from the equations
H = const., a = const., b = const.
656]
ON THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS,
135
we determine p, q, r as functions of x, y, z, the resulting value of ^ da; + g dy + r d^r is
an exact differential, and we have
V =\-\-\{pdx-\-qdy-\-r dz).
a solution containing three arbitrary constants, \, a, b, and therefore a complete solution
of the proposed partial differential equation H = const.
But (as is known) there is a different process of integration, for which the con-
jugate solutions are not required, and which has reference to a system of initial values
o'o, Vo, ^0. i'o. 90. i^o- ■viz. if the independent solutions of (H, @) = 0, are a, b, c, d, e, and
if Uc, b„, Co, do. Co denote respectively the same functions of the initial variables that
a, b, c, d, e are of x, y, z, p, q, r, then if from the equations
a = ao, b = b„, c = Co, d = do, e = eo, H= const.
we express p, q, r as functions of x, y, z and of x^, y^, z^, H, these last being regarded
as constants, we have pdx + qdy +rdz an exact differential, and
V—\+ I (pdx+qdy +r dz),
a solution containing the constants X, x„, y^, Zo (that is, one supernumerary constant), and
as such a complete solution.
It is interesting to prove directly that pdx + qdy + rdz is an exact differential.
I consider first the more simple case where the variables are p, q, x, y. Here p, q
are to be found from the equations
a = a„, b = bc, c = Co, H = const
and it is to be shown that pdx + qdy is an exact differential.
Considering p, q, p^, q„ as functions of the independent variables x, y, then dif-
ferentiating in regard to x, and eliminating -f- , -^, ^, we have
dx' dx ' dx
I
da da dq da da^ da„
dx dq dx' dp' dp„' dq„
db db dq db db„ db^
dx dq dx' dp' dpo ' dq,,
dc dc dq do dc„ dco
dx dq dx' dp' dp„' dq^
dH dHd^ dH ^ ^
dx dq dx' dp ' '
= 0,
186 ON THE THEOBY OF PARTIAL DIFFERENTIAL EQUATIONS. [656
or introducing a well-known notation for functional determinants, and expanding the
determinant, this is
9 (p.
. ?o) [d(p, a>) d{p, q) dx]
But in the same way
9(po. 9o) 19 (q, y) 9 (q, p) dy\
or adding these, attending to the value of (H, c), and observing that x^ ' \ — ~ 3 / ' n
we have
9(ao, 60)
(H-)-l|?:l(l-|)h^=».
9(Po, 9o)
the terms denoted by the &c. being the like terms with h, c, a and c, a, b in place
of a, b, c. We have (H, a) = 0, (H, b) = 0, (H, c) = 0, and the equation in fact is
(_ 9(ao, K) a(H, c)l /^_ f^P\ _ 0 .
r 9(p, ?) 90j, 3)1 [dx Ty) "■
viz. we have ^ — r~ = 0, the condition for the exact differential.
ax dy
Coming now to the case where the variables are x, y, z, p, q, r, and in the six
equations treating p, q, r, p^, q^, r„ as functions of the independent variables x, y, z, —
then differentiating with regard to x and proceeding as before, we find for -y- the
equation
9(Co, dp, 60) (dr d(a, b, H) 8 (a, b, H)| ^ ^ _ q
d(po, qo, U) \dx d(r, p, q) d{x, p, q))
We have, in the same way, for -f- the equation
a(co, dp, Co) (dp d(a, b, H) d(a, b, H)| ^ ^^ ^ ^ .
9 {p„, g-o, n) (d^ 9 (p, r, 3) 9 (z, r, q)] ' '
or, adding the two equations,
9 (Co, dp, Co) {/dr dp\ 3 (a, b, H) 9 (a, 6, H) 9 (a, 6, H)| ^^ _q
9(^0, 9o, ro) (Vda; d^r/ a(»-, p, q) d{x, p, q) d{z, r, q)) ' '
where the terms denoted by the &c. indicate the like terms corresponding to the
different partitions of the letters a, b, c, d, e.
The equation may be simplified ; we have identically
da., „, d6,„ , dU, ^,_9(a., b, H) . d(a, b, H)
-dq^^- ^^-dq^^' ^>- d^ <"' ^)- d(x, p, q) + d (z, r, q) '
656] ON THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS. 137
JIT
or since (H, a) = 0, (6, H) = 0, the left-hand side is simply — ^— (a, b), and the equation
becomes
9(Po, 9o, n) \\d^ dz) d(r, p, q) dq^ ' -'J '^' ~
This ought to give -i ^ ~^' ^°^ ^* ^^^ ^^ ^^ ^^ ^^y
^mc.,d,,e.) )
[d {po, qo, n) )
this is then the equation which has to be proved. By the Poisson-Jacobi theorem, (a, b}
is a function of a, b, c, d, e: if we write
/ . >_ 8(ao, h) d(ao, h) djag, b^)
then (a„, 6o) is the same function of a^, bo, Co, d„, «„; but these are =a, b, c, d, e
respectively, and we thence have (a, 6) = (a,,, b^), and the theorem to be proved is
8(c„rfo, Co) ] ^
But substituting for (a,, 6,) its value, the function on the left-hand is (it is easy to see)
the sum of the three functional determinants
9(ffli), bp, Co, d„ e,) 9(ao, 6,, Cp, tZ,,, Cp) 9(go, 6o, Cp, dp, eg)
9(i»o, ?o, »o, i^o, a:p) a(pp, gp, rp, g„, y„) 9(jOp, g,, rp, r„ ^p)'
each of which vanishes as containing the same letter twice in the denominator, that is,
as having two identical columns; and the theorem in question is thus proved. And in
the same way -f —j^ , j^ —-t~ are each = 0 : or we have pdx + qdy + rdz an exact
differential.
The proof would fail if the factors multiplying t^ — ;/^ ■ &c., or if any one of these
factors, were = 0 ; I have not particularly examined this, but the meaning would be,
that here the equations in question « = «,, &c., H= const., are such as not to give
rise to expressions for p, q, r as functions of x, y, z, x^, y,,, Zq, H, as assumed in
the theorem ; whenever such expressions are obtainable, then we have pdx+qdy + rdz
an exact differential.
The proof in the case of a greater number of variables, say in the next case
where the variable.s. are x, y, z, w, p, q, r, s, would present more difficulty — but I have
not proceeded further in the question.
It is worth while to put the two processes into connexion with each other: taking
in each case the variables to be x, y, z, p, q, r, and the partial differential equation
to be H = const. ;
C. X. 18
138 ON THE THEORY OP PARTIAL DIFFERENTIAL EQUATIONS. [656
In the one case, a, b being conjugate solutions of (H, 0) = 0,
from the equations H = const., a = const., b = const.,
we find p, q, r functions of x, y, z, H, «, 6 :
and then pdx + qdy + rdz is an exact differential.
In the other case, a, b, c, d, e being the solutions of (H, 0) = 0,
from the equations H = const., o = Oo, b = bf,, c = %, d=do, e — e„,
we find p, q, r functions of a;o, yo. ^o. H :
and then pdx + qdy + rdz is an exact differential.
It may be added that, if from the last mentioned equations we determine also
Po, qo, n as functions of x, y, z, x„, y^, z^, then considering only H as a constant, we
ought to have pdx + qdy+rdz—p^dx^ — qady„—i\dz^ an exact differential; I have not
examined the direct proof.
Caiyihridge, 28 Nov., 1876.
657] 13^
657.
NOTE ON THE THEORY OF ELLIPTIC INTEGRALS.
[From the Mathematische Annalen, t. xii. (1877), pp. 143 — 146.]
The equation
Mdy da
Vi -yi. 1 -1^ ~ Vn^Ti - k'a?
is integi-able algebraically when M is rational : and so long as the modulus is arbitrary,
then conversely, in order that the equation may be integrable algebraically, M must
be rational. For particular values however of the modulus, the equation is integrable
algebraically for values of the form M, or (what is the same thing) -j^, = a. rational
quantity + square root of a negative rational quantity, say =-(i + mv' — n), where
I, m, n, p are integral and n is positive ; we may for shortness call this a half-
rational numerical value. The theory is considered by Abel in two Memoirs in the
Antr. Nach. Nos. 138 & 147 (1828), being the Memoirs* XIII & XIV in the (Euvres
Completes (Christiania 1839). I here reproduce the investigation in a somewhat altered
(and, as it appears to me, improved) form.
Putting the two differentials each =du, we have x=an{u + a), y = sn (iiT + ^]) ^^^
the question is whether there exists an algebraical relation between these functions,
or, what is the same thing, an algebraical relation between the functions a; = sn m and
u
y=8n^.
Suppose that A and B are independent periods of snu; so that sn(M+ .4)=snM,
sn (u +B) = sn u, and that every other period is = mA + nB, where m and n are
integers. Then if u has successively the values u, u + A, u+2A, etc., the value of x
[* They are the Memoirs xix. and xx. in the (Euvres Complitei, t. i., Christiania, 1881.]
18—2
140 NOTE ON THE THEORY OF ELLIPTIC INTEGRALS. [657
remains alwajrs the same, and if x aad y are algebraically connected, y can have
only a finite number of values : there are consequently integer values jj', p" for which
sn ■Y^(« + p'^) = 8n -t^(m+P"j4.): or writing v,—f'A for w and putting /j" — p' = p, there
is an integer value ^ for which sn ,j(u+/)^) = sn jj.m.
Similarly there is an integer value 5 for which an ■p(u + g'JS) = 8n -jj^m; and we are
at liberty to assume q = p; for if the original values are unequal, we have only in
the place of each of them to substitute their least common multiple.
We have thus an integer p, for which
sn-jj^(w+^^) = 8n TjvM,
There are consequently integers m, n, r, s such that
■—- = mA + nB,
M
■equations which will constitute a single relation -p. = m, if m = s, r = n = 0; but in
every other case will be two independent relations. In the case first referred to, the
modulus is arbitrary and M is rational.
But excluding this case, the equations give
B (mA +nB) = A (rA + sB),
or, what is the same thing,
rA^ - (m -s)AB-nB' = 0,
an equation which implies that the modulus has some one value out of a set of
given values. The ratio A : B oi the two periods is of necessity imaginary, and hence
the integers m, n, r, s must be such that {rn — sf + nr is negative.
The foregoing equations may be written
A+ nB = 0,
(-J)-
r
whence eliminating A and B we have
(»-i)('-f)-«-°.
657] NOTE ON THE THEORY OF ELLIPTIC INTEGKALS. 141
that is,
and consequently
E-:^(m + s)±^ V(m - sY + nr ,
where, by what precedes, the integer under the radical sign is negative: and we have
thus the above mentioned theorem.
As a very general example, consider the two rational transformations
I^dz dx
z = {x, u, v); mod. eq. Q{u, v) = 0;
y = {z, V, w) ; mod. eq. P(v, w) = 0 ;
'S/I - 2!' . 1 - ifz* Vl-^.l-M«iC='
Mdy _ dz
Vl-t/'. l-tw'y' Vl-^M -»««=■
viz. z is taken to be a rational function of x, and of the modular fourth roots
u, v; and 3/ to be a i-ational function of z, and of the modular fourth roots v, w;
the transformations being (to fi.x the ideas) of different orders. We have 1/ a rational
function of x, corresponding to the differential relation
MNdy _ dx
Vl - y= . 1 - 'u^y'' >Jl-a?.l- u^a? '
Suppose here vfi = «', or say w = 0u, 6 being an eighth root of unity : we then have
Q(u, v) = 0, P{v, 6u) = 0, equations which determine u. The differential equation is then
MNdy _ dx
Vl - yM - «y Vl - ar" . 1 - v?3f '
an equation the algebraical integral of which is y = a rational function of x as above :
hence, by what precedes, we have
a half-rational numerical value, as above.
To explain what the algebraical theorem implied herein is, observe that the
equations Q (u, v) = 0, P (v, du) = 0, give for u an algebraical equation. Admitting 6 as
an adjoint radical, suppose that an irreducible factor is <^(w), and take u to be
determined by the equation <^m = 0 ; then v, and consequently also any rational function
-Tj^ of M, V, can be expressed as a rational integral function of u, of a degree which
is at most equal to the degree of the function <^u lesw unity. The theorem is that,
in virtue of the equation ^m = 0, this rational function of u becomes equal to a half-
rational numerical value as above. Thus in a simple case, which actually presented
itself, the equation <f>u = 0 was li'— 4<it. + 1 = 0; and -j^-j^ had the value u — 2, which
in virtue of this equation becomes = ± V — 3.
142 NOTE ON THE THEORY OF ELLIPTIC INTEGRALS. [657
Thus if the second transformation be the identity z = y, w = v, M =\: we have
V = Oil ; and the equations are
y = (x, u, 6u), Q (m, 6u) = 0, . ^ = . _--^^^
* ^ '• ^^ ' Vl-3/M-My Vl-a^.l-wV
In particular, if the relation between y, x be given by the cubic transformation
V + 2m'' m' ,
x + — a?
_ V jr
so that the modular equation Q{u, v) = 0 is u* — v* + 2uv(l—uV)==0; then, writing
herein v = 6u, and taking 0 a prime eighth root of unity, that is, a root of ^ + 1=0,
we have
Q (u, 6u) = - 16^ iC" {Ou- + 6-- + u*) ;
viz. disregarding the factor it", the equation for w is u* + 0u' + 0^ = 0; or, if w be an
imaginaiy cube root of unity (to- + m+l =0), this is (u- — m0) (u- — m^d) = 0 ; so that a
value of u^ is «'- = — <od.
Assuming then ^ + 1=0, v = 6u and u^ = — aO, we have {v + 2u^) v = &^w (1 + 2ft)),
= d'-'o) ((o — lo-) : = &) — ft)''; - = ft)=, (v + 2u^)vu- = — a)-((o — m-), u^ = m*0* = — ti}: and
the formula becomes
giving
_ (ft) — ft)^) X + cc^a^
^ ~ 1 -&)==(&) -&)=)«=' '
<fy _ (ft) - ft)*) cfe
Vl - 2/M + ft>y^ Vl - a^ . 1 + 1
where as before ft)- + ft) + 1 = 0, a result which can be at once verified. We have
(ft) — ft)-)- = — 3 ; or the coeflScient &) — ft)* in the differential equation is = V — 3, which
is of the form mentioned in the general theorem.
We might, instead of z = y, have assumed between y and z the relation cor-
responding to any other of the six linear transformations of an elliptic integral, and
thus have obtained in each case, for a properly determined value of the modulus, a
cubic transformation to the same modulus.
Cambridge, 10 April, 1877.
658] 143
658.
ON SOME FORMULAE IN ELLIPTIC INTEGRALS.
[From the Mathematische Annalen, t. Xll. (1877), pp. 369 — 374.]
I REPRODUCE in a modified form an investigation contained in the memoir,
Zolotarefif, "Sur la mdthode d'integration de M. Tchebychef," MatJiematische Annalen,
t. V. (1872), pp. 560—580.
Starting from the quartic
(a, b, c, d, e)(a;, ly, =a.x — a.x—^.x—'y.x — S,
we derive from it the quartic
(a,, 6,, c, d,, e,) (iCi, 1)* = a, . a;, - a, . «, - A . a;, - 7, . a;, - S,,
where, writing for shortness
X = -a + /3 + 7-S,
fj,= a-^+y — B,
V = a + ;8 — 7 — 8,
the roots of the new quartic are
»-'*%■
0 being arbitrary: the differences of the roots Sj, /9,, 71, 8, are, it will be observed,
functions of the dififerences of the roots 0, /9, 7, S.
144 ON SOME FORMULAE IN ELLIPTIC INTEGRALS. [658
We assume a, = a = l, nevertheless retaining in the formulae a, or a (each mean-
ing 1), whenever, for the sake of homogeneity, it is convenient to do so. The relations
between the remaining coefficients 6,, c,, d,, ej, and 6, c, d, e, are of course to be
calculated from the foi-mulas — 46 = Sa, 6c==2a/3, &c., and the like formulae — 46i = 2a,,
6c, = So,/3,, &c. We thus have
-46, = 4^ +i S't",
A.
- 4d, = 4^ + f ^ 2 ^ + J^SX= + ^Xfiv,
A,
where 1, ^ = -^ IX'u?.
Writing, for shortness,
C = ac — 6",
D = a'd-Sabc +2b\
E = a^e - Wbd + 6a¥c - 36* = a-I - 3C,
I =ae — ibd + 3c-,
J = ace— ad" —b-e +2bcd — c',
-a^I + 12G'
B =
W
we have
2\ =-4(6 + 8),
2V = - 48C,
tXfi = 24a^-8(6 + S)^
Xfjiv = 32i),
S\V= 64(-a^/+12(7'),
where the last equation may be verified by means of the formula
(2 V)» = SXV + 2\fip S\.
And we hence obtain
a, = 1,
6,=-^- B,
c, = 6^ + 236 - 20,
d, = -d'-SBd'+ 6Ce -D,
e, = e* + 4jB^ - 12C^ + 4Z)5.
658] ON SOME FORMULA IN ELLIPTIC INTEGRALS. 145
And consequently
(a„ b„ c„ d„ e,)(x„ iy = {l, - B, -2C, -D, 0){x,-e. 1)«.
Hence also
7, = a,ei - 46,rfi + Scj' = - iBD + 12C= = a=7 ;
J, = OiCie, - a,d,« - 6i^e, + 26,Cirfi - Ci' = - D' +8C» - 4>BGD
= -J> + 8C' + C,{a'I - 12C^)
= a'CI - W - D'
= a?J;
where, as regards this last equation a^CI — 4C' - D^ = a^J, observe that G and D are the
leading coefficients of the Hessian H and the cubicovariant <I> of the quartic function
U, and hence that the identity -4)= = J"f7^-/f/»H + 4H', attending only to the term
in a", becomes —Ifi = a? J — a^CI + 4C, which is the equation in question.
We thus have 1^=1, Ji = J; viz. the functions (a, b, c, d, e){x, 1)*, (a,, 6j, Ci, cZi,ei)(a'i, 1)*,
are linearly transformable the one into the other, and that by a unimodular substitution
Xi = pa;+a; y, = p'a; + o-', where pa' — p'a = 1. It may be remarked that we have
(a, b, c, d, e)(x, 1 )*=(!, 0, C, £>, E)(x + b, 1)*; and hence the theorem may be stated
in the form : the quartic functions (1, 0, C, D, E){x, 1)*, and (1, -B, - 2C, -D, 0) («,, l)^
are transformable the one into the other by a unimodular substitution: or again, sub-
stituting for E its value a'/ — 3C", = — 45i) + 9C', the quartic functions
(1, 0, C, D, -4,BD+9C')(x, 1)*, and (1, - B, - 2C, - D, 0)(x„ ly
are linearly transformable the one into the other by a unimodular substitution. In
this last form B, C, D are arbitrary quantities ; it is at once verified that the invariants
/, J have the same values for the two functions respectively ; and the theorem is thus
self-evident.
Reverting to the expressions for a,, /9,, 71, Sj we obtain
a — S./9 — 7
Oi - 8,
y3_ S .7 — a
7 — 8.a — yS
7, - S,
a — 3./3— 7, /3— S.7— a, 7— 8.0— j8
= a, -8, .^1-7,, /S, -81.7, -a,, 7i-Sj.a, — ^1,
which agrees with the foregoing equations A = / and J, = /, since /, J are functions
of the first set of quantities and /j, J^ the like functions of the second set ; in fact,
I = ^{F'+Qf'-^R'), and J = ^^(Q - R)(R- P)(P - Q), if for a moment the quantities
are called P, Q. R.
C. X. 19
".-«-fI^
'^■"'^'=2/./''°"^'^'
ft-8, = g,
^' «'-2rx(^' '^>'
Hence also
--^'-t-
''■-^' = 2Xm(^=-^^)'
146 ON SOME FORMULA IN ELLIPTIC INTEGRALS. [658
We consider now the differential expression . ; to transform
sx — a.x — fi.x — '^.x — h
this into the elliptic form, assume
jj_ a—ff.y— 8 J _7 — a
7— a. p — 6 7 — 6
■(where a is of course not the coefficient, = 1, heretofore represented by that letter :
as a will only occur under the functional signs an, en, dn, there is no risk of ambiguity).
And then further
a sn' u — B sn'' a
no = ;; ; •
sn' w — sn' a
Forming the equations
>. , a — yS ,, . 7 — a.a — /9
we deduce without difl&culty
7 — a sn' M x—h
svl^ a = — K^ , — r— = ,
7 — 6 su' a x — a
cn'a =
dn'a =
a — B en' u _x — y
y — B' en" a x—a'
a — B dn^ u _x — fi
W^B ' dn'a ~ x-a '
1 icsna- ^_g^_8 -^_g.,y_g'
the use of which last equation will presently appear.
We hence obtain
2 sn M en M dn M dw = — (a — B) sn' a
(^-ay
, . 'Jx — a.x — ^.x — y.x—B
sn « en « dn « = sn a en a dn a ~, r^ >
{x — af
and consequently
„ J (a — S) sn a dx
zdu = - ^ 'j , ,
en a an a >Jx—a.x-^.x — y.x — B
or, reducing the coefficient,
dx -2 ,
= __________ du,
Vx — a.x — ^.x — y.x — S vy — a.ff — B
which is the required formula.
We next have
J „ _ 4 sn' a en' a dn' a_4i8 — S.7 — a_7i — Oj
658] ON SOME FORMULA IN ELLIPTIC INTEGRALS. 147
in virtue of the foregoing values
7,-ai = ^(/3-S)(7-a) and y,-S^ = J^.
Moreover
a-/3.7-8_ ai-/8i.7i-S,
A> = -
7-a.y3-S 7i - «! ■ /3i- Si '
Hence the like formulse with the same value of k^, and with 2a in place of a, will
be applicable to the like differential expression in x^ : viz. assuming
_ a, sn" Ui — Bi sn^ 2a
^' 8n»Mi-sn'2a
we have
dxt -2
Va;, - a, . a;, - /9i . a?! - 7, . a^i - Sj V71 - a, . A - Sj
We have thus the integral of the differential equation
diCi dx
dui.
Va;, — «! . a;, — /8, . «, — 7, . a;, — 8, 'Jx — a.x — ^.x — '^.x — h
(the two quartic functions being of course connected as before); viz. assuming x, x,
functions of ti, v^ respectively as above and recollecting that 7, — aj . /S, — Sj = 7 — a . ;8 — S,
we have du^ = du ; and therefore ?t, = u +/ {f an arbitrary constant) ; the required
integral is thus given by the equations
sn'w x-i sti'iu+f) a^-Si , . ,, . . <• • x .- x
— J— = — — ; ro~^ ~ > (/ "^^ constant of integration).
oil CL X ^~ OL SIl ^U iCj ^ OL^
Using the formula
. . ,. 8n*M— sn'/
sn (m +/) = TTj — ? 7. J — ,
"^ snttcn/dn/ — sn/cnwdnw
we obtain
^~^3n°2a= Ka;-8)sD'a-(a;-a)8n'/}' ^
^ — «i iVa; — a . a; — S sn a cn/dn/ —Vx — ^.x — y sn/cn a dn a}''
which is the general integral.
We obtain a particular integral of a very simple form by assuming f= a, viz.
this is
a^-a, cn^adn'a {Va;-a.a!-S-Va;-/3.a;-7}''
this is
<Ci-8i 7i-«i ^ 7-a.;5-S
a^i — «! 7i ~ ^1 IVa: — a.x — S — '^x—/d.x — y]'
19—2
148 ON SOME FORMULA IN ELLIPTIC INTEGRALS. [658
or writing 7 — a.)8 — 8 = 7, — a,.^, — 8,, reducing and inverting, we have
which may also be written in the equivalent forms
a^-A 1
{ Va; — 0.x — S — ^x — y,x — a]*.
a^i — S, 7, — S, . a, — Si
"^^ = s^o — F Ka!-7.a!-S - V« - a . a: - /3}».
a;, — 61 Oi — S, . /3, — 5i ' '^'
In fact, from the first equation we have
«.-gi-A-ga.7.-g. ^ (^^ _ g^) (^^ _ gj _ {^a!-a.x-8 - '/x-^.x-yY,
Xi — O]
where the expression on the right-hand side is
81' - S, (a, + /9i + 7,) + ajS, + A71 - 2a^ + «; (a+ /3 + 7 + S) - «S - /37 + 2 VX,
X having here the value
X=x — a.x — ^.x—y.x — S.
Writing for a moment
P = a8 + yS7, Pi = aaSi + /8,7i.
then, by what precedes, Qi— Ri, Ri — Fi, Pi-Q, are equal to Q — R, R—P, P — Q
respectively ; that is, P, — P = Q, — Q = i2i — i2, = (suppose) H, a function symmetrical in
regard to a,, /8j, 71; a, /8, 7 : the equation therefore is
''~^"^'~f"'^'"^' = ai(5i-a.-A-7i)-2^ + ^(« + ^ + 7 + g) + 2VX + fl,
Xi — O]
or the relation is symmetrical in regard to a,, /9i, 71; a, ^, 7: and the first form
implies therefore each of the other two forms.
Cambridge, 8 May, 1877.
659] 149
659.
A THEOREM ON GROUPS.
[From the Mathematische Annalen, t. Xlli. (1878), pp. 561 — 565.]
The following theorem is very simple ; but it seems to belong to a class of
theorems, the investigation of which is desirable.
I consider a substitution-group of a given oi-der upon a given number of letters ;
and I seek to double the group, that is to derive from it a group of twice the order
upon twice the number of letters. This can be effected for any group, in a manner
which is self-evident and in nowise interesting : but in a different manner for a
commutative group (or group such that any two of its substitutions satisfy the condition
AB = BA) : it i.s to be observed that the double group is not in general commutative.
Let , the letters of the original group be abcde ..., we may for shortness write
U= abode...; and take U as the primitive arrangement: and let the group then be
I, A, B,... where A, fi, ... represent substitutions: the corresponding arrangements are
II, AU, BU,... and these may for shortness be represented by 1, A, B,...; viz.
1, A, B,... represent, properly and in the first instance, substitutions; but when it is
explained that they represent arrangements, then they represent the arrangements
U, AU, BU,....
For the double group the letters are taken to be aJ)iCidje, ... and a.bx.^^.i . . . ,
= Ui and Ui suppose, and U-^U.^ is regarded as the primitive arrangement; Ai and A^
denote the same substitutions in regard to Ui and U-^ respectively, that A denotes in
regard to U: and so for fi,, B^, etc.; moreover 12 denotes the substitution (aifflo)
(6,6,)(c,Cj) (djtii) (e,e,) ..., or interchange of the suffixes 1 and 2. The substitutions
A^, Ai, or any powers of these Ai', A/, are obviously commutative; applying them to
the primitive arrangement U,U, we have Ai'A/U,U. and AfA^'^UU., each =Ai'UiAfU.,.
But -4,', A/ are not commutative with 12: we have for instance 124i" . U^Uj
= 12A,*U,.U, = A,'U,.U, but A,'l2U,U, = Aj'.U,U=U,.A,-U. If instead of the
substitutions we consider the arrangements obtained by operating upon UiU.j, then we
150 A THEORKM ON GROUPS, [659
may for shortness consider for instance AjA^ as denoting the arrangement AiUi.AM~.
But observe that in this use of the symbols the Aj, A3 are not commutative, A^Ai
would denote the diflFerent arrangement AiU^.AiUj: in this use of the symbols, 1
would denote UjUt, and 12 would denote UjUi, but it would be clearer to use 12, 21
as denoting U1U2 and fTaf/*, respectively.
These explanations having been given, I remark that in every case the substitution-
group 1, A, B,... gives the double group
1, A,A,. BA,...
12, 12A,A„ 12JSA, ...
as is at once seen to be true: but further when the original group 1, A, B, ... is
commutative, then if m be any integer number, such that m" = 1 (mod. the order of
the original gi'oup), we have also the double group
1, A.A,"", B,Br,...
12, 12^,^0'", \2BiBr,...
where of course if the order of the original group (=/i suppose) be prime, we have
m = 1 or else m = — 1 (mod. /x), say m=\ or /* — 1 ; but if the order /i be composite,
then the number of solutions may be greater.
The condition in order to the existence of the double group of course is that,
in the system of substitutions just written down, the combination of any two sub-
stitutions may give a substitution of the system. And this is in fact the case in virtue
of the formulae
1». ^,il,'» . BA"" = A,B, {A, B^T,
2°. A.A^.l 2B,Bi"' = 12.4,'»£, (ArB^T,
3°. 12^,4/' . B,Br = 12 {A,B,) (^, A)"*,
4°. 12^,^,'" . UB^B^"" = ^ ™5, (A^'^B^f,
inasmuch as 1, A, B,... being a group, AB and A"'B are each of them a substitution
of the group, =C suppose; we have of course in like manner A^Bi^C^, A.iB« = C.,,
etc., and the right-hand sides of the four formulaj are thus of the forms CiCf",
12(7iCj'", 12(7iC,"', OjOa™ respectively, viz. these are substitutions of the system.
To prove for instance the formula 2", considering the arrangements obtained by
operating upon ?7,£/j, we have
B,BrU,U3 = B,Br, 12B^B,'»U,U3 = B^B,'", A^A^ 12B,B3"'U,U^ = A^^B, A,Br.
where of course the expressions on the right-hand side denote arrangements. By
reason that the original group is commutative (^ '"£)"' is = A"*^B"* or since m* = 1 (mod. fi)
this is = AB^ ; hence also {A^'^B^)'" = A^B./" : hence, considering as before the arrange-
ments obtained by operating on l^ii/a, we have
659] A THEOREM ON GROUPS. 151
{Ar'B.y U,U,= 1. A,Br ; A^B, (ArB,)"' U, U, = A,'»£,4,B/»,
and
12A,'»B, {Aj'^B^T U, U^ = A^"B^A^Bi"^,
where of course the right-hand sides denote arrangements. Hence in the formula 2°,
the two substitutions operating on UiU« give each of them the same arrangement
A.i"B^AiBi", that is, the two substitutions are equal. And similarly the other formulae
1°, 3°, 4° may be proved.
By interchanging A and B, in the formulae I obtain
1°. .^i^s™. B^B.r' = ^,Si(^„£,)"';
A^r . A,A.r= B,A,{BiA,y- = A,B,{A,B,y\
which is
= A^A.r .B,B.r;
2° and 3°. A-^A^^ . 12B,B^'" = 124i'»5, {A^'^B^y ;
125,5,™ . A.A,"" = 125,^, (B.A,)"' = 124,5, {A,B,y>\
which is not
= A.A,"' . UBrBn"" ;
3° and 2°. 12-4,4^™. 5,5/'' = 124,5, (^j^,)™;
B^B;" . 124,4j"' = 12AiBr{A^Bi"'y" = 124,5i''' A^'^B^,
which is not
= 124,4s'». 5,5j'»;
4°. 124,4," . 125.5„'» = 4,'»5,(4„"'52)'»;
125,5,"' . 124,4,"' = A.B,"' (A^B^^y^ = (4i'»5,)"' 4„'»5,,
which is not
= 124,4,"*. 125,5,"'.
That is, in the double group any two substitutions of the form 4,42™ are commutative,
but a substitution of this form is not in general commutative with a substitution of
the form 125,5,'", nor are two substitutions of the last-mentioned form 124,42'" in
general commutative with each other; hence the double group is not in general
commutative.
In the formula 4°, writing 5 = 4, we have
(124,4,"'>' = 4,'»+« 4,'"^+^ = 4,™+' . 4,™+' ;
hence, if X is the least integer value such that
\ (m + 1) = 0 (mod. /[*),
we have (124,4,™)"= 1, viz. in the double group the substitutions of the second row
are each of them of an order not exceeding 2\, the substitution 12 being of course
of the order 2. In particular, if m = /a — 1, then X = 1 : and the substitutions of the
second row are each of them of the order 2.
152 A THEOREM ON GROUPS. [659
As the most simple instance of the theorem, suppose that the original group is
the group 1, (abc), (acb), or say 1, B, 0-, of the cyclical substitutions upon the 3 letters
abc. Here m" = 1 (mod. 3) or except m = 1 the only solution is m = 2, and thence
X=l. The double group is a group of the order 6 on the letters aibiCiO^b^c-.: viz.
writing €) = {abc), and therefore 0i=(aj6iCi), ©i' = (a,Ci6,), ©j = (ajtjCj), ©j" = (ojCaio), also
writing 12 = a, the substitutions are
1, 0,0/, 01*0.,
a, a0,0.=, a0,=0„,
the arrangements corresponding to the second row of substitutions are ajijCjOitiCi,
fcjCjOsCiaiti, CjastiiiCiOi, viz. the substitutions are ((ha^) (bib^) (CiCj), {(hhi) (biC.,) (cia.,),
(oid) (biOi) {Cib^), each of them of the second order as they should be.
I take the opportunity of mentioning a further theorem. Let fi be the order of
the group, and a the order of any term A thereof, a being of course a submultiple
of fi: and let the term A be called quasi- positive when ^(1 1 is even, quasi-
negative when fi(l 1 is odd. The theorem is that the product of two quasi-
positive terms, or of two quasi-negative terms, is quasi-positive; but the product of a
quasi-positive term and a quasi-negative term is quasi-negative. And it follows hence
that, either the terms of a group are all quasi-positive, or else one half of them are
quasi-positive and the other half of them are quasi-negative.
The proof is very simple : a term A of the group operating on the fi terms
(1, A, B, G,...) of the group, gives these same terms in a different order, or it may
be regarded as a substitution upon the fi symbols 1, A, B, C, ...; so regarded it is
a regular substitution (this is a fundamental theorem, which I do not stop to prove),
and hence since it must be of the order a it is a substitution composed of - cycles,
each of a letters. But in general a substitution is positive or negative according as
it is equivalent to an even or an odd number of inversions; a cyclical substitution
upon a letters is positive or negative according as a — 1 is even or odd ; and the
substitution composed of the - cycles is positive or negative according as -(a — 1),
d CL
that is, /ifl ), is even or odd. Hence by the foregoing definition, the term A,
according as it is quasi-positive or quasi-negative, corresponds to a positive substitution
or to a negative substitution ; and such terms combine together in like manner with
positive and negative substitutions.
Cambridge, 3rd April, 1878.
660] 15a
660.
ON THE CORRESPONDENCE OF HOMOGRAPHIES AND
ROTATIONS.
[From the Mathematische Annalen, t. xv. (1879), pp. 238 — 240.]
It is a fundamental notion in Prof. Klein's theory of the " Icosahedron " that
homographies correspond to rotations (of a solid body about a fixed point): in
such wise that, considering the homographies which correspond to two given rotations,
the homography compounded of these corresponds to the rotation compounded of the
two rotations.
Say the two homographies ' are A + Bp+ Cq + Dpq=:0, Ai + Biq + Cir + Diqr = 0,
then, eliminating q, the compound homography is A^ + B^p + C^r + D^pr = 0, where
A,, B„ C„ D, = B,A-A,C, B,B - A,D, B.A-Q.G, D,B-C,D;
and the theorem is that, corresponding to these, we have rotations depending on the
parameters (X, fi, v), (Xj, ^, Vi), (X«, fi^, p^) respectively, such that the third rotation
is that compounded of the first and second rotations. The question arises to find the
expression for the parameters of the homography in terms of the parameters of the
corresponding rotation.
The rotation (\, fj,, p) is taken to denote a rotation through an angle ^ about
an axis the inclinations of which to the axes of coordinates are /, g, h, the values.
of X, fi, p then being = tan ^^ cosy, tan J^^ cos 5^, tan ^^ cos A respectively: (Xj, /i,, Pi}
and (X,, fi2, P2) have of course the like significations ; and then, if (X, ft, p) refer to
the first rotation, and (Xj, ^, Vj) to the second rotation, the values of (Xa, fi^, p^
for the rotation compounded of these are taken to be * :
Xj = X + Xi + fiPi — fj-iP,
/^ = /* + /tj + i/Xj — i/jX,
Vi = 1/ + J/] + X/tti — Xi/x,
* The nnmeratoTS jnight equally well have been X + Xj- (/u/i-^ji/), etc., but there is no esseDtial difference:
we pass from one get of fomiulee to the other by reversing the signs of all the symbols: and hence, by
properly fixing the sense of the rotations, the signs may be made to be + as in the text. Assuming this
to be so, if we then reverse the order of the component rotations, we have for the new compound rotation
the like formnls with the signs - instead of + ; but this in passing. The formulaj, virtually due to
Bodriguea, are given in my paper "On the motion of rotation of a solid body," Camh. Math. Journal,
t. m. (1843), [6].
C. X. 5iO
154 ON THE CORKESPONDENCE OF HOMOGRAPHIES AND ROTATIONS. [6G0
each divided by
1 — X\i — [ifii — vvi ;
and if we then write for \, /*, v, the quotients x, y, z each divided by w, and in
like manner for Xj, /*,, v, and X,, /t-j, v^, the quotients .r,, y,, z, each divided by Wi,
and iCj, ^a, ^j each divided by Wj, the formulae for the composition of the rotations are
Xi = xWi + XiW + 3/^1 — 2/1^,
Vi = yw] + yiw + zi^- Z\<e,
z^ = ^Wi + ZiW + a^, — iCiy,
Wj = wWi — xx^ — yyi — zz^ ;
and the question is to express A, B, C, D as functions of {x, y, z, w), such that
Ai, Bi, Ci, Di denoting the like functions of {x,, y^, z^, Wj), A^, B., C.^, D« shall
denote the like functions of {x^, y«, z^, Wj).
It is found that the required conditions are satisfied by assuming
A, B, G, D=%x — y, — iz + w, — iz — w, —ix — y,
{where i = V- 1 as usual) : in fact, we then have
A, = B,A-A,C
= (- izi + w,) (ix-y)- (ixj - y,) (- iz - w)
= i {xwi + XiW +yzi~ yiz) — (ywi + y^w + zxi - z^x)
as it should be : and we verify in like manner the values of B.,, C^ and D^ respectively.
The result consequently is that we have the homogi'aphy
{ix — y) + {—iz+w)p + (— iz — 7v)q + {—ix — y}pq = 0
corresponding to the rotation ( - , - , - ) : where - , - , - are the parameters of
•^ ° \w 10 wj w tv w ^
rotation, tan ^^ cos/, tan ^^ cos ^r, tan ^^ cos A.
I remark as regards the geometrical theory that, if we consider two lines J and K
fixed in space, and a third line L fixed in the solid body and moveable with it ;
then, for any given position of the solid body, the three lines J, K, L are directrices
of a hyperboloid, the generatrices whereof meet each of the three lines: and these
generatrices determine, say on the fixed lines / and K, two series of points corresponding
homographically to each other: that is, corresponding to any given position of the
solid body we have a homogiaphy. But it is not immediately obvious how we can
thence obtain the foregoing analytical formula.
Cambridge, 3 April, 1879.
661] 155
661.
ON THE DOUBLE ^-FUNCTIONS.
[From the Proceedings of the London Mathematical Society, vol. IX. (1878), pp. 29, 30.]
Prof. Cayley gave an account of researches* on which he is engaged upon the
double ^-functions. In regard t<J these, he establishes in a strictly analogous manner
the theory of the single ^-functions, the process for the single functions being in fact
as follows: — Considering m, a; as connected by the differential relation
s, Sir
m =
V a — x.h — x.c — x.d — X
then, if A, B, C, D, il denote functions of u, viz. for shortness, the single letters are
used, instead of writing them as functional symbols, A (w), B (w), &c., then, by way of
definition of these functions (called, the first four of them ^-functions, and the last
an «a-function), we assume •
A, B, C, D=£l\/a -x, fl -Jb^^, £1 '•Jc-x, fi ^d-x
respectively, together with one other equation, as presently mentioned. Without in any
wise defining the meaning of fl, we then obtain a set of equations of the form
AlB-BhA = ^^^c-x.d-xlu,
(mere constant coefficients are omitted), or, what is the same thing,
AlB-B%A = GI)hu,
which are differential equations defining the nature of the ratio-functions A : B : C : D.
If, proceeding to second differential coefficients, we attempt to form the expressions for
Ah^A-ihAf, &c., these involve multiples of il^n-{hilf; in order to obtain a con-
[• See paper, number 665.]
20—2
156 ON THE DOUBLE ^FUNCTIONS. [661
venient form, we assume nSTl — (SQ)' = fl'iltf {^uf, where M ia a, function of x. We
thus obtain an equation AS'A — (BAf = il'% (SuY, where the value of St depends upon
that of M. The value of M has to be taken so as to simplify as much as may be
the expression of 21, but so that M shall be a symmetrical function of the constants
a, b, c, d: this last condition is assigned in order that the like simplification may
present itself in the analogous relations BS'B — {BBy — 11^33 (Buy, &c. The proper
expression of M is found to be
M = -2id' + x{a+b + c + d) + a^ + b^ + c° + d^-2bc-2ca-2ab-2ad-2bd- 2cd,
viz. M having this value, the one other equation above referred to is
ns'ii - (Biiy = n'MiBuy ;
and we then have a system of four equations such as
AS'A-iBAy^il'^iBuy,
where 21 is a linear function of x, and where consequently fl^2l can be expressed as
a linear function of any two of the four squares A'^, B', G\ B^.
To establish the connexion with the Jacobian H and © functions, the differential
relation between u, x may be taken to be
-. Bx
bu = , ;
'Jx.l-x.l-ld'x
viz. we have here d= oo , and to adapt the formulae to this value it is necessary to
u
write —.J instead of u, and make other easy changes. The result is that fl diffei-s
from D by a constant factor only, so that, instead of the five functions A, B, C, D, n,
we have only the four functions A, B, C, D. The equation ilSr-n - (Bny = Cl^ {Buy
is replaced by an equation DB!'D-{BDy = I>'S)iBuy, or say S^ (log Z)) = 2) (Sit)=, which
gives a result of the form
showing that D differs from Jacobi's © (w) only by an exponential factor of the form
Oe*"". And it then further appears that A, B, G differ only by factors of the like
form from the three numerator functions H (m), H (u + IT), © (m + K), so that, neglecting
constant factoi-s, the functions
i ^ £ are equal to ^^> "^"M) ®Oi±iQ.
that is, to the elliptic functions an u, en u, dn u.
662]
157
662.
ON THE DOUBLE ©-FUNCTIONS IN CONNEXION WITH A
16-NODAL QUARTIC SURFACE.
[From the Journal far die reine und angewandte Mathematik (Crelle), t. Lxxxiii. (1877),
pp. 210—219.]
I HAVE before me Gopel's ipemoir, " Theoriae transcendentium Abelianarum primi
oi-dinis adumbratio levis," Crelle's Journal, t. xxxv. (1847), pp. 277 — 312. Writing
P„ P., P,, etc., in place of his F, P", F", etc., also a, /3, 7, S, X', Y', Z', W, in
place of his t, u, v, w, T, U, V, W, the system of 16 equations (given p. 287) is
(1) P==
(4) P.' =
(9) P,' =
(12) P,'' =
(3) <? =
(2) Q^ =
(11) &^ =
(10) Q,' =
(13) B? =
(16) ii,» =
(5) R,- =
(8) i?/ =
(1.5) 6" =
(14) >S,» =
(7) S,' =
(6) -S,^ =
a, -p, -7, h)iX', Y',Z', W),
a, /3, -7, - B){X', ¥', Z', W),
a, -A 7. - S)(^'. I^', ^'. ^').
a, A 7. «)(^', 3^', -^'. TT'),
A -a, -S, 7)(^'. i''. -^'. >»").
A a, -S, - 7)(Z', F, -^', If'),
/3, -a, S, - 7)(Z', F, .^', W),
A a, 8, 7)(^', Y', Z', W),
-B, -a, /3)(Z', ¥', Z', W),
S, -a, -^)(Z', r, .?', TT').
-8, a, -y8)(Z', F, Z', F'),
8, a, /3)(Z', F, Z', TF),
- 7, - A «) (^', F', Z', W),
7, - /3, - a) (X', F, .^', TT').
- 7, A - «) (X', F, Z', W).
/3, a)(Z', F, 2", W);
158 ON THE DOUBLE ©-FUNCTIONS IN CONNEXION [662
viz. we have P'^aX' — ^Y' — yZ' + SW', etc. The reason for the apparently arbitrary
manner in which I have numbered these equations, will appear further on. I recall
that the sixteen double ©-functions, that is, ©-functions of two arguments u, u', are*
■t^t ■t li ■* S> -'8)
iQ, Qu iQ., Q.,
iR, iRi, Ri, Ri,
S, iSj, iSi, 83,
the factor i, = *J— 1, being introduced in regard to the six functions which are odd
functions of the arguments. But disregarding the sign, I speak of P^, P,^, . . . , Q^, etc.,
as the squared functions, or simply as the squares ; a, /9, 7, S are constants, depending
of course on the parameters of the ©-functions ; X', Y', Z', W, which are however
to be eliminated, are themselves ©-functions to a different set of parameters: the
16 equations express that the squared functions P", P,°, etc., are linear functions of
X', Y', Z', W, and they consequently serve to obtain linear relations between the
squared functions: viz. by means of them, Gopel expresses the remaining 12 squares,
each in terms of the selected four squares P,^, P^^, Si', S.J', which are linearly inde-
pendent : that is, he obtains linear relations between five squares, and he seems to
have assumed that there were not any linear relations between fewer than five squares.
It appears however by Rosephain's "Memoire sur les fonctions de deux variables
et k quatre p^riodes etc.", Mem. Sav. Etrangers, t. xi. (1851), pp. 364 — 468, that
there are, in fact, linear relations between four squares, viz. that there exist sixes of
squares such that, selecting at pleasure any three out of the six, each of the
remaining three squares can be expressed as a linear function of these three squares.
Knowing this result, it is easy to verify it by means of the sixteen equations, and
moreover to show that there are in all 16 such sixes: these are shown by the following
scheme which I copy from Kummer's memoir " Ueber die algebraischen Strahlensysteme
u. 8. w." Berlin. Abh. (1866), p. 66 : viz. the scheme is
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
9
10
11
12
13
14
15
16
1
2
3
4
5
6
7
H
13
14
15
16
9
10
11
12
5
6
7
8
1
2
3
4
8
7
6
5
4
3
2
1
16
15
14
13
12
11
10
9
7
8
5
6
3
4
1
2
15
16
13
14
11
12
9
10
6
5
8
7
2
1
4
3
14
13
16
15
10
9
12
11.
* The same functioos in Bosenhain's notation are
00, 02, 20, 22,
01, 03, 21, 23,
10, 12, 30, 32,
11, 13, 31, 33;
viz. the fiRores here written down are the sufiSxes of his ^-functions, 00=%g,g (v, to), etc.
662]
WITH A 16-NODAL QUARTIC SURFACE.
159
In fact, to show that any four of the squares, for instance 1, 9, 13, 8, that is,
P^, Pi, B?, R,-, are linearly connected, it is only necessary to show that the determ-
inant of coefficients
a,
-A
-7.
s
«,
-/8,
7,
~s
7.
- s,
-a,
^
7.
s,
*,
yS
is =0, or what is the same thing, that there exists a linear function of the new
variables {X, Y, Z, W), which will become =0 on putting for these variables the values
in any line of this determinant : we have such a function, viz. this is
or say
pX + aY-hZ-ryW,
[1] (^, a, -S, -7)(X, F, Z, W).
This function also vanishes if for (X, F, Z, W) we substitute the values
S, -7- ^> -a.
5, 7, ^, o,
which belong to 7, 6, that is, S-? and S^ respectively. It thus appears that 1, 9,
13, 8, 7, 6, that is, P*, P^, IC, R3', S,^, S,^, are a set of six squares having the
property in question. I remark that the process of forming the linear functions is
a very simple one ; we write down six lines, and thence directly obtain the result, thus
1
a,
-a,
-7.
S
9
a,
-A
7.
-s
13
7.
-s.
— «,
/3
8
7.
8,
a,
/3
7
8,
-7.
a,
— a
6
8,
7>
A
a
A a, -S, -7:
viz. /3, a, 8, 7 are the letters not previously occurring in the four columns respect-
ively: the first letter fi is taken to have the sign +, and then the remaining signs
are determined by the condition that, combining the last line with any line above it
(e.g. with the line next above it yS8 + 07 — 8/3 — 7a), the sum must be zero.
We find in this way, as the conditions for the existence of the 16 sixes respectively,
[1] (a, a, -8, -7)(X F, Z, W} = 0,
[2] (a, -/3, -7, 8)(Z, Y,Z, W) = 0,
[3] (a, A -7. - 8)(X, F Z, W) = 0,
[4] (0, -a, -8, y)(X, F, Z, W) = 0,
160
ON THE DOUBLE 6-FUNCTIONS IN CONNEXION
[662
squares, viz. these are
[5]
(S.
7.
A
a)(X,
F, ^. W) = 0,
[6]
(7. -
■S,
a,
-/3)(Z.
Y, Z, W) = 0,
[7]
(7.
s,
a,
y8)(^,
Y. Z, W) = 0,
[8]
(S. -
-7.
/9,
- a)(X,
Y, Z, W) = 0,
[9]
(/3,
«,
s.
y){X,
F, Z, W) = 0,
[10]
(«, -
-/3,
7.
-S)(X,
Y. Z, W) = 0,
[11]
(«.
/9.
7.
S){X,
Y, Z, W) = 0.
[12]
(A -
■ a,
8,
-y){X,
Y, Z, W) = 0,
[13]
(S,
7.
-A
- «)(^,
Y, Z, W) = 0,
[14]
(7. -
■S,
- a,
y3)(X,
Y, Z, W) = 0.
[15]
(7.
S,
- a,
-y9)(X
Y, Z, W) = 0,
[16]
(8. -
-7.
-^.
«)(Z,
Y, Z, W) = 0.
new
order
th(
; sets
of coefficients which b
e
(1)
p-..
(«. -
-A -7.
S).
(2)
Qr
(^,
a, - a,
-7).
(3)
Q=
(/3, -
-a, -S,
7),
(4)
Pr
(«.
0, -7>
- 8),
(5)
ii.=
(7, -
- S, a,
-/3).
(6)
s^
(S.
7. ^.
«).
(7)
s.?
(8, -
-7, /3,
-«).
(8)
i?a=
(7.
S, «,
/3),
(9)
P.=
(«. -
-A 7.
-8),
(10)
Q3=
(y3.
a, 8,
7).
(11)
Q/
(/8. -
-a, S,
-7).
(12)
Pa^
(«.
A 7.
8).
(13)
J?^
(7. -
-B, -a,
/3),
(14)
-s.^
(8,
7. -A
- «).
(15)
s=
(S. -
-7. -A
«).
(16)
Er
(7.
8. -a,
-/3).
And I remark that, if we connect these with the multipliers (F, —X, W, — Z), we
obtain, except that there is sometimes a reversal of all the signs, the same linear
functions of (X, F, Z, W) as are written down under the same numbers in square
brackets above: thus (1) gives
(a, -A -7. ^)iY, -X. W, -Z), which is (A a, - S, -y){X, Y, Z. W), =[1];
662] WITH A 16-NODAL QUARTIC SURFACE. 161
and so (2) gives
(^, a. -S, -y){y, -X, W, -Z\ which is (-«, /3, 7, -S)(Z, F, Z, Tf),
or, reversing the signs,
(a, -;8, -7, S)(Z, F, Z, W), =[2].
Comparing with the geometrical theory in Rummer's Memoir, it appears that the
several systems of values (1), (2), ..., (16) are the coordinates of the nodes of a 16-nodal
quartic surface, which nodes lie by sixes in the singular tangent planes, in the manner
expressed by the foregoing scheme, wherein each top number may refer to a singular
tangent plane, and then the numbers below it show the nodes in this plane: or
else the top number may refer to a node, and then the numbers below it show the
singular planes through this node.
And, from what precedes, we have the general result: the 16 squared double
©-functions correspond (one to one) to the nodes of a 16-nodal quartic surface, in
such wise that linearly connected squared functions correspond to nodes in the same
singular tangent plane.
The question arises, to find the equation of the 16-nodal quartic surface, having
the foregoing nodes and singular tangent planes. Starting from one of the irrational
forms, say \\
^A [1] [5] + V^l2r[6] -h v^C [3] [7] = 0,
the coefficients A, B, C are readily determined ; and the result written at full length is
V2 (a/3 - 78) (aS + ySy) (y3Z + aY -BZ- yW) {SX + y7 + &Z + aW)
+ -/(a" - ^ -y' + S')(ay - ^B)(aX - ^Y^ -yZ -\- SW)iyX - SY+ aZ - ffW)
+ -^{a.* + ^ -rf -S')(a,y + ^B){aX + ^Y -yZ - BW)iyX + BY + aZ + ^W) = 0.
It is a somewhat long, but nevertheless interesting, piece of algebraical work to
rationalise the foregoing equation: the result is
(^-Y- - a'S' ) (fa' - ^8') (a»/3= - y'S') (X' +Y* + Z*+ W*)
+ (7V - ^B"-) (a-/3^ - y^B") {a> + B* - ^* - y") ( Y'Z' + X' W^)
+ (a'/S^ - 'fS' ) {^Y- - a^S" ) (0* +B*-y*- a') (Z'X' + Y- W)
+ (^rf - a'B' ) (rfa.^ - y3»8') (7* + 8* - a* - ^) (X^F» -I- Z"^ W^)
- 20/378 (a^ -(- yS^ + 7» + 80 (a= + 8^ - /9= - r) (/S= ->rB^-o?- 7') (7' + 8= - a= - /S^ -^ ^^^ = 0 ;
or, if we write for shortness
L =^-cPB-, F =a- +B^ -ff'- 7=,
M^y'a'-^B', G=^ + B' -r-a\
N = a'^- - 7=8- , H = r + B'-a'- /3^
A = a'' + /3» + 7= H- 8%
C. X. 2^
162 ON THE DOUBLE 6-FUNCTIONS IN CONNEXION [662
then the result is
+ MN(FA + 2L)( Y^Z" + X'-W-)
+ NL(G^ + 2M) (Z^X' + Y'W)
+ LM(H^ + 27V) (X'Y' + Z'W)
- 2a8yBFGH^ X YZW = 0.
It may be easily verified that any one of the sixteen points, for instance (a, 0, y, B),
is a node of the surface. Thus to show that the derived function in respect to X,
vanishes for X, Y, Z, W= a, 0, y, B; the derived function here divides by 2o, and
omitting this factor, the equation to be verified is
LMN . 2a« + MN (FA + 2L) S*" + iVZ (GA + 2M) 7= + LM (ZTA + 2iV) /?> - ^'-S^FQHA = 0,
viz. the whole coefficient of LMN is 2(a^ + ^ + y' + S'), = 2A ; hence throwing out the
factor A, the equation becomes
2LMN+ MNFB' + NLGy" + LMHfi' - ^B^FGH=0.
Writing this in the form
L {2MN + NGy- + MH^) = i?'8= (C?.^/3V - MX),
we find without difficulty GH^'f — MN = - (ff' — y-f L ; hence throwing out the factor L,
the equation becomes
N (2 if + Grf) + MH^ + F&- O - 7^)-^ = 0 ;
we find
MH^- + FB' (8' - rfy = (a'^ - 7^g=) {28'B' - 7= (a^ + /3- + 8=) + 7^)
= iV(2/S^S» _ 7^ (a» + ^ + S^) + Y),
or throwing out the factor N, the equation becomes
2ilf + Gy- + 2^-S' - y' (a= + 8- + 8-) + 7-' = 0,
which is at once verified : and similarly it can be shown that the other derived
functions vanish, and the point (a, /3, 7, 8) is thus a node.
The surface seems to be the general 16-nodal surface, viz. replacing X, Y, Z, W
by any linear functions of four coordinates, we have thus 4.4 — 1, =15 constants, and
the equation contains besides the three ratios a : /9 : 7 : 8, that is, in all 18 constants:
the general quartic surface has 34 constants, and therefore the general 16-nodal surface
34— 16, =18 constants: but the conclusion requires further examination.
Gopel and Rosenhain each connect the theory with that of the ultra-elliptic
functions involving the radical VJf, ='^x.l —x.l —Ix.l —nucl — nx; viz. it appears by
their formulae (more completely by those of Rosenhain) that the ratios of the 16 squares
can be expressed rationally in terms of the two variables x, as', and the radicals
662] WITH A 16-NODAL QUARTIC SURFACE. 163
VX, VX', X' being the same function of x' that X is of x. We may instead of the
preceding form take X to be the general quintic function, or what is better take it
to be the sextic function a — x.b — x.c — x.d — x.e — x.f—x; and we thus obtain a
remarkable algebraical theorem : viz. I say that the 16 squares, each divided by a
proper constant factor, are proportional to six functions of the form
a — X .a — x',
and ten functions of the form
— 7^ {Va — x.b — x.c — x.d — x'.e — x'.f— x'— ^/a — x'.h — x'.c — x'.d — x.e — X .f— xY,
(X - x'f
and consequently that these 16 algebraical functions of x, x' are linearly connected in
the manner of the 16 squares; viz. there exist 16 sixes such that, in each six, the
remaining three functions can be linearly expressed in terms of any three of them.
To further develop the theoi-y, I remark that the six functions may be represented
by A, B, C, D, E, F respectively: any one of the ten functions would be properly
represented by ABC .DEF, but isolating one letter F, and writing DE to denote DEF,
this function ABC . DEF may be represented simply as DE; and the ten functions
thus are AB, AG, AD, AE, BC, BD, BE, CD, CE, DE.
Writing for shortness a, b, c, d, e, f, to denote a — x, b — x, etc., and similarly
a' , b', c', d', c', /', to denote a — x[, b - x', etc., we thus have
(13) A = aa',
(9) B = bb',
(7) C = cc ,
(8) D = dd',
(6) E = ee' , (= E),
(1) F^ff, (=n
(3) DE = -^ {-J^MfTf' - -JRWde/Y, {= D).
{SO ^~ CC f
(4) CE=.-^^^[^abdA-e'f-'JWd'cefY, (= E),
(2) CD = .-^^^, [^'db^dlf - ^a'b'e'cdfW
(14) BE=- ^^^{'Jacdh7f-'Ja'7d'bef\\ (= B), '
(16) BD = 7-^-7^ {^aceb'd'f - '/^Wbdf}',
(1.5) BC = 7—^., {^adeb'c'f - '/a'd'e'bcf]\
(x — X )
(10) AE= ^--^^{'^b^do^'-'/Wd/E^Y, (=1),
21—2
164
ON THE DOUBLE ©-FUNCTIONS IN CONNEXION
[662
(12) AD = . ^ ,^ {y/bcea'd'f - ^b'c'e'ad/Y,
(11) AC= .-^4^» Wbdea'c'f - Vt'dVoc/}'.
(5) AB = ^ {^/cdea'b'f - VcWot/}',
where the numbers are in accordance with the foregoing scheme; viz. the scheme
becomes
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)
F CD DE CE AB E G D B
B AE AC AD A BE BC BD F
A BE BC BD B AE AC AD AB E C
DC E AB CE DE CD F BD BC BE
G D AB E DE CE F CD BC
E AB D G CD F CE DE DE
AE AG AD A BE BC BD
CD DE CE AB E C D
D F CD DE CE
A AD AC AE B
BE AG AD B AE
BC AE B AD AC.
BD A
A BD
There is of course the six A, B, G, D, E, F; for each of these is a linear
function of 1, x + x', xx', and there is thus a linear relation between any four of
them. It would at first sight appear that the remaining sixes were of two different
forms, A, B, AB, CE, CD, DE, and F, A, AB, AC, AD, AE; but these are really
identical, for taking any two letters E, F, the six is E, F, AE, BE, CE, DE, or, as
this might be written, E, F, AEF, BEF, CEF, DEF, where AEF means BCD . AEF,
etc.; and we thus obtain each of the remaining fifteen sixes. The six just referred
to, viz. E, F, AE, BE, CE, DE, or changing the notation say E, F, A, B, G, D as
indicated in the table, thus represents any one of the sixes other than the rational
six A, B, C, D, E, F: and there is no difficulty in actually finding each of the fifteen
relations between four functions of the six in question, E, F, A, B, G, D. It is to
be observed that every such function as A contains the same irrational part
a
-pr- 'Jabcde/a'b'c'd'e'f',
(x — x'y
and that the linear relations involve therefore only the differences A — B, A — C, etc.,
which are rational. Proceeding to calculate these differences, we have for instance
C-D= ,-^—7^ (ce/a'b'd' + c'e'faM - defa'b'c - d'e'fabc) = ^-^,y, (cd' - c'd){efab' - e'fab) ;
or, substituting for a, a', etc. their values a — x, a— x', etc., we have
cd' — c'd = {x — x) (c — d),
efa'b' - e'fab = (x — x') 1, x + x, xx
1, a+b , ab
1, e+f, ef
= {x — x) [xx'abef\
or say for shortness
662] WITH A 16-NODAL QUARTIC SURFACE.
We have therefore
C-D = (c-d)[xai'abef];
and in like manner we obtain the equations
B-G = (b-c) [xx'adef], I-B = (a-d) [xx'bcef],
G -A = {c-a) [xx'bde/l B-D = (b-d) [xx'caef],
A — B = (a — b) [xx'cdef], C — D = {c — d) \xx'ahef\
It is now easy to form the system of formulae
E F A B G
165
D
ae . a/, bed
—be. bf. cda
+ ee . cf. dab
- de . df. abc
ad . hf. cf
- ad. be.ce
+ «/
-ef
bd.cf.af
- bd.ce .ae
+ «/
-ef
ed.af.hf
-cd.ae.be
+ «/
-ef
hc.af.df
— bc.ae.de
+ «/
-«/
ca.bf.df
-ca.be.de
-¥
+ «/
ab.cf.df
— db . ce.de
+ «/
-ef
-a/, bed
+ be .cd
+ ce.db
+ de.bc
-b/.cda
+ ae.cd
+ ce.da
+ de.ac
-c/.dab
+ ae .bd
+ be. da
+ de . ab
-df.cJbc
+ ae.be
+ be.ca
+ ce. ab
— ae . bed
+ bf. cd
+ cf. db
+ df.bc
— be . cda
+ qf.cd
+ cf.da
+ df.ac
- ce . dab
+ af. bd
+ bf. da
+ df.ab
— df . abc
+ af. be
+ b/.ca
+ cf.ab
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0,
where for shortness ab, ac, etc., are written to denote a—b, a — c, etc. ; also abc, etc.,
to denote (6 — c)(c — a) (a — b), etc. : the equations contain all of them only the
diflferences of A, 5, C, B; thus the first equation is equivalent to
ae.af.bcd(A - D)-be.bf. cde(B- B) + ce.cf. dab{C-D) = 0,
and so in other cases.
Cambridge, 14 March, 1877.
166 [663
663.
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS.
[From the Journal fur die reine und angewandte Mathematik (Crelle), t. Lxxxiii. (1877),
pp. 220—233.]
I CONSIDER six letters
a, h, c, d, e, f;
a duad ab not containing / may be completed into the triad abf, and then into the
double triad at/", cde ; there are in all ten double triads, represented by the duads
ab, ac, ad, ae, be, bd, be, cd, ce, de,
and the whole number of letters and of double triads is = 16.
Taking x, a/ as variables, I form sixteen functions ; viz. these are
[a] =a — x . a — x' ,
J. , , _ 1 I la — X .b — X ./— X la — a! .b — x .f— x'Y
(X— x'f I V c —x' .d — x' .e — x' ~ \ c — x.d — x.e—x) '
where the function under each radical sign is the product of six factors, the arrangement
in two lines being for convenience only: the sign + has the same value in all the
functions, and it will be observed that the irrational part is
2 la — x .b — x .c — x .d — x .e — x .f— x
~ ~ {x — x'y V a — x'.b — x'.c — x'.d—x'.e — x'.f—af'
viz. this has the same value in all the' functions.
The general property of the double ^-functions is that the squares of the sixteen
functions are proportional to constant multiples of the sixteen functions [a], \ah\ ; but
this theorem may be presented in a much more definite form, viz. we can determine, and
663] FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. 167
that very simply, the actual expressions for the constant factors; and so we can enunciate
the theorem as follows; the squares of the sixteen double ^-functions are propoi-tional
to sixteen functions — [a], + \ab} ; where, in a notation about to be explained,
{a} = Va [a], [ab] = Va6 [ab].
Here in the radical Va, a is to be considered as standing in the first place for the
pentad bcdef, which is to be interpreted as a product of differences,
= bc .bd .be.bf.cd.ce .c/.de . df. ef,
(where be, bd, etc., denote the differences b — c, b — d, etc.). Similarly, in the radical
VoA, ab is to be considered as standing in the first instance for the double triad abf. cde,
which is to be interpreted as a product of differences, = ab . af. bf. cd.ce. de, (where ah, af,
etc., denote the differences a — b, a —/, etc.).
It is convenient to consider a, b, c, d, e, f as denoting real magnitudes taken in
decreasing order: in all the products bcdef, etc., and in each term abf or cde of a
product ahf.cde, the letters are to be written in alphabetical order; the differences
be, bd, etc., ab, af, etc., which present themselves in the several products, are thus all of
them positive ; and the radicals, being all of them the roots of positive quantities, may
themselves be taken to be positive.
We have to consider the vilues of the functions [a], [a6], or [a], {ab}, in the case
where the variables x, x' become equal to any two of the letters a, b, c, d, e, f; it is
clearly the same thing whether we have for instance x = b, x' = c, or x = c, x =b, etc. :
we have therefore to consider for x, x the fifteen values ah, ac, ..., af, .... ef; there is
besides a sixteenth set of values x, x each infinite, without any relation between the
infinite values.
Taking this case first, x, x each infinite, and in [oft], etc., the sign + to be +, we
have
^3? x'^
[a-\ = xx', [a&] = ^^_^.),,
or, attending only to the ratios of these values,
[«] = 1, [a6] = ^-^-^„
where - — - — 7^ is infinite, and the values may finally be written
[a]=0, [a6] = l;
whence also, for x, x infinite,
[a] = 0, {ab} = Va6,
the radical Va6 being understood as before.
Suppose next that x, x denote any two of the letters, for instance a, b ; then two of
the functions [a] vanish, viz. these are [a], [6], but the remaining four functions acquire
determinate values; and moreover four of the functions [ab] vanish, viz. these are
[ab], [cd], [ce], [de], for each of which the xx' letters a, b occur in the same triad (the
168
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS.
[663
double triads for the four functions are, in fact, ab/.cde, cdf.abe, cef.ahd, de/.abc)',
but the other six functions [ab], for which the letters a, b occur in separate triads,
acquire determinate values.
It is important to attend to the signs : for example, if x, x = b, e, we have
[c] = ce .cb, = — bc.ce
[ce] =
cb . eb .fb _ cb .fb _ _ bc.bf
{bef ae.be .de' ae.de' ae.de
Table I. of the values of [a], [ab], etc.,
■ ]
a;' =00 00
ab
ac
ad
ae
af
be
bd
w
0
0
0
0
0
0
+ ah .ac
+ ah .ad
m
0
0
— ab .be
— ab . bd
— ab .be
-ah.bf
0
0
w
0
+ ac . be
0
— ac .cd
— ac .ce
— ac .cf
0
-be .cd
[d]
0
+ ad.bd
+ ad.cd
0
— ad.de
-ad.df
+ bd.cd
0
w
0
+ ae .be
+ ae . ce
+ ae . de
0
— ae .ef
+ be . ce
+ be .de
[/]
0
+ a/.b/
+ af.cf
+ af.df
+ af.ef
0
+ bf.ef
^bf.df
[«6]
+ ab/. cde
0
ad. ae
be . cf
ac . ae
'^ bd.df
ac. ad
be.ef
0
ae . bd
be . cf
ad. be
^ be.df
M
+ a<f. hde
ad. ae
'be.bf
0
ab . ae
cd.ef
ab . ad
+ J-
ce . ef
0
^ab.bf
cd.ce
0
[ad]
+ adf. bee
ac . ae
~ hd.hf
ab . ae
cd.ef
0
ah . ae
^ de.ef
0
0
_ab.bf
cd . de
M
+ aef . bed
ac . ae
~be.bf
ah . ad
ce . cf
ab.ae
~de.df
0
0
0
0
[be]
+ bqf .ode
ac . af
~ bd.be
ab . af
cd . ce
0
0
oJ .ac
~ df.ef
0
ab .be
~ cd.df
M
+ bdf.aee
ad. af
be . be
0
^ab.af
cd. de
0
ah . ad
~cf.ef
ab .be
^ cd.ef
0
[be]
+ bef . acd
ae . af
~ bc.bd
0
0
ah . af
ce . de
ab .ae
~ cf.df
ab.bd
ce .cf
ah . be
^ de.df
[cd]
+ cdf. ahe
0
ad.af
be . ce
ac.af
"^ bd.de
0
ac .ad
~hf.ef
ae .bd
bf .ce
ad.be
^ bf.de
M
+ cef . ahd
0
ae . af
be . cd
0
oc . af
be . de
ae .ae
~bf.df
ac . be
* bf.ed
0
[de]
+ def.ohe
0
0
ae.af
be . cd
ad.af
be . ce
ad.ae
~bf.ef
0
ad. be
~ bf. cd
663]
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS.
169
Here the symbols be, ce, etc., denote differences ; [ce] is the product of four differences ;
the arrangement in two lines is for convenience only.
We thus obtain the series of values of [a], [ab], etc., which although only required
as subsidiary to the determination of the corresponding values of {a}, {ab], I nevertheless
give in a table.
The signs are given as they were actually obtained, but as we are concerned only
with the ratios of the functions, it is allowable to change all the signs in any
FOR THE SIXTEEN SPECIAI. VALUES OF W, x' .
be
¥
ed
ce
cf
de
df
"/
+ ab .ae
■i ah .af
+ ac.ad
+ ac.ae
+ ac .af
+ ad .ae
+ ad.af
■V ae .af
0
0
+ be.bd
+ be .be
+ bc.bf
+ bd .he
+ hd.hf
+ he.bf
— be .ce
be .ef
0
0
0
+ cd . ce
+ cd . df
+ ce .cf
- bd.de
-bd.bf
0
— ed.de
-cd.df
0
0
+ de .df
«
-be .ef
+ ce .de
0
— ce . ef
0
— de.ef
0
+ bf.ef
0
+ rf.df
' + cf. ef
0
+ df.ef
0
0
ae . be
0
0
0
ac . be
0
ad. hd
ae . he
^ hd.ef
~ df. «/
~ cf .ef
- ef.df
0
ab . be
ad.be
a«. be
0
0
ad.bf
ae.bf
* df.ef
ce .df
cd . ef
cd . ef
ce .df
0
ab.bd
^ cf.ef
ac . bd
cf.de
0
ac.bf
+ J J-
cd. ef
ae . bd
ed.ef
0
ae .bf
ef . de
ah.bf
ee .de
ab .be
^ ef.df
0
ac . he
^ ef.de
ac . hf
^ ee . df
ad. he
'^ ce .df
ad.bf
cf . de
0
ab.bd
0
ac . bd
ac . he
0
0
af.hd
af.he
ce .ef
" ce .df
" cd.ef
ed.ef
ce .df
ad. be
0
ad. be
0
^ af. he
ad . be
0
af . he
~ de.ef
ef.de
cd . ef
cd .ef
" cf.de
0
0 .
0
a/i . be
■*■ cf.de
^ af.be
ce . df
ae . hd
ee . df
af. hd
* ef. de
0
0
af .be
0
ae . be
0
ad.bd
0
_af.bf
bd.ef
^ de..ef
ce . ef
ce . de
ae.he
'hf.de
af.be
be.df
ac . be
~ de . df
0
0
ae . be
"^ ed.df
cd . de
0
ae . bd
af.U
ad.bd
ae . he
"f.hf
0
0
0
~ bf.ee
he.ef
ee .cf
~ cd.ef
cd . ce
C. X,
22
170 FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS, [663
column : and it appears that there are four columns in each of which the signs are
or can be made all + ; whereas in each of the remaining twelve columns the signs
are or can be made six of them +, the other four — .
Passing to the values of {a}, [ab], etc., we have for example, fi*om the ab column
of the foregoing table,
{c} = + '^c.ac. be,
\d} = + '^.ad.bd,
, , / — ac . ae
{'^} = -'^^- bc.bf
where (since the radicals are all positive) the signs are correct: substituting for the
quantities under the i-adical signs their full values, and squaring the rational parts in
order to bring them also under the radical signs, this is
{c} = + vab .ad.ae. of. bd.be . bf . de . df . ef. axi^ . b&,
[d\ = + ^ab .ac.ae. af. bc.be .bf.ce . cf. ef. ad' . bd^,
[ac] = - '^ac . af. cf. bd .be.de . ac' .ae-.b&. bf,
where all the expressions of this (the aft-coluinn) have a common factor,
ac. ad. ae . af. be .bd .be . bf.
Omitting this factor, we find
\c] = + Vafc .ac.bc.de. df. ef,
{d] = + 'Jab .ad.bd.ce.cf.ef.
{ac} = — ^ad .ae.de .be . bf. cf;
viz. recurring to the foregoing condensed notation, this is
{c} = + Vde,
[d] = + \/ce,
[ac] = — Vftc,
and, in fact, the terms in the several columns have only the ten values \/ab, ^ac,
etc. each with its proper sign. I repeat the meaning of the notation : ab stands in
the first instance for the double triad aJbf . ode, and then this denotes a product of
differences ab.af.bf.cd.ce.de. We have thus the following table in which I have
in several cases changed the signs of entii-e columns.
663]
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS.
171
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22—2
172
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS.
[663
Referring now to Gopel's memoir, Crelle, t. xxxv. (1847), pp. 277 — 312, we have
the sixteen double ^-functions
P, P„ P„ P,; iQ, Q„ iQ„ Q,; iR, iR,, R,, R,; S, i8„ iS„ S„
where the six functions affected with the t(=V— 1) are odd functions, vanishing for
the values w = 0, m' = 0 of the arguments. It is convenient to take xi , oo as the
values of x, x corresponding to these values m = 0, m' = 0 : the expressions {oj will
thus correspond to the six squares — Q*, —Qi, —if, —R\, —S^, —^i, and the ex-
pressions \ah\ to the remaining ten squares P^, P^ iS,' ; and after some tdtonneme)it,
I succeed in establishing the correspondence as follows
/Sf,», SA R,^ i?, Q^ Q,', (2,^ p,», p», -s^ p,», p,». Si', q,\ r/, r,\
= {a}, {b], {c}, {d], {e}, {/}, [ab], {ac}, \ad}. [ae], {be}, {bd}, [be], {cd}, {ce}. {de},
viz. the sixteen squared double ^-functions are proportional to the sixteen expressions
— {a}, + {ab}, as hereby appearing.
Table III. of the sixteen forms of
0
A
B
A+B
K
jK+A
K+B
X + A+B
s,'
- Si = a
-Si=-be
-S" =-ae
- -S? = b
+ Ri-=de
Ri= ce
R^ =-d
Ri = -e
S^'
-S,-=h
-5= =-ae
-Si^^-be
-Si= a
- Ri = c
- A« = d
-Ri = -ce
-Ri = -de
iJ/
- R^^^c
-R^= + d
-Ri:^~ce
~Ri= -de
- Si - b
-S' =-ae
-Si^-be
-Si-= a
^
-JP^d
-Ri = + c
-Ri = -de
- Ri=-ce
S* =ae
Si=-b
Si^-a
Si= be
<?
-«==«
-Qi = -ab
-Qi=^+/
-Qi-^-cd
P' =ad
Pi= ac
Pi= be
Pi= bd
Q.'
-«,» = /
-Qi = -od
-0" =+e
-Qi = ~ab
Pi = bc
Pi= bd
P" ^ ad
Pi= ae
Q."
<2.»=aA
Q'^-e
Qi= cd
Qi=-f
Pi^ae
P' = ad
Pi= bd
Pi= be
P.'
P* = ac
P' = ad
Pi= bd
Pi^ be
Qi = ab
Q' =-e
Qi= cd
Qi = -/
/»
F" =(id
Pi= ac
Pi-r be
/V- bd
-Q'^e
-Qi^-db
-Qi- f
-Qi^-cd
-s»
S^ =ae
Si=-b
6V=-a
Si = be
-- R' =d
-Ri= c
-Ri^-de
-Ri^-ce
/>,«
Pi ■■= be
/V= bd
/» = «rf
/V= ac
-Qi'-f
-Qi^-cd
-<f = e
-Qi = -ab
p^
Pi = bd
Pi= be
P,»= ac
r- = ad
Qi = cd
Qi = -f
Qi= ab
r/=-«
•iV
Si = be
Si=-a
Si=-b
S" = ae
Ri=--ce
Ri= de
Ri= ,:
R' =-d
«.'
Qi=cd.
Qi = -f
Q*-^ ab
qp =-«
Pi^bd
Pi= be
Pi= ac
P' ^ ad
Rz'
Ri=ce
Ri=^ de
III — — c
R^ =-d
Si = be
Si=-a
Si=---b
S" -^ ae
R.'
Ri - de
Ri= ce
R^ =-d
Ri^-c
- Si = a
-Si=-be
-S» =-a«
-Si = -b
ed
^
ah
be
bd
ad
663]
FURTHEK INVESTIGATIONS ON THE DOUBLE ^-FTTNCTIONS.
173
We have, after Gopel {I.e. p. 283), a table showing how the ratios of the double
^-functions are altered, when the arguments are increased by the quarter-periods
A, B, A + B, K, L, K + L,
that is, when u, u' are simultaneously changed into u + A, ii + A' or into u + B, u + B'
etc. If instead, we consider the squared functions, the table is very much simplified,
inasmuch as in place of the coefficients ±1, ±i, it will contain only the coefficients
+ 1 : and we may complete the table by extending it to all the combinations 0, A, B,
A + B, K, K + A, K + B, K + A + B, L, L + A, L + B, L + A + B, K + L, K + L + A,
K + L + B, K + L + A + B of the quarter-periods : we have thus a table included in
the annexed Table III., viz. attending herein only to the capital lettei-s P, Q, R, S, the
sixteen columns of the table show how the ratios of the terms — S^', — Si', etc., of the
first column are altered when the arguments are increased by the foregoing combinations
of quarter-periods, as indicated by the headings 0, A, B, etc., of the several columns.
THE SQtr.\KED DOUBLE ^-FUNCTIONS.
L
L + A
L + B
L + A4-B
K + L
K+L+A
K+L + B
K+L+A+B
-«,'=/
-Q^ = -cd
-QP .. e
Q,'=^-ai>
P,' = bc
l'i= bd
P^ - ad
P^- .. ae
«.'=«*
Q'^-e
Qi= cd
Q.'—f
l\^ = ae
1" = Cld
/V= hd
Pi= be
Pi' = ae
I» = ad
P,^= bd
/V = be
Q;'=ah
Q' =-e
Qf= cd
G./=-/
P" =ad
P,*= ae
P*= he
/»,*= bd
-Q' =e
~Q-^ = ~ab
-e/= /
-Q^'=-ed
S^ =ae
Si'=-b
S,*=-a
.S',» = he
^ R' =d
^R{'= c
-R} = -de
-R^ = -ee
- AV =«
-S,'=-be
-S' =-ae
- 67 = b
R.'^de
«;■'= ce
m =-d
Ri' = -e
-5,«=6
-5» =-ae
-S,'=-be
-S^'= a
,- R,' = c
-R' = d
-R,' = -ce
-R,'= de
- «,' = c
-IP = d
-R,*^-ce
-R^^=-de
-S,^=b
-S^ =-ae
-6V--6e
-67= a
~ JP =d
-R*= c
-R^= de
-R^'^-ee
S" =<U!
6,= = 6
6/=-a
6V-' = be
-<■/=«
-Q^ = -ab
-Q^J- f
-Q,'-cd
P' =ad
^,-^= ac
/V= he
/V= hd
Ji,' = de
B^= ce
IP =-d
R,^ = -c
-67 =«
-67 = -6e
-S" =-ae
-6',= = b
R;' = ce
R,'= de
R^ = -c
K' =-d
S./ ^be
63^= -ft
6? --6
S'' = ae
Q^^cd
Qi—f
«>'= «6
if =-e
l'.i --. hd
/•/= be
P,»= ae
P'= ad
St^=be
S,'=-a
S,'=-b
f? = ae
R/ = ee
/?,»= de
Ri'^-c
R^ --d
P»^ = bd
P,»= be
/»,'= ac
F' = ad
Q,'^cd
Q.' = -/
Q,'= ab
«==-e
P^ = hc
/»,»- bd
I» = nd
1\- = ac
-Qr^f
-Qi^-cd
-Q^ = e
-<?/=- 06
«/
be
¥
de
df
cf
174
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS.
[663
But I have also in the table inserted the values to which - S^', — S,', etc., are
respectively proportional, viz. the table runs —S,' = a, —Si'' = b, etc., (read —52" = {a},
_S,2={6), etc., the brackets [ j having been for greater brevity omitted throughout the
table), and where it is of course to be understood that — S^-, - Si', etc., are proportional
only, not absolutely equal to [a], {b}, etc. And I have also at the foot of the several
coluDins inserted suffixes <x> x , ah, cd, etc., which refer to the columns of Table II.
Comparing the first with any other column of the table, for instance with the
second column, the two columns respectively signify that
.5,= («)={al.
.S,»(«)={6},
-Si'{ti + A)=-lae\,
Q,''(u + A) = -[e},
where, as before, the sign = means only that the terms are proportional ; u is written
for shortness instead of (u, u), and so u+A for (u + A, u' + A'), etc.: the variables in
the functions [a}, [be], etc. are in each case x, x. But if in the second column we write
u — A for A, then the variables x, x will be changed into new variables y, y' , or the
meaning will be
X, X
-5,»(m) = (61,
Q,« («) = {ab\.
y> y
-Si'{u)^-{be\,
-S;'{i,) = -[m\,
so that, omitting from the table the terms which contain the capital letters P, Q, R, S,
except only the outside left-hand column — &", — S^, etc. , the table indicates that these
functions —8^, —S^, etc., are proportional to the functions [a], [b], etc., of x, x given in
the first column; also to the functions — (ie), —[o>e], etc., of y, y given in the second
column ; also to the functions — [as], — (6e), etc., of z, z given in the third column ; and
so on, with a different pair of variables in each of the 16 columns.
Thus comparing any two columns, for instance the first and second, it appears that
we can have simultaneously
X, x' y, y
[b] = - [ael
{ail =-{«).
(fifteen equations, since the meaning is that the terms are only proportional, not absolutely
equal), equivalent to two equations serving to determine x and x' in terms of y and y ,
663]
FURTHEK INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS.
175
or conversely y and y' in terms of x and x . The functions in each column form in fact
16 sixes, such that any four belonging to the same six are linearly connected; and in
any such linear relation between four functions in the left-hand column, substituting for
these their values as functions in the right-hand column, we have the corresponding
relations between four functions out of a set of six belonging to the right-hand column,
or we have an identity 0=0. I will presently verify this in a particular case.
If in any column we give to the variables the values x , x we obtain for the
terms in the column the values which the terms of the fii-st column assume on giving to
JO, x the values shown at the foot of the column in question ; thus, in the second column
giving to the variables the values « , x , the column becomes
-V6e, -Vae, 0, 0, -Vat, - Vcrf, 0, 'J ad, "Joe, 0, 'Jbd, s/hc, 0, 0, Vde, Vce
which is, in fact, the crf-column of Table II. : this is of course as it should be, for the
values in question are those of the functions — S.^, — S{-, etc., on writing therein
X, X = c, d.
The formulae show that
Vat, "^ac, "Jad, 'Joe, Vie, "Jbd, 'Jhe, Vcri, Vce, Vrfe,
are, in fact, proportional to
/■2
W,-,
^i,
W3-, 0-3,
P-^.
pi-.
(jfc,, ki, ... are Gopel's Ic , k",...). This gives rise to a remarkable theoi'em, for the
ten squares are functions of only four quantities a, 0, 7, 8 (Gopel's t, u, v, w). For
greater clearness, I introduce single letters A, B, ..., J and write
A=abc. def= {*Jdef = p.;
B = abd.cef=('Jcey=p,'
C =abe.cdf=-('/cdf=k,
D = abf. cde = (V<i6)- = k,
E = acd . bef = ('^bef = <Ts
F =ace.bdf= {'Jhdy = ■a.,
G = acf. bde = ('^acy = w
H = ade.bcf={Shcf=m..
I = adf. bee = (Vad)" = w
J =aef.bdc = ('Jaey =
= (OL' - 0-- + r - ^-y,
= i (ay + 0By,
= 4 (a8 + 0yy,
= (a' - 0' - r + B^y,
= 4 (a/8 -f 78)-,
= id' + 0' + y' + SJ,
= 4 (a8 - 0yy,
= 4 (07 - 0By,
= 4 (a/3 - 78)-,
= (cc^ + 0^-r-8'y;
viz. it has to be shown that A, B, ..., J, considered ajs given functions of the six letters
a, b, c, d, e,f, are really functions of foui- quantities a, 0, y, 8; or, what is the same
thing, that A, B, ...,J, considered as functions of a, b, c, d, e, f satisfy all those relations
which they satisfy when considered as given functions of a, /3, 7, 8.
176
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS.
[663
Now considering them as given functions of a, ^, y, B, they ought to satisfy six
relations; and inasmuch as, so considered, they are, in fact, linear functions of
a* + 0* + Y + S\ a^^ + y'ST; aV + ^'S^ o'S» + /8V. a/37S,
five of these relations will be linear: there is a sixth non-linear relation, expressible in a
variety of different forms, one of them, as is easily verified, being
VZ7±VC(? + VnF=0.
Now considering A, B, ..., J as given functions of a, b, c, d, e, f, there exist
between them linear relations which may be obtained by the consideration of identities
of the form
ahcd = 0,
ahcdef
where the lefl-hand side is used for shortness to denote the determinant
= 0.
1.
1,
1,
1
a.
b,
c,
d
a?,
b\
c^.
d-'
1,
1,
I,
I,
1,
1
a,
h.
c.
d,
e,
/
a-,
y.
c=,
d',
e=,
/
We thus obtain between them a system of fifteen linear relations, which present them-
selves in the form
(1) A-J+ E-B=0,
(2) -A-I +F -0 = 0,
(3) A-H+G-D = 0,
(4) - B- G + H+C = 0,
(5) B- F + I +J) = 0,
(6) C -E+J-D = 0,
(7) -E-D-H+E = 0,
(8) E-C- I +G = i.),
(9) F-B-J-G = 0.
(10) H-A + J -I =0,
(11) -J + D-G+I=0,
(12) J + C-F+H = 0,
(13) I + B-E -H = 0,
(14) G + A+E-F = 0,
(16) D-A + B-C=^0,
I
663] FURTHER INVESTIGATIONS ON THE DOUBLE ^FUNCTIONS. 177
and these are all included in the equations (10), (4), (12), (15), (6), which serve to
express G, B, E, F, I in terms of D, H, C, A, J, i.e. ac, ce, eh, bd, da in terms of
ab, be, cd, de, ea, if for the moment we write G = ac, etc. But the five linear relations
in question are, it is at once seen, satisfied by A, B, ..., J considered as given functions
of a, /3, 7, 8.
The equation \/ AJ ± \/ DF ± VCG = 0, substituting for A, B, ..., J their values in
terms of a, b, c, d, e, /, becomes
"Jabc . def. aef. bed ± '/abf. cde . ace . bdf± 'Jabe . cdf. acf. bde = 0,
which (omitting common factors) becomes Vit? . ef'- ± 'Jif^ . cc' + Vie^ . cf- = 0 ; or, taking
the proper signs, this is the identity be . e/+ be .fc + bf.ce = 0.
It is to be noticed that
S^ + a=-/3-^-7=, 2(a/S-78), 2(7a + /3S),
2(«/S + 78), S' + ^-'-r-a\ 2i0y-aS),
2(7a-y3S), 2(0y + ah), B"- + rf - cC' - ff',
each divided by 8- + a" + /S'- + y, form a system of coefficients in the transformation
between two sets of rectangular co9rdinates. We have therefore
"i/ab, \ad, Vce,
Vie, 'Jde, ^ac,
•J be, ^cd, 'Jae,
each divided by ybd and the several terms taken with proper signs, as a system of
coefficients in the transformation between two sets of rectangular axes: a result which
seems to be the .same as that obtained by Hesse in the Memoir, " Transformations-
Formeln ftlr rechtwinklige Raum-Coordinaten " ; Grelle, t. LXiii. (1864), pp.247 — 251.
The composition of the last mentioned system of functions is better seen by writing
them under the fuller form "Jabf. cde, etc. ; viz. omitting the radical signs, the terms are
abf. cde, adf. bee, abd . cef,
bef . acd, def . abc, acf . bde,
bcf . ode, edf . abe, aef . bed,
each divided by bdf.aee; or, in an easily understood algorithm, the terms are
bf.d df.b bd.f
a .ce
bf.d
df.b bd.f
e.ac
bf.d
df.b bd.f
c.ae
bf.d
df.b bd.f
each divided by bdf. ace.
C. X.
23
178 FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. [663
Rtiverting to the before-mentioned comparison of the iirst and second columns of
Table III., four of the equations are
X, x y, y X, X y, \f
[c!= \A\, that is, Vc[c]= 'Jd\d\
\d\ = (cj , that is, Vd [d] = \/c [c],
\e\ = - \ah\, that is, V^ [e] = - Va6;[a6],
(/} = - {cd}. that is, </[/] = - Vcd [cd] ;
viz. the four terms on the left-hand side are not absolutely equal, but ouly proportional,
to those on the right-hand side. Substituting for Vc, Vd, etc., their values, and in-
troducing on the right-hand side the factor
'Jojc .bc.ce. cf. ad . bd . de . df,
the equations become
xx' yy'
[c] = ac.bc. ce . ef [d],
[d] = ad .bd.de. df[c],
[e] = — ce.de [ab],
[/]=- cf.d/[cd].
The functions on the left-hand satisfy the identity
^f[c] - «/c [d] +fcd [e] - cde [/] = 0,
or, as this may also be written,
de/[c] - cef[d] + cd/[e] - cde [/] = 0.
Hence substituting the right-hand values, the whole equation divides by ce.de. cf. df;
omitting this factor, it becomes
ef.ac.bc[d] — ef. ad . bd [c] — cd {[ab] — [cd]] = 0,
where the variables are y, y : it is to be shown that this is in fact an identity, and (as
it is thus immaterial what the variables are) I change them into x, x'.
We have
ac . 6c [d] — ad . 6d [c] = {a — c)(b — c)(d — x) (d — x)
— {a — d)(b — d) (c — x) (c - x)
= (c — d) 1, x+x', xx'
1, a + b , ab
i, c + d, cd
= cd \xx'abcd\,
suppose.
663] FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. 179'
We have moreover
where for the moment a, 6, a', etc., are written to denote a~ x, h—x, a — x', etc. ; we
have then
e'f- ef = (e - x') {f-x)-(e- x) (/- x)
= - (e -/) (« -«') = - e/(x - x),
and
aZw'rf— a'6'cd = (a -a:)(6 -a;)(c — a!')(d — a;')= — (a; — a;') 1, x + x, xx'
— {a — x')(b — x')(c — x)(d — x) 1, a + b, ah
1, c + rf, erf
= — (x — x) [xx'abcd].
Hence [ab] — [cd] = ef[xx'ahcd], and the equation to be verified becomes
(ef. cd—cd. ef) [a;ar'a6crf] = 0,
viz. this is, in fact, an identity.
Cambridge, 14 March, 1877.
I
23—2
180 [664
664.
ON THE 16-NODAL QUARTIC SURFACE.
JFrom the Journal fur die reine und angewandte Mathematik (Crelle), t. Lxxxiv. (1877),
pp. 238—241.]
Prof. Borchardt in the Memoir " Ueber die Darstellung u. s. w." Crelle,
t. LXXXill. (1877), pp. 234 — 243, shows that the coordinates x, y, z, w may be taken as
proportional to four of the double ^-functions, and that the equation of the surface
is then Gopel's relation of the fourth order between these four fimctions: and he
remarks at the end of the memoir that it thus appears that the coordinates x, y, z, w
of a point on the surface can be expressed as proportional to algebraic functions,
involving square roots only, of two arbitrary parameters f, f.
It is interesting to develope the theory from this point of view. Writing, as in
my paper, "Further investigations on the double ^-functions," pp. 220 — 233, [663],
[a] = aa',
[b]=bb',
[c] = cc',
[d] = dd',
[e] = ee'.
t"*] = (P-~py (^«¥c'<^'«' - '^a'b'f'cdey.
etc.,
where on the right-hand sides a, b,...,a',... denote a — f, 6 - f, ..., a — ^', ... (f, f'
being here written in place of the x, x' of my paper), then the sixteen functions
664] ON THE 16-NODAL QUARTIC SURFACE. 181
are proportional to constant multiples of the square-roots of these expressions; viz.
the correspondence is
^2 = ^13> 'Sli = ^34, it, =^3, il=^(u, V=^l> V2=^03.
i^s/[al i\^\/[bl i\/~cV[6\, i\/d\/\dl, i\/e\/[^l i\/f\/]jy,
\/abV\aS\, \/^\/\a6\, \/'^d\/\ad], \/'^\/\a^\ \/hc\/\b6\, \/U\/\bdy,
O3 ^= ^23, Va = ^0 ) ^3 ^ •J4> ^2 ^ ^03>
</be\/\be], \/cd\/\cc[\, x/ceVJce], \/de\/[de];
where, under the signs ^, a signifies bcdef, that is, be .bd.be. bf.cd .ce.cf.de .df.ef,
and ab signifies abf.cde, that is, ab.af.bf.cd.ce.de, in which expressions be, bd, ...,
ab, af, ... signify the differences b — c, b — d,...,a — b, a—f,... But in what follows,
we are not concerned with the values of these constant multipliers.
Prof. Borchardt's coordinates x, y, z, w are
a; = %=P; y=^^ = S,; z = %, = -S; w=% = F,;
viz. P, S, Pi, S3 are a set connected by Gopel's relation of the fourth order — and
this relation can be found (according to Gopel's method) by showing that Q' and j^
are each of them a linear function of the four squares P\ P./, S% S.^^ and further
that QR is a linear function of PS and P3S3; for then, squaring the expression of
QR, and for Q' and R^ substituting their values, we have the required relation of
the fourth order between P, 8, P3, S3.
Now we have P, S, P3, S3, Q, R = constant multiples of \/[ac], ^[aF], V[cd],
y/[bd], V[6], Vfc] respectively : and it of course follows that we must have the like
relations between these six quantities; viz. we must have [6], [c] each of them a
linear function of [ac], [ab], [cd], [bd] ; and moreover V[6] Vfc] a lineai' function of
^^[ac] V[ai] and ^[60?] Vp].
As regards this last relation, starting from the formulae
V[^ = j~t' {"^a^b'd'e + -^a'cf'bde],
V[6d] = ~y {'/bdfa^'e' + 'Jb'df'ace] ,
Vp] = J," p {y/abf^d'e + ^/a^'cde}.
V[cd] = p^ {'^cdfctFe' + -Jc'df'abe],
182
we have at once
ON THE 16-NODAL QUARTIC SURFACE.
[664
V[m\ V[a6i = (Y~fY f <«/'''*' + "/'<^) '^^^' + ^^' + ^'"^ ^<w^'<?e'}.
46d]V[cd] = -
{(d/aV + d/'ae) Vftcft'c' + (be' + 6'c) Vodea'dV} ;
the difference of these two expressions is
where substituting for a, d, e, f, a', ... their values a — |, d — f, e-f, /— f, a — f, ...
we have ad'-a'd = (a- d)(f -f), /e'-/'e = (/- e)(f-f) ; also ^/636'c' = \/[6] V^ ; and
we have thus the required relation
V[oc] \/[^] - V[6rf] \/[crf] = - (a - d) (e -/) V[6] ^^].
As regards the first mentioned relation, if for greater generality, d being arbitraiy,
we write [d^^Off, that is, = (,6 — ^) {_B — ^), then it is easy to see that there exists
a relation of the form
V [6] = A [ab\ + B {ac\ + G[hd\ + D [cd],
where A-^B+G+D=Q. The right-hand side is thus a linear function of the
differences [a6] — \ac\, [ah\ - \hd\, [ah] — [cd] ; and each of these, as the irrational
terms disappear and the rational terms divide by (^ — ^'Y, is a mere linear function
of 1 , f + f ', ?^' ; whence there is a relation of the form in question. I found
without much difficulty the actual formula; viz. this is
{a-d)(h-c){e-f)
= 1, e, f, ef
1, h, c, 6c
1, d, a, ad
1, e, e, &>
1, e + f, ef \[e]
1, 6+ c, be
1, a + d, ad \
[oc]-
1,
e, f, ef
[ah]-
1,
c, b, be
1,
d, a, ad,
1,
e, e, ^
1,
e, f, ef
[ed] +
1,
b, c, be
1,
a, d, ad
1,
e, e, ^
1. e, f ef
1, c, b, be
1, a, d, ad
1, 0, e, &-■
[bd].
where observe that on the right-hand side the last three determinants are obtained
from the first one by interchanging 6, c : or a, d : or b, c and a, d simultaneously : a
single interchange gives the sign — , but for two interchanges the sign remains +.
664] ON THE 16-NODAL QUARTIC SURFACE.
Writing successively 0 = b and 0 = c, we obtain
188
(a-d)(e-f)
[b]
1, e + f, ef
1, b+ c, be
1, a + d, ad
= (a -/) (b -d)(b- e) [oc] -(a-b) (6 -/) {d - e) [iib]
+ (a-b){b- e) {d -/) [cd] -{a-e){b- d) {b -f) \_M\ ;
{a-d){e-f) 1, e + / ef [c]
1, 6 + c, 6c
1, rt + d, ad
= -{a - c)(c-f){d- e)[ac]+{a-f)(c - d)(c - e)[ab]
-(a-e)(c-d)(c-/)[fid]+(a-c)(c-e)(d-/)[6d];
which values of [6] and [c], combined with the foregoing equation
(a - d) (e -/) V[6] \/\c] = - V[ac] Vp] + \/[c£Z] V[iW],
give the required quartic equation between V[ac], \^[a6], v^[cd], ^[ftd].
Cambridge, 2 August, 1877.
184 [665
665.
A MEMOIR ON THE DOUBLE ^-FUNCTIONS.
[From the Journal fur die reine und angewandte Mathematik (Crelle), t. LXXXV. (1878),
pp. 214—245.]
I RESUME my investigations on these functions; see my two papers, Crelle,
t. LXXXlii. (1877), pp. 210—233; [662] and [663]. But it is proper in the first
instance to develope in a corresponding manner, the theories of the circular (or
exponential) functions, and of the single ^-functions.
Part I. Preliminary investigatimis.
Starting from the differential relation
dx
du =
'/a—x.b
between the variables u and x, I write for shortness the single lettei-s A, B, 12,
instead of functional forms A(u), B (u), n{u), to denote functions of u; and 1
assume as definitions the equations
A =il •^a — x,
B=n »/b^x,
and another equation to be presently mentioned : these two equations imply between
A, B, il the algebraical relation
A-'-B'^il^ia-b).
Differentiating, we obtain
3i4 = 3n . Va — a; — \^a — x.b — xdu,
2 vo— ar
665] A MEMOIB ON THE DOUBLE ^-FUNCTIONS. 185
that is,
n
dA = ~dn-iBdu,
and similarl}'
whence
dB = ^dn-^Adu,
AdB-BdA = -^{A^-B')du,
Proceeding to a second differentiation, we find
d'A = '^d'n-^dndii+iA{duy,
d'B = ^d'n-^ an att + iB{duy,
and thence
A d'A -{dAy = ^' {n c^n - (dny} + ka-- - b^) (di,)\
Bd'B- (dBy = ^, {n 3»n - (dny} + hb^ - A') (Buy.
To simplify these we assume (as the third equation above referred to)
D.d'n-(dny=o.
The last-mentioned two equations then become
A d'A - (dAy = i(A'- B') (duY,
B d'B - (dBy = i(B'-A'>) {duy,
which several equations contain the theory of the functions A, B, D,: we have as
their general integrals
JB = - i Ae*« Va-6 {e*<«+>" - e-*<"+'" j,
n = Ae*«
where A, X, v are arbitrary constants. Forming the quotients A : D,, B : VL, and
introducing the notations cosh, sinh, of the hyperbolic sine and cosine, also writing
for simplicity v = 0, the equations give
\fa — x= 'Ja — b cosh ^ u,
»jh—x = — 'J a — h sinh J u,
which express the integral of the differential relation
dx
8m = :.
\a — x.b — X
c. X. 24
186 A MEMOIR ON THE DOUBLE &- FUNCTIONS. [665
Instead of considering in like manner the i-adical '^/a — x.b—x.c — x, I pass at once
to the radical '^a — x.b — x.c—x.d — x; and starting from the diiferential relation
dx
du =
•Ja — x.h — x.c — x.d — x
and using the single letters A, B, C, D, SI to denote functions of u, I assume as
definitions
A^Sl *ja — x,
B = n Vb^^,
C = n Vc - a;,
Z) = n -^cT^,
and another equation to be presently mentioned ; A, B, G, D are called ^-functions,
and il is called the to-function.
But before proceeding further I introduce some locutions which will be useful.
In reference to a given set of squares or products, I use the expression a sum of
nquares to denote the sum of all or any of the squares each multiplied by an
arbitrary coeflScient ; and in like manner a sum of products to denote the sum of
all or any of the products each multiplied by an arbitrary coefficient: in particular,
the set may consist of a siugle square or product only, and the sum of squares or
products will then denote the single term multiplied by an arbitrary coefficient. In
the present case, we have the quantities Va — x, \/b — x, Vc — x, Vd — x, and the squares
are a — x, b — x, etc., which belong all to the same set ; but the products (meaning
thereby products of two quantities) ^/a — x.b — x, etc., are considered as being each of
them a set by itself A sum of squares is thus a linear function X + ^, and
conversely any such function is a sum of squares ; a sum of products means a single
term v'Ja — x.b — x (or v va — x.c — x, etc., as the case may be), and conversely any
such function is a sum of products : the coefficients \, fi, v may depend upon or
contain fl, and in differential expressions (8m being therein considered constant) the
coefficients X, /i, v may contain the factor du or (3m)^ — and if convenient we may of
course express such factor by writing the coefficients in the form \du, or \(9m)=
etc., as the case may be.
We may now explain very simply the form, as well of the algebraical relations,
as of the differential relations of the first and second oixlers respectively, which
connect the functions A, B, C, D.
The functions A'', B', G-, Df are each of them a sum of squaies, and hence there
exists a linear relation between any three of these squares. But the products AB,
AG, etc., are each of them a sum of products (meaning thereby a single term, as
already explained); and hence there is not any linear relation between these products.
Considering the first derived functions dA, dB, etc., these each contain a term
in 3n, which however disappears (as is obvious) from the combinations AdB — BdA,
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 187
etc. ; and, without in any wise fixing the value of fl, we in fact find that each of
these expressions is a sum of products; the form is, as will appear,
AdB-BdA = afl^ ^c-x.d-x = v CD, etc.*
Passing to the second derived functions and forming the combinations Ad-A—{dA)-,
etc., each of these will contain a multiple of fl d'Q, — (9fi)-, but if we assume this
expression fi d'Q. — (diiy = il^M, where ilf is = (duf multiplied by a properly determined
function of x, then it is found that each of the expressions in question Ad' A —{dAy, etc.,
becomes equal to a sum of squares, that is, to a linear function fl- (\ + fix) : viz. it
is equal to a sum of squares formed with the squares A'', B-, G\ D-.
The foregoing equation
where M has its proper value, is the other equation above referred to, which, with
the equations J. = fl Va — a;, etc., serves for the definition of the functions A, B, C, D, fl;
it may be mentioned at once that the proper value is
M=\(d%i,f[-2af-\-x(a-\-h + c + d)+K\,
where « is a constant, symmetrical as regards a, h, c, d, which may be taken = 0,
but which is better put
= a^ + lr+ c^ + d^—ab-ac — ad-bc — bd— cd.
For the proof of the formula, I introduce and shall in general employ the
abbreviations (a, b, c, d) to denote the differences a — x, b — x, c — x, d — x: the
differential relation between x, u thus becomes dx = du Vabcd. I use also the ab-
breviations fl d'n - (dny = Afi, etc.
We have _ _ _
^85-534 =fl»(VaaVb-VbaVa),
the teiTQs in 3fl disappearing : viz. observing that 9a = 8b = — dx, this is
or observing that a — b = a — 6, and writing for dx its value = vabcd du, this is
AdB- BdA = _ ^(a - 6) n^ -Jcddu,
= - ^ (a - 6) il^'Jo-x.a-xdu,
which is the equation expressing AdB — BdA as a sum of products : it is further
obvious that the value is
= -i(a-6)CZ>aM.
* It ia hardly necessary to remark that a, v contain each of them the factor ou ; and the like in other
cases.
24—2
188 A MEMOIR ON THE DOUBLE ^FUNCTIONS. [665
Proceeding next to find the value of AA, =Ad'A-(dAy, =A*^\ogA, it is to
be remarked that we have in general
APQ = P»AQ + Q'AP,
and therefore also AP* = 2P^AP, and consequently A VP = ^ AP. Hence starting from
A =0. Va, we have
AA = aAQ + n' i Aa,
a
where Aa = -&d'x-(dxy. I assume that we have An = il^M={il'S(duy, where S
denotes a function of x which is to be determined : the equation thus becomes
AA = ifl» {a^ (duY - 2d'x - 2 {dxf} ;
we have (dxy = abed (duY, and thence, differentiating and omitting on each side the
factor dx, we obtain
2d'x = - (abc + abd + acd + bed),
and the equation becomes
AA = ^n^ [a (S + be + bd + ed) - bed) (duY,
which is to be simplified by assuming a proper value for S; in order that the same
simplification may apply to the formulae for AB, etc., it is necessaiy that S be
symmetrical in regard to a, b, c, d.
Writing for the moment b', c, d' to denote b — a, c — a, d—a respectively, we have
b', c', d' = b — a, c — a, d — a, and thence
b'c'd' = bed - a (be + bd + cd) + a= (b + c + d) - a^
and consequently
a (be + bd + cd) - bed = - i'c'd' + a* (b + c + d - a) :
hence, in the expression of A.4, the factor which multiplies ^fl'^(3u)' is
a{S + a(b + c + d-a)} -6'c'd',
viz. the expression added to S is
(a—x){b + c+d — a— 2x),
= a(b+c+d-a)-x{a+b + c + d)+2x'.
Hence assuming
S=-2a^ + x(a + b+c + d) + K,
K being a constant symmetrical in regard to a, b, c, d, which may be at once taken
to be =a* + l^ + (^ + d^ — ab — ac — ad — bc — bd—cd; then writing also
\ = ¥ + c^ + d" — be — bd — cd, /a = — b'c'd' = a—b .a ~c.a — d,
the expression a {/S + a (b + c + d — a)] — b'c'd' becomes = a\ + /i ; and the sought for
equation thus is
AA^Ad-'A- (3^1 )• = in" (a\ + /*) (3«)',
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 189
the equation in il being of course
Afl = n d^n - (dny = i n^ (-, 2a» +x{a + b+c + d)+K} (duf.
The theory in regard to the second derivatives is thus completed.
To adapt the formulae to elliptic integrals, and ordinary H and ® functions, the
radical must be brought to the form Va; . 1 — <» . 1 — te. Writing for this purpose
a, b, c, d = -M\ 0, 1, ^, (/=oo),
2m
substituting also -j for u, and iki . A, iB (i = v— 1 as usual) for A, B respectively,
we find 'Ja — x.h — x.c — x.d—x = I \x .\—x.l— k?x ; and then
dx
2du =
^x.l-x.l-kfx'
and
^ = n, 5=nV^, c=ftVF^, D = J n Vi - i^x.
n is in this case = A, a, ^-function : and in the equation for All, writing A in
place of CI, the equation becomes '
Ad'A-idAy^ iA'- 1- 2af' + x(- M') + k] ^^^^' ,
viz. replacing j^ by a new constant, = \ suppose and finally putting / = oo , this is
Ad'A- {dAf = A^{X- li?x) (duy.
The differential equation is satisfied by ir = sn*M, giving 1 — a: = cn-w, 1— fe=dn^tt:
and the equation for A then ia
3= log A=(\-k^ sn'M) (9w)^
or say
A=Le Jo Jo J
viz. by properly assuming the constants L, \, we shall have A = Jacobi's function 0i< :
and then sn it = -^ , en m = -j , dn m = -i- , which will give the ordinary expressions of
^ ^ ^
sn, en, dn in terms of H, 0.
Part II. The double ^-/mictions.
Passing now to the double S-functions, and writing for a moment
yX = Va — x.b — x.c — x.d — x.e — x .f— x,
^/Y='Ja-y.b-y.c-y.d-y .e-y.f-y.
190 A MEMOIR ON THE DOUBLE a-FaNCTI0N8. [665
the differential equations which connect u, v with x, y are
du =
dx
■ 9.V
^ xdx ydy
There are here sixteen S>-function8 A, B, C, D, E, F, AB, AG, AD, AE, BC, BD, BE.
CD, GE, DE, and an associated w-function fl, where for shortness I use the single and
double letters A, B, ..., AB, ...,fi, instead of functional expressions A{u, v), B(u, v),..,
AB(ti, «),.., fl(M, v), to denote functions of the two letters u, v. Writing also
(a, b, c, d, e, f) for the differences a — x, b — x, etc., and (a,, b,, c,, dj, e^, {,) for the
differences a — y, h — y, etc., whence vX = v'abcdef and Vy = VaibiC,die,f'i, and 6 for
the difference x—y, we have sixteen a;y-functions which are represented by
oja, i^h, i/c, *jd, tje, V/. "Jab, 'Jac, '/ad, '/cue, 'Jhc, '/hd, "/he, 4cd, Vce, Vrfe,
the values of which are
\/a = Vaai, (six equations),
Va6 = -n {Vabfcid,ei — Vaibifjcde), (ten equations),
and the definitions of the sixteen ^-functions and the m-function are
^ = fl Va. (six equations),
AB = n */ab, (ten equations),
and one other equation to be afterwards mentioned.
I call to mind that, in a binary symbol such as wab, it is always / that accom-
panies the two expressed letters a, b: the duad ab is, in fact, an abbreviated expression
for the double triad ahf. cde : and I remark also that I have for greater simplicity
omitted certain constant factors which, in my second paper above referred to, were
introduced as multipliers of the foregoing functions *Ja, ..., ^ab, ... I remark also that,
to avoid confusion, the square of any one of these functions Va or Va6 is always
written (not a or ab, but) (V")" or {*/aby.
I use 9 as a symbol of total differentiation : thus
^ . dA dA ^ ., . d^A d-A ,-,--., d'A ,- ,„ ^
^^ = d^ ^" + dv ^' ^ ^ = -dk^^ (^"^ + 2 d«^. ^^ ^> + dv^ ^^'>' ^''-
Moreover I consider du and dv as constants, and use single letters \, L, etc., to
denote linear functions adu+^dv, or quadric functions a(dii)' + 20dudv + y(dvy (as the
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 191
case may be) of these differentials; thus, in speaking of AdB — BdA as a sum of
products, it is implied that the coefficients of the several products are linear functions
of du, dv; and so in speaking of Ad^A — (dA)- as a sum of squares, it is in like
manner implied that the coefficients of the several squares are quadric functions of
&«, dv.
An a;2/-function is simplex, such as Va, or complex, such as Voi; the square of
the former is aa, = a= — a (a; + ^) + xy, which is of the form X + /* (a; + y) + vxy ; the
square of the latter is
= ^ {abfc,d,e, + a,b,f,cde -2's/XY],
2
where observe that the iiTational part — ^ '^XY is the same for all these squares :
so that, taking any two such squares, their difference is =^ multiplied by a rational
function of x>/ : this rational function in fact divides by ^, the quotient being a rational
and integral function of the foregoing form X + fi(x + y) + v xy. Hence selecting any one
of the complex functions, say \de, the square of any other of the complex functions
is equal to the square of this plus a term \ + fi (x + ?/) + vxy, and hence the square
of any function simplex or complex is of the form \ + fi(x + y) + vxy + p {'Jde)^ ; this
being so, the squares of the a;y-functions may be regarded as forming a single set ;
ever>- sum of squares is a function of this form \ + fi{x + y) + v xy + p {'Jdef ; and
conversely every function of this form is a sum of squares. A sum of squares thus
depends upon four arbitrary coefficients \, fi, v, p: and we may, in an infinity of
ways, select four out of the 16 squares such that every sum of squares can be
represented as a sum of these four squares each multiplied by the proper coefficient ;
say as a sum of the selected four squares : in particular, each of the remaining
squares can be expressed as a sum of the selected four squares. It appears, by the
first of my papers above referred to, that there are systems of four squares connected
together by a linear equation : we are not here concerned with such systems ; only
of course the four selected squares must not belong to such a system.
We have the products of the a;y-functions, where by product is meant a product
of two functions. The number of products is of course = 120, but distinguishing these
according to the radicals which they respectively contain, they form 30 different sets.
Thus we have
Vi ^^o6 = a (^ Vafb,c,die, — b, Vaifjbcde},
VcVac = ^{c „ -c, „ j,
>Jd'^ad = ^{d „ -d, „ },
Vc ^ae = z {® " ~ ^» " )'
192 A MEMOIR ON THE DOUBLE ^FUNCTIONS. [665
which four expressions form a set, and there are 15 such sets. The set written
down may be called the set af : and the fifteen sets are of course ab, ac, etc.
Again, we have
VaVt= Vabajb,,
'Jac '^bc = ^ {(cfdjei + Cjfide) Vaba,bi — (ab, + ajb) VcdefCidiCif,},
VcidV6d = ^{(dfc,e, + d,f,ce) ^ „ - „ „ },
Vae Vie = ^ {(efc,d, + eiftcd) „ - „ „ },
which four expressions form a set, and there are 15 such sets. The set written down
may be called the set abajb, : and the fifteen sets are of course abajbi, acajCi, etc.
The 15 and 15 sets make in all 30 sets as mentioned above.
The expression, a sum of products, means as already explained a sum of products
belonging to the same set; and there are thus 30 forms of a sum of products. The
products of the same set are connected by two linear relations, so that, selecting at
pleasure any two of the products, the other two products can be expressed each of
them as a linear function of these ; hence a sum of products contains only two
arbitrary coefficients.
Reverting now to the equations A = il V«, etc., we see at once the form of the
algebraical equations which connect the 16 ^-functions. Every squared function
A', ..., {A By, ... is a sum of squares, whence selecting (as may be done in a great
number of ways) four of these squared functions, each of the remaining 12 squares is
a sum of these four squares each multiplied by the proper coefficient ; or say it is a
sum of the four selected squares. And in like manner the 120 products of two of
the 16 functions form 30 sets, such that selecting at pleasure two of the set, the
remaining two of the set are each of them a linear function of these.
Considering the first derived functions dA, dB, ..., dAB, ..., each of these contains a
term in 9fl; but 912 disappears (as is obvious) from the several combinations IdJ — Jdl
(I write / and similarly J to denote indifferently a single letter -4 or a double letter
AB) : and, without in any wise fixing the value of Cl, we in fact find that each of
these expressions is a sum of products.
Passing to the second derived functions, and forming the combinations Ad^A—{dAy,
etc., or to include the two ca-ses of the single and the double letter, say Id'I — {dif,
each of these will contain a multiple of fiS'O— (30)'; but if we assume this expression
il 9''n — (9n)^ = £l^M, where M \s & quadric function of 9m, dv, the coefficients of
(9u)', dudv, {dvy being properly determined functions of xy, then it is found that each
of the expressions in question I d^I — {bFf becomes equal to a sum of squares.
It is to be observed that M is not altogether arbitrary: the equation as con-
taining terms in (9m)', dudv, and (9t;)', in fact represents three partial differential
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 193
equations, which for an arbitrary value of M would be inconsistent with each other :
it is therefore necessary to verify that the value assigned to M is such as to render
the three equations consistent with each other, and this will accordingly be done.
The foregoing equation
n d'-£i - on)= = D.-M,
where M has its proper value, (or say the three partial differential equations into which
this breaks up), constitutes the other equation above referred to, which with the original
equations A = rL <Ja, etc., serve to define the sixteen ^-functions and ft.
The remainder of the present memoir is occupied with the analytical investigation
of the foregoing theorems. Although the mere algebraical work is very long, yet it
appears to me interesting, and I have thought it best to give it in detail.
The equations
give
The analytical theory : various subheadings.
a. dx dy
-jy =dv — ydu, — jy = dv— xdu,
\
which determine dx, dy in terms of 9m, dv. A different form is sometimes convenient ;
writing diir = dv — a du, and recollecting that a, ai denote a — x, a — y respectively, the
equations become
ddx ^ o ^^y a , o
— -= = 3«r + a, OM, — j^ = OCT + a du.
Expression for d ^a.
We have
9 Va = 9 "^aai = — ;= (a 9a, + a, 9a) = j=- (a dy + a, dx)
2 V aa, 2 vaai
^ ^ {a \/Y{dv-x du) - a, VZ {dv - ydu)] ;
2V^^'
substituting for 'JX, \/Y their values Vabcdef, Va,b,c,die,f, , this is
9 Va = I {VabiC,die,f, {dv — a;9w) — Va,bcdef (9» — y du)],
and by the mere interchange of letters we can of course find 9 \/b, etc.
C. X. 25
194 A MEMOIR ON THE DOUBLE ^-FUNCTIONS. [665
Expression for d 'Jab.
We have next to find
9 >Jab = 3 ^ (VabfcidiC, — Va,b,f,cde} ;
here
dO = dx — dy, = ^ { Vabcdef (9t) — ydu) + VaibjCidiBif, (dv — x 3m)},
and consequently 3Va6 contains a term
— ^ {VabfcidiSi - Vajbificde},
which is
~ S { ( ~ *^f Vcdecid,e, + ode '/al)fa,bifi) (dv — y du)
+ (— CidiCi Vabfajbjf, + ajb/j VcdeCidiei) {dv — x du)],
or, what is the same thing,
1 ( / . J . .. / ode - C]d,ei , , - ode y + Cid,e, x \
= ^, j Vabfa,b,f, {^ 0 ^ + 0 ^J
/-j — J— /— abf+a,b,f, „ , abfy -aib,fia;- \"|
+ V cdecjdie, f 0 ^ + g j | '
Now
-^^^^^-^ =-{cd + ce + de) + {c + d + e)ix + y)-x-'-xy-y\
— cde V + Cid,e, x , / . . x . / . \
a =cde-(c + d+ e)xy+xy(x-\-y);
o
with the like formulae with a, b, f instead of c, d, e. Hence the foregoing, or say
the first, part of 3 Va6 is
= ^ [Vabfa,b,f, {{-{cd + ce + de) + {c + d + e){x + y)-icr - xy- y^} dv
+ { cde — (c + d + e)xy + xy(x + y)} du)
+ N/cdeCid,ei ({ ab + af+ hf -{a + b +f) (x + y) + x^ + xy + f}dv
+ [- abf+ (a + & +/) xy-xy(x + y)] du)].
The other or second part of 3 Va6, using for shortness an accent to denote diSerentiation
in regard to a; or to y, according as it is applied to a function of x or of y, is
readily found to be
= i [Vabfa,b,f, ({- (cde)' - (c,d,e,)') 3f + { y (cde)' + x (c,d,e,)') 3m)
+ \/cdec,d,e, ({ (abf)' + (a,b,f,)'} dv +{-y (abQ' - « (a,b,f,)'} 3m)].
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 195
Hence uniting the two terms so as to form the complete value of 9 'J ah, we have
first, a term 5; Vabfa,b,fi 9y, the coeflScient of which is
= — {cd + ce + de) + {c + d+ e){x-{-y) — af — xy— y-
-i{(cdey + (c,dA)'}:
this second line is
= erf + ce + de - (c + (i + e) (a; + y) + 1 a^ + 1 2/»,
or the coefficient is — ^ a^' + a;y — ^ y^ = — i ^' ; the term is thus
= J Vabfaibifi dv.
Secondly, a term in ^ VcdeCidje, dv which is in like manner found to be
= — ^ Vcdecidje, dv.
Thirdly, a term ^ Vabfa,bif, 9m, the coefficient of which is
= cde — {c -¥ d + e) xy ■\- uPy + xy^
+ ^{T/(cde)' + a;(c,d.e,)'}:
this second line is »^
= — (cd + ce + rfe) i (a; + y) + (c + rf + e) 2xy —%a?y-% xy^,
and the coefficient is thus
= i{2cde-(cd + ce + de)(« + y) + (c + d + e)(a:2 + y»)-iB' -f
— {c+d + e){x —yf + a?— a?y — xy"^ + ]f\,
which is
= \ [cde + c,d,e, - (c + d + e) ^ + (« + y) ^},
or the term is
= \ V£bfaM ^^^ y ''^'^' - (c + rf + e) + a; + yl 9if.
1
And, fourthly, a term in ^ VcdeCid,ei 9m, which is in like manner found to be
= - J V^d^^Ai: 1^^^— ^ - (a + 6 +/) + «> + 2/} aw.
Hence combining these several terms, we have finally
9V(* = iV^b£kM[ 9^ + (^^^^~^-c-d-e + a;+y)9itl
+ \ Vcdecd.e. \-h>- i^^ "^^'^'^' -a-h -f+ x + y) 9m] ;
and by the mere interchange of letters we can of course find 9 "Jac, etc.
25—2
196
A MEMOIR ON THE DOUBLE ^-FUNCTIONS.
[665
Expression for A dB— BdA.
Starting now from the equations ^ = fl ^/a, 8=0, v'6, we obtain
AdB -BdA = a* ya ^^Jb-^/b^ Va}
lOa
= -a- ( Vaa, {Vba,Cid,e,f, (3d — a; 9m) — Vbjacdef (9« - y du)]
-Vbb,fv'abAd,e,f,( „ )- \/a,bcdef ( „ )}).
ins
= ^-w- {(a, - b,) Vabc,d,e,fi (dv — x du) — (a — h) Va,b,cdef (dv — y 9«)) ;
or since a,— b, = a— b = a-6, this is
AdB-BdA = ^-^{a-b) {Vabc,d,e,f, {dv - xdu) - V'a,b,cdef(at) - y du)],
which is a sum of products of the set ab: in fact, the four products of this set ai*e
V/ Vaft = ^ { f Vabc,d,eif, — fj Vaibicdefj,
Vc Vde = ^ {— c
+ c,
» i<
'Jd'Jce=-A-di „ +d, „ },
Ve Vcd = 3 |— e
+ e,
}:
choosing any two of these at pleasure, for instance the first and second, multiplying by
dv — c 9m, dv — f du and adding, we have
(dv — c du) \/f "Jab
+ {dv-fdu)sjc Vde
(dv — c 9tt) f VabcidiCif, — (dv-c du) ft Va^b,cdef|
g{-(9t)-/9w)c „ +(9y-/a«)c, „ },
where the coefficients {(dv — cdu)—c(dv-fdu), and f, (9t) — c 9m) — c, (9*) — /9m), by sub-
stituting for f, c, f], c, their values, become =(f—c)(dv — xdu) and (f—c)(dv — ydu);
and the expression is thus
= "^ ^ {Vabc,d]e,f, (dv — x du) — Va]b,cdef (9y — y du)].
Reverting to the original expression for AdB — BdA, it may be remarked that, if
we write dv — adu — dnt, dv — bdu = da, then
(a— b) (dv — x du) = a 9ff — b dvr, (a — b) (dv — y du) = a, 9o- — b, 9«,
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 197
and the formula thus becomes
AdB-BdA =^' {Vabc,d,e,f,(aacr -hdzr)- Va,b,cdef(a, da - b^ 8ct)}:
but I shall not in the sequel use this formula, or the notation dv — b du = da introduced
for obtaining it.
Expi-ession for AdAB ~ ABdA.
Starting from the equations A = n «Ja and AB = O, \/ab, we have
A dAB - ABdA = n= [^Jad'^/ah -'Jabd >Ja],
where the term in { ) is
= -J^,\ jVabf£;;b:f, I dv ^i^^^^^^ -c- d - e ^- x + r^du\
+ \ Vcdec,d,e, I- dc - /?M+f^f" _a-h-f+x+y)^
— —, (Vabfcid,e, — Va,bif,cde) [Vab,c,die,f, {dv — x du) - Vajbcdef (3^ — ydu)].
To reduce this, I write 9?; — a du =i3cr, and therefore
dv — xdii = dxs ->t- & du, dv —ydu=dvr +aridu;
then for convenience multiplying by 2,6", the term is
= aa, Vbfbjf, j ^ Dtir + [(a - c - rf - e + a; + y) ^ + cde + c,die,] du]
+ Vacdea,c,d,ei [ - ^ Sor + [ (h +/- x-y)&^ - abf - aib,f, ] du}
• — (Vabfcid,e, - Vaibifjcde) lVab,c,dieif, {d^ + a 9h) - Vajbcdef (9to- + ai du)\.
The last line hereof is
= Vbfbif] {— acidie, (3xt + a du) — a,cde (3or + a, 9m)J
+ Vacdea]C,d,ei { bif, (9w + a 9m) + bf (9or + a, 9m)}.
Hence we have first, a tenn in Vacdea,Cidie, , the coefficient of which is
= - ^- 9i3- + [(6 +/- X - y) ff' - abf - a,b,f,] 9m + b,f, (dm + a 9m) + bf (dw + a, du),
viz. this is
= d^{-&^+ b,f, + bf) + 9m [- (a - a,) (bf - b.f,) + {b +/- x-y) 0'],
where (b- b,)(f-f,)= ^, that is, bf + b,f, - ^ = bf, + b,f, also
(a - a,) (bf- b,f,) = (6 +/- * - y) ^,
or the coefficient is =(bfi+ b,f)9ar: viz. the term in question is
= VacdeaiCid,ei (bf, + b,f) 9W.
198 A MEMOIR ON THE DOUBLE ^-FUNCTIONS. [665
We have then, secondly, a term in Vbfb]f,, the coefficient of which is
= aai{^3sr + [(a— c-d-e + a; + y)^ + cde + Cid,ei] 3m}
— ac,d,e, (9w + a 9«) — ajcde (dvr + a, 9m),
viz. this is
= (aa,^ — ac,d,ei — ajcde) dvr + [aa, (a-c — d — e + x + y)&' + (afide — ac,die,) (a — a,)] 9w.
We have a — a, = — ^ ; also ajcde — aCidjC]
= 6 [cde - a (cd + ce + rfe) + (c + d + e) [a {x+y) — xyl — a (.t= ■¥ xy + 'tf) + xy {x + y)],
where the coefficient of ^ is
= - (a - c) (a - d) (a - e) - (c + d + e) [a- - a {x ■\- y) + xy}^- {a + x -h y) [a= -a(x + y) + xy],
viz. it is
= — (a — c){a—d)(a—e) + a£ki{a — c — d — e + x + y).
Hence the coefficient in question is
= (aa,^ — ac,d,e, — aicde) 9w + (a — c) (a — d)(a — e)ff' du,
and the second term is =Vbfbif,, multiplied by this coefficient.
Hence, observing that the whole has to be multiplied by ^^^, we find
A BAB -ABdA=^D,^ {VacdeaAd^i (bf, + b,f ) 9sr
+ Vbfbifi [(aa#' — acidjej — ajcde) 9or + (a — c) (a — d)(a — e) &' 9m]},
where I retain d-sr in place of its value, =dv— a du.
This is a sum of products of the set bf b/i : we, in fact, have
•^ac side = ^ {(bfi + bjf) VacdeaiCidiCi — (acdjCi + ajCide) Vbfb,f,},
Vod Vce = „ } „ „ — (adciC, + aidjce) „ },
VaeVcd=„{ „ „ - (aecid, + aiCicd) „ },
VW/= ., { +^ „ 1,
and selecting any two of these, for instance the first and the fourth, the coefficient
of ^iV is at once seen to be of the form d'a >J ac 'J de + K ^Jh »Jf ; and for the determ-
ination of K, we have
(- fuxl,e, - aiC,de) dts + Kd"- = (aai^ — aCidiCj — ajcde) 9«r + (a — c) (a —d){a — e)G^ du,
viz. this gives
Ke^ = {aai^- + (c - Ci) (adie, - ajde)) 9or + (a - c) (a -d){a- e) 6' 9m.
( VacfbidiC, — \^aiC,fibde) -
665] A MEMOIR ON THE DOUBLE ^-FUNCTIOXS. 199
We then have
(c - Ci) (ad,ei - aide) = &' {- aa^ + (a - d)(a - e)],
and the whole equation divides by 6'-; substituting for 3w its value, we find
K=(a~ d){a — e) (dv — c du).
Expression for ACdAB — AB BAG.
Starting in like manner from the equations AB=D, "Jab, AC = fl^/ac, we have
ACdAB-ABdAC = lD:\
o
multiplied by
(+ Vcd^d;e, f- S^ + ( b+f-x-y- ^i±^^ 8 J
( Vaefe;^\ r d^ + [a-b-d-e+x + y+^^^^^^^\diM
+ (- -/abfcd.e, - Va,b,^cde) -J -" j- ,
1+ Vbdeb,d,e. T- a« + ( c+f-x- y- ^±^ ) ^^ j
which, omitting the factor ^ Qp^ is
= { af bi Vbca]d,e]f I — ajf ,b Vb,Ciadef } 3«- + (a — c — rf — e + a; + yH ^ — ^— \ du
+ 1 cd,e, VbiCjadef — c,de Vbcaid,e,f i} — 3or + f b +f -x — y ^ — — j du
+ {- afci Vbca,d,e,f 1 + a,fiC VbicTadef ] di!r + (a-b-d-e + x + y+ ' '^M du
+ {- bde, VbAadef + bjde Vbca,die,f,j - aw + ^ c +/-x- y - "'"^"' ' ]du\;
and here the whole coefficient of dv is
= (b, - Ci) (af — de) Vbcajdie,fi — (b — c) (ajfj — d,ei) VbiCjadef,
viz. observing that b, — e, = b — c = 6 — c, this is
= (6-c) [[af- de - (a+f-d-e)x] VbcajdjCif, - [a/- de - (a +/- d - e) ^] Vb,c,adef j,
or, what is the same thing, it is
= (b-c){[-(a-d){a-e) + {a+f-d-e) a] Vbca,djeif ,
- [- (a - d) (a - e) + (a +f-d-e)a^] VbjCiadef}.
200 A MEMOIR ON THE DOUBLE ^-FUNCTIONS. [665
The coefficient of 9m contains the factor Vbcajdieifj, multiplied by
„ / ode + CidiCA
afb, ta-c — d-e-i-a; + y+ g^ 1
-afc(a-6-d-e + . + 3/ + ^^^-+^^)
+ b,de( c+f-.-y-'^it^);
here the terms divided by ^ destroy each other, and the expression of the coefficient
of Vbca,d,e,f, becomes
= (b, — c,) [af(a — d — e + .'s + y) + de(/— a; — y)] + (af— de)(6c, — cb,),
or since b, — Cj = 6 - c, 6ci — cb, = — (6 — c) y, this is
= (b — c)[a.f {a — d - e + X + y) + de(/— a; — y) — (af — de) y],
which is
= (6 - c) [af (a - d - e) + de/+ (af - de) x],
and is readily reduced to
(6 - c) [{a — d)(a — e)f— (a —d){a — e) x], =(b — c) (a — d) (a - e) f :
viz. the coefficient of du contains the term (6 — c)(a — d)(a— e)f Vbcaid,ejf,. There is
a like term — (6 — c) (a — d) (a — e) f, VbiCjadef, and the two terms together form the
whole coefficient of du.
Hence, restoring the outside factor | il", we have
ACdAB-ABdAG
= ^il'{b-c) \{[- (a -d){a-e) + (a +/- d - e) a] Vbca,dxe.f,
— [— (a — d) (a — e) + (a +/— d — e) a,] Vb,c,adef j dis
+ {a — d){a — e) {f VbcajdjCifi — fi VbiCiadef} du ,
where, as before, I retain Sta- instead of its value =dv — a du. This is a sum of
products of the set be: the products, in fact, are
Va v'rfe = ^ {— a
Vbcajdje
,fi + ai VbiCiadef},
VdVae = „ {-d
>j
+ d, „ },
v/e "^ad = „ {— e
}i
+ e, „ },
V/V6c=„{+f
»
- f. „ ).
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 201
whence, observing that a — f=ai — fi=a — /, we have
•J a V5e + •Jf'Jbc = - "^-a^ { VbcaidiCifi - VbiCjadef ) :
it is clear that the term in question is at once expressible as a sum formed with
the products nja 'Jde and nJf'Jhc.
It is to be remarked that there are 15 expressions such as AdB — BdA, and
45 expressions such as AGdAB— ABdAC; and that each of these (15 + 45=) 60 ex-
pressions is a sum of products of a set such as ab : and that there are also 60
expressions of the form AdAB — ABdA, and that each of these is a sum of products
of a set such as aba,bi.
Eicpression of 0. d'O. - (diiy, = \ Mn'.
We assume il 9'fl — (diiy = J Mil-, where M is a, quadric function of du, dv ; suppose
M=^ {duy + 233 au aw + (5 (dvf.
It is to be noticed that the 21, S, (4 are not all of them arbitrary functions of (*•, y)
or (w, v); we, in fact, have {M=-- ^^=S'^logn; and hence 21, S3, S satisfy the
conditions
d2l^d» dS^dS
dv du ' dv du'
Taking 21, S3, 6 as functions of x, y, these become
\dx ^ dxj \dy dyj
/d33 d^\ 1^ /dS dS\ 1^
Ay dy
Putting for the moment
\ = a +6 +c , p = d +e +/, p = \+/3,
fi = ab + ac+bc, a = de + df+ef, q=fi + a;
V = ahc, T = def, r = p +t,
I found it convenient to assume
(5 = -2(a^ + xy + y') + p {x +y),
where observe that p, =a-\-b + c + d + e +/, is symmetrical in regard to the constants
a, b, c, d, e, f. And then, 6 having this value, there exists (as is seen at once) a
value of 33, =2(iio'y + xy^)—pxy, for which
dx ^ dx ' dy dy
and which thus satisfies the second of the above-mentioned conditions.
C. X. 26
202 A MEMOIR ON THE DOUBLE ^FUNCTIONS. [665
Assuming now
SI = — 2**^ + qxy — r{x + i/) — fji<r+&,
where © has to be determined so as that the first of the same conditions may be
ulso satisfied, then substituting this value of 21, we have
that is,
(- a,bA - d,e.f, + ^®) VZ = (- abc - def + ^®) VT,
viz. in the terms independent of © writing for i/X, \/V theii- values, this is
(abc + def) Va,b,c,d,e,f, - (a,biC. + d,e,f,) Vab^(R+ J® \fX - ^® V F= 0,
or, what is the same thing,
- (Vabcaib,c, - VdiM^eif,) (Vdefa,b,c, - Vd,e.f,abc) + ^ V J - ^ V 7 = 0.
But treating 0 as a function of u and v, we have
d& _d@dx d® dyl /dB , „ _ dB jy\
dv ~ dx dv dy dv 0 \dx dy J '
also
\fde = ^ (VdefaibiC, — Vdie,f,abc) ;
u
and we thus reduce the equation to
JOk
- (Vabca,biCi - Vdefd,e,f,) Vde + -t- = 0.
But, referring to the expression for d^ah, we have, by a mere interchange of letters,
T- Vde = — J (VabcaibiC, — Vdefd.eifi),
dv -
and the formula thus becomes
dv dv
consequently
© = - (Vdi)" = -\, (abcd,e,f, + a,b,Cidef- 2 V5T),
and the value of 21 thus is
1 o
21 = ^, {- abcd.eif, - a,b,c,def + 6- (- 2*^ + </«^ - r (a; + y) - /io-)) + ^ VZF,
or, as this may be written,
21 = ^ (abc - a,b,c,) (def - d.e.f.) - 2a;y -\- qxy - r{x-\- y)- p^ -^ ( VZ - V Y)'.
665] A MEMOIR ON THE DOUBLE ^-FUKCTIONS. 203
Here
abc — a,b,c, = {v — ijuc + \a? — a?) — {v — fxy + Xy" — y»)
= 0[-^-{-\{x+y)-(x- + ay + y%
and siniilarlj'
def - diCif, = 0[- <r + p{x + y) -(of + xy + y-)] ;
the expression of 21 contains therefore the terms
[fi — \(x + y) + a;- + xy + y^] [a — p{x + y) + a^ + xy + y^] — na — r(x + y) + qxy — 2a?y",
viz. for r, q substituting their values v + t, fi + p, these terms are
= -{fip+ a-\ + v + T}{x + y) + (fi + <r + \p){x + yf
- (X, +/»)(« + y) (x- + xy + y"-) + {x^ + xy + y^y - 2a?y-.
The coefficients pLp-\- a\-\- v + t, 11 + a ->r \p, \ + p are, in fact, symmetrical functions of
a, b, c, d, e, f, viz. writing
X=a — x.b — x.c — x.d — x.e — X. f— x,
= A — ar + car* — Da;' + Eic* - Far> + a;*,
that is,
A = abcdef, b = S ahcde, c = S abed, D = S abc, E = 2 a6, F = S a,
(f = a + b + c + d + e+f, which has in fact previously been called p), we have
fip + a\ + V + T = V, p, + a + \p = E, \ + p = F,
and the terms are
= - (x + y) [d — e{x + y)+ F(sfi + xy + y")} +x*+ 2a^y + x'y^ + 2ccy' + y* ;
viz. we have
-{x + y){D-E{x + y) +¥{ai' + xy + f)]+{a^ + 2x>y + o^y^ + ^ayy^ + y").
To this I join the foregoing values of %, 6 ; viz. writing F in place of p, these
are
33 = - F a;^/ + 2 {a?y + xy"^),
g = F (a; + y) - 2 (a;- + a;?/ + y%
where it will be noticed that the values of 21, S3, G are all of them symmetrical
in regard to the constants a, b, c, d, e, f.
I recall the original form of 21, viz. this was
21 = — /io- — r (a; + y) + 9' a;y — l3?y- — (ydeY
= - (a6 + oc + be) (de + df+ ef) - {abc + def) {x + y)
+ {ah+ac + hc + de + df+ ef) xy - 2ar'y' - {'J def
= 21. - (Vdey,
26—2
204 A MEMOIR ON THE DOUBLE ^-FUNCTIONS. [665
suppose ; and <!(, 53, 6 denoting as above, we have
M = ^ (duY + 233 du 8v + g (dv)', il d*n - {dOf = i MQ.\
For the subsequent calculation of Ad'A —i^Ay, it is convenient to transform this
expression by introducing therein dvr in place of dv, and a, a, in place of x, y. We
have
JIf = {2(„ - {'Jdef] (?u)- + 233 9m (8^ + a Sw) + 6 (a^sr + a duf
= {?[„' - ( \/dey} (duy + 233' du dw + 6' (durf,
suppose, where
S' =6,
93' =93 +a(S,
2l.' = 3l„ + 2a'»+aMS.
Writing
a; = a — a, y = a— a,,
we find
S = - 6a» + 2aF + (6a - f) (a + ai) - 2 (a» + aa, + a^O,
95 = 4a'-a^ + (-6a» + aF)(a + a,)4-2a(a^ + a,=) + (8a-F)aa,-2(a»a, + aa,*),
% = -(ab + ac+ be) (de + df+ ef)
— (abc + def) (2a — a — a,)
+ (ofc + ac + 6c + de + df-\- ef) [a? — a (a + a,) + aaj]
- 2 [a" - a (a + a,) + aoip,
the developed value of which is
= -2a-' + a'(6 + c) + a^(-6c + rfe + d/+e/)
+ a {- 2def- (b + c)(de + df+ ef)} -bc(de + df+ ef)
+ {4a'-a^(6 + c)-a(de + d/+e/) + de/} (a+a,)
-2o''(a» + ai-^) + |- Sa^ + a{b + c) + bc + de + df+ef]&a.,
+ 4a (a^ + aa,»)
- 2a'a,»,
and thence without diflficulty
6' = - 6a^ + 2aF + (6a - f) (a + a^ - 2 (a- + aa, + a,»),
35' = - 8a' + a^'F + (6a - f) aa, - 2 (a»a, + aa,'),
SJ„' = a' (6 + c) + a» (- bc + de + df-\- ef) + a J- 2def- (b + c) (de + df+ ef)} -bc(de + df+ ef)
+ [-a^ + a^{d + e +/) - a (de + df+ e/) + de/) (a + a,)
+ {4a^ + a (- 6 - c - 2d - 2e - 2/) + 6c + de + df+ ef} aa,
- 2a=a,S
which are the required values.
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 205
Eicpression for A?FA —(^Ay-. several subheadings.
Writing for shortness Ad^A — (dA)- = ^A, as before, and so in other cases: then
in general APQ= R-AQ + Q'AP, and thence Ai^ = 2P=AP or AVP=4aP. Hence
starting from A = (1 -^a^n Vaa,, we have
AA = Afl >/aa, = aa, Aft + ^— (a=Aaj + a,'Aa),
aS]
where
Aa = a 3% — (daf = — a.d'x — (3a;)», Aa, = — a,d^ — (dy)-.
Hence wi-iting
An = \Mn\
we have
i A4 =iaa,J/- I {^d'x+ad'x,)-is jj (9^)' + * (wl ■
But we have
dx = ^(dv-ydu), dy = - -g- (dv - x ^i) ;
squaring the first of these and di£ferentiating, we find
2dxd'x =
X
(--^ + ^)3^ + ^Sy (a»-yau)»-23yaM^(at)-ya«),
where as regards X the accent denotes differentiation as to a; (and further on, as
regards Y, it denotes differentiation to y), viz. this is
= (- -^ + ^) 3«+ ^ 3y i^-yduf -2dydu^ (dv-ydu),
= (- ^ + ^j^"! (dv - y^uY + ^ (dv- ydu- 0du)(dv-ydu)dy,
where the second term is
2X
^ {dv — xdu){dv — ydu)dy,
which is
= 2g — {dv — X duy dx :
hence dividing by 2da;, the equation is
and similarly
^/TF iV Y'\
and we may in these values in place of 9o — y9u and dv — xdu write dm + SL^du and
dv + & du respectively.
206 A MEMOIR ON THE DOUBLE ^-FUNCTIONS. [665
Hence in >,j^-4 the irrational part is
^ l±l- {a, {d^ + a aw)' - a (3^ + a, Mf\
i'^def = Jj {abcd,ejf, + a^bAdef- 2 VZT},
But we have
whence
^ = i (abcd,c,f, + a,b,c,def) - ^ (Vde> ;
and the term thus is
[i (abcd.e.f, + a^bicdef) - J (Vde)^"] {(Sisr)-^ - aa. (duy-].
Joining hereto the rational part of t^ AA, and multiplying the whole by 4, we have
4 ^ . „ r / 2X X'\ 2a, Z"l ,. , , , „
+ r^ (abcd,e,f, + a,biCidef ) - (Vdef~\ {(dury - aa, (Buy},
where M has its foregoing value = {2(„' - (Vde)''} (duy + 233' att a^r + g' Oiir)^
JVrst step of the reduction.
Writing bcdef= U, bjC,d,e,f, = Uu then X = aU, Y=a,iUi, and consequently
X'=:-U + &U\ Y' = -U, + a,U,',
the accents in regard to U, Ui denoting differentiations as to x, y respectively: then
/ 2Z X'\ 2a, Z_ (2&U U-a.U'\ l&.&U _ ( 2U £\_s^U
and similarly
/ 2F Y'\ 2a F_ _ /_ 2J^i _ f^A _ af^.
* r "^ ~ ^ j ~ a, ^ " ~ ^ V ^ ^ y' <?» ■
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 207
The formula thus becomes
4,
- L. (abcdieif, + aib,c,def) - (Vde)4 (3^)^
- ^ (9^ + a, duf - ^' {dvT + a duf
+ j-^(abcd,e,f, + aib,Cidef)-(Vc?e)'4 (3ct)-,
viz. substituting for M its value, the term in {"Jdefiduf disappears, and the formula is
^, Ail = aa, ["sTo' {duf + 253'8«atT + 6' (atsr)'^ + (^^ - ^') (9^ + a, diCf
+ f- ^' - ^J (9«^ + a ^M^ - ^ (abcd,eifi + a,biCidef) (att)^
- ^(ao-H- a, ait)= - *^' (acr + a 9m)=
+ J25 (abcd,e,f, + aib,c,def) - (^^6)4 {disf :
say for shortness this is
4
= aa, S - ^ (atr + a, a«)' - ''^^' {dts + a ?«)- + ^. (abcd.e.f, + a,b,c,def ) - (^/def {d^f.
Second step of the reduction.
In the reductions which follow, we make as many terms as may be to contain
the factor aa,, so as to simplify as much as possible the portion not containing this
factor.
We have a«r + a, aw =(3^+ 08») +aa«<, and consequently
(disr + a, duf ={d'!sr + d duf + aP,
where P = 2 dudzj + (a. + 20) (duf : similarly a^ + a du = (dsr - ddu) + a, du, and therefore
(axT + a duf = {dvT-e duf + a, P, ,
where P, = 2dudu + (&,— 20) (duf : the values may also be written
P=2dudT!J + (2a, - a) (3(0=, P, = 2 3^ aisr + (2a - a,) (duf.
208 A MEMOIR ON THE DOUBLE ^-FUNCTIONS. [665
The formula thus becomes
-^{dm+0 duy -^(diiy-0duy+^ (abcd,e,f, + a,b,c,def ) {dmf
-C^deYidnrf.
The second line here is
+ 2{— ajCT— aI7i+ abcdiCifi + a,b,Cidef j (3w)',
and the coefficient herein of (9w)= is = ^(adie,f, - aidef)(bc-biCi). Writing for the
moment d — a, e — a,/—a = d', e',f', we have
^ (adjCif, — a,def) = (a (d' + a, . e' + aj ./' + aj — a, (d' + a . e' + a ./' + a)}
= - d'e'f + aa, (d' + e +/' + a + a,),
^ (be - b,Ci) = -{b' +c' + a + aO.
The whole term in (8sr)' is thus
= {(6' + c' + a + a,)d'e/' + aa, (6' + c' +a + a,)(d' + e' +/' + a + a,)} (^«^)^
2
The coefficient of dudiir is — ^(aiC/^— af/,): viz. this is
2
= jai (6' + a . c' + a . d' + a . e' + a ./' + a) — a (f + a, . c' + ai . d' + ai . e' + ai ./'+ a,)j,
and if
6' + a . c' + a . d' + a . e' + a ./' + a = b' + c'a + D'a= + K'a? + ¥'&* + a»,
that is,
B'==b'c'd'e'/', c=lb'c'd'e', T)' = lb'c'd', e' = S6'c',
f' = S6' = b' + c' + d' + e' +/',
this is
= 2 {b' — D'aa, — E'aa, (a + a,) — F'aa, (a- + aa, + a,'') - aa, (a' + a^a, + aa,^ + a,')! :
or say for shortness it is = — 2(b' — aa,*) where
O = d' + e' (a + a,) + F' (a= + aa, + a,=) + a» + a'a, + aa,-' + a,' ;
the term in question thus is — 2 (b' — <ia, *) dii dur.
The coefficient of (duY is — (a,f/^+ af/^), viz. this is
— a, (6' + a . c' + a . d' + a . e' + a ./' + a) — a (6' + a, . c' -f a, . d' + a, . e' + a, . /' + a,),
which is = — (a + a,) b' — aa,^, where
^ = 2c' + u' (a + a,) + e' (a* + a,') + f' (a» + a,') + a* + a,*.
665] A MEMOIE ON THE DOUBLE ^-rUNCTIONS. 209
The formula thus is
^, A4 = aa, Is - ^ P - ^P, - (b'+ c'+ a + a,)(rf'+ e +/'+ a + a,) {d^f + 2* Szt 8w - ^ {duy ■
- (a + a,) b' (diif - 2b' du dm + (b' + c' + a.+ aO d'ef (9i!r)= - ('/dey- {duf.
The whole coefficient of aaj, substituting for 2, P, Pj, <I», N?' their values, and arranging
according to d-er, du, is
< 9.TT TT / 9.11 TT'\ 1
= (a«)=|S' + ^-^+(-^'-^fj-(^''+c' + a + a,)(d' + e'+/' + a+aO|
+ d' + e' (a + ai) + f' (a^ + aa, + aj'') + a? + s^&i + aaj'' + ai' •
+ (az.)'{ao' + a,'(^-J')+a=(-^'-^) + (a-2aO^
- g^ (abcd,e,f, + ajbAdef ) - 2c' - D' (a + a,) - e' (a= + a,^) - f' (a' + ai») - a* - sA :
and we have to reduce separately the three coefficients of this formula.
Third step of the redicction.
First, for the coefficient of (9w)- ; recollecting that ^ = a, — a, we have
2 -^—' = - 2c' - 2d' (a + a,) - 2e' (a' + aa, + a,") - 2f' (a' + a=a, + aa," + a,')
- 2 (a* + a'a, + a'a," + aa,' + a,*),
-(U'+ [/■,') = 2c' + 2d' (a + a,) + 3e' (a= + a,'') + 4f' (a' + a,') + 5 (a^ + a,*).
Adding these, the right-hand side divides by (a,— a)', that is, by ^; and the resulting
value is
= e' + 2f' (a + a,) + 3a= + 4aai + 3a,l
The term - (6' + </ + a + a,) (d' + e' +/' + a + a,), attending to the values of e' and f', is
= b'c' + d'e' + d'/' + e/'- e' - f' (a + a,) - a=- 2aa, - a,'';
hence the whole coefficient of (jdraf is
= S' + b'& + d'e + d'f + e'f + f' (a + a,) - 2 (a" + aa, + a,'),
or substituting for b', c, d', e', f their values, this is
= g' + 4a» - a (6 + c + 2d + 2e + 2/) + 6c + de + d/H- e/+ (f - 6a) (a + a,) - 2 {2? + aa, + a,=).
Proceeding next to reduce the coefficient of 2d'Brdi(, observing as before that ^ = ai-a,
we have
2aitr-2atr, ^ ^^ _ ^^,^ _ ^^,^ (a + a,) - 2F'aa, (a" + aa, + a,') - 2aa, (a» + a% + aa," + a,»),
0
C. X.
27
210 A MEMOIR ON THE DOUBLE ^-FUNCTIONS. [665
also
-(aal7'+i7)-(af7,' + fr,) =
- 2b' + d' (- a» + 4aa, - ai') + E' (- a» + 3a»a, + 3aai= - a,') + v' (- a* + 4a'a, + 4aa,' - a,*)
— a* + 5a*a, + 5aa,* — a,' ;
adding these two expressions, the right-hand side divides by (a, — a)', that is, by ^, and
the resulting value is
= - d' - e' (a + a,) - f' (a" + a,') - a' + a=a, + aa,' - a,'.
To this is to be added
+ 2d' + e' (a + ai) + f' (a« + aa, + a,'') + a' + a'a, + aa," + a,» ;
we thus see that the whole coefficient of 29w3sr is
= d' + F'aa, + 2 (a'ai + aa,'),
or say it is
= d' + (F - 6a) aa, + 2 (a-a, + aa,=).
Lastly, for the coefficient of (duy-, we have
2a,'' 77 — 2a? l)
— g ^ = 2b' (a + a,) + 2c'aai - 2E'a''a,2 - 2F'a2a,' (a + a,) - 2a V (a= + aa, + a,-),
and also
- a,=?7' + (a - 2a,) JT" - a?U,' + (a, - 2a) U" =
- b' (a + a,) + c' (2a2 - 4aaj + 2ai2) + d' (a» + a,') + e' (a* - 2a'a, + Ba^'a,'^ - 2aa,' + a,*)
+ f' (a» - 2a*a, + 4aV + 4aV - 2aa,< + a,') + (a« - 2a'ai + 5a*a,= + oa'a,* - 2aa,» + a,'),
whence the sum of these two expressions is
= b' (a + a,) + c' (2a» - 2aa, + 2a,'') + D (a' + a,») + e' (a* - 2a% + 4a2a,- - 2aai' + a,'')
+ f' (a» - 2a*a, + 2a»a,'' + 2a=a,'' - 2aa,« + a,») + a« - 2a»a, + Sa^a," - 2a'a,» + 3a»a,^ - 2aa,» + a,«.
We must to this add the term — (abcd.eif, + a,b,c,def), that is,
— a (6' + a . c' + a . d' + a, . e' + ai ./' + a,) — a, (6' + a, . c' + a, . d' + a . e' + a ./' + a).
Putting for the moment
d' + a . e' + a ./' + a = t' + o-'a + p'a- + a",
that is,
T' = d'e'f', <r'^d'e' + d'/' + e'f', p' = d' + e'+f',
the term is
- b'cV (a + a,) - (b' + c) t' (a' + a,») - t' (a' + a,') - (6'c' + a) (a»a, + aa,»)
- (h' + c' + p) (&W + a'a,') - 2a''a,'
- 26'c'<r'aa, - [(6' + c') a + h'c'p'] (a'a, + aa,=) - 2 (6' + c') /a'a»a,».
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 211
Adding it to the preceding expression, the sum is
= (b' - 6'c't') (a + ai) + {2c' - (¥ + c') t'} (a' + a^^) + (d' - t') (a^ + a^') + e' (a^ + a,*)
+ {- 2c' - 2b'ca'} aai - {(b' + c') a + b'c'p] (a^a^ + aa.^) - (2e' + b'c' + a') (a'a, + aa;')
+ (4E' - 2 (b' + c') p'} a=a,'
+ F'(a» + ai») 4- a« + ai«
- 2F'(a% + aa,*) - 2 (a»ai + aai"*)
+ (2f' -b' -c'- p) (a'ai= + a=a,') + 3 (aV + aV) - 4a»ai».
This is, in fact, divisible by (a, — a)^, that is, by 6^: for we have between the symbols the
relations
V = b' +c' + p,
E' = 6'c' + (6' + c')p' + o-',
d' = b'c'p + {b' + c') 0-' + T,
C' = b'c' a' + {b' + c') t',
B' = b'c'r',
and we thus reduce the expression to
{2c' - {V + c') t'} (a' - 2aa, + a,") + (d' - t') (a» - a'a, - aa,^ + a/)
+ e' (a* - 3a»a, + 4aV - 3aa,» + a,*) + {b' + c) p' (a'a, - 2a=a,= + aa,')
+ F* (a" - 2a*a, + aX" + &W - 2aa,* + a,'')
+ (a« - 2a''ai + 3a%= - 4aV + 3aV - 2aai' + aj"),
viz. eflfecting the division, the quotient is
= 2c' - (6' + c') t' + (d' - t') (a + ai) + e' (a^ + ar) + f' (a' + a,=) + a-" + 2a=a,= + aj* - {b'c' + a') aa,.
To this must be added
-2c' -D'(a + a,) -E'(a» + aj'')-F'(a=' + ai»)-(a* + a/);
and we thus obtain the coefficient of (9m)' in the form
%; - {V + c') t' - t' (a + a,) - {b'c' + o-') aai + 2b,W,
viz. this is
= 2lo' + (6 + c - 2a) (a -d){a- e) {a -/) + {a-d){a- e) {a -f) (a + a,)
+ {- (a - b) (a - c) - (a - d) {a-e)-{a- d) {a -/) -{a-e) {a -/)] aa, + 2a=a,»,
27—2
212 A MEMOIR ON THE DOUBLE ^-FUNCTIONS. [665
or finally it is
= 51,' - 2a* + a^b + c + 2d+ 2e + 2/) + a' {- (6 + c)(d + e +f)-2(de + df+ef)]
+ a[(b + c) (de + d/+ ef) + 2def\ - (6 + c) def
+ \af-a? {d + e+f) + a{de + d/+ e/)-def} (a + a,)
+ {-4a* + a(b + c + 2d+2e + 2f)-bc-de- df- ef\ aa,
+ 2a''a,».
It is to be observed that the investigation thus far has been entirely independent of
the values of 2lo', S3', S': these values are, in fact, such as to make the coefficients
of (3w)^ disr du, (3m)" each equal to a constant, and it was really by such a condition
that the value of (5 (= S') was determined ; but if we had thus also determined the
values of %,' and 33', it would not have been apparent that the values of 2Io', 33'
and S' thus determined would be consistent with each other: the foregoing investi-
gation of these values was therefore prefixed.
Completion of the redv^tion and final expression for A-4.
But now substituting the values of SI,', 33', S', we find
coeff. of (3or)2 = ab + ac + hc + de+df+ef
„ „ 2idvidu = — a^{a—b — c — d — e—f),
„ „ (Buy = -2a* + 2a=(b + c + d + e+f)
- a^bc + bd + be + bf+ cd + ce + cf+ de + df+ ef)
— (bade + bcdf + beef + bdef + cdef),
viz. these coefficients belong to the portion which contains the factor aa, of the
4
expression for t^ ^A : the other portion was
(6' + c' + a + a,) d'ef (Sw)' - ( Vrf^)» (diiry - 2b' du 3tir - (a + a,) b' (duy,
where
b' = b'c'd'ef, V = b — a, etc.
We have thus the complete result, viz. this is .
4
jt; A J: = aai {{ab + ac+bc + de + df+ ef) (^mf
— a^ {a — b — c — d — e —f) 2dis du
(-2a' + 2a'{b + c + d + e+f) "j
+ <- a;^ {bc+bd + be + bf+ cd + ce + cf+ de + df+ ef) i {duy]
[-(bcde + bcdf+ bcef+ bdef + cdef) J
- (- 2a + b + c + a + &,)(a - d) (a - e) (a -f) (dvry - (Vde)» {dvy
+ (a — b){a — c) (a — d){a — e) (a —f) 2dudiir
+ (a + a,) (a - b) (a -c)(a- d) (o - e) (a -/) (du)\
which is obviously a sum of squares.
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 213
As a partial verification, I remark that AA should be symmetrical in regard to
the constants b, c, d, e, f; this is obviously the case as regards the terms in du d-ar
and (du)-, and it must also be so in regard to the term in (d^y. The whole
coefficient of {dvry is
= aa, (a6 + ac + 6c + de + <(/■+ e/)
- (- 2a + 6 + c + a + a,) (a - rf) (a - e) (a -/) - (Vde)^
and if we interchange for instance b and d, this coefficient becomes
= aai (ad + ac + cd + be + bf+ ef)
-(-2a + d + c + a, + At)(a-b)(a- e) (a -f) - ('^bef.
These two expressions must be equal ; viz. we must have
(^/bey - (^dey = -aa,(b-d)(a + c- e-f) + (a-e)(a -/) (6 - d) (- a + c -t- a + a,) :
the left-hand side is
= ^(^1 - ^i^^ (efa,c, - Cifiac),
and we have
bd,-b,d = (6-d)^;
rt (efajC, — e,f,ac) = — aa, (a + c — e — /) + (a — e) (a —f) (— a + c + a + a,).
hence, throwng out the factor b — d, the equation to be verified becomes
Writing
e = e' + a, etc., ^ = aj — a, ,
the left-hand side is
(a + a,) e'f + aa, (e' +/') + c'e'f - c'aa, ,
and the right-hand side is
- aa, (c' - e' -/') + e/ (c' + a + a,),
and these are equal.
There are of course, in all, six expressions such as AA, each of them being by
what precedes a sum of squares. And there are besides ten expressions such as
AAB, =:ABd''AB-(dABy,
each of which should be a sum of squares : but I have not as yet effected the
calculation of this expression AAB.
Cambridge, 1th December, 1877.
214 [666
666.
SUR UN EXEMPLE DE REDUCTION D'INTEGRALES ABELIENNES
AUX FONCTIONS ELLIPTIQUES.
[From the Comptes Rendus de l'Acad4mie des Sciences de Paris, t. Lxxxv. (Juillet —
D^cembre, 1877), pp. 265—268 ; 373, 374 ; 426—429 ; 472—475.]
Je reprends I'investigation de M. Hermite par rapport aux integrales rdductibles
f Jl,x)dw
J ^/x.l — x.l + cuc.l+bx.l — abx '
publide sous ce meme titre : " Sur un exemple, etc.", (Annates de la Society scientifque
de Bricxelles, 1876). »
Nous avons les constantes a, b et les variables x, y, u, v; et en posant
X = X .1 — X .1 + ax .\ -^bx .1 — abx,
Y = y.l—y.l + ay.l + by.l— aby
(et c = Vl + tt.l+6), M. Hermite a efifectu^ rintdgration, par fonctions elliptiques, des
Equations diffi6rentielles
dx dy 2 , , J .
xdx ydy 2,, ,.
-== + ^7=^ = - —j=. (du - dv) ;
11 a en effet trouv^ les expressions, au raoyen des fonctions elliptiques de u, v, des
fonctions sym^triques x + y, xy, et, de 1^, des cinq fonctions a, b, c, d, e dont je vais
parler.
Au cas d'une fonction X du sixifeme ordre, on a dans la theorie seize fonctions,
savoir six fonctions a, b, c, d, e, f, et dix fonctions abf.cde, ..., ou (avec une noUtion
666] SUR UN EXEMPLE DE REDUCTION d'iNTEQRALES AB^LIENNES. 215
plus simple) ab, ac, ad, ae, be, bd, be, cd, ce, de: dans le eas d'une fonction du
cinquifeme ordre, et ainsi dans le cas actuel, I'une des six fonctions, disons f, se reduit
a I'unit^, et Ton a les cinq fonctions a, b, c, d, e, et les dix fonctions ab, ...,de.
Presentement, ces fonctions sont
a. = xy,
h = \ — x.\ —y,
c = 1 + ow; . 1 + ay,
d = 1 + 6a; . 1 + 6y,
e = 1 — abx . 1 — ahy.
ab = ( Va; .\—x.\-\-ay.\-\-by.\— aby —*Jy.\—y.l+ax.\ + hx.l— abxf -i-{x — y^,
ac = (Va;. l-^ax.\—y.l + hy A —ahy— 'Jy.l+ay.l—x.l+bx.l — abx)- -i- (x — yf,
ad = (Va; .1 +bx.l—y .1 +ay.l— aby — '/y.l+by.l—x.l+ax.l— abxf -^(x — yy,
ae = (va; . 1 — abx .1 —y .1 +ay .1 + by — yy . 1 — aby .l—x.l+ax.l+ b.vf -^{x — yY,
be = ( Vl —x.l+ajn.y.l + by.l — aby — '^l—y.l+ay.x.l-*-bx.l- abxf -i-(x — yf,
bd = (Vl —x.\+bx.y.l+ay.\-^ aby — '^\—y.l+by.x.\-\-ax.\— abxY -=- (a; — yf,
be = (v 1 —x.l — abx .y.l+ay.l + by-^/l—y.l- aby . a; . 1 + aa; . 1 — bxy -f- (« — yY,
cd =« (Vl + ax.l + bx.y.l — y.l -aby - ^1 + ay.l + by .x.l — x .1 — abxf -^{x- yY,
ce = (Vl + oa; . 1 — abx .y .1 -y .1 +by — -Jl +ay.l— aby .x.l—x.l+ bxf -^{x- y)"-,
de = (Vl +6a:.l —abx.y.l—y.l +ay-'Jl + by.l —aby .x.l - x .1 + axf -^ {x - yf,
et je remarque que la difiP^rence de deux quelconques des fonctions ab, ac, ... est
uae fonction rationnelle et entiere de x, y. On a, par exemple :
ac — ad = a — 6. 1 —ab xy,
be — bd = a — J . — 1 + ab {x + y) — ab xy,
be — cd = l+a. 1+6 — l + a6 xy,
ce — de = a — 6 . — 1 + (a; + 2/) — a6 ary.
En faisant, comme auparavant, c = Vl + a . 1 4- 6, et puis
ck ='/a + '/b, cl = */a — ^,
chf =1- 'Jab , cl' = 1 + "Jab ;
o- = sn (m, k) , 0-1 = sn (v, I),
7 = en (m, k), 7, = en {v, I),
S = dn (m, k) , hi = dn {v, I),
216 SIJB UN EXEMPLE DE REDUCTION D'iNTlfojRALES AB]£lIENNES [666
(oil j'6cris sn, en, dn pour sin am, cos am, A am), et, pour un moment,
f = "Jab (7<»-,8, + 7iffS), 17 = c (— A'ff7i8, + VaiyS), f = 70-181 — 7iO-S *,
X, y sont donnas au moyen des fonctions elliptiques o-, 7, S, o-,, 7,, 81 de ^^ ?; par
les Equations
^^ + 2/ = ^ f ' ^ = |i'
ou, ce qui est la meme chose, on a identiquement
de mani^re que x, y sont les racines de I'^quation quadrique
?^' - (f + r - 1?') ^ + ?= = 0.
On a I'identit^ (due k M. Hermite)
(Pi' + Qz^ + Rz + Sf- &>&%- (o-» - 0-,=)^ Z
= [jr" (1 + az) (1 + bz) - e'z] [<r,» (1 + az) (1 + bz) - cFz] x [f 5^ - {^' + f ■= -v°-)z + ^l
ou
^=2.1— i.l + a^^.l + 6^.1 — a6^ ;
et alors les valeurs de P, Q, It, 8 sont
P = — ab ^abffai (7o-,8i + 7i<7S),
Q = 'Jab cro-i [ - (a + 6 - Va6 ) 70-181 - (a + 6 + Voft ) 7,0-8] + c* Vat (8o-,7, + 8,0-7),
R = atTi [(a+b — "Jab ) 70-181 —{a + b-\- 'Jab ) 71O-8] + c- (80-171 — 8,07),
)S = aa^ (70-181 - 71O-8),
lesquelles peuvent aussi s'dcrire comme il suit : '
P = — abaai^,
Q = — abaa-^^— d' 'Jabtxffi (l-ya^Bi + k^iaS) + c^ Va6 (80-17, + ^i*'^).
R = a-ai^ + c-o-Oi (/-70-181 — ^108) + c'^ (8o-i7i — 81^7),
/S = o-o-if,
et je remarque I'^quation
P + Q + iJ + /Sf = 0*771 (- A;'o-7i8i + ro-i78)
= cVyii?.
En dcrivant successivement z = x, z = y, et en choisissant convenablement les signes
des radicaux, on obtient
Pa?+Qa? + Rx + S = ch\{a^-a,^)'JX,
Py' + Qy' + Ry + S = c88i (a' - a,') ^Y;
on con9oit sans peine que c'est a cause de ces expressions rationnelles des radicaux
que I'int^gration des Equations differentielles rdussit.
* En (icrivant
on a _
i=-Jab^, r, = cr,', f=r;
je me sera, dans la suite, de ce symbole
{i'=7(riJi + 7i(r«.
ou
et puis
666] AUX FONCTIONS ELLIPTIQUES. 217
[Pp. 373 — 376* ; 426 — 429.] Les valeurs de x-^y, xy donnent sans beaucoup de
peine celles de a, b, c, d, e; mais les reductions pour obtenir les valeurs des dix
fonctions ab, ..., de sent trfes p^nibles; je donne seulement les r&ultats. Ces valeurs sont
Vb = -7^— r, ■ — k'ayiSi + Va^yh,
\^c = -j^ . IBa-iyi - kBiay,
Vd = -p= . iSo-,7, + k'B,ay,
Ve = p . k'cryiBj + t'o-^yB,
^ = ya-A + 7i<^S ;
V ab = p> .yy,BB, — k'l'acr,,
^ = n^.^^™^) • * <'" + ^"^'^^ "-"^ + ' ^^" + ^^ '^''^'^■'
Vb^ = ^l" . ^•'S,» + i:S'-kl {k'a^,' + To-iY),
. k'Bf 4 rs^ + ^-^ (k'a'yi- + ra-.y-),
. -CTffjgg, +7^1.
1 1 + * /, •. 1 + « 7 - , 7 7 ■■ "
. 1 J- lea- j^ io"i' + kl(j-(Ti,
c\a c\a
. 1 - ^^ A;cr» + ^~ la,^ - klaW-
cJb cJb
' Voir la note, p. 426 du volume. — Dans la seconde Communication (p. 37.S), une erreur de composition
a fait placer, k la saite de la treizieme ligne de la page 874, deux pages et demie de texte qui ne devaient
trouver place que dans la Communication suivante. Nous rfitablissons integralement oette seconde Communi-
cation : la troisi^me sera ins^r6e dans le prochain num^ro.
C. X. , 28
Vbd =
r
Vbi =
c
r
V8d =
c
f
Vce =
c
^56 =
c
218 sua UN EXEMPLE DE REDUCTION d'iNT^RALES AB^LIENNES [66G
Les valeure de a, b, c, d, e donnent
V3'VF=VaVbVcVdV^
X (iSo-,7, — kS,<Ty) (i8ffi7i + kS^ay) {k'a-y,S, + I'a^yB),
= 7^^,. {r<^,%' - 7. V^O (- k'^a^yrS,^ + I'^a.yS') {P^afy;' - L^S,^a^) ;
j'ai v^rifi^ que le signe s'accoi-de avec celui de la valeur obteniie au moyen des
expressions mtionnelles de VX, VF.
On v^rifie en partie les valeurs des fonctions '/ah. Vac, .... en cousiddrant les
differences des carr^s de ces fonctions; mais ce calcul n'est pas toujours facile. Par
exemple, nous avons
ac — ad = (a — 6) (1 — «A xy)
et cette valeur doit ainsi ^tre egale k
- ai^hA^yf^ t* (^' + ''^''^ '^'^^ + ^ (^" + ^> '^•'^■^ ■
Pour voir cela, j'ecris pour le moment
A=kil''+ i V) "y^' B = l (A'» + ky) «7,7,8, ,
a = iSo-,7, , /3 = A:S, 0-7 ;
I'dquation devient ainsi
Ulaa,yy, SB, (a^ - ^f = (a + /9)= {A - Bf - (a - /9)' (.1 + 5?,
= 4 [o^ (^^ + 5^) - ^fi (tf + /S»)] ;
or, en remarquant que AB et a^S contiennent chacun le facteur ^•/ o-cr, 77, S81, cette
equation devient
(a' - y3--=)2 = ^2 (r + iv)' <^vs- + 1' (k'" + kyy o-i^,='S,=
c'est-^-dire
666] AUX FONCTIONS ELLIPTIQUES. 219
or les deux facteurs a droite se reduisant I'un et I'autre a
c'est-a-dire a (a- — ;8-), la verification est ainsi compldt^e.
La difference be — cd donne un exemple beaucoup plus simple ; on a
be — cd = 1 + o . ] + 6 (— 1 + a6 xy)
= F2(~*'^'^iTyiS8i);
r^quation k verifier est ainsi
- 4a-o-i77,8Si = (- cro-,S8i - 771)° - (- o-c7,SS, + 771)",
ce qui est juste.
[Pp. 472 — 475.] Je donne quelques autres formules dont je me suis servi dans le
cours de cette recherche. Partant des expressions de ^, rj, ^, on a
d^ = Xdw + Xidv = Va6 { ' [- aSa.i, + 77, (1 - 2i-V=)] du
+ [77, (1-2ZW)- 0-^,88, ]dv},
drj — fidu + fjiidv — c{ [ — ^-'787,8, + I'acri ( — 1 —k--\- 2A;V^)] du
+ [k'<T<T, (1 + P - 2lW) + I'yhA ] dv},
df = fdu + Vidv = { [— a-Ba-iB, — 771 (1 — 2AV^)] du
+ [77,(l-2^-0 + <^8<r,S, ]dv};
en prenant pour A, B, G des fonctions telles que
Ad^+Bd7] + Cd^=du + dv,
on a
A\ +Bfi +Cv =1,
A\,+B/i, + Gv, = 1.
Je pose aussi
A^ + Br,+ C^ = 0,
et au moyen de ces Equations, j'obtiens pour A, B, G les valeurs
GV^^{-U-W),
28—2
220 8UR UN EXEMPLE DE REDUCTION d'tNT^RALES AB^LIENNES [666
oil
U = W (8ff,7, + 8,<T7) + ^•''^'V.' (l'S<^>y^ + k%<Ty),
W = fc'S,= (So-,7, + 8,0-7) + ^'o-.V (''^o-i7i + ^"S,o-7).
F = 2 [(r^ + z-^7,0 <77S + (^-'^ + ky) <7-,7,8,],
V = (A;'o-7,S, + I'aifB) (i8o-,7, — ^8,0-7) (/8<7i7i + ^•8,(77) ;
«t de \k aussi
_ £/■ + TT = (88,77, - k'l'aa^) (7<r,8, + 7,0-8),
c
i;- + Tf = ? ( jl + fVo-.-^ - V^ [(1 + k'l') <7^ - iV.=]} 8ff,7.
C
+ {1 - /fc»o-V,= + V^ [(1 + Z;7') o-,^ - A^ff']) Sjo-y).
En admettant I'equation
^fX ^IY C^ "
on obtient sans peine les relations
7; a;-2/VVZ Vy/'
,^ _ c f-x+1 - y + 1
et, en multipliant par
et dans les seconds niembres, au lieu de
d'SS, (ct2 - a,-) V Z, c=88, ( 0-- - 0-,=) V F,
substituant les valeurs
Pa-' + Qaf + Rv + S, Pf+Qy' + Ry + S,
on obtient, apres quelques reductions simples, les equations
C*a688, (o-- - o-,-) V^ = aba-ari^T)^— crai^-T] + c^yi^%
VB = a6c7<7, f ( f^ + r^' - r;:) + a<r, f» - Qf ?,
V C = aiaa, 7; (- 2r^ - f"' + »?=) + Qf^ - c»77. f.
lesquelles satisfont, comme cela doit etre, a la condition
Reciproquement, en verifiant ces identites, ce qui est assez pdnible, on obtient une
d^monsti-ation de I'equation difiP^rentielle
dx dy 2 ,, , , ,
7= + -A = — -(du + dv).
666] AUX FONCTIONS ELLIPTIQUES. 221
En ^rivant, pour plus de simplicity,
c c c
les valeui-s de 21', S3, (i sont
21' = 77i88i — k'l'aai ,
33 = (l'-^ 4- ^/) 0-78 + {k'-' + ^•Y) o-i7iSi .
6 = [l - /Vo-r - \/a6 [(1 + i'Z') G- - f-o-,^]) So-,7,
+ [1 - LV-a;' + Va6 [(1 + k'l') o-,- - i^or^]} 8i<r7 ;
et des trois equations pour A^, B~, Cy, on deduit
oil
et c'est au moyen de ces Equations que j'ai trouve k'S valeurs ci-dessus donndes pour
v^ab, Vac, ... ; on a, par exemple,
* ~ViVb(a;-2/)' VX Vf'/ \^Vb(a.--^/)n V^'
ce qui se leduit sans peine k Vab = p 2i'. Les dix fonctions contiennent de cette
maui^re les facteurs suivants: '
V^, {l + ay^-^Wv,
Vad, (l + 6)«-^~2r7,,
Vae, (l-«6)'iB + V(^2l'i7,
Vbc. -(s + y/|2('?,
222 SUR UN EXEMPLE DE REDUCTION D'iNT^GRALES AB^LIENNES. [666
Vbd, -g + A/^r?'.
V ct
c
Vce, - [o Va6 17 + r ? (1 + a) (1 - ab) 33?- 617],
c
Vd^, - [b '^abv^' + {l + 6) (1 - at) 35?- 617]:
c
mais il y a des d^nominateurs variables qui contiennent des facteurs dont quelques-
uns divisent les nnmdrateurs, et la reduction aux formes ci-dessus donn^es m'a
coAt^ assez de peine.
667] 223
667.
ON THE BICIRCULAR QUARTIC : ADDITION TO PROFESSOR
CASEY'S MEMOIR "ON A NEW FORM OF TANGENTIAL
EQUATION."
[From the Philosophical Transactions of the Royal Societi/ of Lotidon, vol. CLXVii.
Part II. (1877), pp. 441—460. Received January 24 —Read February 22, 1877.]
K
Profe&sor Casey communicated to me the MS. of the Memoir referred to, and he
has permitted me to make to it the present Addition, containing further developments
on the theory of the bicLrcular quartic.
Starting from his theory of the fourfold generation of the curve. Prof Casey
shows that there exist series of inscribed quadrilaterals ABCD whereof the sides AB,
EG, CD, DA pass through the centres of the four circles of inversion respectively ;
or (as it is convenient to express it) the pairs of points (^A, B), (B, C), (C, D), (D, A)
belong to the four modes of generation respectively, and may be regarded as depending
upon certain parametei-s (his 6, 0', &' , &", or say) w,, Wo, 0)3, «<j respectively, any
three of the.se being in fact functions of the fourth. Considering a given quadrilateral
ABCD, and giving to it an infinitesimal variation, we have four infinitesimal arcs
AA', BE, CC, DD' ; these are differential expressions, A A' and BB' being of the form
M,d<o„ BB' and CC of the form M,d(o,, CC and DD' of the form M,dco„ DD' and
A A' of the form Mdto; or, what is the same thing, A A' is expressible in the two
forms Mdm and M,d<i),, BB' in the two forms i¥,f?a), and M.,d(o„, &c., the identity of
the two expressions for the same arc of course depending on the relation between
the two parameters. But any such monomial expression Mdeo of an arc AA' would
be of a complicated form, not obviously reducible to elliptic functions; Casey does
not obtain these monomial expressions at all, but he finds geometrically monomial
expressions for the differences and .sum BH - AA', CC-BF, DD' + CC, DD' - AA'
(they cannot be all of them differences), and thence a quadrinomial expression
AA' = N,dm, + N,da>, + N,d(o, + Nd(o (his ds' ^ p dO + p' dd' + p" dO" + p" d6"') ; and that
without any explicit consideration of the relations which connect the parameters.
224 ON THE BICIRCULAR QUARTIC. [667
I propose to complete the analytical theory by establishing the monomial equations
AA' = Mdo) = Midtoi, &c., and the relations between the parameters a>, w,, 10.2, lo^ which
belong to an inscribed quadrilateral A BCD, so as to show what the process really is
by which we pass from the monomial form to a quadrinomial form
A A' (or dS) = Ndta + N^dw^ + N^dw^ + Nsdwt,
wherein each term is separately expressible as the differential of an elliptic integral ;
and further to develop the theory of the transformation to elliptic integrals. We
require to establish for these purposes the fundamental formulae in the theory of the
bicircular quartic.
I remark that in the various formulae /, g, 0, 0^, 0.,, 63 are constants which enter
only in the combinations /+ 6, f— g, 0, — 6, 0., — 0, 0:,— 0: that X, Y are taken as
current coordinates, and these letters, or the same letters with suffixes, are taken as
coordinates of a point or points on the bicircular quartic: and that the letters (x, y),
(^1, yi), {*2i yj). {^»> y>) *re used throughout as variable parameters, viz. we have
{f+0)x- +{g + 0)y^ =1,
{/+0^}w^' + (g + 0,)y,'=l,
{f+0,)x.? + {g+0,)y,^ = l,
(f+0.)x:r + (g + 0.)yj' = l;
so that X, y= -7— , . , are functions of a single parameter w, and similarly
•J '^ if
(^1. yO' ("'i' yd> ("^S' ys) ^^^ functions of the parameters coi, co.,, m^ respectively. We
sometimes use these or similar expressions of {x, y), &c., as trigonometrical functions
of a single parameter; but we more frequently retain the pair of quantities, considered
as connected by an equation as above and so as equivalent to a single vaiiable
parameter.
Formulce for the fourfold generation of the Bicircular Quartic. Art. Nos. 1 to .5.
1. We have four systems of a dirigent conic and circle of inveraion, each giving
rise to the same bicircular quartic : viz. the bicircular quartic is the envelope of a
generating circle, having its centre on a dirigent conic, and cutting at right angles
the corresponding circle of inversion ; or, what is the same thing, it is the locus of
the extremities of a chord of the generating circle, which chord passes through the
centre of the circle of inversion, and cuts at right angles the tangent (at the centre
of the generating circle) to the dirigent conic; the two extremities of the choi-d are
thus inverse points in regard to the circle of inversion. The four systems ai"e
represented by letters without suffixes, or with the suffixes 1, 2, 3 respectively; and
we say that the system, or mode of generation, is 0, 1, 2, or 3 accordingly.
2. The dirigent conies are confocal, and their squared semiaxes may therefore be
represented by /+^, g + 0: f+0u g + 0i: f+03, g + 02- f+S>, g + 03, (which are, iu
667] ON THE BICIRCULAR QUARTIC. 225
fact, ftmctions of the five quantities f+6, f—g, 0^ — 6, 0«—6, 6^—6); and we can
in terms of these data express the equations as well of the dirigent conies as of
the circles of inversion ; viz. taking X, Y as current coordinates, the equations are
+ -T--/J =1' {X-ay + {Y-^f-r =0, or X"-+ F=-2aZ-2/S Y + k =0,
^' + i'/j = 1' (-^ - "i)" + (^ - ^^f - y' = 0, or X^ + Y^- - 2a,Z - 2/9, Y+k, = 0,
7?/. +^^ = 1- (^-«:)= + (F-/3,)=- 7,^ = 0, or X'+ Y' -2a„_X - 20,Y+h = O,
j + tfj g + tfn
-^a +:rTV = ^' (^-a.)= + (l'-y33)=-7:r = 0, or X"- -^Y- -2a,X -2^,Y -\-k, = 0,
/ + t's g + Oi
where
^/■
f±lJ±^^±lld^A^ = (/+ 6) a = (/+ e.) a, = (/+ 0,) «, = (/+ ^3) a.
g J u,
f+e .g + e .y" =e -e,.e -e,.e -e„
/+e,.g + ^. .7.-" = ^. -d .e,-d,.e,- e,,
/+ 0,.g+0,.y,'^0,-d .0, -0,.0,- 0,,
/+0,.g + 0,.yi' = 0,-0 .0,-0,. 0,-0,,
f+g + 0 + 0, + 0., + 0, = k-\-20 = k, + 20, = k, + 20, = k, + 2d,.
3. The geometrical relations between the dirigent conies and circles of inversion
are all deduciblc from the foregoing formulae ; in particular, the conies are confocal,
and as such intersect each two of them at right angles ; the circles intersect each
two of them at right angles. Considering a dirigent conic and the corresponding
circle of inversion, the centres of the remaining three circles are conjugate points in
regard as well to the first-mentioned conic, as to the first-mentioned circle; or,
what is the same thing, they are the centres of the quadrangle formed by the
intersections of the conic and circle.
4. The centre of the conies and the centres of the four circles lie on a
rectangular hyperbola, having its asymptotes parallel to the axes of the conies. Given
the centres of three of the circles (this determines the centre of the fourth circle)
and also the centre of the conic, these four points determine a rectangular hyperbola
(which passes also through the centre of the fourth circle); and the axes of the
conies are then the lines through the centre, parallel to the asymptotes of the
hyperbola^
C. X. 29
226 ON THE BICIRCULAR QUARTIC. [667
5. The equation of the bicircular quartic may be expressed in the four forms
(X' + F» - hy - 4 [(/+ e,) (x-a,y + (g + e,)(Y- ^,y] = o,
(Z^ + F« - k,y - 4 [(/ + 0,) (X - a,y + (g + e,)(Y- ^,y] = o,
the equivalence of which is easily verified by means of the foregoing relations.
Determination iis to Reality. Art. Nos. 6 and 7.
6. To fix the ideas, suppose that f—g is positive; then in order that the centres
of the four circles of inversion may be real, we must have /+ 6 ./+ 0, ./+ 0« ./+ 0.,
positive, but g + 0 .g+0^. g + 0s.g+ 0j negative; and this will be the case if f+0,
/+01, f+0-2, f+ 03 are all positive, but g + 6, g + 0i, g + 0i, g + 03 one of them
negative, and the other three positive. In reference to a figure which I constructed,
I found it convenient to take ^3, ^,, ^„, ^„ to be in order of increasing magnitude:
this being so, we have /+ 03 positive, g + 03 negative ; and the other like quantities
/+ 0i, f+00, f+02, g+01, g+00, g+02 all positive: we then have 7,- and 7," each
positive, 7o^ negative, 7,^ positive : viz. the conies and circles ai-e
Hyperbola H3, corresponding to real circle 0,,
Ellipse El, „ real circle Cj,
„ E^, „ imaginary circle C„,
(viz. the radius is a pure imaginary),
„ E-2, „ real circle C^,
and the confocal ellipses E^, E„, E„ are in order of increasing magnitude. The
centre Co is here a point within the triangle formed by the remaining three centres
Ci, Ci, Cj. It will be convenient to adopt throughout the foregoing determination
as to reality.
7. It may be remarked that a circle of a pure imaginary radius 7, —iX, where
\ is real, may be indicated by means of the concentric circle radius X, which is the
concentric orthotomic circle ; and that a circle which cuts at right angles the oi'iginal
circle cuts diametrally (that is, at the extremities of a diameter) the substituted
circle radius \; we have thus a real constiiiction in relation to a circle of inversion
of pure imaginary radius.
Investigation of dS. Art. Nos. 8 to 17.
X^ Y-
8. The coordinates of a point on the dirigent conic v^^H a~^ '"*y ^
taken to be {f+0)x {g + 0)y: and we hence prove as follows the fundamental
667] ON THE BICIRCULAR QUARTIC. 227
theorem for the generation of the bicircular quartie. Consider the generating circle,
centre (f+0)x, {g-\-0)y, which cuts at right angles the circle of inversion
If for a moment the radius is called Z, then the equation of the generating circle is
the condition for the intersection at right angles is
(a -/+ 0xy + (^-g + Oyf = 7= + 8^
and hence eliminating 8°, the equation of the generating circle is
r-+Y"--k-2{X-a){f+e)x-2{Y-^){g+e)y = 0;
and considering herein x, y a& variable parameters connected by the foregoing equation
{f-\-6)a?+(^ + 6)y- = \, we have as the envelope of this circle the required bicircular
quartie.
9. It is convenient to write i2 = ^ (X' +Y- — k). The equation then is
R-{X-a){f+d)x-{Y-^){g + e)y^O-
the derived equation is ^
{X-<i)(f+e)dx + {Y-^){g + e)dy=0-
and from these two equations, together with the equation in {x, y) and its deriva-
tive, we find X — a = Rx, Y— /8 = Ry ; from these last equations, and the equations
R = ^{X-+ F= - k), (f+0)se' + (g + d)y^=l, eliminating x, y, R, we have
(f+d)(X-ay + (g + e){Y-0r^R;
that is,
(X'+Y"--ky-i[<i/+d)(x-ay + (g + e){Y-0y] = o,
the required equation of the bicircular quartie.
10. We have thus X — a= Rx, Y — ^ = Ry, as the equations which serve to
determine the bicircular quartie : if from these equations, together with R = ^ (X^ + Y^ — k),
we eliminate X and Y, we have R expressed as a function oi x, y ; and thence also
X, Y expressed in terms of x, y; that is, in effect the coordinates X, F of a point
of the bicircular quartie expressed as functions of a single variable parameter. The
process gives 2R + k ={a+ Rxy + {^ ■{■ Ry)', viz. this is
or putting for shortness
this is •
R^ {x' + y')-2(l-ax- 0y) E + 7= = 0,
n = (l-aa:- 0yy -y'(x' + y').
P_l-aa;-/93/+ Vn
29—2
228 ON THE BICIRCULAR QUARTIC. [667
or say the two values are
P_l-aLC-)9y+VQ „,_l-aa;-/3y- Vn
a? + f ' ~ a?-^y^ '
to preserve the generality it is proper to consider Vxi as denoting a determinate
value (the positive or the negative one, as the case may be) of the radical.
11. Considering the root R, we have X = a-\- R'x, Y=fi + R'y; from these equa-
tions we obtain
dX = R'dx + X dR,
dY = Rdy + ydR.
But from the equation for R we have
[R (x' + f) - (1 - ax - ySy)] dR + R- {xdx + y dy) + R(adx + ^dy) = 0,
that is,
- Vn dR + R {Xdic + Ydy) = 0 ;
whence
dX = Rdx + ~1 (Xdx + Ydy),
vO
dY = Rdy + ^I {Xdx + Ydy).
12. The differentials dx, dy can be expressed in terms of a single differential dto,
viz. writing
cos ft) sin ft)
and
S = {f+d){g+d),
then we have
dx=—'' 7=- y rfft), dy =-'—;=^ x da.
It is to be observed that, when the dirigent conic is an ellipse, o) is a real
angle, and 0 is positive (whence also V® is real and positive); but when the dirigent
conic is a hyperbola, ta is imaginary, and H is negative ; we have, however, in either
case
d^^df = ^f±l'^^^^^l±?T^d<o^
and we may therefore write
dft) _ ds
where '^(/+dya^+(g + dyy^ is positive; ds is the increment of arc on the conic
(/+ 0) x--\-{g + 6) y^ = 1, this arc being measured in a determinate sense, and therefore
da being positive or negative as the case may be : -p^ has thus a real positive or
negative value, even when m is imaginary, and it is convenient to retain it in the
formulae.
667] ON THE BICIRCULAR QUARTIC. 229
13. It may further be noticed that, if v denote the inclination to the axis of x
of the tangent to the dirigent conic at the point V/" f 0 cos m, ^g + 0 sin ca, where
V is Casey's 0, then
cos V sin V , TT / ^ /i\ „ , /,. • „
«= ^, y = -y-^, where l/ = (/+ ^)cos''if + ((/ + ^)sin=u,
viz. we have
cos ca _ cos V sin a sin v
giving, as is easily verified, ^ = -v= ; we have therefore
dm dv ,
= dv,
or
dm
which is another interpretation of -r= .
14. Substituting for dx, dy their values, the formulae become
ll- dto.
dX = ^\^-{g + 0)y^^{-{g + 0)yX^{f+0)xY)
^^^^\ ^-^^ ^^ '■ ^ h ^~ ^^ + ^^ ^^ + ^-^^ ^^ "'^^l '^"'-
We have
that is,
xX -^ yY = ax -ir &y +{0^ + y^) R
= 1 - Vn,
,_\-xX-yY_
Vn '
and consequently the foregoing expressions of dX, dY become
dX = ^^[(g^e)y{xX + yY-l) + x{-{g + 0)yX-^(f+0)xY)]
R'd,
= -^^{i9^Of+f-^0ai^)Y-{g + 0)y\,
dY = ^^^[<,f+0)x{l- xX -yY) + y{-{g+ 0)yX + {f+0)xY)]
= ^^{{f+0)a:-{{f+0)a?+{g + 0)y^X],
230 ON THE BICIRCULAR QUARTIC. [667
or finally
15. We have
{R'x + o -/+ e xY + {R'y + ^-g+eyy
= Ji" {x' + f) -2R'{l-ax- 0y)
viz. this is
= {a-f+exy + i^-g+dyy-r
= 8'^ the radius of the generating circle.
Hence if dS, =^/dX'-+dV'-, be the element of arc of the bicircular quartic, this
element being taken to be positive, we have
,(, e'R'Bdo)
do = —7=: 7=r ,
VnVe
where e' denotes a determinate sign, + or — , as the case may be.
16. I stop to consider the geometrical interpretation; introducing dv, the formula
may be written
^^^€'.R'{af + y^)Bdv
Vii
and we have (aP + y-)R' = 1 —ax — ^y — Vfl, or
(x" + f-) R' ^l - axj- fiy ^
Here — . - is the perpendiculai- from the centre of the circle of inversion upon
the tangent to the dirigent conic, and is the half-chord which this perpendicular
Var' + y'
forms with the generating circle. Hence 7=- — - — 1 = (perpendicular — half-chord)
-r- half-chord, the numerator being in fact the distance of the element dS (or point
X, Y) from the centre of inversion : the formula thus is
dS=±^~-dv,
where h is the radius of the generating circle, p the distance of the element from
the centre of the circle of inversion, and c the chord which this distance forms with
667] ON THE BICIRCULAR QUARTIC. 231
the generating circle. If we consider the two points on the generating circle, and
^vrite dS' for the element at the other point, then we have
2 ^
which is Casey's formula ds' -ds= 2p d<^ (273).
17. The foregoing forms of dX, dY are those which give most directly the required
value of dS: but I had previously obtained them in a different form. Writing
^ = ^x-ay + {f-g)xy,
then
or since
{f^e)a-=l-{g+d)f-,
this is
x^ = pa?-axlJ + [\-{g + ff){a? + ^f)^\ = y{\-ouc-^^J) + (a?-\.y''){^-{g + &)y)
= (a:-' + y^){yR' + ^-(g + e)y} + y\/n,
that is, _
a;^-y'/n = (x'-\-y'){yR' +0-(g + e)y];
and similarly
-yA-x \/f! = (*•» + y') \xR' + a-(/+0} x}.
We have therefore
dX ^'^^ ^ (xA - y Vn),
dY = — ^_^= (yA + X Vn),
and thence a value of dS which, compared with the former value, gives
n + A- = iaf + rf) h\
an equation which may be verified directly.
FormtdcB for the Inscribed Qiutdrilateral. Art. Nos. 18 to 22.
18. We consider on the curve four points, A, B, G, B, forming a quadrilateral,
ABCD. The coordinates are taken to be {X, Y), (Z„ F,), (Z„ K), {X„ F,) respect-
ively. It is assumed that (.4, B), (B, G), (G, D), {D, A) belong to the generations
1, 2, 3, 0, and depend on the parameters (xi, y,), {x., y^), {xj, y-j), (x, y) respectively.
We wiite
fi = (1 - a a; - Byf-y- {ai' + f ),
fl, = (1 - a, a;, - /3,!/,)' - Ji' (*'i' + I/'),
n, = (1 - a,x, - /3,y,r - r^ (a;/ + yi),
n» = (1 - a^a;, - ^^y^f - 7/ (x^' + y/) ;
232 ON THE BICIRCULAR QUARTIC. [G67
and then, Vft denoting as above a determinate value, positive or negative as the case
may be, of the radical, and similarly Vll,, VTl^, Vfl, denoting determinate values of
these radicals respectively, each radical having its own sign at pleasure, we further
write
(a? +f)R' =l-ax -^y --Jil, («,= + y,') E, = 1 - «,«, - /8,y, + Vfi;,
(«,» + y,») iJ,' = 1 - a,ai - /S.y, - Vft, , {x* + y,») iJ, = 1 - <l,x, - 0,y, + Vft,,
(iPi' + Vi^) ii-' = 1 - 0,*" - /S^ys - "^^2, («j= + y»') fij = 1 - 9,a;3 - /3sya + Vn^,
(a^' + yj') J?»' = 1 - a^x, - 0,y, - Vn,, (x' +f)R = 1 - a rf - /3 y + Vfl ;
and this being so, we must have
X =a +R'x =a,+R,x„ Y =^+R'y =y3,+-R,y,, R' =U^' + F' -k ), iJ.=i(Z= + F' -A;,),
Z,=a, + B,'a;,=a,+ i2,a;,. Y, = ^, + R,'y,=l3,+R,y,, i?,'=i(Z,»+ F,'-fc,), ft,=i(Z,=+F,'-U
Z,=a,+iJ,'a;,=a,+i?,a;„ ¥, = &,+ R,'y,=^,+R,y„ R/^^iXJ'+Y.'-k,), R,=i(X,'+Y^-h),
X,=a,+R,%=^a +Rx, F,= ^,+R,'y,=^ +R y , i?/=i(Z,»+ F,'-*,), ft =H^»'+ F,»-A;) ;
and then from the values of X, F, R', R, we have
a — a, + JR'a; — iJia-, = 0,
{e-e,) + R' -R, =0,
givmg
and similarly
(^ -y8,)(«^ -^,)-(« -«i)(y -yi) + (^ -^,)(a;y,-a;^)=0:
O. - A) (a;, - .r„) - (a, - a,) (y, - y,) + (<?i - (?,) (a^y, - a;^,) = 0,
Os - /3s) («» - X,) - (a, - a,) (y, - y,) + (0, - 0,) (x.^, - x^y^) = 0,
(/3» - /3 ) (a^s - a; ) - (a, - a ) (y^ - y ) + (^3 - ^ )(a;3y - ajy,) = 0,
which are the relations connecting the parameters (x, y), (x,, y^, (x^, y^, (a;,, y,) of the
quadrilateral.
19. We have thus apparently four equations for the determination of four quantities,
or the number of quadrilaterals would be finite ; but if from the first and second
equations we eliminate (a;,, y^), and if from the thiid and fourth equations we eliminate
(a^.i, yO. we find in each case the same relation between (x, y), (a;,, y^), viz. this is
found to be
nn, = (l-ax,-^ y.^ (1 - a,^ - ^.^y ;
and we have thus the singly infinite series of quadrilaterals. We have, of course, between
{o'i> Vi)' {""*> y*) *^*^ ^^^^ relation,
n.n, = (1 - a,.r, - /S,y,)» (1 - <M:, - ^,y,y.
667] ON THE BICIRCULAR QUARTIC. 233
20. The relation between {x, y), {x^ , y^ may be expressed also in the two forms :
a?+y'^
l-o (a; + a;,)-/9 {y + y,) + {f+e,)xx, + {g + e,)yy,+ - — ^ (a - a^y. - ^ - Aa^i) = 0,
l-a,{x-\-x,)-^,{y-^y,) + {f+e)xx, + {g + e)yy, + f'^y£(<x,-ay -K^oo)=Q.
In fact, the first of these equations is
{l + (f+0i)a!x^ + (g+ 6^) yyi} {xy, - x^y) -{a.{x + x^) +^(y + yO} (xy, - x^y)
+ {(a - «:) yi - (/3 - ^,) X,} {a? + f) = 0,
which, by virtue of the original form of relation, is
- (l + (/ + ^i) a^afi + (£f + t'l) yyij j—^
-{a{x + x,) + ^{y + y,)]{xy,-x,y) + {{a-a,)y,-{^-^,)x,]{u? + f) = 0;
or, in the first term, writing
^-6', (7 + 0,' (9-6', /+6'/
and in the third term
this is
In this equation the coefiBcients of a and of /9 are separately = 0 : in fact, the coefficient
of )3 is
^4^ "^"^ +^! '^^ (a; - a,) + (a; - a;,) yy, - (y + y,) (^ry. - x^) + -^ a:, (?? + y'^)
= ^ {^ - (/+ ^') '"•' - (5^ + ^0 y.'} - -x^ {1 - (/+ ^) ^ - (Sr + 9) y»} = 0 ;
y T c, g + vi
and similarly the coefficient of a is = 0.
And in like manner the second equation may be verified.
21. The two equations are:
l-ax -ySy -(x' +y^)R' =axi + ^y,-(f+0,)xx,-(g + e,)yyu
1 - a,x, - Ayi - (ai' + y:») R, = a,x+l3,y-(f + d)xx,-ig + d) yy, ;
or, substituting for R' and iJ, their values, these are
Vn = oar, + /3y, - (/+ ^,) a;a:, - (^r + 6,) yy,, Vil, = - a,x - fty + (/+ 6) xx, + {g + e) yy,;
C. X. 30
234 ON THE BICIRCULAR QUARTIC. [GG7
and similarly
Vn^ = a^x:, + /8^s - (/+ Bi) a^a;, - (fir + 0^ y^^, VH; = -a^-^^i + (/+ ^,) a^a;, + (fir + tf,) yj^,
Vn, = ot^s + y9^, - (/+ 0,) x^3-(g + ^,) y^„ Vfl, = - a,ar, - ^sj/j + (/+ 0i)'>V«3 + (g + 0^) Ms.
Vfr, = o^ +$^ -(f+0)x^ -{g+0)yzy , -^^ =-ax,-0y3 + (f+03)x^ +{9 + 03)yiy-
Differentiating the equation
(j8 - /SO (a; - X,) - (a - a,) (y - y,) + (^ - 0,) (a;^, - x,y) = 0,
we have
[(/9 - ySO + (5 - ^0 y>] da; - [(a - a,) + (^ - ^0 a;,] dy
- [(/3 - A) + (^ - ^Oy ] (fe: + [(« - ai) + (0 - ^,) a? ] rfy: = 0 ;
and writing herein
V0 V0,
we find
-^^{(^ + e)(/8-y90t/ + (/+^)(«-«i)«' +(^-^0((/+^)^x + (fi' + ^)yy.)}
+ ^ {(^ + 0,) (/8 - A) 2/1 + (/+ ^i)(« - ai) *i + (^ - ^i) ((/+ ^i) <cx, + (.gr + 0,) yy,)\ = 0 ;
viz., dividing by 6-0^, this becomes
,_ da> ,— da, n ^\. j. ■ da , dco^ .
- Vn, -7 Vn -1=- = 0. that IS, ,_ ^ + ,_ '■= = 0 ;
or, completing the system, we have
da — da-i da^ — da^
VeVn VHiVfii -/©jVOj VejVfij'
which are the differential relations between the parameters a, w,, a^, a^, or {x, y),
(«i, 2/i). («2. 2/2). («3, yO-
22. From the equations X = a+ R'x, 7 = /3 + R'y, we found
the new values, X = ai + RjXi and F= j8] + i^it/,, give in like manner
667] ON THE BICIRCULAR QUARTIC. 235
in virtue of the relation just found between da) and dasi, these two sets of values will
agree together if only
R'{Y-{g + e)y]=R,{Y-{g + e,)y,},
R [X - (/+ e)x] = R,[x- (/+ e,) X,].
These are easily verified : the first is
R'Y-ig + e){Y-^) = {R-e + e,)Y-{g + 9My-^.),
viz. this is {g + 0) ^ -{g + 6^^i=(i, which is right; and similarly the second equation
gives (/+ d)a—{/+ 6^) aj = 0, which is right.
From the first values of dX, dY, we have, as above,
,„ e'R'Sdw
dS= ,— ,— ;
and the second values give in like manner
eiRiBi dwi
dS =
VfliV©! '
where e, is = + 1. It will be observed that we have in efi"ect, by means of the relation
(/3 — y3,) (x - Xi) — {a — a^(y— y^) +(6 — 0,) {xy^ — x^y) = 0, proved the identity of the two
values of dS.
Considering the quadrilateral ABGD, and giving it an infinitesimal variation, so as to
change it into A'B'G'iy, then dS is the element of arc AA'; and writing in like manner
d/Sfj, dSj, dS, for the elements of arc BB', CC, DD', we have, of course, a like pair of
values for each of the elements dS^, dS^, dS,.
Formvlce for the elements of Arc dS, dSi, dS^, dS,. Art. Nos. 23 to 27.
23. The formulae are
dS =e'RB -jJ^ =6,iJ.S,^i^_ ,
dSi = €i Roi —J— — T= = ejt^^
dSi = Cj'iJa'Ss -;= — y=- = eRS
where the e's each denote i 1. Supposing as above that y" is negative, but that
7i'. 7a'. 73' are positive ; then R', R have opposite signs : but E,', iJ, have the same sign,
30—2
236 ON THE BICIRCULAR QUARTIC. [667
as have also i2,' and R,, and i2,' and i2,. We may take B, S,, Bt, and 8, as each of them
positive: the signs of
dm doD-i da>t dto,
T^^S^ 8,re +, -, +, -, or -, +,-,+:
v'nVe' Viij^^i' Vi2,\/0,' Vna\/e,
hence to make dS, dS^, dS^, dS, all positive,
6 , Ci , 6j , €3 , ei , €2, €3 , e,
must have either the signs of
R', —R\, Ri, — Ra'y ~ Rit Ri, ~ Rii R,
or else the reverse signs: hence in either case e' = — e, ei' = ei, €3 = e^, €3 = e,; or the
equations are
d8=-eRB-J^=e,R,B,^J^,
dSi = e-iRi 03 -T^r — T=^ = €3x1303
(2&)3
.3= c3..3-3^^~-r^ -^"^ ^^.
d/S^ = esZij'oj -T= — ;-^ = € RS
VfijVe/
(2q>,
VX23V03'
(2(i>
24. But we have R' — R= , &c. ; and hence, putting for shortness
^.2 ^ yS r o
r
8j S3
TT- — -t > ■'I ) -« S > -Is I
(^ + y»)\/0' (ir,» + y,»)Ve;' (a;,^ + i/,»)V0;' (a;3' + 2/3=) V^
dS +dS, = + 2ePd(o,
dS, -dS =- 2eiP,d«„
dS, -dS, = - 2e,R,dio„
dSs-dS =-2e,P3dw3,
and consequently
dS = ePdft) + eiPjdiOi + e-iP^dco^ + esPsdooj,
dSi = ePdco — eiPida, + e^P^dco-i + e^Padcos,
dSi = ePdw — CiPidcoi — e^PidtOi + esPsdwj,
d/S', = ePdas — eiP^dcoi — eaPjdwa — esPsdwj,
which are the required formulae for the elements of arc.
25. The determination of the signs has been made by means of the particular
figure ; but it is easy to see that the pairs of terms could not for instance be
dS—dSs, dSi — dS, dS^ — dSi, dSa — dS, or any other pairs such that it would be
possible to eliminate dS, dS^, dS^, dS,, and thus obtain an equation such as
ePda> + eiP]d<»i + e^P-idco^ + eaP^dcOi = 0 ;
667] ON THE BICIRCULAR QUARTIC. 237
this would, by virtue of the relations between da), dioi, dm^, do),, become
S'/n, v^i gjVn, gaVn; _
an equation not deducible from the relations which connect w, toi, Wj, Wj, and which
therefore cannot be satisfied by the variable quadrilateral.
26. The diiferentials of the formulae are, it will be observed, of the form Pdio
Sda
where V0, = V/+ ^ . ^ + ^, is a mere constant,
_ cos m sin a>
and
S-^ = {(/+ ^) a; - aj= + {(5r + ^) 2/ - ^p- r ;
viz. the form is
v/(cos m \{/'+ e-af-V (sin w Vqr + 5 - /S)' - y" .
V0.(^" +
sin" ft)
which is, in fact, the same as Casey's form in ^, equation (300), his ^ being
= 90° - 0).
Writing as before v in place of his 6, the differential expression becomes simply
= Bdv: but S^ expressed as a function of v is an irrational function M + N "^ U,
and 8 would be the root of such a function; so that, if the form originally obtained
had been this form idv, it would have been necessary to transform it into the first-
mentioned form ^^ , in which 8 is expressed as a function of (x, y), that
(a^ + 2/»)V0 ^ ^ "^
is, of o).
27. The . system of course is
dS = eZdv + 6] hidvi + e^h„dv^ + 6383^113,
dSi = thdv — eiSidu] + e^B^dv^ + 6383^^3,
dSj = eSdu — 61 81 dvi — 62 82 ^''s + ^s 83 c^vs ,
dS, = e8di/ — eiSidvi — e282di'2 — f383dv3,
where dv = ,-^ , &c. : and this is the most convenient way of writing it.
(«» + 2/»)V0 ^
Reference to Figure. Art. No. 28.
28. I constructed a bicircular quartic consisting of an exterior and interior oval
with the following numerical data: (/+(9, = 48, f+0, = 5&, /+6'„ = 60, f+ff, = 80;
^ + 0, = - 6, ff + 01-2, g +6„=6, g+02= 26),— not very convenient ones, inasmuch as
238
ON THE BICIRCULAR QUARTIC.
[667
the exterior oval came out too large. The annexed figure shows 0, 1, 2, 3, the
centres of the circles of inversion, the interior oval, and a portion of the exterior
oval, also the origin and axes; it will be seen that the centres 0, 2 lie inside the
interior oval, the centres 1, 3 outside the exterior oval: I add further the values
V7+"(?, = 6-93, V- (7+ 6*7) = 2-45, a, = 1018, /3, = - -98,
\//T^, = 7-48, s/gVe, =1-41, a,= 8-73, /3: = 4-2-94,
V7:5r^. = 7-75, 'Jg + e, =2-45, 0,= 8-15, /So = + -98,
\7+'^, = 8-94, -/g + e, =509, a,= 6-10, ^8, = + 23.
We thus see how there exists a series of quadrilaterals ABCD, where A, B are
situate on the interior oval, C, D on the exterior oval. Considering the sides as
667] ON THE BICIRCULAR QUARTIC. 239
drawn in the senses A to B, B to C, G to D, D to A: and representing the in-
clinations, measured from the positive infinity on the axis of x in the sense a; to y,
by Wi, i»2, Vj, V respectively: then, in passing to the consecutive quadrilateral A'B'CD',
we have Vi and Vj decreasing, v, and v increasing, that is, dvi and dv^ negative, dvi
and dv positive; so that, reckoning the elements A A', BE, CC, DD', that is, dSj, dS^,
dS,, dS, as each of them positive, we have
dS„-dS, = -2S,dv„
dS, - dS, = - 2Ldv,,
dS -dS, = + 2B,dv„
dS, + dS = + 2Bdv ,
and thence
dS = Bdv — Bidvi — Sodfo + S^du^,
dSi = Bdv + S,du, + B^dvs — Sjdvj,
dS., = Bdv — Bidvi + Bndv^ — B^dv,,
dSf = Bdv — Bidui — B.,dv„ — Sjdws,
which are the correct signs in regard to the particular figure.
r ^ /7
Reduction of I ,— to Elliptic Inteqrals. Art. No. 29.
29. The expression in question is
, ^(cos o) Vf+~0 -aY + (sin w -^J+d -fif-'f
fcos»« sinvo,)^-
where V® ia a mere constant; and we may apply it to the Gaussian transformation,
_ a + a' cos r + a" sia T
. _b + h' oosT+ 6" sin T
.smo,- g + c'cos2' + c"sin2"
where the coefficients a, b, c, a, b', c', a", b", c" are such that identically
cos» CO + .sin» 0, - 1 = — -^ „?^ „ ■ ^,, Icos^ r+ siti'^ T - 1) :
(c + c cos I +c sm 2 )'
and also
(cos to >//+ 6 - af + (sin (o'Jg^ 6 - ^f- y-,
240 ON THE BICIRCULAR QUARTIC. [667
that is,
cos' a (/+ 6) + sin' a {g ■>!■ 6) - la -Jf+O cos «■> - 2/8 V^r + ^ sin w + ^,
= (c-fo-cosAc-siny)' (^' - ^' '''' ^- ^' '''' ^>-
30. It is found that G,, G^, G, are the roots of a cubic equation
{G + e - e,){Q-\- e -e,){G ■{■ 6 - d,\
which being so, we may assume 0^ = 6^ — 6, Gi = 0. — 6, G3=^d,-6; the second condition,
in fact, then is
(/+ 6) cos' 03 + (g-\-d) sin' © - 2a s/f+d cos « - 2/3 V^ + (9 sin w + A
-(c->-c'cosr\c-lI^{^'-^-^^--^>-°-'^-(^'-^>-^°'^}'
and this being so, we find without difficulty the values
a»= g + ^.-/+g../+g. _ j,^ f+0,.g + e,.g+0, ^ ^^^ f+0,.g-¥0, ^
f— g. 01 — 02.01 — 03' g —f. 01 — 02-01 — Oz' 0^ — 0^.0^-0,'
a'^ = _ 9_+AJ+A^f±A^. y.^ f+02.g + 0i-g + 0, ^,,_ f+0..g + 0,
f— g .0^- 0\.0i— 03 g — f ■ 02—01.02-03' 02 — 0i. 02— 0,'
a"« = _ g + g../+gi./+g. _ j^,„ ^ _ f+0s.g + 0,.g+02 ^ ^,„ ^ _ f+ 0,.g + 03 ^
f— g .0% — 0\ .0% — 02' g — f ■ 0%- 0\.03 — 02' 03 — 0\-0% — 0%
To make these positive, the order of ascending magnitude must, however, be not as
heretofore 0,, 0i, 02, but 0,, 02, 0i, viz. we must have f+0i, f+ 02, f+03, g+0i,
g + 02, -(g+03), 01-03, 0i-0i, 02-03 all positive.
31. The above are the values of the squares of the coefficients; we must have
definite relations between the signs of the products aa, bb', ab, &c., viz. we may have
1
02 — 03
"" ~ f-g. 02-03 y 03-01.0.-02' f-g.03-0iy 01-0.. 0^-0.'
"'' -g^f,e2-03y " ' g-f.0.-0iy
' _ /+^» / - e.eT'
"'^ ~f-g. 0i-02y 02-03.03-01'
W = g + ^' - ./
g-f. 0i-0.y
'^'^ " '07-^2 y
667] ON THE BICIRCULAR QUARTIC. 241
and further
III -^2}: J h'r' — /+ ^2 /
-1
f—g. 6^^-03. 63— 61 " ' 63
f+0. r
0,-0,.d,-e,V f-g
v - ^ + ^' 7
~ 0^-0,.d,-e,v
."a"= ^ + ^' ■ /
0,-03. 0,-0,V
and also
y-» I .v_2ff + g,+g, / g+0r.f+0..f+03 .11^,1 I 2f+0,+03 I f-^0,.g+0,.9+0,
"''+'"'- ^,-^3 V g-fA-0.A-0.' '" ^'''~ ^.-^a V f:^e^0~0'^0\:
b"c + &r" - ?£+^±f' . / g+0,./+03./T0[ „ ., _1f±03±0y ^ I f+0..g+03.g+0r
"'+''' - ^.-^. V - g-/A-0.A-0.' "^ " ^s-^, y~/-g.0^-0.A-0.'
be' +h'c -^l±Mli , / g+03.f+0,.f+0, , . _2/+g,+g, / f+03.g+0,.f^,
32. These values, in fact, satisfy the several relations which exist between the
nine coefficients ; viz. the original expressions of cos o), sin &>, in terms of cos T, sin T'
give conversely expressions of cos T, sin T in terms of cos ca, sin w, the two sets being
a + o' cos T + a" sin T _, a' cos w + 6' sin w — c'
cos <0 — ; = JT-. 7n , COS J = ; ; ,
c+c cos i+c sm jt acos <a + 6sma> — c
6 + 6' cos r^- 6" sin T . „ a" cos « + 6" sin « - c"
sin ta = ; 7i= r;—. — ^ , sm jf = —
c + d cos 2* + c" sin 2" a cos w + 6 sin w - c
and we have then the relations
cos' o) + 8in= 6) - 1 = ; ; =5 „ . „„ (cos» T + sin' T - 1),
(c + c cos T + c ' sm T)" ^ '
COS' r + sin' r - 1 = 7 J—. r, (cos' w + sin' <a - 1).
(a cos CD + 0 sm ft) — c)'
(^ +/) cos' ft> + (^ + g) sin' &> - 2a ^W+fcos a)-2^'j0 +gain(o + k
= (c.fc-cosyH-c"sinr)'K^--^>-^^--^)^»^'^-<^'-^>^^"'^i'
(^1 -0)-{0i-0) cos' T-(0,-0) sin' r
{(^ +/) cos' 0) + (^ + ^) sin' 0) - 2a 'JFVfcoa co - 2/3 V^+g' sin o) + A},
(a cos ft) + 6 sin &> — c^
C. X. 31
242 ON THE BICIRCULAR QUARTIC. [667
giving the four sets each of six equations
„!! +62 _c= =-1, a'a" + 6'6" - c'c" = 0,
a'* + 6'» - c'' = + 1, a"a + b"b - c"c = 0,
a"= + 6"a_c"»= + l, aa' +bb' -cc' =0,
-a" + a'" + a"»= + 1, - 6c + 6'c' + b"c" = 0,
- 6^ + b'' + 6"'= + 1, - ca + c'a' + c"a" = 0,
- c» + c'^ + c"= = - 1, -ab + a'V + o"6" = 0,
{e+f)a? +(5 + ^)6» -^a'JW+fcui -ifi-JF+gbc +kd' = (?,+^,
(^+/)a'» +(0+^)6'» -2a'j0Tfa'c' - 2/3 V^T^ 6'c' +A;c'' =-6,+ 9,
{0 +f) o"» + (^ + ir) 6"» - 2 a 'Jf+fa"c" - 2/8 V^T^r 6"c" + kc"" =-d,+d,
(0 +/) a'a" + (e + g) b'b" - a ^F+f{a'c" + a'V) - ,5 ^WVg {b'c" + b"c') + kc'c" = 0,
{6 +/) a"a +(d-¥g) b"b - a \feTj'(a"c + ac" ) - /3 VgTy (6"c + 6c" ) + A;c"c = 0,
{0 +f) aa' +id + g) bb' - a s/e+f^ac' +a'c)- /3 'JdTg (be' +b'c ) + kcc' = 0,
{9, -9) a'- (9, - 9) a'» - {9, - 9) a"» = 9 +f, or say {9, +/) "■' " (^^ +/) «" -(^3+/)a"»=0.
(0, _ ^) 62 _ (^^ _ 0) 6'» - (^3 - 9) b"' = 9 + g, „ {9, +g)b' - (9, + g)b'' -{9,+g )b"*=0,
(9,-9)c-- (9, -9)c''-{9,-9) c"' = k, „ 9,c- - 9,c'' - 9,c"' =k+9,
- (9, -9}bc + (9, - 9) b'c' + {9, - 9) b"c" = -^^WVg,
-{9,-9)ca+(9,-9)c'a+(9,-9)c"a"=-a\/9Tf,
-{9,-9)ah+{9,-9)a'b' + {9,-9)a"b"= 0;
all which formulae are in fact satisfied by the foregoing values of the expressions
o^ 6^ a'\ &c.
33. We then have
, dT
d(o-
c + c'cosr + c"sin2"
the radical which multiplies da> being
=^c + c'cosJ'+c"siny^^--^''^»«'^^-^'«^»'^'
the differential becomes
dT V^, - 9^ cos' T-9» sin= T
/COS^ O) sin' 0>\ , , m „ ■ m^» /7i
that is,
dT -J 9^ - 9^ cos" T-9t sin' T
1 i- ^ -^fl (« + a' cos r + a" sin T)* + ^ (6 + 6' cos T + b" sin T)'
(/ + 1/ g + 0
• Ve
The denominator could, of course, be reduced to the form (»51, cos T, sin T)^ ; but
the actual form seems preferable, inasmuch as it puts in evidence the linear factors
^ (a + a' cos 2*+ a" sin T) ± ^ (6 + b' cos 2'+ b" sin T),
and there seems to be no advantage in further reducing the integral
668] 24a
668.
ON COMPOUND COMBINATIONS.
[From the Proceedings of the Lit. Phil. Soc. Manchester, t. xvi. (1877), pp. 113, 114;
Memoirs, ib., Ser. ill., t. vi. (1879), pp. 99, 100.]
Pkof. Clifford'.s paper, "On the Types of Compound Statement involving Four
Classes," [volume of Proceedings quoted, pp. 88 — 101 ; Mathematical Papers, pp. 1 — 13],
relates mathematically to a question of compound combinations ; and it is worth while
to consider its connexion with another question of compound combinations, the application
of which is a very different one.
Starting with four symbols, A, B, G, D, we have sixteen combinations of the
five types 1, A, AB, ABC, ABCD, (1+4 + 6 + 4 + 1 = 16 as before). But in Prof.
Clifford's question 1 means A'B'CJy, A means AB'C'D', &c. ; viz. each of the symbols
means an aggregate of four assertions ; and the 16 symbols are thus all of the same
type. Considering them in this point of view, the question is as to the number of
types of the binary, ternary, &c., combinations of the sixteen combinations; for,
according as these are combined,
Wn nf tvT,P« - 1.2.3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15
^ 1, 4, 6, 19, 27, 47, 55, 78, 55, 47, 27, 19, 6, 4, 1
together.
In the first mentioned point of view the like question arises, in regard to the
sets belonging to the five different types separately or in combination with each other;
for instance, taking only the six symbols of the type AB, these may be taken 1, 2,
3, 4, or 5 together, and we have in these cases respectively
M t, 1. 2, 3, 4, 5
No.oftype8 = j-^-2-2;3.
31—2
244 ON COMPOUND COMBINATIONS. [668
as is very easily verified; but if the number of letters A, B,... be greater (say this
= 8), or, instead of letters, writing the numbers 1, 2, 3, 4, 5, 6, 7, 8, then the question
is that of the number of types of combination of the 28 duads 12, 13,..., 78, taken
1, 2, 3,..., 27 together, a question presenting itself in geometry in regard to the
bitangents of a quartic curve (see Salmon's Higher Plane Curves, Ed. 2 (1873),
pp. 222 et seq.): the numbers, so far as they have been obtained, are
T^ f, 1,2,3, 4,.... 24, 25,26,27
No. of types = i, 2, 5, 11, ..., 11, 5, 2, 1 '
It might be interesting to complete the series, and, more generally, to determine
the number of the types of combination of the ^«(« — 1) duads of n letters.
669] 245
669.
ON A PROBLEM OF ARRANGEMENTS.
[From the Proceedings of the Royal Society of Edinburgh, t. ix. (1878), pp. 338 — 342.]
It is a well-known problem to find for n letters the number of the arrangements
in which no letter occupies its original place ; and the solution of it is given by
the following general theorem: — viz., the number of the arrangements which satisfy
any r conditions is
(l-l)(l-2) (1-r),
= 1 -2 (1) + 2 (12)- + (12.. .r),
where 1 denotes the whole number of arrangements ; (1) the number of them which
fail in regard to the first condition ; (2) the number which fail in regard to the
second condition ; (12) the number which fail in regard to the first condition, and
also in regard to the second condition; and so on: 2(1) means (l)4-(2)4- ... +(/•):
2(12) means (12) + (13) + (2r) + ... + (r- 1, r); and so on, up to (12.. .r), which denotes
the number failing in regard to each of the r conditions.
Thus, in the special problem, the first condition is that the letter in the first
place shall not be a ; the second condition is that the letter in the second place
shall not be h\ and so on; taking r = n, we have the known result,
No.=n.-fn(n-i)+-^n(«-2).-H...±'4=i^\
= 1.2.3...n{l-l + ji^-^4-3 + ...±3_2:L_}.
giving for the several cases
n = 2, 3, 4, 5, 6, 7,...
No. = l, 2, 9, 44, 265, 1854,...
I proceed to consider the following problem, suggested to me by Professor Tait,
in connexion with his theory of knots : to find the number of the arrangements of
n letters abc.jk, when the letter in the first place is not a or b, the letter in
the second place not b or c, ..., the letter in the last place not Ic or a.
246
ON A PROBLEM OF ARRANGEMENTS.
[669
Numbering the conditions 1, 2, 3, ..., m, according to the places to which they relate,
a single condition is called [1] ; two conditions are called [2] or [1, 1], according as
the numbers are consecutive or non-consecutive : three conditions are called [3], [2, 1],
or [1, 1, 1], according as the numbers are all three consecutive, two consecutive and
one not consecutive, or all non-consecutive; and so on: the numbers which refer to
the conditions being always written in their natural order, and it being understood
that they follow each other cyclically, so that 1 is consecutive to n. Thus, re = 6, the
set 126 of conditions is [3], as consisting of 3 consecutive conditions; and similarly
1346 is [2, 2].
Consider a single condition [1], say this is 1 ; the arrangements which fail in
regard to this condition are those which contain in the first place a or 6; whichever
it be, the other n — 1 letters may be arranged in any form whatever ; and there are
thus 211 (n — 1) failing arrangements.
Next for two conditions; these may be [2], say the conditions are 1 and 2: or
else [1, 1], say they are 1 and 3. In the former case, the arrangements which fail
are those which contain in the first and second places ab, ac, or be : and for each of
these, the other n — 2 letters may be arranged in any order whatever ; there are thus
3n (re — 2) failing arrangements. In the latter case, the failing airangements have in
the first place a or b, and in the third place c or d, — viz. the letters in these two
places are a.c, a. d,b. c, or b.d, and in each case the other re — 2 letters may be arranged
in any order whatever : the number of failing arrangements is thus = 2 . 2 . 11 (ra — 2).
And so, in general, when the conditions are [a, ^, 7,...], the number of failing arrange-
ments is
= (a + l)(^ + l)(y + l)...Ui7i-<x-^-y...).
But for [n], that is, for the entire system of the n conditions, the number of failing
arrangements is (not as by the rule it should be =re+l, but) =2, — viz. the only
arrangements which fail in regard to each of the »i conditions are (as is at once
seen), abc.jk, and bc.jka.
Changing now the notation so that [1], [2], [1, 1], &c., shall denote the number
of the conditions [1], [2], [1, 1], &c., respectively, it is easy to see the form of the
general result. If, for greater clearness, we write n = 6, we have
1 -2(1)
No. = 720 - {([1] = 6) 2} 120 -I-
+ S(12)
([2] =6)3
+ ([1,1] = 9) 2. 2
24-
+ 2 (1234)
= 6)5
6) 4 . 2
[-|-([2,2] = 3)3.3^
+[ (W ='
- +([3,1] = <
- 2 (12345)
-{([5] = 6)6}1
-2(123)
([3] = 6) 4 ^ 6
| + ([2, 1] =12) 3.2.
[+([1,1,1]= 2)2.2.2
+ (123456)
+ {([6] = 1)2};
669] ON A PROBLEM OF ARRANGEMENTS. 247
or, reducing into numbers, this is
No. = 720 -14.40 + 1296 -672 + 210 -36 + 2, = 80.
The formula for the next succeeding case, 7i = 7, gives
No. = 5040 -10080 + 9240 -5040 + 1764 -392 + 49 -2, =579.
Those for the preceding cases, n = 3, 4, 5, respectively are
No.= 6- 12+ 9- 2 =1,
No. = 24- 48+ 40- 16+ 2 = 2,
No. = 120 -240 + 210 -100 + 25 -2 = 13.
We have in general [l]=n, [2] = n, [1, l]=^n{n — S); and in the several columns
of the formulae the sums of the numbers thus represented are equal to the coefficients
of (1 + ly : thus, when n = 6 as above, the sums are 6, 15, 20, 15, 6, 1. As regards
the calculation of the numbers in question, any symbol [a, /3, 7] is a sum of symbols
[a — q:' + /9 — /3' + 7 — 7'+ ...], where a' + ^' + y'+... is any partition of n — (a +/3 + 7+ ...);
read, of the series of numbers 1, 2, 3,..., n, taken in cyclical order beginning with any
number, retain a, omit a', retain ^, omit 0', retain 7, omit 7', Thus in particular,
» = 6, [1, 1] is a sum of symbols [1—3+1 — 1] and [1 — 2 + 1 — 2] ; it is clear that
any such symbol [a — a' + y8 — /3' + ...] is =n or a submultiple of n (in particular, if n
be prime, the symbol is always = n) : and we thus in every case obtain the value
of [a, /3, 7,...] by taking for the negative numbers the several partitions of
n-(a + ^+y +...),
and for each symbol
[a-a' + /3-/3'+7-7' + ...],
writing its value, = n or a given submultiple of n, as just mentioned. There would,
I think, be no use in pursuing the matter further, by seeking to obtain an analytical
expression for the symbols [a, y3, 7,...].
For the actual formation of the required arrangements, it is of course easy, when
all the arrangements are written down, to strike out those which do not satisfy the
prescribed conditions, and so obtain the system in question. Or introducing the notion
of substitutions*, and accordingly considering each arrangement as derived by a
substitution from the primitive arrangement abcd...jk, we can write down the substitu-
tions which give the system of arrangements in which no letter occupies its original
place : viz. we must for this purpose partition the n letters into parts, no part less
than 2, and then in each set taking one letter (say the first in alphabetical order)
as fixed, permute in every possible way the other letters of the set ; we thus obtain
* In explanation of the notation of substitutions, observe that {abcde) means that a is to be changed
into b, b into c, c into d, d into e, e into a ; and similarly (ab) (cde) means that a is to be changed into b,
b into a, c into d, d into e, e into c.
248 ON A PKOBLEM OF ARRANGEMENTS. [669
all the substitutions which move every letter. Thus when » = 5, we obtain the 44 sub-
stitutions for the letters abcde, viz. these are
(abcde), &c., 24 symbols obtained by permuting in every way the four letters
b, c, d, e;
(ab){cde), &c., 20 symbols corresponding to the 10 partitions ab, cde, and for each
of them 2 arrangements such as cde, ced.
And then if we reject those symbols which contain in any ( ) two consecutive letters,
we have the substitutions which give the arrangements wherein the letter in the
first place is not a or b, that in the second place not b or c, and so on. In
particular, when n = 5, rejecting the substitutions which contain in any ( ), ab, be, cd, de,
or ea, we have 13 substitutions, which may be thus arranged : —
(acted), (ac)(bed), (acebd), (adbec), (aedbc),
(aedbc), (bd){aec),
(acedb), {ce){adb),
(aecbd), (ad) (bee),
(adceb), (be) (ode).
Here in the first column, performing on the symbol (acbed) the substitution (abcde),
we obtain (bdcae), = (aebdc), the second symbol ; and so again and again operating
with (abcde), we obtain the remaining symbols of the column ; these are for this
reason said to be of the same type. In like manner, symbols of the second column
are of the same type ; but the symbols in the remaining three columns are each of
them a type by itself; viz. operating with (abcde) upon (acebd), we obtain (bdace),
= (acebd); and the like as regards (adbec) and (aedbc) respectively. The 13 substitutions
are thus of 5 different types, or say the arrangements to which they belong, viz.
cebad, ceabd, cdeab, deahc, eabcd,
edacb, edabc,
caebd, daebc,
edbac, debac,
daecb, deacb,
are of 5 different types. The question to determine for any value of n, the number
of the different types, is, it would appear, a difficult one, and I do not at present
enter upon it.
670] 249
670.
[NOTE ON MR MUIR'S SOLUTION OF A "PROBLEM OF
ARRANGEMENT."]
[From the Proceedings of the Royal Society of Edinburgh, t. ix. (1878), pp. 388 — 391.]
I
The investigation may be carried further: writing for shortness 113, M4, &c., in place
of ^ (3), "¥ (4), &c., the equations are
«,= 1,
«S = 3W4+ 6M3+ 1,
ti, = 4i<,+ 8m4 + 12m3,
v^ = but+ IOM5 + law, + 18«3 + 1.
Hence assuming
we have
M = M3 + U^fC + U^ + U^ + U-^ + ...,
\— or
+ Ut{^a?+ 8a^+15a:'+22.r'+ ...)
+ M, (4a:»+ 10«* + 18a;» + 2Qofi + ...)
+ M,(5ir*+12a^+21»«+ 30af+...);
so that, forming the equation
w'--^,= Ut{x^+ 2a? ■^- 3ar'+ 4ar'+...)
(l-a:)»
+ W5(2«»+ 4«*+ 6a;»+ 8a;''+...)
+ Ms(3a;*+ 6a;»+ 9a;«+ 12a;'+ ...),
C. X.
32
250 NOTE ON MK MUIR'S SOLUTION OF A "PROBLEM OF ARRANGEMENT." [670
where w' denotes -3- , we have
\l — xf I —Or '
+ u (2a! + 6a? + I2sc> + 180^ +...);
or, what is the same thing,
, a!» 1 ^
U-U ., r- = Tj i + M
(1 — a;)' 1 — ar
tbat is.
2a; 2ar'
Kl-a;)» (l-a:)»(l
+ 4'
2a;* ) a;» , 1
■ M — .', r. U =
I- 2a;
I (l-xy'^{l-xy(l+x)\'^ (l-a;)"'" ~1 -ar"'
This equation may be simplified : write
then
and the equation is
f l-x' 2 1+a; 2 4 1 i_ 1 o 1 + a; q, _ 1
is,
Ll.l 2 2 2 2 I 1 + a; 1
I a;* ar" a;» (1 - a;)" ^ «» (1 - «)» ^ a; (1 - a;)= (1 - a;)»P ^ (1 - a;) a,'^ ^ l-ar"'
that is,
{-
viz. this is
that is,
or
or finally,
giving
and thence
(l-a;")' ,1-«^Q>_1-^.
a;* ^"^ a;" ^ 1+a;'
e(l-^)^-e' = (l^.
-(«+')/• a;» a:4 ,
a;^-! -(».!) f a;» (»■.])
" a;* * i(^ + l)»^ '
'(a; + l)»
which is the value of the generating function
U = «s + U^ + Ud)? + &c.
670] NOTE ON MR MUIR's SOLUTION OF A "PROBLEM OF ARRANGEMENT." 251
But for the purpose of calculation it is best to integrate by a series the differential
equation for Q: assuming
we find
q* = 4g'3 - 2,
^5 = 5g'4 +q3 +3,
9e = 6^5 + 9-4 - 4,
97 = Tq^ + ?» +5,
qn = nq,^t + ?»-2 + (-)""' (n - 2).
We have thus for q^, qt, q^,... the values 1, 2, 14, 82, 593, 4820,..., and thence
M = (1 - «=) (1 + 2a; + 14«= + 82a,-' + 593a;* + 4820a;» +...),
viz. writing
1 2 14 82 593 4820...
-1 -2 -14 -82
the values of M3, M4, ... are 1, 2, 13, 80, 579, 4738,...,
agreeing with the results found above.
I
In the more simple problem, where the arrangements of the n things are such
that no one of them occupies its original place, if m„ be the number of arrangements, we
have
«j =1 =1,
u, = 2 U2 =2,
Ml =3 (Ms + Mj) = 9,
lij =4 (M4 + M3) = 44,
W7.+1 = W ("n + Mn-l),
M = 1 + (2a; + Sx') u + {x'+ai') u;
(- 1 + 2a; + 3a;') M + (ar" + oc^) u' = - 1,
or, what is the same thing,
0/ J_ I 1 ,1/ =;
and writing
we find
that is,
, /3 1\ _ 1
whence
u = ar^ e " I ■:, — ^ e * da;.
jl+a;'
But the calculation is most easily performed by means of the foregoing equation of
differences, itself obtained from the differential equation written in the foregoing form,
(- 1 + 2a; + 3a;=) M + (a;» + a;') m' = - 1.
32—2
252 [671
671.
ON A SIBI-RECIPROCAL SURFACE.
[From the Berlin. Akad. Monatsber., (1878), pp. 309—313.]
The question of the generation of a sibi-reciprocal surface — that is, a surface the
reciprocal of which is of the same order and has the same singularities as the original
surface — was considered by me in the year 1868, see Proc. London Math. Sac. t. il.
pp 61 — 63^ [part of 387], where it is remarked that if a surface be considered as the
envelope of a quadric surface varying according to given conditions, then the reciprocal
surface is given as the envelope of a quadric surface varying according to the reciprocal
conditions ; whence, if the conditions be sibi-reciprocal, it follows that the surface is a
sibi-reciprocal surface. And I gave as instances the surface which is the envelope
of a quadric surface touching each of 8 given lines ; and also the surface called the
" tetrahedroid, " which is a homographic transformation of Fresnel's Wave Surface and
a particular case of the quartic surface with 16 nodes.
The interesting surface of the order 8, recently considered by Herr Kummer, Berl.
Monatsber., Jan. 1878, pp. 25 — 36, is included under the theory. In fact, if we consider
a line L, whereof the six coordinates
a. b, c, f, g, h,
satisfy each of the three linear relations
/la -1- gib + /t,c -1- ai/-H b^g + cji = 0,
/jffl + gj) -\-kfi + a,f+ b^ + cji = 0,
/jO + gJ) + A3C -I- 0,/+ bag + cji = 0,
the locus of this line is a quadric surface the equation of which is
T = (agh) x" + (bhf) y» -|- {cfg) z' + (abc) iv'
+ [{abg) - (cah)] xw + [{bfg) + {chf)] yz
+ [{bch) - {abf)] yw -f [{cgh) + (afg)] zx
+ [{caf) - (beg) ] zw + [{ahf)+ (bgh)] xy = 0.
671] ON A SIBI-RECIPBOCAL SURFACE. 253
where (agh) is used to denote the determinant a^, g^, h^ , and so for the other
^t ffat "■3
symbols. Considering the reciprocal of the line L "in regard to the quadric surface
X-+ Y-+ Z^+ W- = 0, the six coordinates of the reciprocal line are
/. g, h, a, b, c,
and it is hence at once seen that the locus of the reciprocal line ia the quadric surface
obtained from the equation T=0 hy interchanging therein the symbolical quantities a, b, c
and /, g, h: viz. \vriting also (f , 17, f, a) in place of (x, y, z, w), the new equation is
r = {fbc)^ + (gca)v' + (hab)^ + (/gh)oy'
+ [(fgh) - Qifc) ] r« + {{fab) + (to)] vK
+ {{ghc) - {fgaj\ 7,« + [{gbc) + (fab)] ?f
+ [(¥«) - ighh)] ?« + [(Aca) + {gbc) ] f>? = 0 ;
or, what is the same thing, this equation 2" = 0 is the equation of the original quadric
surface (the locus of L) expressed in terms of the plane-coordinates f, ?;, f, w.
Now considering each of the quantities a,, 61, Cj, /,, gi, h-^, a^, bo, etc., Oj, 63, etc., as
a given linear function of a variable parameter \, say 01 = 0/ + a "\, 6, = 6i'4-6i"X, etc.,
the equation 2'=0 takes the form
A\^ + SBX' + 3C\ + D = 0,
where A, B, C, D are given quadric functions of the coordinates x, y, z, w; and the
envelope of the quadric surface 2'=0 is Herr Kummer's surface of the eighth order
{AD - BCf - ^ {AG -&) {30-^)^0.
In like manner the equation 2" = 0 takes the form
ilV + ^E\^ + 3C"X + D' = 0,
where A', R, C, U are given functions of the coordinates ^, 77, t„ (n; and we have
{A'D' - B'CJ - 4 {A'C - B'^) {B'D' - C) = 0,
as the equation of the reciprocal surface ; or (what is the same thing) as that of the
original surface, regarding f, 7;, f, m as plane-coordinates.
In regard to the foregoing equation T = 0, it is to be noticed that, if a,, bi, Cj,
fu g\, K', (h> h, etc., cis, b„ etc., instead of being arbitrary coefficients, were the
coordinates of three given lines L^, L^, L, respectively; that is, if we had
«i/i + b,g, -I- Ci^i = 0,
as/2 + b^i + C'iK = 0,
Oj/a -h b^gi + C3A3 = 0,
254
ON A SIBI-RECIPROCAL SURFACE.
[671
then the three linear relations satisfied by (a, 6, c, /, g, h) would express that the line L
was a line meeting each of the three given lines X,, Zj, Zj: the locus is therefore
the quadric surface which passes through these three lines; and I have in my paper
"On the six coordinates of a Line," Gamb. Phil. Tram., t. xi. (1869), pp. 290—323,
[435], found the equation to be the foregoing equation T=0. But it is easy to see that
the same equation subsists in the case where the three equations a,/, + fti^r, + c,A, = 0,
etc., are not satisfied. For the several coefficients being perfectly general, any one of
the three linear relations may be replaced by a linear combination of these equations ;
that is, in place of a,, 6,, c,, /,, jr,, A,, we may write a,', 6,', c/, //, gi, W, where
a/ = ffiOi + Offti + OaOs, bi = Oibi + 6j).i + 6J)3, etc.; and these factors ^,, 6^, 63 may be
conceived to be such that the condition in question a,'/,' + ft/^'i' + c/Ai' = 0 is satisfied.
Similarly the second set of coefficients may be replaced by a^', 6/, c/, //, g^, h^, where
Os' = f^iOri + ^jOii + <|)3as, etc., and the condition Oa'/,' + b^g,' + Cj'^' = 0 is satisfied : and the
third set by a/, 63', C3', /,', gi, h^, where aj' = 1^,01 + 1^202 + V^aas. etc., and the condition
Oj'/s' + ^igz + Cs'V = 0 is satisfied. We have therefore an equation 0 = {a'g'h') a? + etc.,
which only differs from the equation ^ = 0 by having therein the accented letters in
place of the unaccented ones : and, substituting for the accented letters their values,
the whole divides by the determinant (6^-^), and throwing this out we obtain the
required equation T = 0.
But it is easier to obtain the equation T = 0 directly. We have
hy—gz + aw=0,
-luc . +fz + bw = 0,
gx-fy . +cw = 0,
— ax — by — cz . = 0 ;
viz. in virtue of the equation af+ bg + ch = 0 which connects the six coordinates, these
four equations are equivalent to two independent equations which are the equations
of the line (a, b, c, f, g, h) : or, what is the same thing, any three of these equations
imply the fourth equation and also the relation af+bg + ch = 0.
We might, from the three linear relations and any three of the last-mentioned
four equations, eliminate a, b, c, f, g, h and so obtain the required equation T = 0; but
it is better, introducing the arbitrary coefficients a, /8, 7, S, to employ all the four
equations. The result of the elimination is thus given in the form
a, w,
/3, w,
-z, y
Z, —X
y, w, -y, X ,
S, X, y, z,
/i. <7i. *ii <h, ii, c,
/■it git "3> ^> "2> ^2
fit 9t, fht <ht bf, c»
= 0,
671]
ON A SIBI-BECIPROCAL SURFACE.
255
viz. the left-hand side here contains the factor — (ax + $y + yz + Sw) ; throwing this out,
we obtain the required quadric equation y=0. If for the calculation of T we compare
the terms containing B, we have
Tw= w, —z,y
W, W, Z, —X
w, -y, X.,
/., gu fh, «!. ti. c,
/■2, 93, K, di, h, Cl
Jsi ffsi "3i 0-3> Osi Cs
where observe that, writing w = 0, the right-hand side vanishes as containing the factor
-z, y
Z, —X
-y, X,
Hence the right-hand side divides by w; and one of its terms being evidently w^{ahc),
T contains as it should do the term {abc)w^: the remaining terms can be found
without any difficulty, and thfe foregoing expression for T is thus verified.
256 [672
672.
ON THE GAME OF MOUSETKAP.
[From the Quarterly Joumcd of Pure and Applied Mathematics, vol. xv. (1878),
pp. 8—10.]
In the note "A Problem in Permutations," Quarterly Mathematical Journal, t. i.
(1857), p. 79, [161], I have spoken of the problem of permutations presented by this
game.
A set of cards — ace, two, three, &c., say up to thirteen — are arranged (in any order)
in a circle with their faces upwards ; you begin at any card, and count one, two,
three, &c., and if upon counting, suppose the number five, you arrive at the card
five, the card is thrown out; and beginning again with the next card, you count
one, two, three, &c., throwing out (if the case happen) a new card as before, and so
on until you have counted up to thirteen, without coming to a card which has to
be thrown out. The original question proposed was : for any given number of cards
to find the arrangement (if any) which would throw out all the cards in a given
order; but (instead of this) we may consider all the different arrangements of the
cards, and inquire how many of these there are in which all or any given smaller
number of the cards will be thrown out ; and (in the several cases) in what orders
the cards are thrown out. Thus to take the simple case of four cards, the different
arrangements, with the cards thrown out in each, are
672]
ON THE GAME OF MOUSETRAP.
257
2 3 4
2 4 3
, 3, 2, 4
, 3, 4, 2
, 4, 2, 3
, 4, 3, 2
3, 4, 2,
2, 3, 4,
2,
1,
3,
4
3,
4,
2,
1,
4,
3
—
2,
3,
4,
1
—
2,
3,
1,
4
4,
2,
4,
1,
3
—
'2,
4,
3,
1
3,
2,
3,
1,
2,
4
4,
3,
1.
4,
2
—
3,
2,
1,
4
4,
2. 1,
3,
3,
2,
4.
1
2,
3,
3,
4,
1,
2
—
3,
4,
2,
1
—
4,
1,
2,
3
—
4,
1,
3,
2
3,
4,
2,
1,
3
2,
1, 3,
4,
4, 2, 3, 1
4, 3, 1, 2
4, 3, 2, 1
3, 1, 2, 4,
C. X.
33
258
ON THE GAME OF MOUSETBAP.
[672
Classifying these so as to show in how many arrangements a given card or permutation
of cards is thrown out, we have the table
No.
Thrown ont
9
none
4
1
1
3
2
4
1
1
3,
2,
2
3
1
3,
4
1
1,
3,
4,
2
1
1,
2,
3,
4
1
4,
2,
1,
3
1
2,
1,
3.
4
1
3,
1,
2,
4,
24,
viz. there are nine arrangements in which no card is thrown out, four arrangements
in which only the card 1 is thrown out, one arrangement in which only the card 3
is thrown out, and so on.
It will be observed that there are five arrangements in which all the cards are
thrown out, each throwing them out in a different order; there are thus only five
orders in which all the cards are thrown out.
The general question is of course to form a like table for the numbers 5, 6,...,
or any greater number of cards.
673] 259
673.
NOTE ON THE THEORY OF CORRESPONDENCE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878),
I pp. 32, 33.]
If the point P on a given curve U of the order m, and the point Q on a
given curve V of the order m', have a (1, 1) correspondence, this implying that
the two curves have the same deficiency; then if PQ intersects the consecutive line
P'Q' in a point R, the locus of ii is a curve W of the class m + m, and the point
R on this curve has, in general (but not universally), a (1, 1) correspondence with
the point P on CT or with the point Q on V. For, considering the correspondence
of the points P and R, to a given position of P there corresponds, it is clear, a
single position of jR ; on the other hand, starting from R, the tangent at this point
to the curve W meets the curve U m m points and the curve V in m! points, but
it is in general only one of the m points and only one of the m' points which are
corresponding points on the curves U and F; that is, it is only one of the m points
which is a point P; and the correspondence of (P, R) is thus a (1, 1) correspondence.
But the curves U, V may be such that the correspondence of (P, R) is not a
(1, 1) but a {k, 1) correspondence; viz., that to a given position of P there
corresponds a single position of R, but to a given position of R, k positions of P.
To show that this is so, imagine through P a line 11 having therewith a, {k, 1)
correspondence ; P being, as above, a point on the curve U, the line in question
envelopes a curve W; and the correspondence is such that, for any given position
of P on the curve U, we have through it a single position of the line : but, for a
given tangent of the curve W, we have upon it k positions of the point P, viz. k
of the m intersections of the line with the curve U are points corresponding to the
line ; this, of course, implies that the curve U is not any curve whatever of the
order m, but a curve of a peculiar nature.
33—2
260 NOTE ON THE THEORY OF CORRESPONDENCE. [673
Imagine now that we have on the line 11 a point Q, having with P a (1, 1)
correspondence of a given nature: to fix the ideas, suppose P, Q are harmonics in
regard to a given conic : since on each of the lines 11 there are k positions of P,
there are also on the line k positions of Q, and the locus of these k points Q is a
curve V, say of the order m'.
The point P on the curve U and the point Q on the curve V have a (1, 1)
correspondence. For, consider P as given : there is a single position of the line IT
intersecting V in m' points, but obviously only one of these is the point Q. And
consider Q as given : then through Q we have say /m tangents of the curve W ; each
of these tangents intersects the curve U in m points, k of which are points P, but
for a tangent taken at random no one of these is the correspondent of Q ; it is, in
general, only one of the fi tangents which has upon it k points P, one of them
being the point corresponding to Q; that is, to a given position of Q there corresponds
a single position of P; and the correspondence of the points (P, Q) is thus a (1, 1)
correspondence.
We have thus the point P on the curve U and the point Q on the curve V,
which points have with each other a (1, 1) correspondence ; and the line 11 is the
line PQ joining these points ; this intersects the consecutive line in a point R ; and
the locus of R is the curve W. To a given position of P there corresponds a single
line n, and therefore a single position of R; but to a given position of R there
correspond k positions of P, viz. drawing at R the tangent to the curve W, this is
a line 11 having upon it k points P, or the correspondence of (P, Q) is, as stated,
a {k, 1) correspondence.
The foregoing considerations were suggested to me by the theory of parallel
curves. Take a curve parallel to a given curve, for example, the ellipse ; this is a
curve of the order B, such that every normal thereto is a normal at two distinct
points; and the curve has as its evolute the evolute of the ellipse, or, more
accurately, the evolute of the ellipse taken ttuice; but, attending only to the evolute
taken once, each tangent of the evolute is a normal of the parallel curve at two
distinct points thereof, and the points of the parallel curve have with those of the
evolute not a (1, 1) but a (2, 1) correspondence.
674] 261
674.
NOTE ON THE CONSTRUCTION OF CARTESIANS.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), p. 34.]
If p = o + 6 cos ^, and r = ^ {p ± VCp" — c^)}. then obviously r* — rp + Jc* = 0, that is,
r2-r(a + 6cos^)+ic' = 0,
which is the equation of a Cartesian. Here p = a-\-h cos 6 is the equation of a
lima9on or nodal Cartesian, having the origin for the node ; and for any given value
of 0, deducing from the radius vector of the lima9on the new radius vector r by
the above formula r=\{p ± \/(p-—<?)], we obtain a Cartesian, or by giving different
values to c, a series of Cartesians having the origin for a common focus. The con-
struction is a very convenient one.
262 [675
675.
ON THE FLEFLECNODAL PLANES OF A SURFACE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 49 — 51.]
If at a node (or double point) of a plane curve there is on one of the bi-anches
an inflexion, (that is, if the tangent has a 3-pointic intersection with the branch),
the node is said to be a flecnode ; and if there is on each of the branches an
inflexion, then the node is said to be a fleflecnode. The tangent plane of a surface
intersects the surface in a plane curve having at the point of contact a node ; if
this is a flecnode or a fleflecnode, the tangent plane is said to be a flecnodal or a
fleflecnodal plane accordingly. For a quadric surface each tangent plane is fleflecnodal ;
this is obvious geometrically (since the section is a pair of lines), and it will
presently appear that the analytical condition for such a plane is satisfied. In fact,
if the origin be taken at a point of a surface, so that z = 0 shall be the equation
of the tangent plane, then in the neighbourhood of the point we have
z = {x, yf + {x, yy + &c.;
and the condition for a fleflecnodal plane is that the term {w, y)- shall be a factor of the
succeeding term (as, yy. Now for a quadric surface the equation is
z = ^ {cue" + 2Jucy + by' + 2 (fy + gx)z + cz») ;
that is,
« (1 -Jy - 9^ - h^) = i («^ + 2A^ + by').
or developing as far as the third order in (ar, y), we have
z = ^{aa^ + 2hxy+hy') (1 +fy + gx),
so that the condition in question is satisfied.
675] ON THE FLEFLECNODAL PLANES OF A SURFACE. 263
In what follows, I take for greater simplicity h = 0, (viz. a; = 0, 2/ = 0 are here the
tangents to the two curves of curvature at the point in question), and to avoid
fractions write 2/", 2g in place of f, g respectively ; the developed equation of the
quadric surface is thus
z = ^{aa?+ bf) + (ow^ + hy') {gx +fy).
I consider the parallel surface, obtained by measuring off on the normal a
constant length k. If, as usual, p, q denote t- and j- respectively, then, in general,
(X, Y, Z) being the coordinates of the point on the parallel surface,
Z = z +
X — x —
Y=y-
kp
kq
But in the present case
p = ax + Sagx^ + 2afxy + bgy^,
'q = by+ afa? + Ibgxy + Zbff,
whence
X = x—k {ax + ^aga? + 2afxy + bgy%
Y = y-k{by+ afx' + 2bgxy + 3bff);
or, putting for convenience,
X = {l-ka)l Y = il-kb)v,
then, for a first approximation x = ^, y = rj; whence, writing
P = Sag^ + 2af^r,+ bgrf,
we find
^ kP kQ
^ = ^ + l31te' 2/ = ^+iT:^.6'
and thence
^="(^ + iT^^)+^='^^+r
■ka'
Q
' =^^+i--Tb-
Hence
or, finally,
Z-k^i{a{l- ka) p + 6 (1 - kb) V'} + (a^ + bv') (g^ + fn).
264 ON THE FLEFLECNODAL PLANES OF A SURFACE. [675
where, changing the origin to the point a; = 0, y = 0, z=k on the parallel surface, the
coordinates of the consecutive point are Z — k, X, ={\—ka)^, and F, ={\ — kh)f).
We cannot, by any determination of the value of k, make the plane Z — k = 0
a fleflecnodal plane of the parallel surface ; but if
a?p + 6y '
then
^ '^-oi'p + hy ^ ""-a-'p + by
and the equation becomes
^ - ^ = * |^r^= (^'^ -•^'^'> + <"^ + ^'''> ^^ +-^''> ;
viz. the term of the second has here a factor g^+frj which divides the term of the
third order, and the plane Z—k = Q is a flecnodal plane of the parallel surface.
676]
265
676.
NOTE ON A THEOREM IN DETERMINANTS.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XV. (1878),
pp. 55 — 57.]
It is well known that if; 12, &c., denote the determinants formed with the matrix
a, /3. 7, S
a', /3', 7', S'
12.34 + 13.42 + 14.23 = 0.
The proper proof of the theorem is obtained by remarking that we have
0 =
then, identically,
1, yS,
7. •
a', /3',
7. •
a, /3,
7. S
a'. /3',
7. S'
as at once appears by subtracting the first and second lines from the third and
fourth lines respectively; and, this being so, the development of the determinant
gives the theorem. The theorem might, it is clear, have been obtained in four
different forms according as in the determinant the missing terms were taken to be
as above (S, S'), or to be (a, a'), (/8, ;S'), or (7, 7'); but the four results are equivalent
to each other.
There is obviously a like theorem for the sums of products of determinants
formed with the matrix
i«, /S, 7. S, e, ?
«', ^' , i , ^' , e , ?'
I O", ^', 7", S", e", r"
C. X.
34
266
NOTE ON A THEOREM IN DETERMINANTS.
[676
viz. the theorem is obtaioed by development of the determinant in an identical
equation, such as
0 =
/3,
/3".
7
7
7
7
7
7
8,
S',
8".
S,
S',
S".
r
but we thus obtain 15 results which are not all equivalent.
If, for shortness, we write
^ = 123 . 456,
- 5 = 124 . 356,
- 0=125.346,
i) = 126 . 345,
-.&= 134. 2.56,
- J* =135. 246,
(? = 136 . 245,
-ir=145.236,
7=146.235,
/= 156. 234,
then the fifteen results are
A + B-C -D = Q,
A + B -E- J = 0,
A-C +F - I =0,
A-D + G-H = 0,
A-E + F + G = 0,
A-H- I - J = 0,
' B-C-G +H = 0,
B -D-F + I =0,
B -E + H + I =0,
B -F-G- J = 0,
G +D-E - J = 0,
C -E + G+ I =0,
C -F-H- J = 0,
D-E + F+H=0,
D-G- I - J = 0,
which are all satisfied if only
A= . . +H + I + J,
B = F+G . .+J,
C =F . +H . +J,
D= . G . +r + j,
E = F+G+H + I + J;
and we thus have these five relations between the
A, B, C, D, E, F, G, H, I, J.
ten products of determinants
677'\ 267
677.
[ADDITION TO MR GLMSHER'S PAPER "PROOF OF STIRLING'S
THEOREM."]
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 63, 64.]
It is easy to extend Mr Glaisher's investigation so as to obtain from it the more
approximate value
nn = V(27r)ft''+ie ^2».
We, in fact, have
yK.^ — gSiu!+a«»+6ii!'+...
where a, b, ... are given functions of n, viz.
1
_ (1 1
"-^t3' + 5=+-^(2« + l)=j'
&c.
And hence writing x=\, we have
that is,
t (1) = i 2»>n.(^) (2« + 2)-+' = e-+«+^+--,
n«=(^^-.j 2 e-«-l(a+6+...)
= (n + l)"+i e-"-l«»+''+-)
/ 1 \ "+i
= «"+* ( 1 + - ) e-»-i««+6+").
34—2
268 ADDITION TO MR GLAISHER's PAPER " PROOF OF STIRLING'S THEOREM." [677
1 + -J writing e ^ "'. the whole exponent of e is
(n + J)log(l + -)-n-i(a + 6 + ...)
We have
i4._l i.Al+Ai-
r»+^'+-+(2;rW'=^''^«'-i +*'"■""" ^" I' ^»' ^'-
(the constant is in fact = ^tt^ but the value is not required), hence a = const. —^
+ terms in — , — , &c. ; as regards b, c, &c., there are no terms in - , but we have
n' n' n
b'= const. + terms in — , &e., c = const. + terms in — , &c. Hence the whole exponent
n n'
of e is
= — H+ C*+, ,-+ terms in ,, &c.
12n rr
As in Mr Glaisher's investigation, it is shown that 6"'^'= \/(27r), and hence neglecting
the terms in -, &c., the final result is
n« = v'(27r)w"+ie i«».
678] 269
678.
ON A SYSTEM OF QUADRIC SUEFACES.
[From the Quarterly Journal of Pure and Applied Mailiematics, vol. xv. (1878), pp. 124, 125.]
The following theorem was communicated to me by Dr Klein; "given in regard
to a quadric surface two sibi-reciprocal line-pairs, the two tractors (or lines meeting
each of the four lines) form a sibi-reciprocal line-pair." This may be presented under
a more general form as a theorem relating to the tractors of any two line-pairs. In
fact, if a given line-pair is taken to be sibi-reciprocal in regard to a quadric surface,
we thereby establish only a four-fold relation between the coefficients of the surface,
and the surface will still depend on five arbitrary parameters. Whence if two given
line-pairs are taken to be each of them sibi-reciprocal in regard to one and the same
quadric surface, we thereby establish only an eight-fold relation and the surface will
still depend upon one arbitrary parameter. The theorem thus is: given any two line-
pairs, then each of these, and also the pair of tractors, are sibi-reciprocal in regard
to a singly infinite system of quadric surfaces.
The question arises, what is this system of quadric surfaces ? It is, in fact, the
system of surfaces having in common a skew quadrilateral constructed as follows :
starting from the two given line-pairs, construct the two tractors, each of them
intersected by the given line-pairs in two point-pairs; and on each tractor construct
the double or sibi-reciprocal points of the involution thus determined ; these double
points are the vertices (those on the same tractor being opposite vertices) of the
skew quadrilateral; which is consequently at once obtained by joining the two double
points on the one tractor with the two double points on the other tractor. The
construction is an immediate consequence of the following theorem : consider a skew
quadrilateral, and drawing its two diagonals, take a pair of lines cutting each diagonal
harmonically; these will be sibi-reciprocal in regard to any quadric surface through
the skew quadrilateral.
The condition of passing through a skew quadrilateral is that of passing through
a certain system of eight points ; in fact, the eight points may be taken to be the
four vertices and any four points on the four sides respectively. But observe that
the system of the quadric surfaces through any eight points has the characteristics
(1, 2, 3); viz. there are in the system 1 surface passing through a given point, 2
touching a given line, 3 touching a given plane ; the system of surfaces through
the same skew quadrilateral haa the characteristics (1, 2, 1).
270 [679
679.
ON THE KEGULAR SOLIDS.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 127—131.]
In a regular solid, or saj' in the spherical figure obtained by projecting such
solid, by lines from the centre, on the surface of a concentric sphere, we naturally
consider 1° the summits, 2° the centres of the faces, 3° the mid-points of the sides.
But, imagining the five regular figures drawn in proper relation to each other on
the same spherical surface, the only points which have thus to be considered are 12
points A, 20 points B, 30 points ©, and 60 points ^. These may be, in the first
instance, described by reference to the dodecahedron ; viz. the points A are the
centres of the faces, the points B are the summits, the points 0 are the mid-points
of the sides, and the points <i> are the mid-points of the diagonals of the faces
(viz. there are thus 5 points 4> in each face of the dodecahedron, or in all 60
points ^). But reciprocally we may describe them in reference to the icosahedron ;
viz. the points A are the summits, the points B the centres of the faces, the points
0 the mid-points of the sides, (viz. each point 0 is the common mid-point of a
side of the dodecahedion and a side of the icosahedron, which sides there intersect
at right angles), and the points O are points lying by 3's on the faces of the
icosahedron, each point ^ of the face being given as the intersection of a perpendicular
A@ of the face by a line BB, joining the centi-es of two adjacent faces and inter-
.secting A® at right angles.
The points A lie opposite to each other in pairs in such wise that, taking any
two opposite points as poles, the relative situation is as follows:
Longitudes.
0°, 72°, 144°, 216°, 288°,
36°, 108°, 180°, 252°, 324°,
1
5
5
1
where the points A in the same horizontal line form a zone of points equidistant
from the point taken as the North Pole. And the points B lie also opposite to
679]
ON THE REGULAR SOLIDS.
271
each other in such wise that, taking two opposite points as poles, the relative situation
is as follows :
B
Longitudes.
1
—
3
0°,
120°, 240°
6
(0°,
120°, 240°) + 22° 14',
6
(60°,
180°, 300°) + 22° 14',
3
60°,
180°, 300°
1
—
>
where the points B in the same horizontal line form a zone of points equidistant
from the point taken as the North Pole. Neglecting the 3+3 points B which lie
adjacent to the poles, the remaining 14 points B may be arranged as follows (/8 = 22° 14'
as above):
B
Longitudes.
1
6
6
1
A 120° + /3,
60° + 13, 180° + /3,
240° + )8
300° + 0
60°
-A 120° -/3, 240°-/?,
-/3, 180° -yS, 300°-/?.
And taking the two poles separately with each system of the remaining poles, we
have 2 systems each of 8 points B, which are, in fact, the summits of a cube
(hexahedron); each point B taken as North Pole thus belongs to two cubes; but
inasmuch as the cube has 8 summits, the number of the cubes thus obtained is
20 X 2 -7- 8, = 5 ; viz. the 20 points B form the summits of 5 cubes, each point B
of course belonging to 2 cubes.
It is to be added that, considering the 5 points B which form a face of the
dodecahedron, any diagonal BB of this dodecahedron is a side of a cube. We have
thus 12 X 5, = 60, the number of the sides of the 5 cubes.
It is at once seen that the centres of the faces of a cube are points 0, and
that the mid-points of the sides of the cube are points ^.
To each cube there corresponds of course an octahedron, the summits being
points 0, the centres of the faces points B, and the mid-points of the sides points
$ ; thus, for the five octahedra the summits are the 5x6,= 30, points © ; the
centres of the faces are .5x8, = 40, points B (each point B being thus a centre
of face for two octahedra), and the mid-points of the sides being the 5x12, =60,
points 4>.
Finally, considering the 8 points B which belong to a cube, we can, in four
different ways, .select thereout 4 points B which are the summits of a tetrahedron ;
272
ON THE REGULAR SOLIDS.
[679
the remaining 4 points B are then the centres of the faces, and the mid-points of
the sides are points 0 : there are thus 5x4,= 20, tetrahedra having 20 x 4 summits
which are the 20 points B each 4 times ; 20 x 4 centres of faces which are the 20
points B each 4 times ; and 20 x 6 mid-points of sides which are the 30 points 0
each 4 times.
It thus appears that, as mentioned above, the five regular figures depend only
on the points A, B, 0, and 4>.
We might take as poles two opposite points A, B, 0, or ^; and in each case
determine in reference to these the positions of the other points; but for brevity I
consider only the case in which we take as poles two opposite points A. We have
the following table :
Poles two opposite points A.
N. P. D.
Longitudes.
5^1
5 J,
A,
0°
63° 26'
116° 34'
180°
0°, 72°, 144°, 216°, 288°
36°, 108°, 180°, 252°, 324°
5B,
37° 22'
79° 12'
100° 48'
142° 38'
36°, 108°, , 324°
5B,
36°. 108° 324°
5B,
0°, 72° , 288°
bB.
0°, 72° 288°
50,
31° 43'
58° 77'
90°
121° 43'
148° 17'
0°, 72°, 288'
5©j
36°, 108°, , 324*
10®,
5©.
( 0°, 72°, , 288°)+ 18*
0°, 72° 288°
5®<
36°, 108°, , 324°
5^,
13° 16'
52° 52'
68° 10'
76° 42'
103° 18'
111° 50'
127° 8'
166° 44'
36°, 108°, , 324°
10l»o
(0°, 72°, 288°)+ 9° 44'
10*,
( 0°, 72° , 288°)+ 13° 35'
5*.
0°, 72° , 288°
5*.
36°, 108°, , 324°
10*.
(36°, 108°, , 324°)+ 13° 35'
10*7
(36°, 108°, , 324°)+ 9° 44'
5*8
0°, 72°, , 108°.
679]
ON THE REGULAR SOLIDS.
273
I add for greater completeness the following results, some of which were used in
the calculation of the foregoing table. Considering successively (1) the tetrahedral
triangle, summits 3 points B, centre a point B; (2) the hexahedral square, summits
4 points B, centre a point 0 ; (3) the octahedral triangle, summits 3 points @,
centre a point B; (4) the icosahedral triangle, summits 3 points A, centre a point
B ; (5) the dodecahedral pentagon, summits 5 points, centre a point B; and (6),
what may be called the small pentagon, summits 5 points 4> lying within a dode-
cahedral pentagon, and having therewith the common centre B; we may in each case
write s the side, r the radius or distance of the centre from a summit, p the
perpendicular or distance of the centre from a side. And the values then are
Tet. A
Hex. square
Oct. A
Ices. A
Dod. pentagon
Small pentagon
109° 30'
70 30
90
63 26
41 50
15 30
70° 30'
54 45
54 45
37 22
37 22
13 16
54° 45'
45
35 15
20 55
31 43
10 48
C. X.
35
274
[680
680.
ON THE HESSIAN OF A QUARTIC SURFACE.
[From the Quarterly JoumcU of Pure mid Applied Mathematics, vol. xv. (1878),
pp. 141—144.]
The surface considered is
U=k-w'(^^^ + f^+'^y(a- + f + zj = 0,
or say
U=tw^P-Q' = 0,
viz. this may be considered as the central inverse of the ellipsoid
x' y" z^ ^ ^
The values of the second derived functions, or terms of the Hessian determinant
a, h, g, I
h, b , f, m
g, f, 0, n
I, m, n, d
are
k'
-2Q-
-4a^,
— 'ixy , - 4>xz
2k'
- wx
-4>xy
,
gj^_2Q-43/S -4yz
2k'
- 4^z
)
-iyz , |m;»-2Q-
c
42r^ -J- WZ
2Jt»
— wx
2^ 2B ,,„
^wy , c' "'^ ' ^
680] ON THE HESSIAN OF A QUABTIC SURFACE,
and we thence have
whence, forming the analogous quantities ca — g^, &c., it is easy to obtain
abc - ap - hg- - civ' + 2fgh
275
20/'— — — ^ 4/' — 4. 2'! ^S\
-24Q»,
which is to be multiplied by rf, = i-^P. And
- [P {be -f) + m' (ca - g') + n' {ab - h")
+ 2mn {gh - a/) + 2nl (hf- bg) + 2lm {/g - ch)]
— ^h^vf
+ 4i-*w^
which is
4fc««AP
+ &w|-48g + |;+^)(? + 32P»Q|.
35—2
276
ON THE HESSIAN OF A QUARTIC SURFACE.
[680
Hence, uniting the two parts, we have
3
24
,2 ^
+ ifc^»
\ + 24P2(3
+ ifc=' {-24PQ3).
Writing herein Q'—k^'^P - U, and transposing all the terms which contain U, we
have
= I<ftv*P
27Jfc» ,
■48
ar» W" ^'^
+
where, in the term in { ), the last four lines are
Hence, writing for shortness
680] ON THE HESSIAN OF A QUARTIC SURFACE. 277
we have
Hence, recollecting that U=l<Fw^P — QP, the Hessian curve of the order 32 breaks
up into
U =0, MT* = 0, that is, Q^ = 0, w* = 0, or the nodal conic,
tu = 0, Q=0, 8 times (order 16),
U =0, F =0, that is, Q- = 0, P = 0, or the quadriquadric,
P = 0, Q = 0, 2 times (order 8),
and into a curve (order 8) which is
; IV
Hw^h^^^^At^h'^^''
viz. this, the intersection of the surface with a quadric surface, is the proper Hessian
curve.
278 [681
681.
ON THE DERIVATIVES OF THREE BINARY QUANTICS.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 1.57—168.]
Fob a reason which will appear, instead of the ordinary factorial notation, I
write {o012| to denote the factorial a . a + 1 . s + 2, and so in other cases ; and I
consider the series of equations
(1) = ^.
(2) = ({a0}, [/30}5F, - F),
(3) = ({a01j, 2 {al} {/31}, {/801)][^, -Z', Z"),
(4) = ({a012}, 3 {al2} {y32], 3 {a2} {/S12), {/3012)5;F, - W, - W", - W"),
&c.
where
X=Y + Y',
' Y=Z +Z' , Y' = Z' +Z",
Z=-W+W', Z'=W'+ W", Z" = W" + W",
&c.
We have thus a series of linear equations serving to determine X ; Y, Y' ; Z, Z', Z" ;
W, W, W", W"; &c. We require in particular the values of Z; Y, Y' ; Z, Z" ;
W, W"; &c., and I write down the results as follow:
X = (1),
(I) (^)
{a + m]Y =(^0), +1,
{ ., ]y' ={aO},-l;
681] ON THE DERIVATIVES OF THREE BINARY QUANTICS.
{a + ^2}(l), {a + ;Qi;(2), {a + y801(3).
[a + ^0l2]Z = [m] , +2i/81} , +1
{ „ ]Z" = {aOl) , -2{al} , +1 ;
{a + y334}a), {8 + ^4} (2), {a + ^03|(3), {a 4-^01} (4);
{a+/901...4}F = {/SOI 2)
279
]W"'= {a012}
{a+/34o6j(l>
{a+jS01...6}?7 = {^801231
\U""= {00123}
f
I
&c.
read
+ 3(^12) , +3{;82} , +1 ,
-3{al2) , +3{a2) , -I ;
a+y8156}(2), {a+y8036}(3), {a+;S01.5j(4), [a+/3012)(5);
+ 4{/9123} , +6(^23} , + 4 {^3J
-4{al23i , +6{a23} , -4{a3}
+ 1
+ 1
a+/3.F=^(l) + (2).
„ .F'=a(l)-(2),
a + ;8.a + /9+l.a + /3 + 2.Z =/3./9 + l.a + /3+ 2.(l) + 2.y9 + l.a +yS + l.(2) + a + y9.(3),
.-^" = a.a +l.a + ^+2.(l) + 2.a +l.o + ^ + l.(2)+a + /3.(3),
&c.,
the law being obvious, except as regards the numbers which in the top lines occur
in connexion with a + ;S in 'the { } symbols. As regards these, we form them by
successive subtractions as shown by the diagrams
34
34
456
456
5678
5678 &c.;
2
14
3
156
4
1678
11
03
12
036
13
0378
2
01
21
015
22
0158
3
012
31
4
0127
0123
and the statement of the result is now complete.
In part verification, starting from the F-formulas (which are obtained at once),
{a + ;821(l). {« + /3l](2), {«+;80}(3),
{o + /3012}^ =
assume
we must have
that is,
{ „ } Z' = V , H-' , v
„ J Z = X , M . "
(1M2)
{a + /3012} . ^ + Z' = {a + /3012} 7 , = {« + /312) ({ySO}, +T)
{ „ }.Z' + Z"={ „ }F', ={ „ }({aOj.-l)
!a + ;82).X +\' ={a + iS12J{y80},
{ „ ).\' + \"={ „ ){aO),
280 ON THE DERIVATIVES OF THREE BINARY QUANTICS.
and further
{a + /32} ({aOi;, - 2 {al} {^1], l/801j$\, X', X") = 0,
or, what is the same thing,
X + X' = {« + ^1) {$0],
X' + X"={ „ }{aO}.
({«01), -2{al] (ySl). 1/901}]1X, X', X") = 0.
And in like manner we have
>+/={a + /82}. 1,
fi + fi" ={ „ } . - 1,
({aOl}, - 2 {«!) {^1}, {y801)][M. /, H-") = 0 ;
and
V +v' =0,
((oOl), -2{al)(/81), {^eOlj^i-, v', 0 = 0.
We hence find without difficulty
\,,i,v=^.^ + l, 2./3+1, +1,= {;801} , 2 {/31}, + 1,
X',/,/= a./S , a-^ , -l, = {aO}!yS0), a-/S, -1,
X", fi", I/" = a . a + 1 , - 2 . a + 1, + 1, = laOl) , 2 {al} , + 1 ;
viz. for verification of the X-equations we have
/3./3+1.+ a./3 , that is, a + /3 + 1 . /3, = {a + /31} (/30},
a.yS. + a. a 4-1, „ a + l+/S.a, ={ „ ){aOj,
(a.a + 1, -2.a + l./3+l, /8. /3 + l$/3. /8+ 1, a./3, a.a + l) = 0,
a.a + l./3./S + l.-2.a+l./3 + l.a./3. + /3./3 + l.a.a+l=0;
and similarly the /i- and i/-equations may be verified.
We have thus for the Z's the equations
{a + /32](I), (g-f^l](2), {« + ;80](3).
{a + ^012}Z= {^Olj . 2{/31} , +1 ,
{ „ }Z' = (a0}{/30} , a-/8 , -1 ,
{ „ }Z"= (aOl} , -2{al} , +1 ,
which include the foregoing expressions for Z and Z".
We may then take the expressions for the W's to be
{a + j834l (1), {« + ^14] (2), {« + /303} (3), {a +/801} (4>
{a + (80123) F =
[681
and
that is,
{
j W =
) W" =
} W" =
X
X'
X"
X'"
//
p"
681] ON THE DERIVATIVES OF THREE BINARY QU ANTICS. 281
and we obtain in like manner the equations
X +\' ={a + /3234}{/301},
\'+V'={ „ }{aO} {/SO},
\" + \"'={ „ }{a01},
({o012}, -3{al2){^2}, +S{a2]{^U}, - {y8012}5:\, \', \", \"') = 0;
^ +f^' = (a + ,8134}. 2{;81},
fi'+fi"={ „ }. a-/9,
m" + /"={ „ }.-2{al},
({a012}, -3{al2}{^2}, +3{a2}{^12}, - {^012}$^, m', /', m"') = 0;
V +v' ={a + /9034}. 1,
u'+v"={ „ }.-l,
v"-^v"'={ „ }. 1,
({a012}, -3{ol2}{y82}, + 3 (a2} {/312}. -{001^1^. v', v", 0 = 0;
P +p' =0,
P'+P"=0,
p" + p" = 0,
({a012}, - 3 {al2} (;82}, + 3 {a2} {/312}, -{y8012}$p, p', p", p"')= {a + /901234}.
These give for the \p"' square the values
{/3012} , 3 {^12} , 3 {^2} . + 1,
{aO} {ySOl}, 2a - /3 . {/91}, a - 2/9 - 2, - 1,
{aOl} {/SO}, a - 2/8 . {al }, - 2a + ,3 - 2, + 1,
{a012} , - 3 {al2} , + 3 {a2} , - 1,
and 80 on; the law however of the terms in the intermediate lines is not by any
means obvious.
Consider now the binary quantics P, Q, R, of the forms (•$;», y)*", (*$«, y)',
(*^x, yY ; we have for any, for instance for the fourth, order, the derivates
P(Q, RY, (P, {Q. Ryy, (P, (Q, Ryy, (P, (Q, Ryy. (p, qry;
and it is required to express
Q(P, Py and P(P, Qy,
each of them as a linear function of these.
c. X. 36
282
ON THE DERIVATIVES OF THREE BINARY QUANTICS.
[681
I recall that we have (P, Qf = PQ, so that the first and the last terms of the
series might have been written (P, (Q, R)*)" and (P, (Q, Ryy respectively; and,
further, that (P, Qf denotes d^P .dyQ- dyP .d^Q; (P, QY denotes
4'P . d/Q - 2d:,dyP .d^dyQ+ dy^P . d^^^Q ',
and so on.
I write (a, h, c, d, e) for the fourth derived functions of any quantic U, =(»'^x, y)";
we have, in a notation which will be at once understood,
U= (a, b, c, d, e^x, yf -r [m]« ,
{dz, dy) U= (a, b, c, d), (b, c, d, e)(x, yf -i-[m-l]»,
K, dyY U = {a, b, c), {b, c, d), (c, d, e){x, y)=--[m-2p,
(d„ dyY U^ {a, b), (6, c), (c, d), (d, e) (a;, y)' ^[m-3]>,
(djs, dj,y fr= (a, 6, c, d, e);
and then, taking
(Oj, Oi, C], (Xj, Cj), (Ctj, Oa, Cj, ttj, Cj), (ttj, O3, C3, Cts, 63},
to belong to P, Q, R, respectively, we must, instead of m, write p, q, r for the
three functions respectively.
If we attend only to the highest terms in x, we have
U = ax* -i- [m]* ,
(dz, dy) U={a, b)af> h-[to-1]»,
{d^,dyyU = (,a,b.c)a? -[m-2p,
(4, dy)' U = (a, 6, c, d) a; 4- [hi - 3]',
(dx, dy)* U = {a, b, c, d, e).
Consider now P (Q, iJ/, (P, (Q, ii)'^, &c. ; in each case attending only to the
term in O], and in this term to the highest term in x, we have
(1) [p]* P (Q, R)* = a^e, - 4:b,d, + CcjC, - 4d363 + e^ch (X),
(2) [p - IJ [q - 3]' [r - 3J (P, (Q, RfY = [q- 3]' . b,d, - Sc,c, + Sd.b, - e,a, (- F),
+ [r-3p.aae8- 36ads + 3c3C3-dj6s(F),
(8) [p - 2]» [q - 2]^ [r - 2]= (P, (Q, ij)=)» = [q- 2]» . 0^03 - 2d,A + e,a, (Z"),
+ 2 [9 - 2]' [r - 2]' . tjd, - 2cjC3 + dj>, (- Z'),
+ [r-2]'.ches-2b.d3 + c^C3iZ),
(4) [p - 3p [3 - 1? [r - 1]» (P, (Q, P)')» = [g- ip . d,63 - c.a, (- W"),
+S[q-lY[r-lY.c,c,-dA iW"),
+3[q-lY[r-l]\b,ds-c,c, i-W),
+ [r-l]'.a^,-b,d, (W),
681]
(5)
ON THE DERIVATIVES OF THREE BINARY QUANTICS,
+ i[qY[rf.dA
+ 6 [qY [rp . c,c,
+ 4 Iq} [rf . hA
283
iU"").
(- U'").
(U"),
(- u'),
Thus, for the second of these equations,
(P, (Q, Ryy = d^P . dy {Q, Bf - &c. ;
the term in a, is dy(Q, Ry,=(dxQ, Ry+{Q, dyRy, the whole being divided by [p-1]';
where attending only to the highest terms in x, the two terms are respectively
(bid, — ScsCs + 3^263 - e^tts) -=- [r - 3]S
and
(0363 — Sb^da + SCaCs - d^ba) -^[q- 3]S
which are each divided by [p — ip as above ; whence, multiplying by
b-ip[5-ip[r-lp,
we have the formula in question; and so for the other cases.
Writing now (1), (2), (3), (4), (5) for the left-hand sides of the five equations
respectively ; and
-F', F:
Z", Z', Z:
- W", W", -W, W:
U"", - U'", U", -U', U:
for the literal parts on the right-hand sides of the same equations respectively ;
then we have
X=Y+Y',
Y=Z+Z', Y' = Z' + Z",
&c.,
and the equations become
(1)= X
(2) = [r-lpF-l [9-3]> Y'
(3) = [r - 2]» Z - 2 [r - 2]' {q - 2]' Z' +\ [q- 2]» Z"
(4) = [r - 1]' F - 3 [r - 1]» [q - 1]' W' + S[r- 1]' [q - 1]» W" -l[q- I]' W"
(5)= [r]«f7-4 [r]» [?]• ^-' + 6 [r]' [5]' U" -^{rjiqj U'" -^[qf U"",
which are, in fact, the equations considered at the beginning of the present paper,
putting therein o = r — 3 and ^ = q — 3, they consequently give
{q+r-6, 456)(1), {q+r-6, 156)(2), {q+r-6, 036}(3), [g' + r--6, 015}(4), {q+r-6, 012)(5),
{g+r-6, 01...6lCr = {2-3,0123} , +4 [^-3, 123} , +61g'-3,23}, +4(5-3,3}
j „ }U""= {r-3,0123} , -4{r--3, 123} , +6(r-3,23}, -4{r--3,3}
36-
+ 1
-hi
-2
284 ON THE DERIVATIVES OF THREE BINARY QU ANTICS. [681
Also, attending as before only to the terms in a, and therein to the highest
power of X, we have
R{P, Q)* = a,e,-5-[r]«;
that is,
[qj Q (R, Py = U, [rf R (P, Qy = U"" ;
and, observing that {g + r — 6, 01...6) is =[q + ry, and that {q + r—6, 456}, &c., may
be written {q—r, 210}, &c., where the superscript bars are the signs — , the formulae
become
{q+r, 210}(1), {q+r, 510(2), {g+r, 630}(3), {q+r, 651}(4), {q+r, 654}(5),
[q+ry[qyQ{P, Ry= [qY . +HqY . +6W , +4 [9? , +1
[q+rY[ryR(P,Qy= [r]* , -4[rp , +6[r]« , - 4 [r]' , +1
Written at full length, the first of these equations (which, as being the fourth in
a series, I mark 4th equation) is
[q+rY[q]*Q{P, Ry= l.q+r .q+r-l.q+r-2. [p]' [q]* .P,(Q,Ry (4thequat.)
+4.3+7- .q+r-l.q+r-5.[p-lY[qf[q-SY [r- ip.(P, (Q, Rff
+6.q+r .q+r-S.q+r-6.[p-2J'[q]'[q-2Y[r-2]' .(P, (Q, Ryy
+^.q+r-l.q+r-5.q+r-6.[p-SJ[q]'[q-iy[r-iy.{P, (Q, Ryy
+l.q+r-l.q+r-5.q+r-6. [q]* [rf . P, {Q, Ry ,
and the other is, in fact, the same equation with q, Q, r, R interchanged with
r, R, q, Q; the alternate + and — signs arise evidently from the terms
{R, Qy. =(Q. Ry; (R. Qy. — (Q. Ry; &c.,
which present themselves on the right-hand side.
It will be observed that the identity has been derived from the comparison of
the terms in a, which are the highest terms in as, the other terms not having been
written down or considered; but it is easy to see that an identity of the form in
question exists, and, this being admitted, the process is a legitimate one.
The preceding equations of the series are
[q + rY[qYQiP,Ry= 1. [p]' [?? P (Q, Ry (1st equation)
+ 1. [qY M' (P.QRy-,
[q + rY[q]'QiP,Ry= I. q+r . [p]' [qY P, (Q, Ry (2nd equation)
+ 2.q+r-i.[p-iY [qY [q-n[r- 1? (P, (Q, P)7
+ l.q+r-2. [qY [rY (P.QRy-,
[q + rY[qYQ{P,Ry= I -q+r .q+r-1. [pY [qY P, {Q. Ry (3rd equation)
+ 3. g+r .q+r-3.[p-lY[qY[q-mr-'2Y (P. (Q. ^)')'
+ 2.q+r-l.q+r-4,.[p-2Y[qY[q-lY[r-lY(P.QRy
+ l.q+r-3.q+r-4!. [qY [rp (P.QRy
681]
ON THE DERIVATIVES OP THREE BINARY QUANTIC8.
285
From these four equations the law is evident, except as to the numbers subtracted
from q + r. These are obtained, as explained above, in regard to the numbers added
to a + ;8 in the { } symbols ; transforming the diagrams so as to be directly applicable
to the case now in question, they become
0
0
01
1
1
2
1
2
11
2
01
012
012
0123
0123
03
3
015
4
0127
14
21
036
31
0158
34
12
156
22
0378
3
456
13
4
1678
5678
showing how the numbers are obtained for the equations 2, 3, 4, 5 respectively.
The first equation is
it + qr) Q (P, R) = pqP (Q, R) + qr [Q (P, R) + R (P, Q)],
viz. this is
0 =pq P (Q, R)-qrQ (RP) +qrR (P, Q)
+ (q' + qr)Q{R,P);
or, dividing by q, this is
0 =pP (Q, R) + qQ (R, P) + rR {P. Q),
which is a well-known identity.
We may verify any of the equations, though the process is rather laborious, for
the particular values
P = a;i(P+«) yi(p-«), Q = a^(«+P) yi(«-« H = a;4(»-+Y) yUr-y) .
thus, taking the second equation, we have, omitting common factors,
{Q,Ry= q + ^.q + ^-2.r-y.r-y-2
-2 .q + ^ .q-0.r + y.r—y
+ .q-^.q-0-2.r + y.r + y-2
= ^(r' - r) + y\q^ - q) - 20y (q - l){r -1)- qr (q + r - 2),
(P,(Q,Ryy = (q + ^.r-y.-.q-^.r + y)(p + a.q + r-0-y-2.-.p-a.q+r + 0+y-2)
= (fir -qy){a.q + r-2.-p.fi + y)
= o/3r (r + q—2) — ayq (q + r~2) —pr^ + p{l-r)fiy +pqy',
and from the first of these the expressions of Q{P, Ry and (P, QRy are at once
obtained. The identity to be verified then becomes
[q + r]» [q]^ {o' (f-r) + rf (p'-p) - 2a7 (p -l)(r- 1) -pr (p + r- 2)}
= {q+r)[qf[py{fi'(r^-r)+rf(q'-q)-2fiy(q-l)ir-l)-qriq + r-2)}
+ 2(q + r-l) [qY (p-l)(r-l) {afir (q + r - 2) -ayq(q + r -2)
- prfi' +p(q-r)0y+ pqy'^]
+ {q + r-2) [q]^ [rp [a? (q + r){q + r - I) + (0 + yy (p'-p)
-2aifi + y)(p-l)(q+r-l)-p(q + r)(p+q + r-2)},
286 ON THE DERIVATIVES OF THREE BINARY QU ANTICS. [681
which is easily verified, term by term ; for instance, the terms with a, jS, or 7, give
[q + ry[qYpr(p + r-2)= (q + r) [q]' [pY qr (q + r - 2)
+ {q+r-2) [q]^ [r]' p{q + r)(p + q + r-2),
which, omitting the factor (q + r) (q + r — 2) [q]* pr, is
(q + r-l)(p + r-2) = (p--i)q + (r-l)(p + q + r + 2);
viz. the right-hand side is
(p-l)q + (r-l)q + {r-l)(p+r-2), =(q + r-l)(p + r-2),
as it should be.
The equations are useful for the demonstration of a subsidiary theorem employed
in Gordan's demonstration of the finite number of the covariants of any binary form
V. Suppose that a system of covariants (including the quantic itself) is
P.Q,R,S,..;
this may be the complete system of covariants; and if it is so, then, T Jind V
being any functions of the form P'^^Ry..., every derivative (T, F)' must be a term
or sum of terms of the like form P'^Q^Ry...; the subsidiary theorem is that in order
to prove that the case is so, it is sufficient to prove that every derivative (P, Qy,
where P and Q are any two terms of the proposed system, is a term or sum of
terms of the form in question P'^C^Ry
In fact, supposing it shown that every derivative {T, F)* up to a given value
^0 of 0 is of the form P'^Ry..., we can by successive application of the equation
for Q{P, -B)*"^^ regarded as an equation for the reduction of the last term on the
right-hand side (P, QRy+\ bring first (P, QRy+\ and then (P, QRS)^+\.., and so
ultimately any function (P, F)'+\ and then again any functions (PQ, F)*+',
{PQR, F)*+',.., and so ultimately any function {T, Vy+\ into the required form
P°-Q^Ry...: or the theorem, being true for 6, will be true for ^+1; whence it is
true generally.
682] 287
682.
FOEMUL^ RELATING TO THE RIGHT LINE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XV. (1878),
pp. 169—171.]
1. Let \, /i, V be the direction-angles of a line; a, y8, 7 the coordinates of a
point on the line ; and write
a *= cos X, f = & cos V — 7 cos ytt,
6 = cos /i, g = 'i cos \ — a cos v ,
c =co8 V, h = a COS fi— /3cos \,
whence
a^ + b-^ + c^ = 1,
af+ bg + ch = 0,
or the six quantities (a, b, c, f, g, h), termed the coordinates of the line, depend upon
four arbitrary parameters.
2. It is at once shown that the condition for the intersection of any two lines
(a, b, c,f, g, h), {a, b', c', /', g', h'), is af' + bg' + ch'+a'f+b'g + c'h = 0.
3. Given two lines (a, b, c, f, g, h), (a', b', c', /', g', A'), it is required to find their
shortest distance, and the coordinates of their line of shortest distance.
Let
Ax + By + Cz + D = 0,
Ax + By + Cz + D' = 0,
be parallel planes containing the two lines respectively; then the first plane contains
the point a + rcosX, 0+rcosfi, 7-1-rcosi', and the second contains the point
a + / cos \', /8' + / cos fi', 7' + r' cos v' ; that is, we have
Aa+B^ +Cy +D =0,
Aa! + B^' + C7' + iy = 0,
J^ cos X + B cos fi +G cos v =0,
.4 cos X' + .B cos n' + Ccoa v' = 0,
288
FORMULA RELATING TO THE RIGHT LINE.
[682
which last equations may be written
Aa +Bb +Cc = 0,
Aa' + Bb' + Cc' = 0,
giving
or, if we write
and assume, as is convenient,
then
A, B, G=
A : B : C = bc'-b'c : ca'-c'a : ab'-a'b,
0 = aa' + bb' + cc',
A'- + B' + C'-==l,
be' — b'c cal — c'a ab' — a'b
where 0, = cosine-inclination, = aa' + bb' + cc'.
Hence, shortest distance =D — U
= A{ci-Oi') + B(0-^)+C(y-y')
1
1
V(i-^)
1
{(be' - b'c) (a - a') + (ca' - c'a) (/3 - /3') + {ab' - a'b)]
{a' (c/S - fry) + 6' (07 - ca) + c' (60 - a^)
+ a (c'^' - b'y') + b (ay' - c'a) + c (b'a - a'/S')}
(af + bg' + ch' + a'/+ b'g + c'h), = B suppose.
"V(l-^)
The six coordinates of the line of shortest distance are A, B, C, F, G, H, where
A, B, C denote as before, and F, G, H are to be determined.
Since the line meets each of the given lines, we have
Af +Bg +Ch +Fa +Gb +Hc = 0.
Af' + Bg' + Ch' + Fa'-\-Gb' + Hc' = 0,
and we have also
FA+GB + HG^O,
which equations give jP, G, H. Multiplying the first equation by b'C — c'B, the second
by Be — Cb, and the third by be' — b'c, we find
Here
(b'C - c'B) (Af+ Bg + Ch) + (Be - Cb) (Af + Bg' + Ch') + F
b'C - c'B = _ ^ {V (ab' - a'b) - c' (ca' - c'a)}
a,
b.
c
a'.
h'.
c'
A,
B.
C
= 0.
: ,^ _ g, {a (a'' + 6'= + c'-) - a' (aa' + bb' + ce')}
V(i-^)
(a - a' 6),
682]
and similarly
FORMULA RELATING TO THE RIGHT LINE.
289
cB-bG=P
Also, putting for shortness
V(l - ^)
{a' -ad).
n=
a, b, c
, n' =
a , b , c
a', b', c'
a', b', c'
f,9,h
/'. 5-'. h'
we have
Af+Bg + Ch = ^y.^^ n, Af + £/ + GK = ^^.^ Xi',
V(i-^)
and finally, the determinant which multiplies F is
V(i-<
-^^j^^ {(6c' - Vcf + (ca' - c'a)^ + (ab' - a'bf] = ^^^^^^ 0- - ^), = V(l - ^^^
We have thus the value of F; forming in the same way those of 0 and H, we
find
F =
6 =
-1
(1 - ^)»
-1
(1-^)5
-1
{(a - a'0) D, + {a'- ad) n'}.
{(b - b'e) il+(b'- be) D,'],
^ = (jziW^i ^^^ ~ "'^^ " + («'- c^) "'!>
which, with the foregoing equations for A, B, C, give the six coordinates of the line
of shortest distance.
C. X.
37
290 [683
683.
ON THE FUNCTION axe sin {x + iy).
[From the Quarterly/ Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 171—174.]
The determination of the function in question, the arc to a given imaginary
sine, is considered in Cauchy's Exercises d' Analyse, &c., t. iii. (1844), p. 382 ; but it
appears, by two hydrodynamical papers by Mr Ferrers and Mr Lamb, Quarterly
Mathematical Jownal, t. Xlli. (1874), p. 115, and t. xiv. (1875), p. 40, that the question
is connected with the theory of confocal conies.
Taking c = 'J{a'^ — If) a positive real quantity which may ultimately be put = 1,
the question is to find the real quantities f, r), such that
^-\-ir) = arc sin -{x + iy),
or say
a; + ty = c sin (f + ii}),
so that
a; = c sin f cos itj, iy — c cos f sin irj.
It is convenient to remark that if a value of f + ti? be f' + tV> then the general
value is 2mir + ^' + ir)' or {2m, + 1) tt — (^' + ir)') ; hence, r) may be made positive or
negative at pleasure ; cos irj is in each case positive, but - sin it) has the same sign
as 1] ; hence cos f has the same sign as x, but sin ^ has the same sign as y or the
reverse sign, according as 17 is positive or negative ; for any given values of x and y,
we obtain, as will appear, determinate positive values of sin* f and cos'' f ; and the
square roots of these must therefore be taken so as to give to sin f, cos ^ their
proper signs respectively.
683] ON THE FUNCTION ATC sm (x + ly). 291
Suppose that \, fi are the elliptic coordinates of the point (x, y); viz. that we
have
a?
a' + X
+ 6-
= 1,
a^
+ ,,
f
= 1.
a« + /i h^ + fi
where a^ + \, h- + X, and a^ + z* are positive, but h' + jx is negative. Calling p, a the
distances of the point x, y from the points (c, 0) and (— c, 0), that is, assuming
p = ^{{x-cy + y%
<r = sl[{x + c-f + y%
then we have
VCa^ + X) = ^ (o- + p), whence also ^/{b^ -\-\) = ^ i,J{{<t + pf - ^c^],
>j{a? + ^) = i (<r - p), „ V(6= + m) = 4 V{(<r - P)-' - 4c=}>
which equations determine \, /i as functions of «, y.
Now we have
p<7 = VlCa^ + 2/^ - c=)= - 40^^=} = VK«^ - 2/' - c')' + ^oi?y^],
^» + ff' = 2 (;»= + y + c^) ;
substituting herein for x, y their values
c sin ^ cos tT;, — ci cos f sin t?;,
we find
a? — y'^ — & = c' (sin" f cos' it) + cos" ^ sin'' it] — (sin° f + cos' f ) (sin' i^j + cos' %t\)\
= — c' (sin' ^ sin' ii; + cos' f cos' it;),
whence
{a?-y^- c'y = c* (cos' ^ cos' ir) + sin' f sin' irjy
+ "Wy' — ic* sin' f cos' ^ sin' irj cos' i?;
= c* (cos' ^ cos' t'lj — sin' f sin' t'lj)'.
Hence
2p<r = 2c' (cos' f cos' m; — sin' f sin' t'lj),
and
p' + 0-' = 2c' (sin' f cos' i?; - cos' f sin' t?? + 1) ;
hence
(p + of = 2c' (cos' it} — sin' ir) + 1), = 4c' cos' i?;,
(p - (7)' = 2c' (sin' f -cos'^ +1), =4c'sin'^.
Consequently
a' + \ = c' cos' ii/, and thence 6' + \ = — c' sin' it;,
a' + /* = c' sin' f „ 6' + /it = - c' cos' |,
values which verify as they should do the equations
aP ?/'
a' + \ 6' + \
= 1,
^-+-^ =1
a' + /* 6' + /i
37—2
292 ON THE FUNCTION are am (x + iy). [683
viz. these become
+ ■ . , ■ = sin' f + cos' f = 1,
c* cos' I'lj — c" sin' i»7
■ . ■ . ^ + —^ — T-i, = cos' it; + sin' irj, = 1.
tf'sin'f -c*cos'^
The same equations, or as we may also write them,
\ = — a' sin' irj — 6" cos' it),
yit = - a' cos' f - 6' sin' ^,
determine 77 as a function of \, and f as a function of fi ; \ fi being by what
precedes, given functions of x, y.
Or more simply, starting from the last-mentioned values of \, (i, and substituting
these in the expressions
^ _ «' + X ■ g' + /it , _ 6' + \ ■ 6' + /M.
^ - a'-6» ' ^ ~ 6'-a' '
we find
a? = c^ sin' ^ cos' it;, y' = — c' cos' ^ sin' i?;,
or say
a; = c sin f cos irf, iy = c cos f sin it),
whence
a; + ty = c sin (f + ii;),
the original relation between x, y and f, 7;.
684]
293
684.
ON A KELATION BETWEEN CERTAIN PRODUCTS OF
DIFFERENCES.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 174, 17.5.]
CoNSiDEK the function
where
3 / abc . de
I + bed . ea
J + cde . ah
+ dea . be
+ eab . cd
V <
abd . ce \
+ bee . da
+ cda . eb
+ deb . ac
+ eac . bd
abc = (a — b)(b — c) (c - a),
ab =(a-b){b-a), =-(a-by,
&c. ;
therefore
abc = bca = cab = — bac, &c. ;
ab =ba.
It is to be shown that the function vanishes if e = d. Writing e = d, the value is
3 (bed . da + dab . cd) — abd . ed
— bed . da
— eda . db
— doc . bd,
294 ON A RELATION BETWEEN CERTAIN PRODUCTS OF DIFFERENCES. [684
viz. this is
3 bed . ad — abd . cd
+ 3 abd . cd — bed . ad
— 2acd . bd
= 2 bed . ad — 2acd . bd + 2abd . cd
= 2 (bed . ad + cad . bd + abd . cd),
which is easily seen to vanish; the value is
(b-c)(e-cr){d- b) (a -dy = -{b- c) (a - dy (b -d) {e-d)
+ (c -a)(a-d){d-e){b-df -(c-a)(a-d) (b-dy(c-d)
+ (a -b)(b - d){d - a) (c - dy -(a-b){a-d) (b-d) (c-dy-.
viz. omitting the factor (a — d)(b — d) (c — d), this is
= -{b-c)(a-d)
- (c - a) (b - d)
— (a — 6) (c — d),
which vanishes. Hence the function also vanishes if e = a, or a = 6 or 6 = c, or c — d;
and it is thus a mere numerical multiple of {a —b){b — c) (c — d){d — e) (e — a), or say it
is = Mabcde.
To find M write e=c, the equation becomes
Sabc . de — eda . eh = Mabcde, = Mabe . de,
+ Sbed . ca — ac
+ Sdea . be
+ Scab . ed,
Qabc . dc + 4:dbc . ac + 4eade .be = M . abc . de,
giving M = 10. In fact, we then have
— 4a6c . dc + 4d6c . ac + 4^adc .bc = 0,
that is,
— abc . dc — bde . ac — doc .be = 0,
which is right. And we have thus the identity
viz. this is
or say
3 / abc . de\ — I abd . ce\ = 10 . abcde,
3 [abcde] — [acebd] = 10 {abcde}.
abc . de
+ bed . ea
J+ede. ab
^ ^
+ dea . be
+ eab . cd
abd . ce \
+ bee . da
+ eda . eb
+ deb . ac
+ eac . bd
685] 295
685.
ON MR COTTERILL'S GONIOMETRICAL PROBLEM.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 196—198.]
The very remarkable formulae contained in Mr Cotterill's paper, "A goniometrical
problem, to be solved analytically in one move, or more simply synthetically in two
moves," QuaHerly Mathematical Journal, t. vii. (1866), pp. 259 — 272, are presented in
a form which, to say the least, is not as easily intelligible as might be ; and they
have not, I think, attracted the attention which they well deserve.
Using his notation, except that I write for angles small roman letters, in order
to be. able to have the corresponding italic small letters and capitals for the sines
and cosines respectively of the same angles, we consider nine angles
a, b, c,
d, e, f,
X, y, z,
which are such that the sum of three angles in the same line, or in the same
column, is an odd multiple of tt. Of course, any four angles such as a, b, d, e are
296 ON ME cotterill's goniometrical problem. [685
arbitrary, and each of the remaining angles is then determinate save as to an even
multiple of it.. And it may be remarked that these angles a, b, d, e may represent
the inclinations of any four lines to a fifth line, and that the remaining angles are
then at once obtained, as in the figure. The small roman letters are here used to
denote as well angles as points, being so placed as to show what the angles are
which they respectively denote ; the points *, * are constructed as the intersections of
the lines ac, be by the circle circumscribed about fxy, and the angle z is the angle
which the points *, * subtend at x or y. It will be observed that the sum of the
three angles in a line or column is in each case = it.
But this in passing : the analytical theorem is, first, we can form with the sines
and cosines of the angles in any two lines or columns a function (S presenting itself
under two distinct forms, which are in fact equal in value, or say S is a symmetrical
function of the two lines or columns, viz. for the first and second lines this is
SQ' ^' ''^ = d?Ahc + ^Bca+fCah
= a^ Def + h-Efd + d'Fde,
where, as already mentioned, a, A denote sin a, cos a, and so for the other letters.
Secondly, if to the S of any two lines or columns we add twice the product of
the six sines, we obtain a sum M which has the same value from whichever two
lines or columns we obtain it ; or, say M is a symmetrical function of the matrix of
the nine angles. Thus
M = sh ^' J) + 2a6cde/;
which is one of a system of six forms each of which (on account of the two forms
of the S contained in it) may be regarded as a double form, and the twelve values
are all of them equal. There are, moreover, 15 other forms, of M, viz. 3 line-forms,
such as
hcdx + caey + ahfz (belongs to line a, b, c),
3 column-forms, such as
dxbc + xaef+ adyz (belongs to column a, d, x),
and 9 term-forms, such as
e^^= +f^y^ + lefyzA (belongs to term a),
and the 12-hl5, =27 values are all equal.
The several identities can of course be verified by means of the relations between
the nine angles, or rather the derived sine- and cosine-relations
C^ab -AB,
c =aB + bA, &c.
685] ON MR cotterill's goniometrical problem. 297
/a, b, c\
as, as regarus tne two loiiiis oi o (
written
(a b c\
' ' f), the identity to be verified may be
c (dM6 + e'Ba - cFde) =f(a'De + h^Ed -/Cab).
Proceeding to reduce the factor a'^De + li^Ed —fCah, if we first write herein f=eD + dE,
it becomes
a-'De + h^Ed-{eB + dE)Cah,
which is
= aDe{a-hC) + bEd (6 - aC),
and then writing C=ab-AB, we have a-bG = a(l-b^) + bAB, =B(aB + bA), =Bc;
and, similarly, b — aG = Ac; whence the term is =c{aeBB + bdAE); or, in the equation
to be verified, the right-hand side is =cf{aeBD + bdAE), and by a similar i-eduction,
the left-hand side is found to have the same value.
The paper contains various other interesting results.
C. X. 38
298 [686
686.
ON A FUNCTIONAL EQUATION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XV. (1878), pp. 315 —
325 ; Proceedings of the London Mathematical Society, vol. ix. (1878), p. 29.]
I WAS led by a hydrodynamical problem to consider a certain functional equation ;
viz. writing for shortness a;, = '—j , this is
, , , .Ax + B
I find by a direct process, which I will afterwards explain, the solution
, _A V{(a - dy + 46c} (AD - BO) p sin ^^ sin t;^ d^
9^- C'"'^ CidG-cD) Jo sinfisinhTTi '
where f is a constant, but f, rj are complicated logarithmic functions of x (^, t), ^
depend also on the quantities a, b, c, d, G, D) ; sinh irt denotes as usual the hyperbolic
sine, ^ (e"' — e~^).
The values of f, -q, f are given by the formulae
1 _ g' + d^ + 26c
\ ad — be '
a = ax + b, h = - dx + b,
c = ca;+d, d= ex —a,
W=Ca + Bc,
Z = Ch + Dd,
R= \c + Xd,
S = - c- d,
ie'= w+^z,
S' = -W-\Z,
686] ON A FUNCTIONAL EQUATION. 299
which determine X, R, S, R', S' and then
There is some difficulty as to the definite integral, on account of the denominator
factor sin ^t, which becomes = 0 for the series of values t = -rr , but this is a point
which I do not enter into.
I will in the first instance verify the result. Writing x^ in place of x, and
taking ^i, t)^ to denote the corresponding values of ^, tj, it will be shown that
?i = ?, Vi = V + 2?, see post, (1).
Hence in the difference <f)x — <f)Xi we have the integral
/"sin ^t {sin Tjt — sin (77 + 2f) t} dt
sin ^t sinh -irt '
(where and in all that follows the limits are x , 0 as before) ; here, since
sin rjt — sin (17 + 2^) t= — 2 sin ^t cos (17 + ?) <,
the factor sin ^t divides out, and the numerator is
= -» 2 sin ficos (rj+^t,
which is
= sin (7/ + f- ^)« -sin (17 + ^+ ^)t.
Hence the integral in question is
^ r3m(r) + ^-^)tdt f sin (rj + ?+ f ) ( dt
J sinh -Trt J sinh irt
Now we have in general
1 _ , f sin at dt _
1 + exp. a J sinh irt '
(this is, in fact, Poisson's formula
_ 1 _ 1 _ 9 f sin^2nlog 0 + log k)t. dt
in the second Memoir on the distribution of Electricity, &c., M4m. de I' Inst, 1811,
p. 22.3) ; and hence the value is
1 1
1 + exp. (,7 + ?- f) ^ 1 + exp. (17 + ?+ ?) '
or since
<. 1 » .1 RR ^ , , RS
77 + ?=l0g\ + ^l0g-^, , ? = il0g^;^,
we have
'7 + ? - ? = log X + i log ^;, = log \ -^ ,
R^ R
77 + S'+^ = logX + ilog^ =logX-^,
38—2
300 ON A FUNCTIONAL EQUATION. [686
and the value is thus
1 1 {RS'-R'S)\
Hence, from the assumed value of <f>x, we obtain
, _A V{(a - dy + 46c} (AD - BC) jRS' - R'S) \
<px-,px,-^(x X,) C(dC-cD){XR' + S')(KR + S)
We have
RS' - li'S=^^~^^_^^'^^' (dC- cD) {caf + (d-a)a;- b],
RX+ S = (V - 1) (ca; + d), see post, (2),
R'\+ S' = (\-l)(a + d)(Gx + D),
or since
ca?+(d — a)x — b_
this is
. ^ A. , >J{(a-dy + ^bc}(AD-BC) (a + d)\ ^,
But from the value of \,
\ _ . ad — be
y^'^1 ~ {a + d),/{(a-dy + ^bc} '
and the equation thus is
\i^ AD-BC) _ Ax + B
(px - <f>x, - (« - X,) 1^ - Q^Q^_^ ^)| . -(^-'^•■'Oic + i)'
as it should be.
(1) For the foregoing values of ^i, t;,, we require i?,, (Sij, i?i', (Sf/, the values
which R, 8, R', 8' assume on writing therein x^ for x. We have
Ri= \ (cx^ + d)+ (cajj — a),
Si = — (cXi + d) — \ (cxi — a):
substituting for a;, its value, we find
Ri{cx + d) = (a + d)\{cx + d)-{ad-bc){X + l),
or writing herein
, , (a + dyx
this is
R,(cx+d) = ^^^R;
and similarly
S,icx + d) = l^^S.
686] ON A FUNCTIONAL EQUATION. 801
We have in like manner
i?j'= W, + ^Z„ where W^ = C( aa;, + b) + D(ca;, + d),
S,' = -W,-\Zu where Z, = C{-dx, + b) + D (ca;,-a).
Substituting for Xi its value, we find
TTi (ex + d) = C[{a + d) (ax + b)- (ad - be) x] + ]) [(a + d) (ex + d)- (ad - be)],
Z,(cx + d) = C[ -(ad-bc)x]+D[ -(ad- be)] :
hence, substituting for ad — be as before,
W,(cx + d) = ^^^J(\ + iy W-(a + d)X(Cx+ £>)},
^'^'^ + '^^="(^T^f -(a + d)X(0^ + D)},
whence without difficulty
consequently
R^i _ RS' . , , . t _ t
RiRi ,,IiR' 1 ^ , oi- ,
which are the formulae in question.
(2) For the value of RS' - R'S, we have
R8'-R'S=(Xc + d)(-W-XZ)-(-\d-c)(w+~)
= ( - V + -J cZ+(\ + 1) ((d- c) W-dR}
= -(\-l){(l+\ + ^)c^ + (c-d) TT + d^l;
or substituting for X^ + r- , Z and W their values, this is
= ~^-^^ l(a' +d'' + ad + bc)c (hC + dD)
+ (ad- be) [(c - d) (aC + cD) + d (hC + dD)]].
In the term in { }, the coefficient of C is
[(a' +d^ + ad + bc)h+ (ad - 6c) a] c - d (a - b) (ad - be)
= (a + d)(db- bd) c-(a + d)dx (ad - be),
302 ON A FUNCTIONAL EQUATION. [686
and similarly the coeflScient of i) is
[{a*+d^ + ad + be) d + (ad - be) c]c-d{c-d) {ad - be)
= {a + d) (ad -ch)c — (a + d)d (ad — be).
Hence the whole term in { } is
= (a + d:) {[(db -bd)c- d(ad -be) x]C + [(ad - eh) c-d (ad -be)] D},
which is readily reduced to
(a + d) (sA -he) (-dC + cD);
also
ad — be = (a + d) {cx^ + (d — a)x — b] ;
so that we have
RS'-B'S = ^^^^^^^^\dC-cD)[ca^ + (d-a)x-bl
which is the required value of RS' — R'S ; and there is no difficulty in obtaining
the other two formulae,
R\ +S =(X'-l)(cx + d),
R'\ + S' = (X -l)(a + d)(Gx + D);
the verification is thus completed.
To show how the formula was directly obtained, we have
Ax+B _A_AD -BG 1
Cx + D~C G ~ Gx + D
= Y^ + px suppose ;
the equation then is
tj)X — <})Xi = y^ (a; — a;j) + (« — Xi) l3x.
Hence, if Xi, x^, Xa, ... denote the successive functions 'drx, '^'x, '^x, &c., we have
<f)Xi — (f>X„ = ji(Xi — x^ + (x-i — X^) ^Xi,
^Xi — Axi = -^(Xi- X3) + (x^ - Xi) 0Xi ,
whence adding, and neglecting <f>x^ and x„, we have
<f>X = -p X + [(X — Xi) I3x + (Xi — X^) /SiTj + (Xs — X,) ^Xi +...],
where the term in [ ], regai-ding therein x^, x^, x,,... as given functions of x, is
itself a given function of x ; and it only remains to sum the series.
Starting from
— cv _"^ + ^
' CX+ d'
and writing
1 _a'' + d^ + 2be
\ad—be'
686] ON A FUNCTIONAL EQUATION. 303
then the nth function is given by the formula
_ (X"+' -l)(aa; + b) + (X"- \) j-dx + b)
Xn - '*«a; - ^^„+, _ j^ ^^^ ^^^^ ^^„ -\){cx- a)
_(\"+'-l)a + (X."-\)b
~ (\"+i - 1) c + (X" - \) d
_ \» P + Q
ifP = \a + b, Q = — a — \b, and as before i? = \c + d, /Si = — c — Xd.
I stop to remark that \ being real, then if \ > 1 we have X" very large for n
P
R
P . . .
very large, and a" = p which is independent of n ; the value in question is
_X(aa;+b) + {-da;+b)
^"~X(ca;+d) + ( cx-a)'
which, observing that the equation in X may be written
\a-d _b(\+l)
c(X + l)~ \d-a '
i
is, in fact, independent of x, and is =-7- r^r or — ^ -: we have Xn-i=x,., or
^ c(X+ 1) \d — a
calling each of these two equal values x, we have
ax + b
CX + a
which is the same equation as is obtainable by the elimination of X from the equations
_ Xa-rf _6(X + 1)
'^ ~ c (X + 1) ~ Xd-a ■
The same result is obtained by taking X < 1 and consequently a;n = ^ •
We find
_X"-'P + Q X^P + Q
(x»-'i? + S) (x»ij; + /^ '
where
PS- (2i? = - (X^- l)(ad- bc) = - (X^- 1) (a + d) {ca;^ + (cZ - a)«- 6) ;
and therefore
_(\-l)(X'-l){a + d){ca^+(d-a)x-b}V
"'"-' """ X(X»->iJ + S)(X»i2 + ,S)
Also
fl. - ^^-^^ _J:
304 ON A FUNCTIONAL EQUATION. [686
where
„ r,_C (\»-'P + Q) + D (\"-'.R + S)_ J? V + 8'
^, CP DR ^( b\ „/ d\
8'='CQ + DS, =Ci-&-h\) + D{-c-d\);
viz.
R'^W + lz, S' = -W-XZ,
A.
where ^ and W denote &G + cD and hC + dD as before.
We hence obtain
(\ - 1) (X' - 1) (g + d) (ca^ + (d - g) a; - 6} \"
^ X (12X» + S) (ii'X" + -S")
- {AD - .BC)
C
(X - 1) (X= - 1) {a + d) {car" + (d -a)x- h] (RS' - R'S) X»
\(RS'-R'S) (iJX" + ,S)(iJ'X» + -S')'
or, substituting for RS' — R'S its value in the denominator, this is
AD-BC {ad - be) (X" - 1) (RS'-R'S)\^
{Xn_, x^)^xr,- ^ {a + d)X{cD-dC){R\^ + S){R'X^ + S')
'^{{a-df + 46c} {AD - EG) {RS' - R'S) X"
C{cD-dC) (J?X" + S)(i?'X» + ,S')'
and thence
. __A ^/\{a - dy + 46c} {AD- BC) ^ {RS' - R'S) X"
'"^ g'' G{cD-dC) ''(iJX» + fif)(i?'X» + <S')'
the summation extending from 1 to oo .
Now the before-mentioned integral formula gives
1 _ , /"sin {n log X + log k) t dt
ITXX" ~ * ~ i sinh iri '
1 _ , /"sin {n log X + log A;') < dt
1 + A;'X» ~ *~j sinh-Trt '
/? 7?'
Taking the difference, and then writing /c = -^ , k' = -^ , we have under the integral
sign
sin [n log X + log -^J t — sin (n log X + log -^, j t,
686] ON A FUNCTIONAL EQUATION. 305
which is
= 2 sin ^ ^log -^, j t cos [ii log X + ^ log ^^ j t,
which attending to the before-mentioned values of f, t), ^ is
= 2 sin ^t cos (2nf- ^ + v)t,
and the formula thus is
<Sf S' (RS'-RS)\» _ [2 sin ^t cosi2n^-^+ 7)) tdt
rs) v ^ _ rs
2'X» + <S') J "
EX.'^ + S R'X^ + S" (i?\" + /S) (fi'X» + (S') j sinhTTt
We have here
cos (2nf — f-l- 1;) « = cos 2«(f< cos (17 — f) < — sin 2n5i! sin (17 - 5) f,
whence summing from 1 to oc by means of the formulae
cos2f< + cos4f«+... =-i,
sin 2^t+ sin 4?« + ... = ^ cot ?«,
(which series however are not convergent), the numerator under the integral sign
becomes
sin ^t {— cos (v~ Ot — cot ft sin (rj — ^)t},
which is ,
_ sin f < sin r/t
and the formula thus is
_ (RS' ^S)J^ _ _ [ sin fi sin rjt dt .
(RX" + S) (E'X« + S') "" j sin ?< sinh 7r« '
and we therefore find
. _A V{(a -dy + 46c} (^Z) - BG) C sin ^t sin rjtdt
<f>x-^x+ C{cD-dG) j sin ?« sinh 7rr
which is the result in question.
The solution is a particular one ; calling it for a moment (^a;)' then, if the
general solution be <l>a; = ^x + {<f>x), it at once appears that we must have ^x — <I>a;, = 0 ;
DO'
and as it has been shown that -^ra is * function of x which remains unaltered by
Kb
o,„ j , an arbitrary
7? Si'
function of -p,^. Hence we may to the foregoing expression of </>« add this term
(RS'^
■sj'
Postscript. The new formula
_ (X''+' -l)(ax + b) + (X" - X) (- dx + h)
(X»+'-l)(ca; + rf) + (X»-X)( cx-a)'
^ 1 a'' + #-f26c
where ^ + X=""c^^6^'
C. X. 39
306 ON A FUNCTIONAL EQUATION. [68G
for the nth repetition of ^x, = -,, is a very interesting one. It is to be
remembered that, when n is even the numerator and denominator each divide by
X— 1, but when n is odd they each divide by \=— 1; after such division, then further
dividing by a power of X, they each consist of terms of the form X* + — , that is,
X
they are each of them a rational function of X + r^ . Substituting and multiplying
A.
by the proper power of ad — be, the numerator and denominator become each of
them a rational and integral function of a, b, c, d of the order w + 1 when n is even,
but of the order n when n is odd ; in the former case, however, the numerator and
denominator each divide by a + d, so that ultimately, whether n be even or odd, the
order is =n as it should be.
For example, when n = 2, the value is
(X'-l)a + (V-X) b ^ (X'' + X+l)a + Xb ^ (.^"'"x"'" ^j ^"^^
(X'-l)c + (X'-X)d' (v + x+l)c + Xd' ~/^^l^i\^,^j'
or, as this may be written,
fx + -+2ja-a + b
fx + - + 2Jc-c + d
where, observing that
X + - 4- 2 = ^^^ , - a + b = - (a + d) «, -c + d = -(a + d),
the numerator and denominator each divide by a+d, and the final value is
_ (a + d){ax-\- b) — {ad — hc)x _ {a? + be) x + b (a + d)
~ (a + d) {ex + d) - {ad -be) ' ~ c{a + d)x + bc+ d' '
which is the proper value of "^^x. But, when ?i = 3, the value is
X + ^)a + b
and this is
or finally
(X^ - 1) a + (X' - X) b _(X- + l)a + Xb _V X
(X«-l)c + (X»-X)d' ~ (X» + 1 )"c + Xd ' ~T iT,",'
_ {a^ + d^+2bc){aa;+b) + {ad-be){-dx + b)
~ {a- + d' + 26c) {ex + d) + {ad - be) { ex -a)'
_ {a* + 2abe + bed) x + b{a^ + ad + bc +d')
~ c{a'' + ad + bc-\-d')x + {abe + 2bcd + d') '
which is the proper value of ^V.
687] 307
687.
NOTE ON THE FUNCTION '^ = a'{c-x) ^{c{c-x)-b%
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 338—340.]
Starting from the general form
7a; + 6
we have
^ (X"+' - 1) (euc + /3) -f (\» - \) (- g-B + y3)
(X"+>-l)(7a! + S)+(\''-\)(7a;-a) '
where
^ ^ 1 _ a' + 8^ + 2^87
X aS — ^y
For the function in question
a^ (c - x)
aa; =
c(c-a;)-6»'
(a form which presents itself in the problem of the distribution of electricity upon
two spheres), the values of a, jS, 7, S are
a==-a% ^ = a\ y = -c, B^c'-b^;
the equation for \ therefore is
1 _ a* + (c» - ¥y - 2a-c»
or, what is the same thing,
(\ + iy _ {a' + 1'- ey
X ~ a=6»
39—2
308 NOTE ON THE FUNCTION '^x = a' {c-x)-i-{c{c-x) -¥\. [687
Suppose that a, b, c are the sides of a triangle the angles whereof are A, B, C;
then c^ = a' + 6* — 2ah cos C, or we have
^ — r— - =4co8'C;;
or, writing this under the form
V(X.) + -y/^v = 2 cos G,
the value of X is at once seen to be =e"^; and it is interesting to obtain the
expression of the nth function in terms of the sides and angles of the triangle.
The numerator and the denominator are
X»P + Q,
X»i2 + S,
where
P= X(aa; + y3)+ (-&; + /3), R= x(ya; + S)+ r^x-a,
Q = - (aa; + /3)-X(-Sa; + /9), S =- (yx + B) - \ (yx - a).
Hence, writing the numerator and the denominator in the forms
Xi»P + X-i"Q,
these are
(P + Q) cos nC + {P- Q) i sin nC,
(R + S)coanC+(R-S)isinnC;
viz. they are
(X - 1) (a + 8) a; cos nC+{X + 1) {(a -B)x+ 2^} i sin nC,
(X - 1) (a + S) . cos n(7+ (X + 1) {27a; - (a - 3) ) i sin nC,
or, observing that --- =itanC7 and removing the common factor i(X + l), they may-
be written
tan G{a + S)x cos nC +[{a-Z)x-\- 2/3} sin nC,
tanO(a + S). cos»iC'+ j27a;-(a-S) }sinre(7.
Substituting for a, y3, 7, 8 their values, these are
tan C {(c» - a'' - &") a; cos nC] + {(6» - a= - c=) a; + 2a?c] sin nC,
tan C {{d" -a-- 6») . cos nC} + {- 2ca; - (6^ -a' - c=)J sin nC,
= tan C {- ab cos Ca; cos nC } + {— ac cos S . a; + a^c j sin nC,
tan C j— 06 cos (7a; cos 71C } + {- ex + ac cos B } sin nC,
= x \— ab sin C cos nC — ac cos £ sin nC] + a^c sin nC,
— ex sin nO + [ac cos J? sin nC — ab sin Ocos nC] ;
687 J NOTE ON THE FUNCTION ^X = a- {c- x) -^ {c {c - x) -¥}. 309
or, writing herein bsinC = c sin B, these are
— aca; {sin B cos n(7+ cos 5 sin ?iCj +a^c sin nC,
— ex sin nC + ac jcos B sin nC — sin B cos nC},
whence finally
- „ _ a' sin nC - cw; sin (mC + B)
a sin (n(7 — i?) — a; sin reC "
As a verification, writing n = 1, we have
a^sinC — iix sin A
^a; =
or observing that
a sin (C — B) - xsinC
sin A
a'c — acx -. — -r,
sm G
sin(C-£) '
ac ^^ — 7s — — ex
sm U
sin ((7-5) . ,„
ac — \ „ - = c= - 6-,
sm C7
(for this is sin A sin {C — B) = sin'' C — sin' B), we have
c»-6»-ca;
as it should be. If in the formula for ^"a; we write x=0, we have a formula given
in the Senate-House Problems, January 14, 1878 : it was thus that I was led to
investigate the general expression.
310 [688
688.
GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE.
[From the QuaHerly Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 340—347.]
I CONSIDER, from a geometrical point of view, the phenomena of a solar eclipse
over the earth generally; attending at present only to the penumbral cone, the
vertex of which I denote by V. It is convenient to regard the earth as fixed, and
the sun and moon as moving each of them with its proper motion, and also with
the diurnal motion. The penumbral cone meets the earth's surface in a curve which
may be called the penumbral curve; viz. when the cone is not completely traversed
by the earth's surface, (that is, when only some of the generating lines of the cone
meet the earth's surface), the penumbral curve is a single (convex or hour-glass-
shaped) oval ; separated, as afterwards mentioned, into two parts, one of them lying
away from the sun, and having no astronomical significance ; but when the cone is
completely traversed by the earth's surface, then the penumbral curve consists of two
separate (convex) ovals; one of them lying away from the sun and having no
astronomical significance, the other lying towards the sun. The intermediate case is
when the cone just traverses the earth's surface, or is touched internally by the
earth's surface ; the penumbral curve is then a figure of eight, one portion of which
lies away from the sun, and has no astronomical significance : there is another limiting
case when the cone is touched externally by the earth's surface, the penumbral curve
being then a mere point.
It is necessary to consider on the earth's surface a curve which may for shortness
be termed the horizon ; viz. this is the curve of contact of the cone, vertex V,
circumscribed about the earth ; it is a small circle nearly coincident with the great
circle, which is the intersection by a plane thiough the centre of the earth at right
angles to the line from this point to the centre of the sun.
Regarding F as a point in the heavens, capable of being viewed notwithstanding
the interposition of the moon ; the horizon, as above defined, is the curve separating
688] GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. 311
the portions of the earth's surface for which V is visible and invisible respectively.
The horizon does or does not meet the penumbral curve, according as this last
consists of a single oval or of two distinct ovals; viz. in the latter case the horizon
lies between the two ovals, in the former case the horizon traverses the area of the
oval (separating this area into two parts), thus meeting the oval, or penumbral curve,
in two points, or say these points separate the oval into two parts ; from any point
of the one part V is visible, from any point of the other part V is invisible ; and
from each of the two points themselves V is visible as a point on the horizon in
the ordinary sense of the word ; that is, there is an exterior contact of the sun
and moon visible on the horizon. It is to be observed that, in the limiting cases
where the penumbral curve is a mere point and a figure of eight respectively, the
horizon passes through the mere point and through the node of the figure of eight
respectively.
The two points of intersection of the penumbral curve with the horizon may
for shortness be termed critic points. The lines which present themselves in a diagram
of a solar eclipse, (see Nautical Almanac:) are the "northern and south lines of simple
contact," say for shortness the " limits " ; viz. these are the envelope or, geometrically,
a portion of the envelope of the penumbral curve; and the lines of "eclipse begins
or ends at sunrise or sunset," say for shortness the critic lines ; viz. these are the
locus of the critic points.
The point V considered as a point in the heavens is a point occupying a position
intermediate between those of the centres of the sun and moon ; hence referring it
to the surface of the earth by means of a line drawn from the centre, its position
on the earth's surface is nearly coincident with that point to which the sun is then
vertical ; and its motion on the earth's surface is from east to west approximately
along the parallel of latitude = sun's declination, and with a velocity of approximately
15° per hour. For any given position of V on the earth's surface, describing with
a given angular radius nearly = 90"^ a small circle (nearly a great circle), this is the
horizon; as V moves upon the surface of the earth, the horizon envelopes a curve
which is very nearly a parallel, angular radius = sun's declination (there are two such
curves in the northern and southern hemispheres respectively, but I attend only to one
of them in the proper hemisphere, as will be explained), say this is the horizon-envelope;
the horizon in each of its successive positions is thus a curvilinear tangent (nearly
a great circle) to this horizon-envelope. If for a given position of V, and also for
the consecutive position we consider the corresponding horizons, these intersect in a
point K on the horizon-envelope, and the horizon for V is the circle centre V and
angular radius VK ; K is a, point which is very nearly upon, and which may be
taken to be upon, the meridian through V; the horizon may be regarded as a
tangent which sweeps round the horizon-envelope ; to each position thereof there
corresponds a position of V, and consequently also a penumbral curve; and (when
this is a single oval) the horizon meets it in two points, which are the critic points.
It is to be added that, if for a given position of the horizon we consider as well
K as the opposite point K^, (viz., if, lies on the great circle KV), then the points
K and if, divide the horizon into two portions; for any point on one of these portions
312 GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. [688
V (considered as a point in the heavens) is rising, for a point on the other of them
it is setting; and for the points K and K^ respectively it is moving horizontally;
that is, first rising and then setting, or vice versd.
A solar eclipse is of one of two classes ; viz. either the penumbral cone completely
traverses the earth, so that towards the middle of the eclipse the penumbral curve
consists of two separate ovals : or the penumbral cone does not completely traverse
the earth, so that throughout the eclipse the penumbral curve consists of a single
oval only. In the former case, we have to consider the commencement, during which
the penumbral curve passes from a mere point to a figure of eight: the middle,
during which it passes from a figure of eight through two ovals to a figure of eight :
and the termination, during which it passes from a figure of eight to a mere point.
In the latter case, we consider the whole eclipse during which the penumbral curve
passes from a mere point through a single oval to a mere point.
In an eclipse of the first class: for the commencement, the penumbral curve is
at first a mere point (point of first contact) ; it then becomes a convex oval, each
oval in the first instance inclosing the preceding ones, so that there is not any
intersection of two consecutive ovals. We come at last to an oval which is touched
north by its consecutive oval, and to an oval which is touched south by its consecutive
oval (I presume that the contacts north and south do not take place on the same
oval, but I am not sure); and after this, the ovals assume the hour-glass form, each
oval intersecting the consecutive oval in two points north and two points south ; the
ovals thus beginning to form an envelope or limit. There are on each of the ovals
two critic points, and we have thus a critic curve commencing at the mere point
(point of first contact) and extending in each direction from this point. The point,
where an oval is touched by the consecutive oval, is not so far as appears a critic
point; that is, the critic curve does not at this point unite itself with the envelope
or limit. But the critic curve comes subsequently to unite itself each way \vith the
limit ; and, since clearly it cannot intersect the limit, it will at each of these points
touch the limit ; that is, we have a critic curve extending each way from the point
of first contact until it touches the northern limit and until it touches the southern
limit. Observe that the penumbral curve, as being at first a mere point or an
indefinitely small oval, does not at first contain within itself the point K or /f , :
it can only come to do this by passing through a position where the curve passes
through K or /T, ; viz. K or K^ would then be a critic point ; and I assume for
the present that this does not take place. The critic curve at the point of first
contact is a curve " eclipse begins at sunrise," and as not coming to pass through
a point or Ki, it cannot alter its character; that is, the critic curve, as extending
each way from the point of first contact until it comes to touch the northern and
southern limits respectively, is a curve " eclipse begins at sunrise " ; at the terminal
points in question, there is a mere contact of the sun and moon, so that they are
points, where the eclipse begins and simultaneously ends at sunrise. Continuing the
series of ovals until we arrive at the figure of eight, there are on each of them
two critic points, which ultimately unite in the node of the figure of eight; these
constitute a critic curve, extending each way fi-om the node of the figure of eight to
688] GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. 313
the contacts with the northern and southern limits respectively. There is, as before,
no passage through a point K or K^, the curve in question thus retains throughout
the same character ; and by consideration of the two terminal points it at once appears
that it is a curve "eclipse ends at sunrise." The above-mentioned critic curves form
together an oval touching the northern and southern limits respectively ; say this is
the sunrise oval.
The termination of the eclipse is similar to this, only the events happen in the
reverse order; we have a critic line starting from the node of the figure of eight
and extending each way until it comes to touch the northern and southern limits
respectively, viz. this is the line " eclipse begins at sunset " ; and then, extending each
way fiom the points of contact to reunite itself at the point of last contact, this
being the line "eclipse ends at sunset," and the two portions together form an oval
touching the northern and southern limits respectively ; say this is the sunset oval.
It is to be noticed that certain portions of the two limits are generated as the
envelope of the penumbral curve during the commencement and during the termination
of the eclipse.
For the middle of the eclipse; the penumbral curve, in the first instance a
figure of eight, breaks up into two ovals, but only one of these is attended to ;
and ultimately the oval unites itself with another oval so as to give rise to a new
figure of eight. There is thus throughout the middle of the eclipse a single oval ;
this has, north and south, an envelope which joins itself on to the portions enveloped
during the latter part of the commencement and the former part of the termination
of the eclipse, and constitutes therewith the northern and southern limits respectively,
viz. each of these is considered as extending from a point of contact with the sunrise
oval to a point of contact with the sunset oval.
The line KiVK, or say the meridian line through V, travels westwardly, while
the penumbral curve travels eastwardly ; the two come to touch each other, and there
are then two intersections which ultimately come to the northern and southern limits
respectively : the locus of these is a line of " eclipse commences at midday " ; as the
motion continues, the points of intersection move away from the two limits respectively
and ultimately unite at the point where the line KVKi again touches the penumbral
curve ; the locus is the line of " eclipse terminates at midday," the two lines together
forming an oval which touches the northern and southern limits respectively and which
may be termed the midday oval. In all that precedes, no distinction has been made
between the two portions of the horizon-envelope, or the points K and Ki, and either
curve and point indifferently may be alone attended to.
Considering now an eclipse of the second kind, the penumbral curve is at first
a mere point (the point of first contact) and it then becomes an oval, the successive
ovals not at first intersecting each other, but each oval inclosing within itself the
preceding ones. Any oval is met by the corresponding horizon in two points P and P',
at first coinciding with each other at the point of first contact, and then separating
from each other, one of fchera, say P, moving down towards and ultimately arriving
at one of the horizon-envelopes, say to fix the ideas the southern one (which curve
C. X. 40
314 GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. [688
is henceforth selected as being, and is called, the horizon-envelope, and the points on
this curve are taken to be the points K), viz. P is then a point K on the
penumbral curve, I call it Kj. The successive ovals will in the meantime have
begun to intersect each other so as to give rise to a northern limit; this will touch
the critic line (locus of P, P'), and we have a portion of the critic line extending
from the point of first contact, in one direction to the point of contact with the
northern limit, and in the other direction to the point iT, on the horizon-envelope ;
this is the line " eclipse begins at sunrise." As* the horizon continues to sweep on,
the other point P", which has not yet reached the horizon -envelope, will gradually
approach and ultimately arrive at the homon-envelope, say at the point K2 ; we
have thus a second portion of the critic line extending from the contact with the
northern envelope to the point K,; this is the line "eclipse ends at sunrise." The
horizon continuing to sweep on, the point P beginning with the position K,, which is
now on the other side of the point of contact of the horizon with the horizon-envelope,
will trace out a portion of the critic curve extending from iT, to a second point of
contact with the northern limit ; this will be the line of " eclipse begins at sunset."
And, finally, the point P from the last-mentioned point of contact, and the point P"
from its position ^o, which is now on the other side of the point of contact of
the horizon with the horizon-envelope, (that is, P, P' have now each passed through
the point of contact of the horizon with the horizon-envelope, and are both of them
on the same side thereof, viz. the side opposite to their original side), will come to
unite at the point of the last contact; we have thus a fourth portion of the critic
curve extending from ^2 to the second point of contact with the northern limit, viz.
this is the line "eclipse ends at sunset." The description will be more intelligible
by means of the figure, in which 1, 1', 2, 2', . . . , 8, 8' represent successive corresponding
positions of the points P, P', the successive positions of the horizon being given by
the right lines 11', 22', &c., all of them tangents to the dotted circle or horizon-envelope.
The entire critic line is thus a figure of eight, twice touching the horizon-envelope
688] GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. 315
and also twice touching the limit. If we consider, a& before, the intersections of KV
with the corresponding penumbral curve, this will be a curve extending from if, so
as to touch the limit, and thence onward to K^, the portion from K^ to the contact
with the limit being the line " eclipse begins at transit," . and the portion from the
limit to K^ the line " eclipse ends at transit." I say " transit " instead of midday,
since for a circumpolar place the phenomenon may happen at one or the other transit
of the sun over the meridian. It is to be remarked, that the node of the figure of
eight is a point, such that the eclipse there begins at sunrise and ends at sunset;
this point does not appear to be an important one in the geometrical theory.
The two loops of the critic line may be of very unequal magnitudes, and in
particular one of them may actually vanish ; viz. the points K^ and K^ then coincide
together, and the critic curve is a closed cuspidal curve touching the horizon-envelope
at the cusp; moreover, instead of two contacts with the limit there is one proper
contact, and an improper contact at the cusp, that is, the limit simply passes through
the cusp. And through this special separating case, we pass to the case where,
instead of the figure of eight, we have a single oval, not touching the horizon-envelope
(viz. the points if,, K^ have become imaginary), but still touching the limit twice ;
this is a distinct type for an eclipse of the second class.
And, similarly, in an eclipse of the first class, where the points Ki, K^ do not
in general exist (viz. geometrically they are imaginary), these points may present
themselves in the first instance as two coincident points, viz. instead of the sunrise
oval or the sunset oval (as the case may be), we have then a cuspidal curve ; or
they may be two real points, viz. instead of the same oval, we have then a figure
of eight touching the horizon-envelope twice, and also touching each of the two limits.
These are thus the several cases.
When the Earth traverses the penumbral cone, the critic curve is
1. A pair of ovals:
2. An oval and a cuspidate oval:
.3. An oval and a figure of eight.
And when the Earth does not traverse the penumbral cone, the critic curve is
4. A figure of eight :
.5. A cuspidate oval:
6. An oval.
To which may be added the transition case which separates 1 and 4, viz. here the
Earth just has an internal contact with the penumbral cone, and the critic curve is
7. Two ovals touching each other.
But of course 2, 5, and 7 are so special that they may be disregarded altogether;
and 3 and 6 are of rare occurrence. I have not sufficiently examined the conditions
for the occurrence of these forms 3 and 6 ; my attention was called to them, and
indeed to the whole theory, by a question proposed by Prof Adams in the Cambridge
Smith's Prize Examination for 1869.
40—2
316 [689
689.
ON THE GEOMETRICAL REPRESENTATION OF IMAGINARY
VARIABLES BY A REAL CORRESPONDENCE OF TWO PLANES.
[From the Proceedings of the London Mathematical Society, vol. ix. (1878), pp. 31 — 39.
Read December 13, 1877.]
In my recently published paper, "Geometrical Illustration of a Theorem relating
to an Inational Function of an Imaginary Variable," Proceedings of the London
Mathematical Society, t. viii. (1877), pp. 212—214, [627], I remark as follows:— "If
we have v a function of u determined by an equation f{u, v) = 0, then to any given
imaginary value x + iy of u there belong two or more values, in general imaginary,
of v; and for the complete understanding of the relation between the two imaginary
variables we require to know the series of values x ■\-iy' which correspond to a given
series of values x-\-iy oi v, u respectively. We must, for this purpose, take x, y as
the coordinates of a point P in a plane 11, and x, y' as the coordinates of a
con-esponding point P" in another plane 11' " ; — and I then proceed to consider the
particular case where the equation between u, v is u- + «^ = a-, that is, where
(x + iyy+ (a/ + iy'Y = a-.
The general case is that of an equation (*) (u, 1)" {v, 1)" = 0, where to each
given value, real or imaginary, of w, there correspond n real or imaginary values of
v; and to each given value, real or imaginary, of v, there correspond vi real or
imaginary values of u. And then, writing u = x + iy and v = x' + iy, and regarding
(x, y), {x', y') as the coordinates of t^^e points P, P" in the two planes 11, 11'
respectively, we have a real (m, n) correspondence between the two planes ; viz. to
each real point P in the first plane there correspond n real points P' in the second
plane, and to each real point P' in the second plane there correspond m real points
P in the first plane. But such real correspondence of two planes does not of
necessity arise from an equation between the two imaginary variables m, v; and the
689] ON THE GEOMETRICAL REPRESENTATION OF IMAGINARY VARIABLES. 317
question of the real correspondence of two planes may be considered in itself, without
any reference to such origin.
I was under the impression that the theory was a known one ; but I have not
found it anywhere set out in detail. It is to be noticed that, although intimately
connected with, it is quite distinct from (and seems to me to go beyond) that of a
Riemann's surface. Riemann represents the value u, =x + iy, by a point P whose
coordinates are x, y ; but he considers u', ■=x' + ii/, as a given imaginary value
attached to the point P, without representing this value by a point P', coordinates
< y'-
I proceed to consider the general theory of the real {m, n) correspondence.
Points in the first plane are denoted by the unaccented lettei-s P, Q,.. ; and the
corresponding points in the second plane are in general denoted by the same letters
accented ; but there are, as will be explained, special points V, W where the letters
are interchanged ; viz. to the points F or TF in the first plane correspond points
W or V in the second plane.
1. To a point P there correspond in general n distinct points P' ; and as P
varies continuously, each of the points P' also varies continuously.
2. There are certain points V called branch-points (Verzweigungspunkte), such
that to each point V there correspond two united points, represented by (W), and
n — 2 other distinct points W. The points ( W) are called cross-points, and the
number of them is of coui-se equal to that of the branch-points V.
It is throughout assumed that a point denoted by a letter other than V is not
a point V.
3. If the point P, moving continuously, describe a closed curve so as to return
to its original position, then, if this curve includes within it no point V (or all the
points F)*, each of the corresponding points P will describe continuously a closed
curve returning into its original position. Supposing that the curve described by P
is an oval (non-autotomic closed curve), and taking this to be in the first instance
an indefinitely small oval, then the curves described by the points P' will in the
first instance be each of them an indefinitely small oval ; but it is worth while to
notice how, as the oval described by P increases, any one of the ovals described by
a point P' may become autotomic ; viz. if the oval described by P passes through
two points Q, Q of the m points Q which correspond in the first plane to the same
Of in the second plane, then Q' will be a node in the closed curve described by
that point P* which in the course of its motion comes to pass through (^. This
curve is in general an inloop curve composed of two loops, one wholly within the
other (united at the point Q"), and such that they each include one and the same
point V (viz. V is included within the inner loop): as to this, see post, Nos. 9
and 10. It will be observed that this node (/ is not a point (W) nor any other
special point of the second plane.
* The two cages of the closed curve including no point V, and including all the points V, are really
identical, as the discontinuity at infinity may be disregarded. It is to be observed that, this being so, it
follows that the number of the points V must be even.
318 ON THE GEOMETRICAL REPKESENTATION OF IMAGINARY VARIABLES [689
4. Consider, as before, i^ as describing a closed curve which does not include
within it any point V, and the corresponding points P" as describing each of them
a closed curve. As the curve described by P approaches a point V, the curves
described by two of the points P" will approach the corresponding point (W); and
when the curve described by P passes through V, the curves described by the two
points P' will unite together at this point ( W) as a node ; viz. they will form a
figure of eight*, the crossing being at the cross-point (W), which corresponds to the
branch-point V. And, corresponding to the closed curve described by P, we have
this figure of eight (replacing two of the original n closed curves), and w — 2 closed
curves described by the other points P".
5. Supposing, next, that the closed curve described by P (instead of passing
through the point V) includes within it the point V, then the figure of eight
transforms itself into a twice-indented oval*. There are on this curve two of the
points 1-^ which correspond to the given point P; and as P, moving continuously
in its closed curve, returns to its original position, the first of these points P',
moving continuously along a portion of the curve, comes to coincide with the original
position of the second point P" ; while the second point P', moving continuously along
the remaining portion of the curve, comes to coincide with the original position of
the first point P'; viz. the two portions of the curve are described by the two points
P' respectively. The curve may thus be regarded as a bifid curve, belonging to these
two points P". And, corresponding to the closed curve described by P, we have this
bifid curve belonging to the two points P", and n — 2 single closed curves belonging
to the other n — 2 points P" respectively.
6. If the closed curve described by P (including within it a point V) comes
to pass through a second point V, the effect will be a new node at the corre-
sponding point (W); viz. at this point (W) either the bifid curve unites itself
with one of the single curves, or two of the single curves unite together, or the
bifid curve there cuts itself. And, if the curve described by P comes to include
within it this second point V, then in the three cases respectively : — the bifid curve
takes to itself the single curve, so that the system then is a trifid curve and n — 3
single curves; or the two single curves give rise to a bifid curve, so that the
system is two bifid curves and n — 4 single curves ; or, lastly, the bifid curve breaks
up into two single curves, so that the system resumes its original form of n single
curves.
7. We thus see how the closed curve described by P, including within it
certain of the points V, may be such as to have corresponding to it an a-fid curve,
a /3-fid curve, &c., (a + ^ + . . . = n) ; viz. an a-fid curve contains upon it a of the
points P" which correspond to the original position of P; and then, as P describes
* The name figure of eight refers to the case where the two curves which come to unite at (U") are
proper ovals (non-autotomic closed curves). They might have one or both of them a node or nodes, as
explained in No. 3; and the term would then be inappropriate. And so, lower down, the name twice-
indented oval is used to express the form into which a proper figure of eight is changed by the disappearance
of the node.
689] BY A REAL CORRESPONDENCE OF TWO PLANES. 319
continuously its closed curve, returning to its original position, each of these points
i" describes a portion of the a-fid curve, passing from its original position to the
original position of a point P" next to it upon the a-fid curve ; and the like as to
a ^-fid curve, &c. The numbers a, /8, ... are not of necessity unequal, and we may
have sets of equal numbers in any manner. It is hardly necessary to remark that,
if the curve described by P passes through any point or points V, then two of the
curves described by the points P" will unite together, or it may be that one of these
will cut itself at the corresponding point or points (W); and further that, as in
No. 3, if the curve described by P passes through two or more of the points Q
which correspond to the same point (/, then any such point Q' will present itself
as a node upon the curve belonging to some point, or set of points, P. But the
order of succession in which the original w single curves unite themselves together
into multifid curves, or again break up into single curves, cannot, it would appear,
be explained in any general manner, and would in each case depend on the nature
of the particular correspondence.
8. We may consider the case where the closed curve described by P cuts
itself. The curve may here be considered as made up of two or more ovals, or, to
use a more appropriate term, say loops, each such loop being a curve not cutting
itself; and the case is thus reducible to that before considered, where the curve
does not cut itself. Thus, to fix the ideas, let the curve be a figure of eight, the
initial position of P being at the crossing, and let neither of the loops contain
within it a point V. Then, as P passes continuously along one of the loops, re-
turning to its original position, each of the corresponding points P' describes a closed
curve, which will be in the nature of a loop, viz. the initial and final directions of
the motion of P not being continuous with each other, the initial and final directions
of the motions of each point P" will not be continuous with each other, or there
will be at the point P' an abrupt change in the direction of the curve. Similarly,
as P describes the other loop of the figure of eight, each of the points F will
describe another loop ; and the two loops belonging to the same point P' will unite
together so as to form a figure of eight ; viz. to the figure of eight described by P
there will correspond figures of eight described by the n points P" respectively.
9. But consider next the case where the two loops of the curve described by P
include each of them one and the same point V. This implies that one of the two
loops lies inside the other, or that the curve is what has been called an inloop
curve. As P, which is in the first instance taken to be at the node, passes con-
tinuously along one of the loops and returns to its original position, there are two of
the points P" such that the first of these passes from its original position to the
original position of the second, and the second of them passes from its original
position to the original position of the first of them. We have thus two arcs between
these two points P'; but inasmuch as the initial and the final directions of motion of
the point P are not continuous with each other, these two arcs are not continuous
in direction at the two points P', but at each of these points P' the two arcs meet
at an angle. As P describes the other loop, we have in like manner two arcs
between the same two points P', these arcs at each of the points P' meeting at an
320 ON THE GEOMETRICAL REPRESENTATION OF IMAGINARY VARIABLES [689
angle ; but they join on to the first-mentioned two arcs in such manner as to form
two ovals intersecting each other in the two points P'. Corresponding to the inloop
curve described by F, we have this pair of intersecting ovals described by two of
the points P", and n—2 other curves described by the other points P', and being
each of them (I assume) an inloop curve.
10. If we attend only to one of the two intersecting ovals, we have in the
first plane an inloop curve, and corresponding thereto in the second plane an oval
passing through two of the points P' which correspond to the node P of the inloop
curve. Interchanging the two planes, and writing Q instead of P, we have in the
first plane an oval passing through two of the points Q which correspond to a point
Q: ; and corresponding to this oval we have in the second plane an inloop curve
having this point Q for its node, viz. these are the corresponding figures mentioned
in No. 3.
11. Consider a given point Q; and let the corresponding points Qf be called
(selecting the suffixes at pleasure) Q/, Q^, . . , Q„'. Taking then a point 0 indefinitely
near to Q, the corresponding points (7 will be indefinitely near to Q/, Q,', . . , Q„'
respectively, and they will be called 0/, 0/, . . , 0,,' accordingly. It is to be observed
that by the indefinitely near point 0 is meant a point such that the distance from
0 to Q is indefinitely small in comparison with the distance of either of these points
from any point V; so that we cannot have from Q to 0 two indefinitely short paths
including between them a point V; or say so that the indefinitely short path from
Q to 0 is determinate.
Proceeding in this manner from Q to 0, and so through a succession of indefinitely
near points to a distant point S, we seem to determine the suffixes of the corre-
sponding points S' ; but, by what precedes, it appears that such determination for a
point S IB dependent on the path from Q to (S; and consequently that we do not
thus obtain a proper determination of the suffixes of the points S'. In fact, if we
were to pass from Q by a path including one or more of the points V back to Q,
we should obtain for the several points Q' respectively suffixes which are in general
difiterent from the suffixes originally given to these points respectively.
12. The difficulty is got over as follows: — Considering as before the given point
Q, and calling the corresponding points Q/, Q^', .., Qn at pleasure, we pass from Q to
the indefinitely near point 0, and thence, by so many paths chosen at pleasure, to
the several branch-points V; these paths from 0 to the several points V are called
barriers. To fix the ideas, we may consider these as non-autotomic non-intersecting
lines drawn from 0 to the several points V. Consider the barrier from 0 to one of
these points V; as P passes along this barrier from 0 to V, two of the corre-
sponding points P' will pass from two of the corresponding points 0' to the corre-
sponding cross-point {W); the paths of these two points are called the counter-barrier
corresponding to the barrier in question ; and we have thus in the second plane a
system of counter-barriers, each drawn from two points 0' to meet in a point (W).
By what precedes, the points 0' have each of them a determinate suffix; a counter-
barrier is thus drawn from two points with given suffixes, suppose 0/ and 0,', to a
689] BY A REAL CORRESPONDENCE OF TWO PLANES. 321
point (W), and this may be distinguished accordingly as a counter-barrier 12; and
in like manner the cross-point (W) through which it passes will be called a cross-
point (Wis); and the barrier corresponding hereto, and the branch-point V at which
it tei"minates, will in like manner be called a barrier 12, and a branch-point F,.j.
Each barrier and branch-point will thus have a pari' of suffixes; and the corresponding
counter-barrier and cross-point will have the same pair of suffixes. It is to be observed
that two or more of these corresponding figures may very well have the same pair
of suffixes; but that such corresponding figures must be distinguished from each
other; thus, if there are two branch -points F,,, these may be distinguished as the
branch-points oFj^ and ySFia, and the barriers, counter-barriers, and cross-points by
means of these same letters a and ^, (or otherwise), as may be convenient. It would
seem that not only the number of the points V must be even, but the number of
each set of points F^ must also be even (see post, No. 15).
13. It is also to be noticed that the determination of the suffixes of the several
points F, &c., depends first upon the arbitrary choice of the suffixes of the points
Q', and next on the choice of the system of barriers; but that, these being assumed,
the suffixes of the several points F, &c., are completely determinate.
14. Taking now any point S whatever, and supposing that P moves from Q
continuously to /S by a path which does not meet a barrier, the points P' will move
from the several points Q' to the several points S' by paths not meeting the counter-
barriers; viz. to each point S' there will be a path from some point Q ; and giving
to such point S' the suffix of the point Q', the suffixes of the several points S'
which correspond to any point whatever, S, will be completely determined. The
determination depends of course on the assumptions referred to No. 13, but not in
anywise on the position of the point ;S'.
It will be noticed that, as all the points V are connected together by the
barriers, the only closed paths from a point to itself are paths not including any,
or including all, of the points F; and that between such paths there is no real
distinction.
1.5. Consider a point P moving continuously in any manner. The several corre-
sponding points Pi, Pj', . . , Pn will each of them move continuously, but the suffixes
interchange ; viz. when P arrives at and then passes over a barrier a/8, the corre-
sponding points Pa' and P^' will each arrive at the corresponding counter-ban-ier 0/8,
and, on passing over this, P„' will be changed into P^' and P^' into P^, the other
points P* remaining unchanged ; and the like in other cases. This in fact includes
the whole or the greater part of the foregoing theory. Thus, if P describe a closed
curve not cutting any barrier, there will be no change of suffix ; and when P returns
to its original position each of the corresponding points P/, P/, . . , P„' will describe
a closed curve, returning to its original position. But suppose that P describes a
closed curve, cutting once only a barrier 12 ; suppose that the path is from P to
Q, and then crossing the barrier to R, and thence again to P; P/ passes to Qi,
and then crossing the counter-barrier it passes from P/ to P2'; while at the same
time P2' passes to Q^, and then crossing the counter-barrier it passes from R^ to Pj';
C. X. 41
322 ON THE GEOMETRICAL REPRESENTATION OF IMAGINARY VARIABLES [689
viz. we have P,', P,' describing the two portions of a bifid curve. If there were only
a single branch-point F,j, and therefore only a single barrier OVi,, then we might
have through P a closed curve cutting OVj^ once only, and including within it the
point 0, but not including within it the point F,j; and here there ought not to be
a bifid curve, but the points P/, P/ ought to describe each of them a single curve.
But suppose there are two points F,,, and consequently two baniers OF,j (meeting
in 0); then the closed curve, meeting once only a barrier 12, (viz. it meets only one
such barrier, and that once only), must include within one and only one of the two
points Fij; and in this case there ought to be a bifid curve. It is by such reasoning
as this that I infer the foregoing theorem (No. 12), that the number of each set of
points F,i is even.
16. We may consider how the suffixes are affected when, instead of the original
system of barriers, we have a new system of barriers. I suppose that we have in
the two cases respectively the same point Q, and the same suffixes for the points
Qi, Qi't • ■ . Qn which correspond thereto. In the first case, passing from Q to an
indefinitely near point 0, say the red 0, we draw from this point to the several
points F a set of barriers, say the red barriers ; while in the second case, passing
from Q to an indefinitely near point 0, say the blue 0, we draw from this point
to the several points F a set of barriers, say the blue barriers ; and we then proceed
as before, viz. in the first case, drawing from Q to the point S a curve which does
not meet any of the red barriers, we determine accordingly the suffixes (say the red
suffixes) of the several corresponding points S' ; and in the second case, drawing in
like manner from Q to S a, curve which does not meet any of the blue barriei"s,
we determine accordingly the suffixes (say the blue suffixes) of the same points S'.
Now the curve drawn from Q to S so as not to cut any of the red bamers, and
which is used for the deteniiination of the red suffixes of the several points S', will
in general cut certain of the blue barriers ; and, by examining the suffixes of the
blue barriers which are thus cut, we determine the blue suffixes of the same points
S'; the result of course depending only on the situation of S in one or other of
the regions formed by the red barriers and the blue barriers conjointly. In particular,
the point S may be so situate that we can from Q to S draw a curve not meeting
any red barrier or any blue barrier; and in this case the red suffixes and the blue
suffixes are identical.
17. We may imagine the first plane as consisting of n superimposed planes or
sheets, say the sheets 1, 2,..,n. Each barrier 12 is considered as a line drawn in
the two sheets 1 and 2 ; and so on in other cases. The point P is considered as a
set of superimposed points P,, Pt,..,Pn moving in the several sheets respectively; under
the convention that P, moving in the sheet 1, and coming to cross a barrier 12, passes
into the sheet 2 and becomes Pj ; and the like in other cases. And this being so, we
say that to a point P, considered as a point P. in the sheet a, there corresponds in the
second plane one and only one point P.'; and that P moving continuously in any
manner (subject to the change of sheet as just explained), each of the n corresponding
points P" will also move continuously, and so that each such point P,' will return
689] BY A REAL CORRESPONDENCE OF TWO PLANES. 323
to its original position, upon the corresponding point P. returning to its original
position and sheet. This is, in fact, Riemann's theory, only instead of the points P'
we must speak of the values x' + iy' of the irrational function of a; + iy.
18. Everything is of course symmetrical as regards the two planes; we have
therefore, in the second plane, a system of points V and of barriers, and in the first
plane a system of points (TT) and of counter-barriers. To a given point P' in the
second plane there correspond m points P in the first plane ; and we can (the
determination depending on the system of barriers in the second plane) assign to the
m points suffixes, thereby distinguishing them as the corresponding points P,, P^,..,Pm.
And we may imagine the second plane as consisting of m superimposed planes or
sheets, say the sheets 1, 2, 3, ..,m; the general theorem then is that to a point P
or P' in either plane, considered as a point P. or P.' in the sheet a or a', there
corresponds in the other plane one and only one point PJ or P^-; and that the first-
mentioned point in either plane moving continuously in any manner (subject to the
proper change of sheet), the corresponding point in the other plane will also move
continuously, and will return to its original position and sheet, upon the first-
mentioned point returning to its original position and sheet.
19. In all that precedes it has been assumed that, to a branch-point V, there
correspond two united points represented by ( W) and n — 2 distinct points W ; the
cases of a point (W) composed of three or more united points, or of the points W
uniting themselves in sets in any other manner, would give rise to further specialities.
41—2
324 • [690
690.
ON THE THEORY OF GROUPS.
[From the Proceedings of the London Mathematical Society, t. ix. (1878), pp. 126 — 133.
Read May 9, 1878.]
I RECAPITULATE the general theory so far as is necessary in order to render
intelligihle the quasi-geometrical representation of it which will be given.
Let a, /3, . . be functional symbols each operating upon one and the same number
of letters, and producing as its result the same number of functions of these letters.
For instance, a(a;, y, ii) = {X, Y, Z), where the capitals denote each of them a given
function of (x, y, z).
Such symbols are susceptible of repetition and combination ;
a'{x, y, z)=a{X, Y, Z),
or
^OLix,y,z) = fi{X,Y,Z),
in each case equal to three given functions of {x, y, z); and similarly for o', a'^, etc.
The symbols are not in general commutative, a/3 not =/3a; but they are associative,
a/3 . 7 = a . y37, each = a/Sy, which has thus a determinate meaning.
Unity as a functional symbol denotes that the letters are unaltered,
1 (a;, y, z) = («, y, z);
whence
la = al = a.
The functional symbols viay be substitutions ; a {x, y, z) = {y, z, x), the same letters
in a different order. Substitutions can be represented by the notation r~-^ , the
I if
substitution which changes xyz into yzx, or, as products of cyclical substitutions,
'UZ3C ^WU
n—^-Jz. /^ =(xyz)(uw), the product of the cyclical substitutions x into y, y into z,
z into X, and u into w, w into u, the letter v being unaltered.
690]
ON THE THEORY OF GROUPS.
325
A set of sjTnbols a, yS, 7, . . , such that the product a/3 of each two of them
(in each order, a/3 and /3a) is a symbol of the set, is a group. It is easily seen
that 1 is a symbol of every group, and we may therefore give the definition in the
form that a set of symbols 1, a, /8, 7,.. satisfying the foregoing condition is a group.
When the number of symbols (or terms) is =n, then the gi-oup is of the order n;
and each symbol a is such that a" = 1, so that a group of the order n is in fact a
group of symbolical wth roots of unity.
A group is defined by means of the laws of combinations of its symbols. For
the statement of these we may either (by the introduction of powers and products)
diminish as much as may be the number of distinct functional symbols; or else,
using distinct letters for the several terms of the group, employ a square diagram, as
presently mentioned.
Thus, in the first mode, a group is 1, /3, ^', a, a/3, a/3^, (a=' = l, /3^=1, a^ = ^a),
where observe that these conditions imply also o/3^ = /3a.
Or in the second mode, calling the symbols (1, a, /3, a0, ^, a/3=) of the same
group (1, a, /3, 7, S, e), or, if we please, (a, h, c, d, e, /), the laws of combination
are given by one or other of the square diagi-ams :
1 a /3 y 8 «
/3
y
1
a
/3
y
8
c
a
1
y
ys
c
8
y
/3
c
8
a
1
y
8
f
1
a
P
8
y
1
c
)8
a
c
/s
a
8
y
1
a
h
c
d
e
f
b
a
d
c
f
e
c
f
e
b
a
d
d
e
f
a
b
c
e
d
a
f
c
d
b
a
f
c
b
e
where, taking for greater symmetry the second form of the square, observe that the
square is such that no letter occurs twice in the same line, or in the same column (or
what is the same thing, each of the lines and of the columns contains all the letters).
But this is not sufficient in order that the square may represent a group; the square
must be such that the substitutions by means of which its several lines are derived
from any line thereof are (in a different order) the same substitutions by which the
lines are derived from a particular line, or say from the top line. These, in fact, are :
ah .cd . ef,
ace . b/d,
ad. be . cf,
aec . bdf,
af .be . de,
826 ON THE THEORY OF GROUPS. [690
where, for shortness, ab, ace, &c., are written instead of (ab), (ace), &c., to denote the
cyclical substitutions a into b, b into a ; and a into c, c into e, e into a, &c. ; and
it is at once seen that by the same substitutions the lines may be derived from any
other line.
It will be noticed that in the foregoing substitution-group each substitution is
regular, that is, composed of cyclical substitutions each of the same number of letters;
and it is easy to see that this property is a general one; each substitution of the
substitution-group must be regular.
By what precedes, the group of any order composed of the functional sjTnbols is
replaced by a substitution-gi-oup upon a set of letters the number of which is equal
to the order of the group, and wherein all the substitutions are regular.
The general theory being thus explained, I endeavour to form a substitution-
group with the twelve letters ahcdefghijkl ; and I assume that there is one substitution,
such as ahc.def.ghi.jkl, and another substitution, such as agj . b/i . cek . dhl. Observe
that, if the twelve letters are to be thus arranged in two different ways as a set
of four triads, without repetition of any duad, all the ways in which this can be
done are essentially similar, and there is no loss of generality in taking the two sets
of triads to be those just written down. But the substitution to be formed with either
set of triads will be different according as any triad thereof, for instance agj, is written
in this form or in the reversed form ajg. There are thus in all si.xteen substitutions
which can be formed with the first set of triads, and sixteen substitutions which can
be formed with the second set of triads; and the relation of a triad of the first set
to a triad of the second set is by no means independent of tlie selection of the
triads out of the two sets respectively. To show this, take the two substitutions quite
at random ; suppose they are those written down above, say
a = abc . def.ghi . jkl, yS = agj . bfi . cek . dhl ;
and perform these in succession on the primitive arrangement f2 = abcdefghijkl. The
operation stands thus:
/8afl =/egkihlbjcda,
ail = bcaefdhigklj,
n = abcdefghijkl,
whence
/3a, = aflibeijcgl . dk,
is not a regular substitution ; and, by what precedes, a, /3 cannot belong to a group.
But take the substitutions to be
a (ss before) = abc. def.ghi. jkl, ^ = ajg .bif.cek.dhl,
then we have
^ail = iejkbhlfacdg,
ail = bcaefdhigklj,
il = abcdefghijkl,
690] ON THE THEORY OP GROUPS. 327
whence
/9a = ai . be . cj . dJc . fh . gl,
a regular substitution; and, for anything that appears to the contrary, a, y8 may
belong to a group. It is convenient to mention at once that these two substitutions
do, 'u\ fact, give rise to a group; viz. the square diagram is
a
h
e
d
e
f
9
h
i
3
k
I
b
c
a
e
f
d
h
i
9
k
I
3
c
a
h
f
d
e
i
9
h
I
3
k
d
I
h
a
9
3
e
c
k
f
i
b
e
J
i
b
h
h
I
f
a
I
d
9
c
f
k
9
k
c
i
d
b
j
e
h
a
9
f
I
e
i
J
d
b
a
e
h
h
d
I
J
a
y
k
e
c
b
f
i
i
e
i
k
b
h
I
f
a
c
d
9
J
i
e
h
k
b
a
I
f
9
c
d
k
9
f
i
I
c
b
0
J
d
= h
a
e
I
h
d
9
J
a
k
e
i
b
/
and the substitutions, obtained therefrom by writing successively each line over the
top line, are
1 =1,
abc . def . ghi . jkl a,
acb . d/e . gih . jlk a^,
ad .bl.ch.eg .fj . ik fi-a^-,
aeh .bjd.cU. /kg /3a-,
of I . bkh . cgd . eij ^a,
agj . bfi . eke . dlh /9»,
ahe.bdj. cli .fgk ^o?^a?,
ai . be . cj . dk .fh . gl /8a,
ajg . bif . cek . dhl /3,
ak ,bg . cf. di . el . hj /3W,
alf. bkh . cdg . eji ^'a^a.
328 ON THE THEORY OF GROUPS. [690
To explain the theory, I introduce the notion of a hemipolyhedron, or sa)- a
hemihedron, viz. this is a figure obtained from a polyhedron by the removal of
certain faces. In a polyhedron each edge occurs twice (more properly it occurs in
the two forms ah and ba), as belonging to two faces; but in a hemihedron one of
these faces must always be removed, so that the edge may occur once only ; and
again (what is apparently, although not really, a different thing), we may remove two
intersecting faces, leaving their edge of intersection; this edge is, in fact, then considered
as a bilateral face ab = ab. ba, just as abc is a trilateral face abc = ah.bc. ca. Thus, if
in a prism we remove the lateral faces, leaving the lateral edges, and leaving also the
terminal faces, we have a hemihedron : thus, the prism being trilateral, say the faces
of the hemihedron are abc, def, ad, be, cf, where ad, be, cf are the edges regarded as
bilateral faces. And, for the present purpose, abc denotes the cyclical substitution a
into b, b into c, c into a; and ad denotes in like manner the cyclical substitution
(or interchange) a into d, d into a.
But the hemihedron about to be considered has no bilateral faces; it is, in fact,
the figure composed of the 8 triangular faces of the octo-hexahedron or figure obtained
by truncating the summits of a hexahedron (or of an octahedron) so as to obtain a
polyhedron of 8 triangular faces and 6 square faces, representing the faces of the
octahedron and the hexahedron respectively. The faces of the octo-hexahedron may
be taken to be
abc, def, ghi, jkl,
^J9> ^f' ^^^> ^^^^'
cbfe, fihd, hgjl, jack, agib, klde,
(where I observe in passing that the symbols are written in such manner that each
edge lib occurs under the two opposite forms ab in abc and ba in agib). And then,
omitting the square faces, represented by the third line, we have the hemihedron,
wherein as before abc denotes the cyclical substitution a into b, b into c, c into a;
and so for the other faces.
I represent this by a diagram, the lines of which were red and black, and they
a l> f g
a
k fi d
will be thus spoken of, but the black lines are in the woodcut continuous lines, and
the red lines broken lines: each face indicates a cyclical substitution, as shown by
the arrows. The figure should be in the first instance drawn with the arrows, but
without the letters, and these may then be affixed to the several points in a perfectly
arbitrai-y manner; but I have in fact affixed them in such wise that the group given
690] ON THE THEORY OF GROUPS. 329
by the diagi-am, as presently appearing, may (instead of being any other equivalent
group) be that group which contains the before-mentioned substitution
a = abc . def. ghi .jkl, and /8 = ajg . hif. cek . dhl.
Observe that in the diagram, considering the lines to be drawn as shown by the
arrows, there is Jrom any given point whatever only one black line, and only one red
line. Let B denote motion along a black line, R motion along a red line (always
from a point to the next point); then R- will denote motion along two black lines
successively, RR (any such symbol being read always from right to left) will denote
motion first along a red line, and then along a black line, and so in other cases; a
symbol or "route" ...RfiR' has thus a perfectly definite signification, determining the
path when the initial point is given.
The diagram has the property that every route, leading from any one letter to
itself, leads also from every othijr letter to itself; or say a route leading from a to
a, leads also from b to b, from c to c, . . . , from I to I; and we can thus in the
diagram speak absolutely (that is, without restriction as to the initial point) of a
route as leading from a point to itself, or say as being equal to unity; it is in virtue
of this property that the diagram gives a group.
For, assuming the property, it at once follows (1) that two routes, each leading
say from the point a to the same point /, lead also from any other point b to
one and the same point g. Such routes are said to be equivalent, or equal to each
other; and the number of distinct routes (including the route unity) is thus equal
to the numbers of the letters, viz. we have only the routes from a to o, to b, ..., to I,
respectively; (2) a route, leading from a point a to a point f, leads from any other point
6 to a different point g; and (3) two routes, leading from the same point a to different
points 6 and c, lead also from any other point f to different points k and I. Hence a
given route leads from the several points abc... I successively to the same series of points
taken in a different order, or we thus obtain a new arrangement of the points ; and
dealing in this manner successively with the routes from a to a, to b,..., to I, we
obtain so many distinct arrangements, beginning with the letters a, b, c,..,l respectively,
such that in no two of them does the same letter occupy the same place; we thus
obtain a square of 12 such as that already written down, and which is, in fact, the
same square, the several routes of course corresponding to the substitutions of the
square. The hemihedron thus gives the foregoing group of 12.
Observe that the diagi-am is composed of the four black triangles representing
the substitution abc . def . gki . jkl, and of the four red triangles representing the sub-
stitution ajg .hif .cek. dhl] viz. these are independent substitutions which by their powers
and products serve to express all the substitutions of the group ; that they are sufficient
appears by the diagram itself, in that every point thereof is (by black and red lines)
connected with every other point thereof The group might have contained three or
more independent substitutions, and the diagram would then have contained the like
number of differently coloured sets of lines. The essential characters are that the lines
of any given colour shall form polygons of the same number of sides (but for different
C. X. 42
330 ON THE THEORY OF GROUPS. [690
colours the polygons may have different numbers of sides; in particular, for any given
colour or colours, the polygons may be bilaterals, represented each by a line with a
double arrow pointing opposite ways) ; that there shall be frmn each point only one
line of the same colour; that every point shall be connected with every other point;
and finally, that every route leading from one point to itself shall lead also from
every other point to itself When these conditions are satisfied the foregoing
investigation in fact shows that the diagram, or say the hemihedron, gives rise to a
group.
It may be remarked that we can, if we please, introduce into the diagram a set
of lines of a new colour to represent any dependent substitution of the group ; thus,
in the example considered, a substitution is aeh.hjd.cil.fkg, and if we draw these
triangles in green (the arrows being from a to e, e to h, h to a, &c.), then there
will be from each point one black line, one red line, and one green line; any route
...G'*BfiB' will thus be perfectly definite, and will have the same properties as a route
composed of black and red lines only; and the theory thus subsists without alteration.
I remark, in conclusion, that the group of 12 considered above is, in fact, the
group of 12 positive substitutions upon 4 letters abed; viz. the substitutions are 1,
abc, acb, abd, adb, acd, adc, bed, bdc, ab.cd, ac.bd, ad. be; the groups each contain
unity, three substitutions of the order (or index) 2, and 8 substitutions of the order
(or index) 3, and their identity can be easily verified.
691] 331
691.
NOTE ON MR MONRO'S PAPER "ON FLEXURE OF SPACES."
«■
[From the Proceedings of the London Mathematical Society, vol. ix. (1878), pp. 171, 172.
Read June 13, 1878.]
CJONSIDER an element of surface, surrounding a point P ; the flexure of the
element may be interfered with by the continuity round P, and it is on this account
proper to regard the element as cut or slit along a radius drawn from P to the
periphery of the element. This being understood, we have the well-known theorem
that, considering in the neighbourhood of the origin elements of the surfaces
z = li{aa?+'2hxy + hy% and zf = ^(a'x'^ -{-^Kafyf + h'y'%
these will be applicable the one on the other, provided only ab — h^= a'b' — h'-. But
in connexion with Mr Monro's paper it is worth while to give the proof in detail.
It is to be shown that z, z' denoting the above-mentioned functions of {ac, y) and
(ar*, y') respectively, it is possible to find (for small values) x', if functions of x, y
such that identically
da;'' + dyf^ + dz"" =dx' + df + dz^.
The solution is taken to be x' = x + ^, y' = y + Vf where ^, jy denote cubic functions of
X, y. We have then, attending only to the terms of an order not exceeding 3 in x, y,
dai' + dy'+2{dxd^ + dydv) + {{a'x + h'y) dx + {h'x + h'y) dyY
= daf + dy'+ {(ax + hy) dx + {hx + by) dyY,
so that the terms da? + dy' disappear ; and then writing
'^^ = f'^ + i^2/> dv = fjx + ^^dy.
42—2
382 NOTB ON MR MONRO's PAPER " ON FLEXURE OF SPACES." [691
the equation will be satisfied identically as regards dx, dy if only
2^ = {ax + hyy-ia'x + h'yy,
^ + ^ = (ax + hy){}ut + by) - {a'x + h'y) (h'x + b'y),
2^ = {hx + byy-{h'x + b'yy.
Calling the terms on the right-hand side 221, S3, 2S respectively, we have
dy dxdy daf '
that is,
(h^ - h") + (A= - /t'O - {{ab + h') - (a'b' + h'^)} = 0,
or, what is the same thing,
a'b'-h'' = ab-h\
& relation which must exist between the constants (a, b, h) and (a', b', h').
It is easy to find the actual values oi ^, t); viz. these are
f = j(a» -a'' )af + ^iah-a'h')a^ + i(h'' -h'^)a^y + ^(bh-b'h')f,
7i=:i(ak-a'h')a^ + ^{h' -h'' )a?y + ^(bh-b'h')a?y -^^(p" -b'' )f,
or, what is the same thing, we have
g_ dll _ -.dD.
^-^^dx' ■^-^^^'
where
a = {a?- a'") x' + 4>{ah- a'h') x'y + e (h' - h'') a^y' +4>{bh- b'h') xy' + (Iy'- b'^) y*.
= (a*» + 2hxy + bfY - (a'a/' + 2h'xy + b'ff = 4 (^^ _ z'^\
in virtue of the relation ab — h^ = a'b' — h'\ The resulting values x' = x-\-^, y'==y + r]
are obviously the first terms of two series which, if continued, would contain higher
powers of {x, y).
692] 333
692.
ADDITION TO THE MEMOIR ON THE TRANSFORMATION OF
ELLIPTIC FUNCTIONS.
[From the Philosophical Transactions of the Royal Society of London, vol. CLXix.
Part II. (1878), pp. 419—424. Received February 6,— Read March 7, 1878.]
I HAVE recently succeeded in completing a theory considered in my "Memoir on
the Transformation of Elliptic Functions," Phil. Trans., vol. CLXiv. (1874), pp. 397 — 456,
[578], — that of the septic transformation, n = 7. We have here
1 -y _ 1 -_« /a -^x + yar'- Ba^V
1 +y~l+x Va + fix + yijd'+Sa^) '
a solution of
Mdy dx
the
determined by the equations
1 2i9
where tt = 1 H ; and the ratios a : /9 : 7 : 8, and the wv-modular equation are
«' (2a7 + 2ay3 + ^') = «' (y + 27S + 2^S),
7= + 2y87 + 2aS + 2y9S = iflv? {^wy + 2/37 + 2aS + ^),
S=+27S = ir'M"'(ai'+2a/9);
or, what is the same thing, writing a = 1, the first equation may be replaced by
V? .
S = — , and then, o, S having these values, the last three equations determine y8, 7
and the modular equation. If instead of /3 we introduce M, by means of the relation
334 ADDITION TO THE MEMOIB ON THE [692
1 1 . . / 1 u*\
Tjv= 1+2/8, that is, 2^ = ^—1, then the last equation gives 27 = mV(-t> — -^ ) ; and
a, /8, 7, 8 having these values, we have the residual two equations
w«(2a7 + 2a/3+y9')= v' (y' + 2yB + 20S),
7» + 2/97 + 2aS + /3S = wV (207 + 2/87 + 2aS + ^),
viz. each of these is a quadric equation in -p; hence eliminating -j^, we have the
modular equation; and also (linearly) the value of j^, and thence the values of
a, fi, 7, S in terms of u, v.
Before going further it is proper to remark that, writing as above a = 1, then
if S = /87, we have
1 - )8a; + 7«' - Sar- = (1 - /9x) (1 + jx'),
1 + /9a; + 7a? + Sa;' = (1 + /3a;) (1 + 7a?),
and the equation of transformation becomes
1-y _l-a; /I - ^x^
viz. this belongs to the cubic transformation. The value of /9 in the cubic transforma-
tion was taken to be /3 = — , but for the present purpose it is necessary to pay
attention to an omitted double sign, and write /9= + — ; this being so, S = ^y, and
V
giving to 7 the value + u*, S will have its foregoing value = — . And from the
theory of the cubic equation, according as ^= — or = , the modular equation
must be
u*-v* + 2uv(l-uHi'') = 0, or ti* - v* - 2uv {1 - uh^) = 0.
We thus see d priori, and it is easy to verify that the equations of the septic
transformation are satisfied by the values
0 = 1,^= - , 7 = M*, S = - , and tt*-v* + 2uv (1 - mV) = 0 ;
« = 1, /9 = , 7 = -t(«, S = -, and «*-?/«- 2md(1 -mV) = 0;
and it hence follows that in obtaining the modular equation for the septic transform-
ation, we shall meet with the factors u* — v*± 2uv (1 — uV). Writing for shortness
uv = 0, these factors are u* — v* ±20(1 — 0^); the factor for the proper modular equation
is u' + ifi—®, where
8 = 8^- 28^ + 566' - 700* + 5G&> - 28^ + 8^,
692] TBANSFORMATION OF ELLIPTIC FUNCTIONS. 335
viz. the equation (1 — u^) (1 —v^) — (l — uvf = 0 is ?t' + ?;' — 0 = 0 ; aad the modular
equation, as obtained by the elimination from the two quadric equations, presents
itself in the form
Proceeding to the investigation, we substitute the values
.-l,^"i(i-l),7-4«v(^-$),8.^,
in the residual two equations, which thus become
= 0,
2^.(1 -t^) +^(l-uvY{l+uv)
+ |(1_„8)_4(1_md)(i +^'
_^j |- mV (1 - uvf (1 + uv)l + ^\u^ (1 - «') + - (1 + uV) (u* - v*)\
'jiU ,,7 1
+ \^+6-il-uV)-u'V\ =0,
(if V )
the first of which is given p. 432 of the " Memoir," [Coll. Math. Papers, vol. ix., p. 150].
Calling them
(a, b, c][;|^, iy = 0, (a', b',c'][^, l)' = 0.
we have
1 2
jVj : -jr^ : 1 = be' — b'c : ca' — c'a : ab' - aT),
and tihe result of the elimination therefore is
(ca' - c'a)» - 4 (be' - b'c) (ab' - a'b) = 0.
Write as before uv = 6. In forming the expressions ca' — c'a, &c., to avoid fractions
we must in the first instance introduce the factor v^: thus
1/' (c&' - c'a.) = v {v(l - u«) - 4, {1 - e) (v + u')} [- d' {I + 0) (1 - 0y]
- {u'* + 6n'0 (1 - 0") - v^e'} {1 - v"},
= -^(l + 0)(l-0y{i»»(-3 + 40) + ii«(-45+3^)}
- {u" + 6w« (0-0>)- v'0^} (1 - t)') ;
but instead of 0V writing «V, the expression on the right-hand side becomes divisible
by «'; and we find
^, (ca' - c'a) = -(1 + 0){l- 0y [v*(-S + 4:0) + u* (- 4,0' + 30")}
-{u"+6ii*{0-0')-v*}(l-if),
336 ADDITION TO THE MEMOIR ON THE [692
and thence
- -, (ca' - c'a) = w" + M« (6^ - 10^ + 11^ - 6^ - 8^ + 10^ - 4^)
+ V* (- 4 + 10^ - 8^ - 6^ + 11^ - 10^ + 6^) + f".
Similarly we have
^(bc' -b'c) = M»(5- 5^ + 4^- 5^ + 2(9*) + M«(90- 16^ + ^+ 10^ + ^-16^ + 9^0
+ v* (2 -56 + 4,6-' -56' + 50*),
-^{ah'-a.'h) = u*{0 + ff'-0*) + v*{2-59 + 4!0' + S0'-lOd* + 30' + *d'-5e' + 20')
+ v"{-l + 0+e');
say these values are
m" +pu* +qv* + ?;", \m'2 + fiu* + w*, pu* +av* + rv".
The required equation is thus
0 = (v}^ + pu* -i- qv* + v^y - 4 (\rt" + fj-u* + vv*) (fm* + <rv* + -ra"),
viz. the function is
+ m" (2p - 4X/>)
+ M» (2q0* +p^ - iXo-e* - ^ixp)
+ (2^''+2^g^<-4\T^='-4/Ao-^-4i'p^)
+ ifi (2pe* + q-- 4!fiT&' - 4i/o-)
+ v^' (2q - 4vt)
+ V"-*,
or say it is
= (1, b, c, d, e, f, \\xi"\ M", u\ 1, ifi, V", v^).
Supposing that this has a factor m' — 0 + 1;*, the form is
(m>« + £«» + C + D?;* + w") (m» - 0 + 1;») ;
and comparing coefficients we have
B-% =b,
C-&B + e^ =c,
De'-ec+B&> = d,
ex-SD+C =e,
-®+D =/
where 0 has the before-mentioned value
= (8. -28, +5C, -70, +56, -28, +8$(9, 0", ^, 6*, O', 6", ^).
From the first, second, and fifth equations, 5 = 6 + 0, 0 = 0 + 05 — 6^,D =/+ 0 ; and
692] TRANSFORMATION OF ELLIPTIC FUNCTIONS. 337
the third and fourth equations should then be verified identically. Writing down the
coefficients of the different powers of 9, we find
2;) = 0 + 12 0 - 20 + 22 - 12 - 16 + 20 - 8 (^, . . , ^)
- 4\p = 0 - 20 + 20 - 36 + 60 - 44 + 36 - 28 + 8
6 = 0- 8 + 20-56 + 82-56 + 20- 8 0
e = 0+ 8-28 + 56-70 + 56-28+ 8 0
•. j5 = 0 0-8 0 + 12 0-8 0 0
that is.
5 = -8^ + 12^ -8^;
and in precisely the same way the fifth equation gives
We find similarly C from the second equation : writing down first the coefficients of
jf, 2q&*, —iXcrd*, aud — 4/ip, the sum of these gives the coefficients of c; and then
writing underneath these the( coefficients of B@ and of — d", the final sum gives the
coefficients of C : the coefficients of each line belong to (6", 6^,.., 6^').
0 0 36 0-120 + 132+ 28-316 + 361- 20-340 + 396-144-112+164-80 + 16
- 8+20-16-12+22-20 0+12
- 40+140-212 + 140+ 80-188 + 168- 92- 64+176-164 + 80-16
-36+64-40+60-72+28 0+68-100+36
0 0 0+64-208 + 352-272-160 + 463-160-272 + 352-208+ 64 0 0 0
0 0 0 - 64 + 224 - 352 + 224 + 160 - 392 + 160 + 224 - 352 + 224-64 0 0 0
- 1
000 0+16 0-48 0+70 0-48 0+16 0 0 0 0,
that is, • . '
C=16^-48^ + 70^-48(9'» + 16^»;
and in precisely the same way this value of G would be found from the fourth
equation. There remains to be verified only the fourth equation (D + B) 0^ — @C = d,
that is,
2^ (- 8^ + 12^ - 8^) - (h)C = (2 - 4\t) ^= + {2pq - 4fia- - 4i/p) &>,
and this can be effected without difficulty.
The factor of the modular equation thus is
rt'» + 1;" + (- 8^ + 12^ - 8^) (m» + «») + 16^ - 48^ + 70e» - 48(9"' + 166''^
c, X. 43
338 ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [692
viz. this is
(u« + ti")* + (- 4^ + 6^ - 4^) 2 («• + «») + 1 6^ - 48^ + eSi?' - 48^" + 1 6^=
= (u» + e»-4^+6^-4^)>
= {(«'-«*)>- 4^ (l-^)*)*,
that is,
{u^-v*- 20(1 - ^)}' [u* - v' + ^dil - e^)]';
or the modular equation is
{„4_^_ 2^(1 -^)j« {«*-?;* + 2^(1 -^)1Hm' + «^-0) = O;
viz. the first and second factors bielong to the cubic transformation ; and we have
for the proper modular equation in the septic transformation u?-\-if — % = 0, or what
is the same thing {l-i^)0.-i^)-{\-ef = 0, that is, {I -ii?){\ -ifi)-{\ -uvf = 0,
the known result; or, as it may also be written,
The value of M is given by the foregoing relations
1 2
^t' -n ■ 1 = \«" +/*«* + in^ : - {u^' + pu* + qv^ + v"") : pu* + av* + tv''' ;
but these can be, by virtue of the proper modular equation «* + 1;* — 0 = 0, reduced
into the form
viz. the equality of these two sets of ratios depends upon the following identities,
(_ ^ + V») («» +pu* + qv* + v'^) + U{0-2&' + 20'-e') {pu* + o-«* + tv'-)
= {- ^w* + (1 - ^)(- 4 - ^ + 5^- ^'- 4^)1;^ + ««} {u^-% + ^),
-7 {6- m') (pu* + (rv* + Tt)") -(0-if) {\u" + nil* + jjv*)
= {{20 + 5^ + 3^ - 2^ - 20^) u* + (2 + 2d - 3^^ - 50' - 20*) v*] (m' -% + ifi),
-2(0-20*+ 20'- 0*) (Xm" + (lu* + w*) + (tt» - 6) (u'" + pu* +qv* + v")
= {u" +0{l-0) (.3 + 50 + 3^) u* - 0V*} (m« - @ + d»),
which can be verified without difiBculty: from the last-mentioned system of values,
replacing 0 by its value uv, we then have
1 2
^j : jg : 1 = ^u (v - «') : Uuv (1 - uv) (1 - uv + u'v^) : -v(u- y"),
which agree with the values given p. 482 of the "Memoir"; and the analytical theory
is thus completed.
693] 339
693.
A TENTH MEMOIR ON QUANTICS.
[From the Philosophical Transactions of the Royal Society of Lmidon, vol. CLXix., Part II.
(1878), pp. 603—661. Received June 12,— Read June 20, 1878.]
i
The present Memoir, which relates to the binary quintic (*^x, yf, has been in
hand for a considerable time: the chief subject-matter was intended to be the theory
of a canonical form which was discovered by myself and is briefly noticed in Salmon's
Higher Algebra, 3rd Ed. (1876), pp. 217, 218; writing a, b, c, d, e, f, g,..,u, v, w to
denote the 23 covariant'i of the quintic, then a, b, c, d, f are connected by the relation
/= = - aH + a^bc - 4c' ;
and the form contains these covariants thus connected together, and also e; it, in fact, is
(1, 0, c,/, a%-3c^, a'e-2cflx, yf.
But the whole plan of the Memoir was changed by Sylvester's discovery of what
I term the Numerical Generating Function (N.G.F.) of the covariants of the quintic.
and my own subsequent establishment of the Real Generating Function (R.G.F.) of
the same covariants. The effect of this was to enable me to establish for any given
degree in the coefficients and order in the variables, or aa it is convenient to express
it, for any given deg-order whatever, a selected system of powers and products of the
covariants, say a system of "segregates": these are asyzygetic, that is, not connected
together by any linear equation with numerical coefficients; and they are also such
that every other combination of covariants of the same deg-order, say every "congregate"
of the same deg-order, can be expressed (and that, obviously, in one way only) as a
linear function, with numerical coefficients, of the segregates of that deg-order. The
number of congregates of a given deg-order is precisely equal to the number of the
independent syzygies of the same deg-order, so that these syzygies give in effect the
congregates in terms of the segregates: and the proper form in which to exhibit the
43—2
340
A TENTH MEMOIR ON QU ANTICS. [693
^zygies is thus to make each of them give a single congregate in terms of the
segregates : viz. the left-hand side can always be taken to be a monomial congregate
a'V... or, to avoid fractions, a numerical multiple of such form; and the right-hand
side will then be a linear function, with numerical coefficients, of the segregates of
the same deg-oider. Supposing such a system of syzygies obtained for a given deg-
order, any covariant function (rational and integral function of covariants) is at once
expressible as a linear function of the segregates of that deg-order: it is, in fact,
only necessary to substitute therein for every monomial congregate its value as a linear
function of the segregates. Using the word covariant in its most general sense, the
conclusion thus is that every covariant can be expressed, and that in one way only,
as a linear function of segregates, or say in the segregate form.
Reverting to the theory of the canonical form, and attending to the relation
/» = -a»d-t-a»6c-4c',
it thereby appears that every covariant multiplied by a power of the quintic itself o,
can be expressed, and that in one way only, as a rational and integral function of
the covariants a, b, c, d, e, /, linear as regards /: say every covariant multiplied by
a power of a can be expressed, and that in one way only, in the " standard " form :
as an illustration, take
a»A = 6acd + 46c= -I- ef.
Conversely, an expression of the stiindard form, that is, a rational and integral function
of a, b, c, d, e, f, linear as regards /, not explicitly divisible by a, may very well
be really divisible by a power of a (the expression of the quotient of course containing
one or more of the higher covariants g, h, &c.), and we say that in this case the
expression is divisible, and has for its divided form the quotient expressed as a
rational and integral function of covariants. Observe that in general the divided form
is not perfectly definite, only becoming so when expressed in the before-mentioned
segregate form, and that this further reduction ought to be made. There is occasion,
however, to consider these divided forms, whether or not thus further reduced; and
moreover it sometimes happens that the non-segregate form presents itself, or can be
expressed, with integer numerical coefficients, while the coefficients of the corresponding
segregate form are fractional.
The canonical form is peculiarly convenient for obtaining the expressions of the
several derivatives (Gordan's Uebereinanderschiebungen) (a, by, (a, bf, &c., (or as I
propose to write them abl, ab2, &c.), whicii can be formed with two covariants, the
same or different, as rational and integral functions of the several covariants. It
will be recollected that in Grordan's theory these derivatives are used in order to
establish the system of the 23 covariants: but it seems preferable to have the system
of covariants, and by means of them to obtain the theory of the derivatives.
I mention at the end of the Memoir two expressions (one or both of them due
to Sylvester) for the N.G.F. of a binary sextic.
The several points above adverted to are considered in the Memoir; the paragraphs
are numbered consecutively with those of the former Memoirs upon Quantics.
693] A TENTH MEMOIR ON QUANTICS. 341
The Numerical and Real Generating Functions. Art. Nos. 366 to 374,
and Table No. 96.
366. I have, in my Ninth Memoir (1871) [462], given what may be called the
Numerical Generating Function (N.G.F.) of the covariants of a quartic ; this was
A{x) =
1 - ouc* . 1 - a V . 1 - aM - aM - a W
the meaning being that the number of asyzygetic covariants a^af-, of the degree 0
in the coefficients and order /i. in the variables, or say of the deg-order d.fi, is equal
to the coefficient of a^af- in the development of this function. And I remarked that
the formula indicated that the covariants were made up of {ax^, a-cv*, a^, a', aW), the
quartic itself, the Hessian, the quadrinvariant, the cubinvariant, and the cubicovariant,
these being connected by a syzygy a'x^'^ of the degree 6 and order 12. Calling these
covariants a, b, c, d, e, so that these italic small letters stand for covariants,
g-order.
1.4
a,
2.0
b,
2.4
c.
3.0
d,
3.6
e.
then it is natural to consider what may be called the Real Generating Function
(R.G.F.): this is
l-a.l-6.l-c.l-d.l-e'
the development of this contains, as it is easy to see, only terms of the form a'^l^ctd*
and a«6^c»d'e, each with the coefficient + 1, so that the number of terms of a given
deg-order 0.fi, is equal to the coefficient of a^af- in the first-mentioned function: and
these terms of a given deg-order represent the asyzygetic covariants of that deg-order:
any other covariant of the same deg-order is expressible as a linear function of them.
For instance, deg-order 6.12, the terms of the R.G.F. are a% a?bc, &: there is one
more term e^ of the same deg-order ; hence t? must be a linear function of these :
and in fact
e- = - d?d -I- a^bc — 4c ',
viz. this is the equation
342 A TENTH MEMOIR ON QU ANTICS. [693
367. Sylvester obtained an expression for the N.G.F. of the quintic : this is
a' . I
+ a' . x' + a^ + of
+ a* . it* + of
+ a' . a; + ar" + a;' — ir"
+ «" . aP + x*
+ a~ . x+a' — af
+ a' . a^ + x*
+ (i» . af-^- x^-aF
+ a>". oc^ + x^-x^'
+ a" . X -^af— a?
+ a}- . x" — a? — .r'"
+ «'•'. X -x'-sfi
+ a" . x'-ai'-a?
+ a" . — «' — «"
+ a" . a? — 35* — a^°
■\- a" . — X — a?
+ 0'". \ - a^ - a? - x'"
+ o'» . - ar> - a;'
■^ a^ . — a? — of — a?
1 -aaf.l -aV.l-aV.l-aM-aM-«'^;
viz. expanding this function in ascending powers of a, x, then, if a term is NaPaf-, this
means that there are precisely N asyzygetic covariants of the deg-order B.fi.
368. It is known that the number of the irreducible covariants of the binary
quintic is =23; representing these by the letters a, b, c, d, e, f, g, h, i, j, k, I, m,
n, 0, p, <[, r, 8, t, u, V, w, (a the quintic itself), the deg-orders of these, and the
references* to the tables which give them are
[* See also the paper, 143, in the second volume of this collection.]
693] A TENTH MEMOIR ON QUANTICS. 343
Deg-order.
1..5
a
Tab. Mem.
13 2
2.2
b
14
»
,,.6
c
15
>,
3.3
d
16
it
„.o
e
17
)»
,,.9
f
18
)»
4.0
9
19
it
,,.4
h
20
jj
,,.6
i
21
»
5.1
J
22
J»
.,.3
h
23
rt
,,.7
I
24
it
6.2
m
83
8
,,.4
n
84
tt
7.1
0
90* 9
„.o
P
91
a
8.0
Q
25
2 [See also paper 143]
,,.2
r
92
9
9.3
sf
11.1
t
94
9
12.0
u
29
3
13.1
V
95
9
18.0
w
29a
5.
Starting from the foregoing expression of the N.G.F. of the quintic, we can, instead
of each term a*af^, introduce a covariant or product of covariants of the proper deg-order
Q-IJi.: the mode of doing this depends of course on the different admissible partitions
of 6, fi, and it is for some of the terms very indeterminate : for instance, aV is ai,
hf, or ce. I found it possible to perform the whole process so as to satisfy a condition
which will be presently referred to; and I found
[* See vol. vn. of this coUection, p. 348.]
+ See end of Memoir. The S of Table 93 has the value -96 (W, jl/) + 16B0 - 7GA', but it is better to
use the simple value -(D, M); and the S of the present Memoir has this value, say .S'= -(d, m).
344
A TENTH MEMOIR ON QUANT1C8.
[693
B.OJ. of
1 .
qointie:
+ d .
l-ag*
+ e
1-6'
+/ •"
1-6
+ h .
I - ag*
+ i /
l-b'ff
+j •■
l-ag'
+ k .]
l-b'
+ 1 .
l-bg
+ m J
\.~ag'
+ n .
i-b^g
+ 0 .
l-b'
+ p .
l-b'g
+ r .
l-%
+ dj .
1-af
+ s .
l-abg
+ hj .
l-a^
+J' '
l-ag*
+jk.
i-h^g
+ t .
l-6»
+jm.
1 -a<jf'
+jo .
l-bg
+ v .
l-6»
+js .
l-bg
+jt .
1-9
+ w .
I —a
1— a. 1-6.1 —c
where observe that each negai
multiplied by a power or product
this is the condition above
each with the coefficient + 1 ;
« (1 — abg)
Deg-orders.
0.0-10.10
3.3-12. 8
3.5- 7. 9
3.9- .5.11
4.4-13. 9
4.6-12.10
5.1-14. 6
5.3- 9. 7
5.7-11. 9
6.2-15. 7
6.4-14. 8
7.1-13. 7
7.5-15. 9
8.2-16. 6
8.4-17. 9
9.3-16.10
9.5-18.10
10.2-19. 7
10.4-18. 8
11.1-17. 7
11.3-20. 8
12.2-18. 4
13.1-23.11
14.4-20. 6
16.2-20. 2
18.0-19. 5
1-^.1-5.1-M,
ive terra of the numerator is equal to a positive term
of terms a, b, g, contained in the denominator :
referred to. The expansion thus consists only of terms
'or instance, a part of the function is
_ S 1 — nhg
\ — a.\ — bA-c.\-g.l-q.l-u' \-c.\—q.\—u'\—a.\—b.\—g'
693] A TENTH MEMOIR ON QUANTICS. 345
where the first factor is the entire series of terms s(^q'iif, and the second factor is
the series of terms a'^b^g^ omitting only those terms which are divisible by abg: and
in the product of the two factors the terms are all distinct, so that the coefficients
are still each = 1. The same thing is true for every other pair of numerator terms :
and since the terms arising from each such pair are distinct from each other, in
the expansion of the entire function the coefficients are each = + 1. Hence (as in
the case of the quartic) for any given deg-order, the terms in the expansion of the
R.G.F. may be taken for the asyzygetic covariants of that deg-order; and if there
are any other terms of the same deg-order, each of these must be a linear function,
with numeiical coefficients, of these asyzygetic covariants : thus deg-order 6 . 14, the
expansion contains only the terms a-h, acd, bcr; there is besides a term of the same
deg-order, ef, which is not a term of the expansion, and hence ef must be a linear
function of a'A, acd, be'; we in fact have ef=a'h—6acd — 4bd'.
The terms in the expansion of the R.G.F. may be called "segregates," and the
terms not in the expansion " congi-egates " ; the theorem thus is : every congregate is
a linear function, with determinate numerical coefficients, of the segregates of the same
deg-order.
369. I stop to remark that the numerator of the R.G.F. ma}- be written in the
more compendious form
(I -b'){l -v) + (l -b')(o + t) + {l -¥){e + k) + {I -b)/
-(-(] -ag'){d + h+j+m + dj+hj+f+jm)
+ il-bg)(l+jo+js)
H- (1 - b'g) (t + n+p +jk)
+ (1 — abg) s
+ (^-9)jt
+ {l -a)w;
but the first-mentioned form is, I think, the more convenient one.
370. It is to be noticed that the positive terms of the numerator are unity, the
seventeen covariants d, e, f, h, i, j, k, I, m, n, o, p, r, s, t, v, tv, and the products of j by
(d, h, j, k, m, 0, s, t), where j' is reckoned as a product ; in all, 26 terms. Disregarding
the negative terms of the numerator the expansion would consist of these 26 terms,
each multiplied by every combination whatever a'^b^c^g'q'u^ of the denominator terms
o, b, c, g, q, u (which for this rea.son might be called " reiterative ") : the effect of the
negative terms of the numerator is to remove from the expansion certain of the terms
in question, thereby diminishing the number of the segregates : thus as regards the
terms belonging to unity, any one of these which contains the factor ¥ is not a
segregate but a congregate : and so as regards the terms belonging to d, any one of
these which contains the factor ag^ is a congregate : and the like in other cases.
For a given deg-order we have a certain number of segi'egates and a certain
number of congregates: and the number of independent syzygies of that deg-order is
c. X. 44
346 A TENTH MEMOIR ON QUANTIC8. [693
precisely equal to the number of congregates: viz. each such syzygy may be regarded
as giving a congregate in terms of the segregates: we have on the left-hand side a
congregate, or, to avoid fractions, a numerical multiple of the congregate, and on the
right-hand side a linear function, wth numerical coefficients, of the segregates.
371. The syzygy is irreducible or reducible; and in the latter case it is, or is not,
simply divisible: viz. if the congregate on the left-hand side contains any congregate
factor (the other factor being literal), then the syzygy is reducible: it Ls, in fact,
obtainable from the syzygy (of a lower deg-order) which gives the value of such
congregate factor. But there are here two cases; multiplying the lower syzygy by
the proper factor, the right-hand side may still contain segregates only, and then no
further step is required : the original syzygy is nothing else than this lower syzygy,
each side multiplied by the factor in question, and it is accoi-dingly said to be simply
divisible (S.D.). But contrariwise, the right-hand side, as multiplied, may contain con-
gregates which have to be replaced by their values in terms of the segregates of the
same deg-order : the resulting expression is then no longer explicitly divisible by the
introduced factor: and the original syzygy, although arising as above from a lower
syzygy, is not this lower syzygy each side multiplied by a factor: viz. it is in this
case not simply divisible.
For example (see the subsequent Table No. 96, under the indicated deg-oi-ders)
(6 . 6), from the syzygy
9d:' = aj-l>> + 2bh-cg,
we deduce (7.11) the syzygy
da<P = a^j — a6' -|- 2abh — acg,
which (all the terms on the right-hand being segregates) requires no further reduction :
it is a reducible and simply divisible syzygy. But we have (6.8) a syzygy giving
de, and also (6.10) a syzygy giving e*; multiplying the former of these by e or the
latter of them by d, we obtain values of de', but in each case the right-hand sides
contain terms which are not segregates, and have thus to be further reduced ; the
final formula (9 . 13) is
3(£(5> = - 4a'bj + 3a% + 4^- 8ab% + ^abcg - 1 26-crf,
which is not divisible by any factor: the syzygy is thus reducible, but not simply
divisible.
A syzygy, which is not in the sense explained reducible, is said to be irreducible.
372. The number of in-educible syzygies is obviously finite: it has, however, the
large value 179 as appears from the annexed diagram, showing the congregates
determined by these several syzygies, and the deg-orders of the syzygies: —
693]
a S o a. s. oa
1^
1^ CCOCi -^t-CO ^-COt^(M-^i— t*0(MC0^^^^O
e ^. IT) C-l <M (M <M S-l CI T'l IM (M (M (N e-1 <M (M eO CO
— o
rM ■* «3 -- in I— (M -# 00 « o I'l CO ec -*■ (M oi
to «S «6 ts t~ t^' fXi 00 00 CTJ CJ o o •— ' im' -*" to
1^ -t to ^- O t'- CI -^ 00 CO O (M to CC ■* (M
„ t^ -♦'-)• ■«• lO ic ^ o to' to w i-^ Qo' oo' cs o ci
_ to X' — 1-- C5 to to >— O I— ■>*• x in to '
^o ci ci C) cc n o>o 'i- -:)> >o >»' to' to i- x I
to ici^— tow cooo5-*toeoi— f
^*55o' — ' -^ -^ <m" CI ri di ri -r -^ in m" to
o ^ — c-< o i
c^ x-— ^- as»— ' tox-— t^csto^^
a?*»n o o o" -^ — ci C) ci en en •*' rtJ
o
t^ •* to —
tn t'- -t -f X en o (M
"^
en O o o — — SiX 'M ni en en Tj"
en i O —
00 I- o — 00 — lO t-- -- to 00
^*>.^ ci ci oi O O rt -^ ^ c-i (N
5»m' a> ai ai o o 2"^' -^ -- ff'
O ni to i-i en O ■*
bs-^ X' x' x" OS 0» <D O" O
t^ to X -^ t^ m 5^00 to_
!© oi x' X x' OS cj ^ 00 o
)-^ I— «
o o
to OS to — ^ t^ l~
© — ic o m
1-^ OS ^H I— t i-M ^^
o?Sim' i-^ t-^ w 00 x'
OS t- OS ^ 00
"?ucn t- i-I t^ 00
— Cl -f X
^ un to' to' to
% t^ to to
>x to I
«■! to
> o
.i <S
S .
p. -a
347
— "« «j V, "9 ■* •»,-!« «»fteos»,K«-~f>S
o
o
en i.n OS -^ to
en en en -^ ^
--cnt-ni'^^-in'Mcn-— f—to
ift iri lO to' to' w w x' cs" ^ en x'
44 — 2
348 A TENTH MEMOIR ON QUANTICS. [693
Observe as regards the foregoing diagram, that dj* is irred\icible (since neither dj
nor j* is segregate), and similarly j'/i, /, &c., are irreducible: we have thus the last
or j* column of the diagram.
The simply divisible syzygies are infinite in number, as are also the reducible
syzygies not simply divisible. There is obviously no use in writing down a simply
divisible syzygy; but as regards the reducible syzygies not simply divisible, these
require a calculation, and it is proper to give them as far fis they have been obtained.
373. The following Table, No. 96, replaces Tables 88 and 89 of my Ninth Memoir.
The arrangement is according to deg-orders, and the table is complete up to the
deg-order 8 . 40 : it shows for each deg-order the segregate covariants, and also the
congregate covariants (if any), and the syzygies which are the expressions of these
in terms of the segregates. When there are only segregates these are given in the
.same horizontal line with the deg-order; for instance, | 5.9 | at", ah, cd, shows that for
the deg-order 5 . 9 the only covariants are the segregates al/', ah, cd ; but when there
are also congregates, the segregates are arranged in the ssime horizontal line with the
deg-order, and the congi-egates, each in its own horizontal line together with its ex-
5 11 *
pression as a linear function of the segregates: thus " , .
are ai, ce, and there is a congregate bf which is a linear function of these, =—ai + ce.
The table gives the irreducible syzygies and also the reducible syzygies which are not
simply divisible, but the simply divisible syzygies are indicated each by a reference
to the divided syzygy which occurs previously in the table.
374. Any syzygy might of course be directly verified by substituting for the
several covariants contained therein their expressions in terms of the coefficients and
facients of the quintic. But it is to be remarked that among the syzygies, or easily
deducible from them, we have (6.18) the before-mentioned equation /- = — a'd + a'fcc — 4c-',
and also & set of 17 syzygies, the left-hand sides of which are the co%'ai-iants
ff, h,..,u, V, w, each multiplied by a or a", and which lead ultimately to the standard
expressions of these covariants respectively, viz. each covariant multiplied by a proper
power of a can be expressed as a rational and integral function of a, b, c, d, e, /,
linear as regards /. Supposing them thus expressed, a far more simple verification of
any syzygy would consist in substituting therein for the several covariants their ex-
pressions in the standard form, reducing if necessary by the equation f^=—a'd+a'bc—i<f:
but of course, as to the syzygies used for obtaining the standai-d forms, this is only
a verification if the standard forms have been otherwise obtained, or are assumed to
be correct.
ai ce
— r , the segregates
693J
A TENTH MEMOIR ON QUANTICS.
349
The 17 syzygies above refeiTed to are
Deg-ord.
6.10 a?g= 120*^ + 46=0 + 6=,
6 . 14 lek = Ucd + 46c= + ef,
5.11 ai = — hf+ce,
6.6 aj= b'-2bh + cg + 9d',
6.8 ajfc = - 2bi + Sde,
6.12 al= Ici-Mf,
7 . 7 am = - Wd - cj + Sdh,
7.9 an= b'e-Gbl- 2ck -fg,
8.6 ao = 26« + ej,
8.10 a;) = - Icn-fj,
9 . .5 a? = - 26y + id^f - 1 2dj?t + hj,
9.7 ar = 6=A; + 6p - co + Ai-,
10.8 (18= Sbdk + 3d/} + 2im,
12.6 a< = ?y7; +jp — 2mn,
13.5 18aM = 2agq + fc^j + 6bmj - Qdf - ghj + no,
14.6 3aw = 26=9 - 86=/ - 26=5r«t + 66% - 1 26»/i= + Set,
19.5 1 8atu = 36=5r« + b'qo - ib/o - bgnio + 1 86m« + Sdgjo - 1 8djt - Sght - 6 m'o,
the last four of these being, however, beyond the limits of the table : the expressions
of g, h, i are here in the standard fomi : the standard forms of the other covariants
j, k,.., u, V, w, will be given further on.
Table No. 96 (Segregates, Congregates, and Syzygies).
Deg-ord.
Congs.
Segregates.
1. 1
I
3
5
a
2. 0
2
h
4
6
c
8
10
a=
3. 1
3
d
5
e
7
ah
9
f
11
ac
13
15
a'
350
A TENTH MEMOIR ON QUANTICS.
Table No. 96 {continued).
[693
Uk-vA.
Cong*.
4. 0
2
4
6
8
10
12
14
16
18
20
y
b*, h
ad, be
ae
a*b, &
«/
a*e
a*
S.D. 5.11, bf
6. 1
3
S
7
9
11
13
16
17
19
21
23
25
3
k
ag, bd
be, I
a6', oA, cd
*
ai, ce
-1 +1
oV, abc
c^e, cf
a'b, ac'
«y
a'e
6. 0
2
4
6
8
10
12
14
16
by, m
n
*
d».9
aj, I?, bh, eg
+ 1 _ 1 +2 - 1
de.S
ok, U
+ 1 +2
*
c?g, abd, bi'c, eh
+ 1 -12 -4 .
*
d/.3
cdte, al, ci
. -1 +2
*
rt'6», o'A, acd, be'
+1 -6 -4
*
ab/
aH, ace
693]
A TENTH MEMOIR ON QUANTICS.
Table No. 96 (continued).
351
Deg-ord.
i Congs.
Segregates.
6.18
20
22
24
26
28
30
4>
«X «'6c, <^
S D. 6 6 d^
- 1 +1 - 4
a^e, acj
a*b, aV
«'/
a*c
7. 1
3
5
7
9
11
13
15
17
19
21
23
0
bj, dg
hk, eg, p
dk.S
abg, am, Vd, ej
0 +1 +2 +1
*
(re
di.3
eh
an, hi, ck, fg
+1 +6 +2 +1
0 +1 +1
0 +4 +2 +1
ad'
a^, oA", ahh, acg, bed
- 1 +6-6
S.D. 6.8 , de
S D 5 1 1 6/*
*
ode
fh.Z
e^k, obi, bee, el
-1 -2 +3 -6
.... S D. G 10 «^
*
ae'
fi
a?g, a^bd, ait'c, ach, e'd
+1 -1 +1 _6
S.D. 6.12, df
S.D. .5.11, b/
4>
ndf
he/
a*he, a% ad, <?e
S.D. 6.14, ef
c?V, a»A, a^cd, aJbe'
S.D. 5.11, bf
•
o?bf
aH, a'ce, <?f
S.D. 6.18,/'
*
afd, a'bc, acP
352
D«g-otd.
7.25
27
29
31
33
35
8. 0
2
4
10
12
14
16
A TENTH MEMOIR ON QUANTICS.
Table No. 96 {cmtinued).
Conga.
dk.3
bd'
ek
A'. 3
Me
dl.%
fj
Ai.3
adh
be-
cdP
d
fk
adi
aeh
hdf
cde
a'd'
aei
bef
fl.Z
Segregates.
a*e, c?cf
€fb, aV
r
l^g, bm, dj, gh
ao, bn, gi
+1 -3 -1
+ 1 -2
«*i, «<&". b*, b-h, beg, cm
-4 +3 +4 -6
+ 4 _3 -4 +8
1 +12
abk, aeg, ap, bH, en
+ 2
+ 1
+ 3 +1 +3
-1 . -2
. +2 -3
a*bg, ahn, aii'd, acj, 6'c, bch, e'g
-1 -2 +1 -2 +2
+1 . -1 +4 -6 +2
-1 . +1 -2 +1
o*n, abl, ack, a/g, bci
a'j, a'6', a'bh, a'cg, abed, 6V, c'h
+ 1 -1
+ 2
6 +G
[693
S.D. 6.6, «P
S.D. 6.8, (/e
S.D. 7.7, dh
S.D. 6.10, e»
S.D. 6.6, oP
S.D. 7.9, feV
S.D. 7.9, rfi
S.D. 7.9, eh
S.D. 6.12, (^
S.D. 6.8, de
S.D. 6.6, rf"
S.D. 7.11, ei
S.D. 6.14, ef
S.D. 6.10, e»
693]
A TENTH MEMOIR ON QTJANTICS.
Table No. 96 (continued).
353
Deg-ord.
Congs.
Segregates.
8.18
20
22
24
26
28
30
32
34
36
38
40
*
a^de
aby
a/h
cdf
a% a'bi, dbce, act, cS
. . . S.D. 6.8, de
. . . S.D. 5.11, b/
. . . S.D. 7.13,/A
. . . S.D. 6.12, d/
. . . S.D. 6.10, e'
. . . S.D. 7.15,/t
. . . S.D. 6.18,/^
. . . S.D. 6.14, ef
*
a/i
hP
cef
a*g, a'bd, drh'^c, a'ch, acH, b&
. . . S.D. 6.12, df
*
a?df
d?be, aH, a^ci, abcf, ac'e
. . . S.D. 6.14, ef
. . . S.D. 6.18, r
*
a*b\ a% d'cd, aW(?, c*
. . . S.D. 5.11, bf
a?kf
. . . S.D. 6.18,/"
*
cfd, a*bc, a'c'
a'e, <j?cf
9. 1
3
5
7
9
9J
bo, gk, s
*
dm. 12
of, aq, Vj, bdg, hj
. _1 _2 +1 +1
*
b'k.3
dn.3
em . 3
hk.3
V
ar, beg, bp, co, gl
+1 . _5 -1 +1
. -1 -1-1
+2 . +2 +1 -1
+2 . +2 +4 -1
. +1 +1
*
bdh.3
d^.27
en
ik
ai^g, abm, adj, agh, b'd, bcj, cdg
+2 +3 . +1+2-3
-1 . . . -3 +3
a X.
45
354
A TENTH MEMOIR ON QUANTIC8.
Table No. 96 (contintied).
[693
Dcg-^rd.
Ck>n«s.
Segregates. 1
1
9.11
13
»
adk
b*e
bdi
beh
¥9
<Pe.9
Jm.3
hi .3
0*0,
cJm, agi,
W, bek, ceg, cp
,
. S.D. 8.6, dk
. S.D. 8.6, ej
• •
•
+ 1 -1
+6 +2 +1
. S.D. 7.9, bi
- 1
+4 +2 +1
. S.D. 5.11, bf
+ 1
+ 1
+ 1
-3 -1
+ 3 +1
-3 - 1
+ 2 +2
. -3 . -3
+ 2 +5 . -6
'cd, c^j
. S.D. 6.6, d:'
. S.D. 8.8, ek
. S.D. 8.8, A»
. S.D. 7.11, ei
. . S.D. 7.7, dh
•
abd^
aek
aX'
bet
cdh
d«».3
a'bj,
a'dg, ab\
ofi'A, abcg, acm^ b
-4
-2
-1
....
+ 3 +4
+ 1 +2
+ 1
-8+4 . -
-3 +1 -2
-2 +1 -3 -
12
+
3 +
2
3
. . S.D. 6.6, O'
10. 0
2
4
6
8
10
bg',
hq, 9^ i
■«
*
do. 3
br,
gn, jk
bdj, bgh, eg", cq
+ 2
-1
*
dfg
eo
Am. 3
aaj.
b'g, b'm,
+ 1
-1
+ 1
+ 2 +12
+ 1+6
-1-4
-12-2
+ 12 -2 +1 -3
-12 +2 -1
. . S.D. 8.6, dk
. . S.D. 8.6, ej
. . S.D. 6.8, de
♦
bdk
bej
deg
dp. 9
hn.3
M».3
jl .3
abo,
ag^, as.
ft^n, bgi, cr
■ •
-5
-6
+ 1
-5
+ 3
+ 1
'. +3
+ 15 +5 - 6
+ 18 +5 -12
-3-1 +3
+ 15 +5 -12
6cm,
cdj,
cgh
♦
adm
6*. 8
b'd'. 72
bek.2
bh'.G
d'h. 27
^9
%.2
in. 4
iU.4
ay,
a'q, ail'j,
ed>dg, ahj, l^h, b'cg.
S.D. 9.5. dm \
+ 1 +10
-1-2
+ 1+2
-1-2
-1-2
-9 +3 +12 .
+ 9 -3+4 -8
-3 +3 . +4
+ 3 -3+4 -2
+3 . +1-2
+ 32
-32
+ 32
- 8
- 8
-12
+ 12
-12
+ 12
+ 12
-4
+ 4
-4
-4
+ 1
. . S.D. 6.10, e"
+ 1 +' 2
-1-2
+ 1+2
-1 . -2
-3 +7 . +4
+ 3 -5 .
-1 -1 .
-12
+ 24
+ 16
+ 6
-24
+ 12
+ 12
+ 2
-4
693]
A TENTH MEMOIR ON QUANTICS.
355
Table No. 96 (concluded).
Deg-ord.
Congs.
Segregates.
11. 1
3
5
go, t
bgj, df, dq, jm
9P
. . . S.D. 10.4, do
*
dr .18
ho . 3
jn
km. 6
b'o, hgk, bs, eg^, eq.
_2 + 5 - 6 . -3
-2+11-24 . -3
+1 - 3 + 6 . +1
-2+5-12 . -3
+ 3
+ 6
-2
+ 3
12. 0
2
4
gr, jo
S^h, hg
*
ko
m\U
by, h'q, bgm, 6f, dgj,
. _2 -2 -4 +3
. +2 +1 +4 -3
. -3
13. 1
3
fj, jq, V
mq, o'
*
jr .2
nw . 2
bgo, bt, g'k, gs, kg
. -2 . +1-1
. -4 . +1 -1
14. 0
2
4
V. bgq, bu, fm, gj".
*
dgo
dt .18
mr. 12
bgr, bjo, ghi, gjk, js,
nq
+1+2 . -1+6
+1+2 . -1 .
+ 3
+ 3
Theory of the Canonical Form. Art. Nos. 375 to 381, and Tables Nos. 97 and 98.
375. As the small italic letters have been used to represent the covariants,
diflferent letters are required for the coefficients of the quintic : using also new
letters for the facients, I take the quintic to be (a, b, c, d, e, f][f, r])". Considering
a linear transformation of - (a, b, c, d, e, f ][f , rif, viz.
-(a, b, c, d, e, f$f-bi;, avY,
45—2
356
A TENTH
MEMOIB
ON QU
ANTICS.
this is
?
6fS,
10f»7;>
10f»?'
5^V*
V'
1
-b
+ b'
- b>
+ b'
- b»
+ b (
1
-2b
+ 3b»
-4b»
+ 5b^)
+ ac (
1
-3b
+ 6b'
-lOb")
+ a»d(
1
-4b
+ lOb^
+ a'e(
1
- 5b)
+ af (
1).
[693
which is
= (
1
0
ao + 1
a»d + 1
a'e + 1
a*£ + 1
b» -1
abc -3
an)d-4
a'be - 5
b» +2
ab'c + 6
b' -3
a»by + 10
ab'c - 10
b» + 4
U VY:
The values of a, b, c, d, e, f, considered for a moment as denoting the leading
coefficients of the several covariants ultimately represented by these letters respect-
ively, are
a b c d e f
a+1
ae + 1
ac + 1
ace + 1
a»f + 1
a^'d + 1
bd-4
\? - 1
ad» -1
abe + 5
abc- 3
c» +3
b='e - 1
bed + 2
c= - 1
wA + 2
b^'d + 8
bc= - 10
b» -2
satisfying, as they should do, the relation
/" = -a»d + a»6c-4c».
Hence forming the values of afli — Sc" and a'e — ^cf, it appears that the value of
the last-mentioned quintic function is
(1, 0, c,/. a'6-3c^ a»e-2c/$e '?)"•
Writing herein x, y in place of f, 17, and now using a, h, c, d, e, f to denote, not
the leading coefficients but the covariants themselves (a denoting the original quintic,
with f , »; as facients), we have the form
^ = (1. 0, c,/, a'6-3c», o'e-2c/$a;, y)»,
693]
A TENTH MEMOIR ON QUANTICS.
357
a new quintic, which is the canonical form in question : the covariants hereof
(reckoning the quintic itself as a co variant) will be written A, B, G, .., V, W, and
will be spoken of as capital covariants.
376. The fundamental property is: Every capital covariant, say I, has for its
leading coeflBcient the corresponding covariant i multiplied by a power of a: and
this follows as an immediate consequence of the foregoing genesis of A, The
covariant t of the form
-(a, b, 0, d, e, fjl^, v)'
has a leading coeflBcient
= l(a»cf-aMe + &c.),
cL
which, when a, b, c, d, e, f, .., i denote leading coefficients, is = i multiplied by a power
of a: and upon substituting for the quintic the linear transformation thereof
(1, 0, c,/, a'6-3c». a?e-2cf\l vY,
(observing that, in the transformation f, t) into ^ — hrj, a?;, the determinant of sub-
stitution is =a), the value is still =i multiplied by a power of a; or using the
relation a = a, say the value is =i multiplied by a power of a. Now the covariant
i is the same function of the covariants a, b, c, d, e, f that the leading coeflBcient
i is of the leading coeflficients a, b, c, d, e, f; hence, the italic letters now denoting
covariants, the leading coeflBcient still is =i multiplied by a power of a: which
ia the above-mentioned theorem.
377. To show how the transformation is carried out, consider, for example, the covariant
B. This is obtained from the corresponding covariant of (a, b, c, d, e, fjf, i;)", that is.
(
ae 1
af 1
bf 1
bd- 4
be- 3
06 — 4
c» + 1
cd+ 1
d" + 3
1^, vf,
by changing the variables, and for the coeflScients
a, b, c, d, e, f
writing
1, 0, c, /, a^b-^c\ a»e-2c/;
thus the coeflBcients are
First.
l(a'6-3c»)
+ 3c^
a»6
Second. Third.
1 (a'e - 2c/) - 4c {d?b - Sc")
+ 2c/ -K3/'
= a'e
= -4a'6c-|-12c*
+ 3(-a»d + a»6c-4c»)
= a" (- 3ad - be) ;
358 A TENTH MEMOIR ON QU ANTICS. [693
and we have thus the expression of B (see the Table No. 97); and similarly for the
other capital covariants C, D, . . , V, W: in every case the coeflScients are obtained
in the standard form, that is, as rational and integral functions of a, b, c, d, e, /,
linear as regards /
378. It will be observed that there is in each case a certain power of a which
explicitly divides all the coeflScients and is consequently written as an exterior factor:
diffl*egarding these exterior factors, the leading coefficients for B, C, £>, E, F are
6, c, ad, e, f respectively; that for G is \2ahd ■¥ ^h^c + ^, which must he = g multi-
plied by a power of a, and (in Table 97) is given as =a?g\ similarly, that for H is
Qacd 4- 460° + ef, which must be = A multiplied by a power of a, and is given as =a*h:
and so in the other cases. The index of a is at once obtained by means of the
deg-order, which is in each case inserted at the foot of the coefficient.
For A, B, C, E, F there is no power of a as an interior factor: and for the
invariants 0, Q, U we may imagine the interior factor thrown together with the
exterior factor, (G = a'g, &c.) : whence disregarding the exterior factors, we may say
that for A, B, C, E, F, G, Q, U the standard forms are also "divided" forms.
But take any other covariant — for instance, D: the leading coefficient is ad, having
the interior factor a; and this being so it is found that all the following coefficients
will divide by a (the quotients being of course expressible only in terms of the
covariants subsequent to /): thus the second coefficient of D is —bf+ce, and (5.11)
we have — b/+ ce = ai, or the coefficient divided by a is = i ; and so for the other
coefficients of D; or throwing out the factor a, we obtain for D an expression of
the foi-m (d, i,...\a;, yf, see the Table 98: this is the "divided" form of D: and
we have similarly a divided form for every other capital covariant. All that has
been required is that each coefficient of the divided form shall be expressed as a
rational and integral function of the covariants a, b, c, ,.,v, w: and the form is not
hereby made definite : to render it so, the coefficient must be expressed in the
segregate form. But there is frequently the disadvantage that we thus introduce
fractions ; for instance, the last coefficient of D is = — ci+df, where to get rid of
the congregate term df we have (6 . 12), 3d/"= — al + 2ct, and the segi-egate form of
the coefficient is = — i^al + fcx.
379. We have in regard to the canonical form, a differential operator which is
analogous to the two differential operators xdy — {icdy}, ydx — {ydx} considered in the
Introductory Memoir (1854), [139]. Let B denote a differentiation in regard to the
constants under the conditions
Sa = 0,
Sb = e,
ac= 3/,
Bd= l{-b/+ce),{=i),
Se = - Gad - 106c,
Bf = 2a»6 - 18c»,
693] A TENTH MEMOIR ON QUANTICS, 359
which (as is at once verified) ai"e consistent with the fundamental relation
then it is easy to verify that
and this being so, any other covariant whatever, expressed in the like standard form,
is reduced to zero by the operator
d . d f.
"'dy-'^'^da,-^'
and we have thus the means of calculating the covariant when the leading coefficient
is known.
Thus, considering the covariant B, the expression of which has just been obtained,
= (5o, Bi, B^x, yy, suppose : the equation to be satisfied is
X (B^x + 2£jy )
-4cy( 2BoX + B,y)
-a^BBo -xyBB,-y^SB, =0,
viz. we have \,
B, -SBo = 0,
25, - 8cB, - SB, = 0,
- 4c5i - S^s = 0 ;
which (omitting, as we may do, the outside factor a') are satisfied by the foregoing
values B^, B^, B„ = b, e, -Sad — be. And if we assume only Bo = b, then the first
equation gives at once the value B, = e, the second equation then gives £j=— 3ad— 36c;
and the third equation is satisfied identically, viz. the equation is
- 4ce + 8 (3ad + 6c) = 0,
that is,
— 4ce = — 4ce = 0,
4- cSb +c . e
+ 6Se + 6 . 3/
+ 3aSd + 3 (- bf+ ce)
which is right.
Of course every invariant must be reduced to zero by the operation S: thus we
have, see the Table No. 97,
a'g= 12abd
+ 46»c
+ le»,
360 A TENTH MEMOIR ON QUANTIC8. [693
aud thence
ade Iff bee
a*Sg= {\2ad + 8bc)Bb = {12ad + 8bc)e = + 12 +8
+ 46' .8c +46" .8/ +12
+ 12ai> .Bd +12b{-b/+ce) -12 + 12
+ 2e .Be + 2e (- 6ad - lObc) -12 -20,
which is =0, as it should be.
380. As already remarked, the leading coeflBcients of H, I, J, &c., are each of
them equal to a power of a multiplied by the corresponding covariant h, i, j, ..; hence,
supposing these leading coefficients, or, what is the same thing, the standard ex-
pressions of the covariants h, i, j, .. ,v, w to be known, we can calculate the values
of Bh, Bi, Bj, .. , Bv, Bw (= 0, since w is an invariant) : and the operation 8, instead
of being applicable only to the forms containing a, b, c, d, e, /, becomes applicable to
forms containing any of the covariants. The values of Ba, Bb, ., , Bv, Bw can, it is
clear, be expressed in terms of segregates ; and this is obviously the proper form :
but for Br, Bt, and Bv, for which the segregate forms are fractional, I have given
also forms with
integer coeflScients. The entire series is
Deg-order.
2.8
Ba = 0,
3.5
Bb = e.
3.9
Be = Sf,
4.6
Bd= i.
4.8
Be =-6ad-10bc.
4.12
Bf = 2a'b - ISc^,
5.3
^9=0,
5.7
Bh = 2Je-4i,
5.9
Bi =-2ab' + 2ah-18cd.
6.4
Bj =~n,
6.6
Bk =- 2aj +6b^- 9bh + 30^^,
6.10
BI =-Sabd-7b'c + 1ch,
7.5
Bm = — bk—p,
7.7
Bn = 4cj,
8.4
Bo = b^g + 6bm-6dj-gh,
8.8
Bp = 8ahj - 5adg - 10b* + 156"^ - 5bcg + 10cm,
9.3
Bq = 0,
9.5
Br = i{aq + Gb!'j-5bdg-jh), = 2b'j - 2bdg - 6dm,
10.6
Bs =- 2agj + 2f^g + Sb'm + 21bdj - 4bgh + 2c5f'' - 3cq,
12.4
Bt = ^(bgm+ib/-Sdgj-kq), = - b'q + hq + 6m\
13.3
Su = 0,
14.4
Bv = i{-5bgr-10bjo + 5gjk-12js-9nq), = - 6dt - 6mr + nq,
19.3
Sw= 0.
693] A TENTH MEMOIR ON QU ANTICS. 361
It is obvious that for every covariant whatever written in the denumerate form
(/o, Ii,...'^a;, yY, the second coefficient is equal to the first coefficient operated upon
by S; so that the foregoing formulae give, in fact, the second coefficients of the
several covariants.
381. It is worth noticing how very much the formulae of Table No. 97 simplify
themselves, if one of the covariants b, c, d, e vanishes, in particular, if b vanishes.
Suppose 6 = 0; writing also (although this makes but little difference) a = l, we have
a
=
1,
b
=
0,
c
=
c.
d
=
d.
e
=
e,
/■'
= -
-d-
■^,
9
=
e\
h
=
6cd +e/.
t = ce,
j = 9d' + c^,
k = 3de,
I = - 3d/+ 2(fe,
m = 9c<f + 3def- (?e,
n = — Qcde — ^f,
0 = 9d'e + c^,
p =- 9d'/ + 1 2d'de + cef,
q =- 54cd' - 27 d'ef+ 18c=de^ + c^f,
r = gcd'e + 3de=/- cV,
8 = - 27dy + 5*c^d'e + dcde"/- 2c»e',
t = - Sidy- 6dV + 216c'd'e + 5icd'^f- 24c»de» - cV/
u =- 21 d' - IScdV - 4rfV/+ c^de*.
V =- H\d*ef- 6dV + 216c'd'e'' + 54cdV/- 24c»de* - lc»ey,
w (not calculated).
These values are very convenient for the verification of syzygies, &c. Take, for instance,
the before-mentioned relation hv = - Qdt - Qmr + nq, that is, if F = (Fo, V^'^x, y), then
F, = — 6d< — 6mj- + ngj : calculating the three products on the right-hand side, observing
C. X. 46
362
A TENTH MEMOIR ON QUANTIC8.
[693
that /' when it occurs is to be replaced by its value - d - 4c', aud taking their sum,
the figures are as follows:
- 6«tt - 6mr + no Sum
+ 486
+ 486
+ 36
+ 54
- 27
+ 63
-1296
- 486
+ 324
. - 1458
- 324
-324
+ 216
- 432
+ 1
+ 1
+ 144
+ 324
-216
+ 252
+ 6
+ 36
- 24
+ 18
- 6
+ 4
- 2
d'f
dV
e'd't
ed'i^/
i'd^e'
<?df^f
where the last column is, in fact, what Fj becomes on writing therein a = 1, 6 = 0. The
verification would not of course apply to terms which contain 6; thus, (13.3), a
derived syzygy is ^V = 6< + mo; and the foregoing values give, as they should do,
jr = ino: we might for the verification of most of the terms in b use values a, b, c, d,
e,f* = l, b, 0, d, e, —d: the only failure would be for terms containing be.
Table No. 97 (Covariants of .4, in the af~ or standard forms: W is not given).
The several covariants are —
A = (
1
0
c + 10
/+10
a'b+ 5
<? -15
a'e + 1
C/-2
lu>=, y)'
0.0
1.3
5 = o'(
2.6
3.9
4.12
5.15
b + 1
e + 1
ad-3
be - 1
"S*, yf
2.2
3.5
4.8
C=-(
c+1
/+1
d*b+ 3
0*6+ 1
a'd + 6
ay- 3
a«6» -1
c» -15
C/-10
a'bc- 3
e* +15
„ce+3
oV+3
a'cd +2
a»6c> + 4
„«/■ + !
aV -1
l[«.y)'
2.6
3.9
4.12
5.16
6.18
7.21
8.24
693]
D = a'{
A TENTH MEMOIR ON QXJ ANTICS.
Table No. 97 {continued).
ad+ 1
bf-\
aW - 1
««?/ + 1
ce + 1
acrf + 3
a%cf + 1
a»6c» + 4
„c»e - 1
„ e/" +1
4.8
5.11
6.14
7. 17
\x, yf
363
£ = a»(
e+1
orf -
- 6
bf- 12
a'i' - 8
a%e - 5
a%d - 6
a»6c-
- 10
ce + 2
acd - 36
a''6<r'+ 12
„«/- 2
adf -1\
aobcf- 4
„c=« + 2
a^6V- 2
„e» - 1
oc'rf + 18
o»6c»+ 6
„ce/+ 2
3.5
4.8
'• 5.11
6.14
7.17
8.20
\x,yf
F=(
/+1
a»6+ 2
a'e + 1
a'd+ 34
a'bf- 40
a*6« - 16
a*be - 7
a'bd+ 6
aWe -12
a«6» - 2
aV-18
aV-36
o'6c- 42
„ ce + 5
a'crf+ 6
aW+ 8
a*6'c-22
a*bce +11
a»6crf+ 6
aV+168
a»cy- 126
o»6c»+134
a*bc/+ 8
„<^ - 1
„fiy- 9
a^6V+12
.,«/- 5
„ c'e + 65
aVd+54
a»crf/*+ 24
„6«/+ 3
aV -252
aV/-84
a'6c'+66
„ce/+38
aV +72
a''6cy+ 32
„(?e -45
aV/+ 9
„ce^ - 1
aVrf-14
a'bc* -16
„cV- 5
aV - 2
3.9 4.12 0.15
6.18
7.21
8.24
9.27
10. ,30 11.33
12.36
lla;,y)*
6 = a*
abd + 12
a'6'c + 4
„e' + 1
=«v
6.10
46—2
364
A TENTH MEMOIR ON QUANTIC8.
[693
H = tf(
6.14
Table No. 97 {continued).
aed +%
a*be + 2
«Vc+ 4
a*de + 2
0*6* + 2
a''bc>+ 4
ad/ +12
„«• + 1
a»6y+ 4
„<? +6
..?/■ +1
o»6c/- 8
ac>d -36
„bce- 6
o»6«i -2
„c>«- 8
a^bi* - 24
oc<^- 12
a'bV - 8
„«/- 6
o»6cy- 8
„c»« + 8
„ 6e/ - 3
„ ce» +1
ac»flj + 6
a'bii" + 4
= a'A
„<^e/ + 1
7.17
8.20
9.23
10.26
Ji^h yY
I=a?{
bf-\
o«6» -2
ad/ -\b
a»6rf-20
a'tfe - 5
a<6» + 2
a*6*« + 1
ce + 1
acd -6
a'>be/+ 5
ac'd + %Q
a»6y + 5
„d' -12
a'W/+3
o'W+S
„ c"* - 5
„bce — 5
a>6cc;- 2
„cde — 5
„ef-2
acd/ +30
rt«6cy+ 5
.jc'e - 5
a>6»c»- 6
„ce» - 2
oc^rf -30
a'bc* - 8
„Ce/- 2
a'l^c/+ 1
,,6c*«-5
a<^d/-S
a»6cy-l
-ai
„c*e +1
5.11 6.14
7.17
8.20
9.23
10.26
11.29
![«. y)'
J=a*{
a»6' + 1
a»6''« - 1
„flP +9
abd/ - 6
a«6V - 4
„cde + 6
„6«/- - 2
a''6»c/-4
„ce' + 1
„6c'e + 8
,.ey +1
= aV
8.16
9.19
11«, y)'
693]
A TENTH MEMOIR ON QUANTICS.
365
K=a^{
7.13
Table No. 97 (continued).
ode + 3
a?b^ + 4
a%h + 1
aWd+ 6
a"!^/ + 2
„^ -18
abdf -i- 6
a»6»c + 2
,.6c6-2
a6ccZ -18
rtC(is -15
„c</^ -18
a»6V-16
o»6V- 2
aAc^tZ- 30
„6e/- 5
„6c'e- 2
a<fe/- 9
„ce» + 1
,,«!/■ - 1
a»6V- 8
„bcef- 5
= a^A
„cV + 3
8.16
9.19
10.22
\x, yf
L=a'{
ad/ -3
a»6d- 3
o'de -12
a<6» - 6
a*6»e - 1
a'l^d + 15
a'ftf/^ - 7
a«6* - 2
Me/- 2
a'b'c- 7
a»6y- 9
„d' - 39
a'bd/+ 39
a'^b'^c - 9
a*6y - 7
..ftrf" + 3
„c>e +2
ac'rf+42
„bce+ 9
a'6cc;+ 40
„cde— 14
„c(;;= + 18
„6^ce+14
a»6W+ 10
o»6c»+28
acd/ +63
a'bV+ 59
a=6V+ 16
a^bc'd- 33
„dP/ +12
„d^ + 2
„ce/+ 7
o»6cy+42
„6«/+ 7
..fic'e- 12
„de/- 3
a'6c<f/+ 23
a*6V +13
„c'e -42
„ce'- 1
„«!/•+ 1
a'6V+ 15
„c-^(Ze-26
„6V+ 4
'
oc'd -210
ac'd/-l05
„6ce/+ 21
a'bV/+2b
„6ce= - 2
a«6c* -140
a!>b<^/- 70
„cV - 12
„ 6(^6-53
„c'd* -15
„c»«/- 35
„c*e + 70
ad'd +126
a»6c» + 84
„c>e/+ 21
„cey- 7
ac»<i/+21
a«6cy + 14
..c'e -14
a^6c'<i- 28
„CCZ6/- 7
a^ftV -19
„6cV-10
..c^e' + 5
a<!»d - 6
ooftc* - 4
= a'^
„c*«/- 1
7.17 8.20 9.23
10.26
11.29
12.32
13.35 14.38
\x, yf
Af = a*{
a?m -
2
a^bde - 1
a*6*
- 1
a'b^c -
1
aW/ - 1
„6(Z»
+ 3
„cd' +
9
„6^ce + 2
a^b'cd
+ 6
ab(^d +
12
„rfy + 9
„d^
+ 1
„de/ +
3
abcd/+ 12
aWc"
+ 5
aoVi? +
4
„ c'rfe - 12
„b'e/
+ 2
„ftce/+
2
o»6V/+ 4
„bc^
- 1
„c»e»-
1
„6c'e - 8
„e'd'
- 9
„cey - 1
a bc?d
„ cdef
a'b^c*
„h<?e/
- 12
- 3
- 4
- 2
= ahn
,,Ce»
+ 1
\x, yf
10.22
11.25
12.28
366
A'=a'(
A TENTH MEMOIR ON QUANTICS.
Table No. 97 (contimied).
aW» + 1
aVe + 4
Mde + 12
M* + i
a*6»« + 2
a64^+6
„ ed* + 36
aVf + 6
„bd' + 12
„d^e + 9
„«fc -6
a'W - 16
„6'c«- 6
a'6W + 8
a'6»<^+ 6
o«6"c/+ *
,.b<x!f- 8
„dy +64
„de' + 4
„ bcde - 10
,.6A-8
„c««» + 4
a6«^+ 36
a»6V - 12
a'ft'c/- 2
..«y -1
„c»<fc-36
„6V- 4
„c»rf'-108
o6c»rf- 96
.. ««?/•- 24
efib^e' - 16
„6c«?/'- 8
„«;•«!»+ 4
„6Ve-ll
,.6«y- 3
„c<?/-18
„c^ + 1
a6c»<(/'-18
„ c'de + 18
«»6»cy- 4
„bc*e + 8
^a*n
„cey+ 1
9.19
10.22
0:=al>{
11.25
12.28
13.31
aVe + 3
a»6»rf - 6
„rf»e + 9
„d' -54
ab^d/+ 12
a'6*c - 2
„6cde- 12
„6V + 1
aWcf + 8
„bcd^ - 18
„ b^c'e - 20
a 6Vrf + 24
..ft**/- 4
„bde/+ 18
„ce» + 1
..cde* - 12
a'bV + 8
„6V- 8
„6c»e«- 10
= a*o
..-y - 1
\^, yY
[693
\^, yy
11.21
12.24
/' = a*(
afbde - 2
a'^ - 1
a6c<^-12
„c»de + 12
a^6»cy- 4
„ ftc** + 8
„c«y+ 1
11.25
o«6* -
„W» +
a'lr'ed-
„cfe= -
a«6><r -
6V/-
, 6c«'
12
10
5
2
1
3
„c'd*+ 90
0 6(^^+120
„c£fe/+ 30
a'b'e* + 40
„6c*e/'+ 20
,.c»«" - 10
a«6»e -
5
„rf»e -
9
a'b'd/-
24
„ bcde +
44
a'b'cf-
6
„ br'c'e +
36
,Mf +
6
„cdy +
90
„ce» -
1
abc^d/+
120
„ ^de -
120
a»6V/+
40
„6c*e -
80
,.<^«'/-
10
a'6'rf -
8
,,rf' -
72
a*b*c +
4
„6V -
1
„6crf»-
24
a'ftW +
52
„6rf«/+
10
8
o»6V +
4
„b-c,f-
„ bc'e^ +
2
2
„e'd'-
180
abc*d-
1
240
„c'de/-
a»6V -
60
80
40
20
■
a'b'de + 6
a*b*/ + 5
,,6'ce -13
„6c/y+21
„ cd»e - 21
a'b'cd/- 4
„ bc'de+ 10
„cfcy+ 4
oWcy-17
„ bVe + 44
„ bcey+ 13
„ c»rfy- 45
„e'^ - 5
a bc'df- 60
„ c*de + 60
a»6V/-20
„6c»e +40
„Cey- 5
12.28
13.31
14.34
16,37
a'6» + 2
„6»d= +12
oWcrf - 2
„ bde' - 3
„cd» -36
a*6V -12
„ 6 V - 5
„ b^c^ + 4
.,6c"cZ''-66
„(fe/ -18
a'fcVrf - 10
„bcdef- 4
..c'rfe' + 1
a'ftV +14
„6Ve/+ll
„6cV -11
„c<rf» +18
„««!/• - 1
ab&d +26
„ c'rfe/+ 6
aWc* + 8
„c»e» - 2
16.40
l^a;. y)'
693]
A TENTH MEMOIR ON QUANTICS.
367
Q = €f
a*b' -
2
„b'd' +
18
a'b'cd +
22
„bde> +
3
„cd^ -
54
a^b'c^ +
12
„bV +
5
„bW -
4
„bc'd--
108
„d'ef -
27
aVe>d -
72
„ bcdef-
36
„c'd<? +
18
oOfiV -
16
„ 6Ve/-
12
„6c-V +
12
„c«y +
1
=««^
Table No. 97 {continued).
K = a'{
a'bde -
1
a*b' +
2
a'bU + 1
a'b*/ +
„6V« -
1
3
„bW +
aWcd -
6
2
aW(^ + 3
„b'cde - 11
„hcPf -
„ crf^e +
9
9
„bde' -
„ cd^ -
2
54
„d^f -27
„(/e^ - 1
ab^cdf ~
12
a^iV -
8
rt'ftV" + 1
„b<^de +
„«fey +
„6Ve +
„6cey +
„<?^ -
24
3
4
12
3
1
„6»ce= +
„ 6cV/- -
.dP"/ -
ab'c'd -
„bcdef-
„c'de^ +
4
2
72
18
24
12
6
„6Ve - 7
„6V/ - 2
„ 6«P/ - 45
„bve' + 1
„ c^tfe + 45
abVdf- 24
„ 6c»cfe + 48
aob^c/ - 4
„6Ve + 12
„ 6cV/ + 3
„cV - 1
= o»r
1[«, 2/)=
14.30
13.27
14.30
15.33
^ = a«(
a^bd'e + 9
a»6*d + 15
a»fr'<^ -
6
a?b'
2
o'6»<^ + 7
„bd' - 27
„d-'e -
27
„b'd'^ +
9
„6V(fc -12
a'b'e + 3
a*bY -
3
„d' -
27
„rfy -27
..fcVrf' - 99
„ b*ce +
9
a'b'cd +
18
o»6V + 2
.,rfV - 18
„ 6vy -
9
„bW +
6
..fe'fr'e - 6
o'iVrf -114
„bcd'e -
18
„bcd' -
54
„ body - 54
„ b^'def - 33
dWcdf -
9
a*bV +
15
„c'd'e +54
„6c(fc» + 12
„ bVde +
24
„6V +
6
a6V(^-36
..c'rf' +162
„ fccfey +
3
„bVd'-
36
„6<r'cfe +72
o'6V - 24
»cd>/ +
81
„bW -
6
„ cdey + 9
„6'c«/- 9
„cde> -
3
„ bVd^ -
27
a»6'cy - 8
,,6'cV + 9
d'b*cy +
6
., irf V -
9
„6Ve +24
..fcc'rf^ +324
,.6Ve -
18
„ crfV -
9
,.bc'e'/+ 6
„cdY+ 69
„ bW/ -
9
a'bVd -
54
„c»«» - 2
aftVrf +216
„bc'd'/ +
162
„ b'cdef-
27
„6cW«/'+120
„bc'e' +
3
„ bc'de' +
3
„e'de' - 54
„c^d^e -
162
„c-W'' -
54 •
aWc* + 48
abVd/ +
108
„d^/ -
2
„6Ve/+ 36
„ bc*de -
216
d'b*c' -
24
„6c«e' - 36
„ c'd^f-
27
„ b'ce-y -
21
„c=ey - 3
aObVf +
24
„b''cV +
21
„AVe -
72
„6cW^ -
108
„6c»ey-
18
„ bcey +
2
„«*«•' +
6
„c'dV'
abVd -
,, be'def-
„c*de' +
a"ft»c« -
„ 6V«/ -
„b<fe' +
27
72
36
18
16
12
12
= a*«
„<?>?/ +
1
U--> !/r
15.33
16.36
17.39
18.42
368
A TENTH MEMOIR ON QUANTIC8.
[693
Table No. 97 (conti/iued).
T = {
<fh*d«
+ 7
0*6' + 2
„hd*o
+ 27
„6W» + 6
«v/
+ 1
(fVcd - 10
„Vc«
- 2
„ b'd^ - 8
„Vd}f
+ 24
„b^ed' - 54
„b*c(Pe
- 54
.,dV - 27
.,dy
- 81
a'bV - 14
„dV
- 6
„6««/ - 7
a'b*c<if
+ 16
„ b*ce' + 9
„V,?de
- 76
„6Vef - 84
nVcU?/
- 12
„b'd'e/ - 27
„hcd?f
-216
„6cdV + 9
„hcd^
+ 5
„c'd* +162
„ c'lPe
+ 216
a'b*(^d - 8
a^bVe
- 8
„Pcde/ + 4
„bV<P/
-216
„bVde^ + 18
„ fiVe'
+ 2
„b<^d' +432
„ be'd^e
+ 432
„bde>/ + 3
„ ccPe'f
+ 54
„cdY +108
abVdf
- 96
„ccfc^ - 1
„ bVde
+ 288
a'hV + 16
„bc'de'/
+ 72
„6V«/ + 20
„c'd^
- 24
„6Ve» - 24
a'b*c*/
- 16
„bVd^ +432
„We
+ 64
„6W/ - 5
„6Vey
+ 24
„bcWe/+2\6
„6cV
- 16
„ bc'e' + 1
„cV/
- 1
„c»««V -108
otVtf +192
„6V*/ + 144
„bc*de' -144
„cW/ - 12
a'bV + 32
„6»cV + 32
„6W - 48
„6<^«y - 8
= aH
..c^e* + 2
l^a;, y)'
?7=
€fm
_
3
„V>d-
+
14
„d'
-
27
a^Vc
-
1
.,b*cd*
+
34
„b'dv
+
11
„bcd*
-
81
a'bVd
+
32
„b*de/
+
10
„ b'cde-
-
6
„bVd'
-
144
„ b(Pe/
-
18
„ cdV
-
18
aWc'
+
8
„b'ce/
+
4
„bVe'
-
6
„ bVd'
-
152
„ b'cd-ef
-
60
„ bc^d'^
+
6
„«p«y
-
4
ab*c*d
-
80
„bVde/
-
56
„ bVde'
+
48
„ bcd^f
+
2
„c'de*
+
1
a»6V
-
16
„¥c'e/
+
16
„ bVe^
+
24
„iW/
+
4
„ b<^e*
+
1
= a'u
21.45
19.41
20.44
693]
A TENTH MEMOIR ON QUANTICS.
369
Table No. 97 {concluded).
7"(
a«6«
- 4
a^b'e - 2
„ b'd-^
- 12
„ b*d'e - 48
a^b^cd
+ 20
„bd*e - 162
„b*de'
+ 23
a'¥df - 6
„ b^cd'
+ 108
„AVrfe + 8
„bdV
+ 81
„6W/ - 144
a*bV
+ 28
„ Vdt^ + 8
»h'ef
+ 15
„b^cd'e + 324
„6W
- 20
„rfy + 486
„ b*(?d-
+ 168
„ dV + 63
„rav
+ 78
a*U'cf - 2
„ fe'crf'e^
- 72
„bVe + 18
„bc'd'
-324
„6V/ + 7
„ d*ef
- 81
„b*cd^f - 144
„cPe*
- 6
„ b*c^ - 9
d'bVd
+ 16
„ bVdr-e + 648
„b*cdef
+ 8
„b-'cPe'/+ 99
„ bVde"
- 112
„bcdY + 1458
„ bVd'
-864
„bcd'^ - 27
„b'deV
- 18
„ cWe - 1458
„ becPef
-432
a'bVdf - 32
„ bcde*
+ 7
„b*c^de + 208
„(^(fe»
+ 216
„b'cd^/+ 20
a'b'g'
- 32
„bVdy+ 1728
„bVe/
- 40
„ 6Vrfe^ - 40
„6V«»
+ 40
..ftr'rf'e -3456
„bW
-864
„bdeY - 3
„b'ce>/
+ 10
„ccP«y - 432
„ b^c'cPef
-648
„ cde^ + 1
„ ic'rf'e^
+ 648
aWc'ey - 20
„cd'<?f
+ 54
„bVdy+ 1008
ab*(?d
-384
„6Ve» + 20
„ bVdef
-384
„ 6V(/^e - 3024
„ iVoTe^
+ 576
,,^'W/ + 5
„ bc'd^/
+ 96
„b^(Pey~ 756
„c'de'
- 24
„ 6cV - 1
aob^i*
- 64
„c»£/V + 252
„6Ve/
- 80
a6V(^ + 288
,,6'c'e'
+ 160
„bVde - 1152
„bW/
+ 40
„bVdey~ 432
„be'^
- 20
..ic^t/e" + 288
»cV/
- 1
„cW/ + 18
a«6V + 32
„6Ve - 160
„6Vey - 80
„6Ve» + 80
..fttr-ey + 10
= a»r
„c*e» - 2
I*', yr
22.46
23.49
C. X
4.7
870
A TENTH MEMOIR ON QUANTIC8.
[693
Table No. 98. Covariants of A, divided and (except as to a few coeflBcients) segregate.
A and B as given in Table 97 were divided and segregate.
C was divided but not segregate: the divided and segregate form is
C = (
e + 1
/+3
a'b + 3
«»« 4 1
a'd + 6
a't +3
a*f>' - 1
aV-15
a'of- 10
a«6c- 3
„c» +15
aV/+3
„A +1
a^cd - 4
oV - 1
H^.y)'-
2.6 3.9 4.12
D divided and segi-egate is
D=a!>{
5. 15
6. 18
7. 21
-i- 3
8.24
c^+1
i+ 1
a6'^ - 1
al
- 1
„h + 1
afci
- 1
aVti - 3
\=^, yf,
3.3 4.6 5.9 6.12
an integer non-segregate form of the fractional coefficient is
ci — \
df + 1
E was divided but not segregate : the divided and segregate form is
E = {
« + l
ad - &
ai + 12
a'i" - 8
a'fie - 5
«V -1
a''bc~ 10
a'ce- 10
„h - 2
„^ + 8
a^'bd + 6
acd - 24
act - 12
o»6^c + 2
aohc" + 20
oVc + 5
„ cA +2
ac'd + 6
a''h<^ - 2
3.6
4.8
6.11
6.14
7.17
8.20
\«', y)'.
693]
A TENTH MEMOTE ON QUANTIC8.
371
Table No. 98 (continued).
F was divided but not segregate: the divided and segregate form is
-r 3
F={
y+1
a»6+ 2
ah+ 1
<^d+ 34
aH + 40
0*6'' - 16
a*be - 21
a'y - 1
a'k - 4
a«6' - 2
fflV-18
aV-36
a'bc- 42
d^ce - 35
„h - 5
aH - 8
a'Jc^H- 18
a'6i + 1
., 6/i + 3
aV+168
aV/+ 126
aW+ 46
„c/ - 16
a'b^c- 18
a*6c«+ 2
^,C9 - 1
a'6c"+ 155
aVe+189
„cA + 38
„c; - 8
a*6V+ 4
aV +252
aV/- 252
c^c'd- 174
a'6<^- 86
aV + 72
aW-16
aVe-13
aV/+ 9
„c%- 5
aVrf + 16
a»6c* + 4
a»c« - 2
3.9
4.12
5.15
6.18
7.21
8.24
9.27
10.30
11.33
12.36
\^,yr.
where for an integer non-segregate value of the fractional coefficient, .see the original
fonn of F.
6 as an invariant was divided and segi'egate, (r = a' g.
4.0
H divided and segregate is
-H 3
4- 3
H=a'{
A+1
be + 2
a'g + 1
o^/fc + 2
a'j + 2
I -4
abd - 12
abi - 8
a"** + 4
a'ch - 6
a''bce- 6
„c; + 12
„ 6A - 5
,.cp + 1
abcd+ 12
aVA+ 3
U;*'. y)\
4.4 5.7 6.10
where the fractional coefficients are =
7.13
8.16
oofe + 2
a'f^ + 2
a«6y + 4
„ d' + 6
„bce — 6
abed - 2
„cl +4
a°6V - 8
„6e/ -3
„c% + 1
„ce^ +1
47—2
878
/ divided and segregate is
A TENTH MEMOIR ON QUANTICS.
Table No. 98 (continued).
-i- 3
/ = a^(
4.6 5.9 6.12 7.16
where the fractional coefiicients are =
8.18
9.21
10.24
a'de - 5
aW + 2
aWe + 1
aby+ 5
„cP -12
a'bdf+ 3
„ bee - 5
a?bcd- 2
„ ccfe — 5
a'cH - 5
a6V- 6
o6V+ 1
„ cdf+ 30
„c»A- 2
„6c'e- 5
„ce« - 2
,,«y - 1
aVti - 18
aVi + I
„cM/- 3
J divided and segregate is
K divided and segregate is
K = a*(
5.6 6.4
k + l
nj - 2
an + I
o'wi - 3
a«6» + 6
a'ck - 3
acj + I
„bh- 9
a'l^c - 2
„cg + 3
„ bch+ 3
"^x, yf.
[693
-7- 3
» + l
ai« - 2
al + 5
a»6rf - 20
aJ'k - 5
at? - 4
a'6'e + 3
„A+ 2
a'ci- 15
a'c^d + eO
a'bi - 25
a»6» +10
„bl + 9
a'cd- 18
aci -30
a'<H + 45
„6A- 8
,,cg + 4
a^6c(i- 6
a ftc* - 18
„cV.- 6
aVrf-54
„cA - 5
,,/y - 3
a'bci - 8
aftV- 9
„bc'e- 15
„c»i + 3
aVt - 3
IJa;, y)*,
5.3
6.6
7.9
8.12
693]
A TENTH MEMOIR ON QUANTICS.
373
Table No. 98 (continued).
L divided and (as to first six coefficients) segregate is
-=-3 -=-3
L = c^{
l + \
abd ~3
tt'k - 4
a'j -
13
a'« —
3
a*m - 1
a?bde- 7
a*6* - 2
a'b^c - 7
abi + I
aW -
5
am-
45
a'b'^d + 13
aW/- 7
„6tf' + 3
„ch +7
aPcl- 21
„hh -
5
„ck-
20
„cj + 3
„6='c«+14
aWcd + 10
..cjr +
10
abci —
10
aWc - 13
„^/ + 12
„de^ + 2
abed +
30
a'cH +
105
„ bch + 29
o6c(//+23
a=6V +13
o«6V +
105
„cV -16
„c'de-24:
„¥ef + 4
„ch -
105
abc'd- 3
a»6V-21
„ c^A + 21
a'>b\f+ 25
„ 6c°e — 55
„ei - 3
„cyA+ 2
„6ce'' - 2
..c'd" -15
oic'c^ -28
„cdef- 7
a'ftV -19
„6cV-10
„cVt - 1
„c'e'' + 5
II*. y)'.
5.7 6.10 7.13 8.16
where the fractional coefficients are =
9. 19
10.22
11.25
12.28
a»6» - 6
a'i^'e - 1
„c^ + 3
abdf + 39
abed + 26
„ cde - 22
ftOftV + 31
a<'6»c/+ 16
„6e/+ 7
„bch- 4
„c% - 7
„c»/ + 19
„ce' - 1
„efh- 8
,Jl -14
„e'/ + 1
the last two coefficients have not been reduced to the segregate form.
M divided and segregate is
M=a'^( m+X bk-\ abj -1 \x, yf.
m + I
bk-l
abj - 1
P -1
adg + 1
„Ocm- 1
6.2
7.5
8.8
874
A TENTH MEMOIR ON QUANTIC3.
Table No. 98 (continued).
[693
If divided and segregate is
A'=a'(
n+ 1
<j/+4
ap-6
a*bj - 4
a*o + 1
cn-6
„dg + 4
o6* + 8
„6»A-12
„%+ 4
„«?;»- 8
a'bn -1
ocp + 2
a'c^ + 1
aVj - 4
5«, y)'.
6.4 7.7 8.10
0 divided and segregate is
0 = a'o(
9.13
10.16
0+1
6V+1
bm + 6
£^ -6
</A -1
![«> y)'.
7.1 8.4
P divided and (as to first three coefficients) segregate is
/' = o'(
p+1
abj + 8
a'o + 7
a'im + 8
a=6»/fc - 3
„dg - 5
a6« - 2
„dj + 3
„bp - 3
a«6< -14
o> - 14
ab^d + 9
„ej» + 2
„ b% + 15
„6c; +13
ab^de+ 3
„6c<)r- 5
„bdh- 9
„6(« + 3
„ em + 10
..cP -81
ao^c - 5
..b'ch- 8
„6^ + 8
„dH -27
„cfe/t - 3
„d/g - 8
„6V- 1
„6cV- 1
„6c(f-'- 9
„bel - 1
„6/fc+ 1
„ c'm - 12
„dei + 9
,,6"^ + 1
,JP - 2
aob'ce - 7
„bM + 1
„bc^k+ 9
„ 6crft — 9
„bceh + \Q
„ d'eg - 3
„cV + 9
„ c(Pe + 18
„ cfm + 4
„«/■* + 3
a*b^ +
I, *«^ -
a'bcm +
„ feeA —
.. e2> -
ab^cd +24
„6cV - 6
„6crfA -33
„<^dg +15
„ ccP - 54
„dfk - 9
a^V + 8
„6VA -11
„6c-V + 3
„ 6<^rf' - 18
„bcfk - 1
„ (?m — 6
„c;^^ + 2
7-8 8.8 9.11 10.14 11.17 12.20
the last three coefficients have not been reduced to the segregate form.
\«^, yf:
693]
A TENTH MEMOIR ON QUANTICS.
375
Table No. 98 {continued).
Q as an invariant was divided and segregate, Q = a" q.
8.0
R divided and segi'egate is
■r 3
-2
ii = o" (
r+l
aq +1
abo — 1
a/'b'J + 6
„9k- I
„bdff-5
„s +3
„hj -1
a»cr - 3
5-«, y)^,
8.2 9.5
where the fractional coefficients are =
10.8
b'j +2
bdk + 3
bdg-2
bej + 1
dm - 6
cr + 1
rf«/-l
rfj9 +3
S divided and (as to the first three coefficients) segregate is
- 2
« + l
agj - 2
abr —
1
a'bq + 4
aob'g + 2
„do-
1
a b^j + 4
„6»ni + 3
a'cs -
3
„6% - 4
„bdj +21
„ 6(^TO - 31
„6^A - 4
„d!'j - 3
„C9' + 2
aob" + 4
„cq - 3
„6»c;» + 16
„ bd'h - 24
„ den + 4
»> 1
9.3 10.6 11.9 12.12
but the last coefficient is neither segregate nor integer.
\^, yf,
376
A TENTH MEMOIR ON QUANTIC8.
[693
Table No. 98 {concluded).
T divided and segregate is
7'=a»»(
-=-2
t + 1
hgm + 1
^•' +4
dgj -3
liq - 1
\=^, y)\
11.1 12.4
where the fractional coefficient is =
Vq
—
1
hq
+
1
m«
+
6
U as &n invariant was divided and segregate, C/'=a" u.
12.0
V divided and segregate is
^ 6
r=o"(
V + I
bgr — 5
bjo - 10
gjk + 5
js -12
ng - 9
l^a:. y)'.
13.1 14.4
where the fractional coefficient is =
dt
-6
mr
-6
nq
+ 1
W as an invariant was divided and segregate, W=a^ w.
18.0
693] A TENTH MEMOIR ON QU ANTICS. 377
Derivatives. Art. Nos. 382 to 384, and Tables Nos. 99 and 100.
382. I call to mind that any two covariants a, b, the same oi* different, give
rise to a set of derivatives (a, by, (a, b)", (a, by, &c., or, as I propose to write them,
abl, ab2, abS, &c., viz. :
abl ^^d^a.dyb— dya . djb,
ab2 = d^a . dy'b — Id^ dya . d^idy b + dy'a . d^b,
abS = d^'a . dy% - SdJ'dya . d^y'b + Sd^dy'a . d/dyb - dy*a . d/6,
&c. ;
or, as these are symbolically written,
ail = 12ai6j, ab2 = l2''aj):„ a63 = l2»a,62, &c.;
where
19- fc e _ ^ A A ^
the differentiations t— , t— applying to the O] and the j— , ,— applying to the tj,
but the suffixes being ultimately omitted : hence if 6 be the index of derivation, the
derivative is thus a linear function of the differential coefficients of the order 6 of
the two covariants a and b respectively: and we have the general property that any
such derivative, if not identically vanishing, is a covariant. If the a and the b are
one and the same covariant, then obviously every odd derivative is = 0 ; so that in
this case the only derivatives to be considered are the even derivatives aa2, aa4i, &c. :
moreover, if the index of derivation d exceeds the order of either of the component
covariants, then also the derivative is =0: in particular, neither of the covariants
must be an invariant. The degree of the derivative is evidently equal to the sum
of the degrees of the component covariants; the order is equal to the sum of the
orders less twice the index of derivation.
383. It was by means of the theory of derivatives that Gordan proved (for a
binary quantic of any order) that the number of covariants was finite, and, in the
particular case of the quintic, established the system of the 23 covariants. Starting
from the quantic itself a, then the system of derivatives aa2, aai, &c., must include
among itself all the covariants of the second degree, and if the entire system of these
is, suppose, b, c, &c., then the derivatives ail, ab2, &c., acl, ac2, &c., must include
among them all the covariants of the third degree, and so on for the higher degrees ;
and in this way, limiting by general reasoning the number of the independent
covariants of each degree obtained by the successive steps, the foregoing conclusion
is arrived at. But returning to the quintic, and supposing the system of the 23
covariants established, then knowing the deg-order of a derivative we know that it
must be a linear function of the segregates of that deg-order; and we thus confirm,
d posteriori, the results of the derivation theory. I atmex the following Table No. 99,
showing all the derivatives which present themselves, and for each of them the
c. X. 48
378
A TENTH MEMOIR ON QUANTIC8.
[693
covariants as well congregate as segregate of the same deg-order: the congregates
are distinguished each by two prefixed dots, ..bf, &c. No further explanation of the
arrangement is, I think, required. We see from the table in what manner the
different covariants present themselves in connexion with the derivation-theory. Thus
starting with the quintic itself a, we have the two derivatives aa4, oa2, which are
in fact the covariants of the second degree (deg-orders 2.2 and 2.6 respectively)
6 and c. For the third degree we have the derivatives ai2, ail, acb, cuA, ac3, ac2,
ocl : the deg-order of acb is 3.1, and there being no covariants of this deg-order,
acb must, it is cleai-, vanish identically : ah2 and ac4 are each of them of the
deg-order 3.3, but for thi-s deg-order we have only the covariant d, and hence ah2
and ac4 must be each of them a numerical multiple of d; similarly, deg-order 3.5,
oil and ac3 must be each of them a numerical multiple of e ; deg-order 3 . 7, ac2
must be a numerical multiple of ah ; and deg-order 3 . 9, acl must be a numerical
multiple of /: the 7 derivatives, which primd fade might give, each of them, a
covariant of the third degree, thus give in fact only the 3 covariants d, e, f; and
in order to show according to the theory of derivations that this is so, it is
necessary to prove — 1", that ac5 = 0; 2", that ac4 and ahl differ only by a numerical
factor; 3", that ah\ and ac3 differ only by a numerical factor; 4°, that acl is a
numerical multiple of ah : which being so, we have the 3 new covariants. The table
shows that
for degrees
2,3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24
No. o£derivatives=2, 7, 19, 29, 41, 46, 52, 46, 44, 35, 26, 19, 17, 12, 13, 6, 6, 3, 3, 1, 1, 0, 1
so that the whole number of derivatives is 429, giving the 22 covariants 6, c w.
While it is very remarkable that (by general reasoning, as already mentioned, and
with a very small amount of calculation) Gordan should have been able in effect to
show this, the great excess of the number of derivatives over that of the covariants
seems a reason why the derivations ought not to be made a basis of the theory.
It is to be remarked that we may consider derivatives P5'l, P5'2, &c., where p, q
instead of being simple covariants are powers or products of covariants, but that
these may be made to depend upon the derivatives formed with the simple covariants.
(As to this see my paper " On the Derivatives of Three Binary Quantics," Quart.
Math. Journal, t. XV. (1877), pp. 157—168, [681].)
Table No. 99 (Index Table of Derivatives).
Deg.
2
3
Ord.
0
2
4
6
1
3 5
7
9
6
c
d e
ab
/
aa
4
2
ac
5
2 1
4 3
2
1
2 derivB.
7 derivs.
693]
A TENTH MEMOIR ON QUANTICS.
Table No. 99 (continued).
379
Deg.
4
5
Ord.
0
2
4
6
8
10
12
1
3
5
7
9
11
13
ff
6^
A
t
a<2
6c
ae
a^6
J
k
ag
bd
6e
I
a6^
ah
cd
ai
..hf
ce
a^d
abo
ad
ae
5
3
4
2
3
1
2
1
ah
4
3
2
1
«/
5
4
3
2
1
ai
5
4
3
2
I
bb
2
bd
2
1
be
2
1
be
2
1
ce
6
4
2
¥
cd
ee
cf
0
3
4
6
2
3
5
2
1
2
4
1
1
3
2
1
19 derivs.
29 derivs.
Deg.
Ord.
I 6
0
2
4
6
8
10
12
14
hy
n
«/
ak
«v
o&e
a'b^
m
6»
bi
ahd
a^
a'h
bh
..lU
6V
ci
acd
«9
cA
..df
bc^
..d»
..«»
..ef
aj
1
ok
3
2
1
al
a
4
3
2
1
bh
2
1
bi
2
1
ch
4
3
2
1
ci
6
5
4
3
2
1
dd
2
de
3
2
1
df
3
2
1
ee
4
2
«/
5
4
3
2
1
//
8
6
4
2
41 derivs.
48—2
380
A TENTH MEMOIR ON QU ANTICS.
[693
Table No. 99 {continued).
Deg.
7
Ord.
1
3
5
7
9
11
13
0
hj
hk
ahg
an
«V
a%
dg
eg
am,
..b^e
a6»
obi
p
bM
..dh
bl
ck
..di
..eh
fg
ahh
acg
.. ad'
bed
. . ei
.. ode
..by
hce
cl
..fh
am
2
1
an
4
3
2
1
hj
1
bk
2
I
bl
2
1
cj
1
ck
3
2
1
cl
6
5
4
3
2
1
dh
3
2
1
di
3
2
1
eh
4
3
2
1
ei
5
4
3
2
1
fh
4
3
2
1
fi
6
5
4
3
2
1
46 derivs.
693]
A TENTH MEMOIR ON QUANTICS.
381
Table No. 99 {continued).
Deg.
8
Ord.
0
2
4
6
8
10
12
14
f
r
6»!7
fiM
abj
o6/fc
a?bg
a^n
9
bm
6n
adg
aeg
ahn
. . ab'^e
dj
..dk
¥
ap
ab^d
abl
gh
..ej
bVi
bS
acj
ack
9*
beg
.. bd'
cm
..ek
. . bde
en
. .dl
■■fj
..hi
. . adh
b'c
beh
..be'
e'g
..ed-"
. . adi
. . aeh
a/g
bci
.. bdJ
. . cde
\
. . d
■ ■fk
ao
1
ap
5
4
3
2
1
bm
2
1
bn
2
1
cm
2
1
en
4
3
2
1
d}
1
dk
3
2
1
dl
3
2
1
«;■
1
ek
3
2
1
d
6
4
3
2
1
fj
1
fk
3
2
1
fl
7
6
6
4
3
2
1
hh
4
2
hi
4
3
2
1
n
6
4
2
52 derive.
382
A TENTH MEMOm ON QUANTICS.
Table No. 99 (contimied).
[693
Deg.
9
Otd.
1
3
6
7
9
11
SJ
bo
<^
ar
ah^g
a'o
gk
aq
..6%
ohm
ahn
»
by
beg
adj
..adk
bdff
bp
agh
■ .aej
. . dm
CO
bH
agi
hj
. . dn
. . em
gi
. .hk
. .ij
bcj
..bdh
cdg
.. cf
. . en
..ik
. .6»e
bH
bek
..bdi
..beh
■hfg
ceg
cp
..d'e
..fin
..hi
or
2
1
bo
1
bp
2
1
eo
1
ep
5
4
3
2
1
dm
2
1
da
3
2
1
6fn
2
1
en
4
3
2
1
fin
2
1
fi^
4
3
2
1
hj
1
hk
3
2
1
hi
4
3
2
1
V
1
ik
3
2
1
a
6
5
4
3
2
1
46 derive.
693]
A TENTH MEMOIR ON QUANTICS.
888
Table No. 99 (continued).
Deg.
10
Ord.
0
2
4
6
8
10
12
bf
br
<^3
abo
ay
aV
bq
..do
b'g
agk
a^q
. . ab'k
gm
gn
bhn
as
ab-'j
abeg
'
f
jk
bdj
bgh
of
eq
..cPg
. . eo
. . hm
b-hi
..bdk
■ • bej
bgi
cr
. . deg
. . dp
..hn
. . im
..jl
abdg
. . adm
ahj
..b'
¥h
b\g
. . b\P
bcm
..bek
..bh'
cdj
cgh
abp
aco
. . adn
. . aem
agl
..ahk
■ • aij
bH
..b'de
ben
. . bdl
. ■ bfj
{
..(Ph
..ep
..fo
. . in
..kl
..bhi
..cdk
■ . cej
cgi
..dH
..deh
..dfg
as
3
2
1
br
2
1
cr
2
1
do
1
dp
3
2
1
eo
1
ep
5
4
3
2
1
fo
1
fP
6
4
3
2
1
hm
2
1
hn
4
3
2
1
im
2
1
■
in
4
3
2
1
Jk
1
jl
1
kk
3
kl
3
2
1
U
6
4
2
44 derivs.
384
A TENTH MEMOIR ON QUANTICS.
[693
Table No. 99 (continued).
Deg.
11
Ord.
1
3
6
7
9
go
bgj
6'o
ahg-
(^
t
dg^
bgk
abq
. . ado
dq
bs
agm
agn
jm
..dr
ef
yp
b-j
b'dg
. . bdm
ajk
..b'k
..b-'eg
Wp
..ho
. .jn
. . km
bhj
CffJ
■ dPj
. . dgh
. . er
. . io
.. kn
beo
. . bdn
. . bem
bgl
.. bhk
. . bij
cjk
C8
..d'k
. ■ dej
. .dgi
. . egh
ff
fq
. . hp
. , Im
bs
2
1
or
3
2
1
dr
2
1
er
2
1
fr
2
1
ho
1
hp
4
3
2
1
*o
1
*P
5
4
8
2
1
jm
1
jn
1
km
2
1
kn
3
2
1
hn
2
1
In
4
3
2
1
35 deriva.
693]
A TENTH MEMOIR ON QUANTICS.
Table No. 99 (continued).
385
Deg.
12
Ord.
0
2
4
6
8
10
^
gr
by
ago
abgj
ab''o
^
jo
¥q
at
. . adg-
abgk
u
bgm
b^r
adq
abs
hf
. . bdo
ajm
. . adr
dgj
bgn
b*g
aeg^
fh
bjk
¥m
aeq
lui
..dgk
b'dj
agp
..ko
..ds
b'gh
. . alio
..to'
■ ■ egj
..hr
..jp
. . mn
beg'
bcq
. .bdJ'g
. . beo
. . bhm
.. bF
cgm
. . ajn
. . akm
b^n
. . bHk
. ■ b-'ej
. . V'gi
bcr
{
. . d^m
. . dhj
. . egk
. . es
. . ir
..kp
..lo
. .»'
. . bdeg
. . bdp
. .bhn
. . bim
. . bjl
. . cdo
cgn
cjk
. . d'n
. . dem
..dgl
. . dhk
. . dij
. . ehj
• -fffj
. . ghi
at
1
ds
3
2
1
es
3
2
1
/s
3
2
1
hr
2
1
ir
2
1
jo
1
jp
1
ko
1
kp
3
2
1
lo
1
Ip
6
4
3
2
1
26 deriva.
C. X.
49
386
A TENTH MEMOIR ON QUANTIC8.
Table No. 99 {continued).
[693
Deg.
13
Ord.
1
3
5
7
^
bgo
ag*
agr
k
bt
<^9i
ajo
V
Si'k
au
. . 6'o
v«
by
. . b'gk
• -jr
bd^
b-'s
hq
bdq
. . bdr
. . ino
bjm
. . dgm
. . dp
..hr
. . no
beg'
beg
bgp
..blio
• • ijn
. . bhn
ego
ct
. . d?o
. . dgn
. . djk
. . egm
..«/
..ghk
■ ■ yij
..hs
..Iq
. . mp
bt
1
ct
1
hs
3
2
1
is
3
2
1
jr
1
hr
1
Ir
1
mo
1
mp
2
1
no
1
np
4
3
2
1
19 derivs.
693J
A TENTH MEMOIR ON QUANTICS.
387
Table No. 99 (continued).
Deg.
14
. _
15
Ord.
(
) 2
4
6
8
1
3
5
bg^
bgr
• • <^j
abgo
ff'o
bstj
. . b'go
bgq
bjo
m
abt
S*
bjq
bH
bu
. . dgo
av
ag'k
n
In
bfk
g^m
..dt
by
ags
<¥
bgs
af
g'n
b'q
. . ajr
dgq
■ ■ bjr
tnq
9J^
Vgm
akq
. . du
bkq
. . o'
js
by
. . amo
yjm
. . bnw
. . mr
bdgj
6V
■ ■f
. . dgr
nq
bgVi
bhq
..bko
cf
cgq
..cPg^
. . djm
. . ego
. . et
. . ghm
..gkl'
..b^do
. . b^gn
b'jk
..bdgk
..bd8
■ ■ begj
bgH
..bhr
biq
..bjp
. . bvm
cgr
cjo
..d'r
. . or
. . djo
eg'
eyq
. . eu
fP
. . g/io
■ ■ gjn
. . gkm
. . ht
..fk
. . Ills
pq
bv
1
..hf
. . def
1
..ks
. . deq
cv
ht
1
'
. . nr
• .dgp
1
1
. . op
..dho
. . djn
. . dkm
it
ma
ns
2
3
1
2
.
or
1
. . ejm
..ghn
pr
2
1
. .gim
..gjl
. . hjk
■■if
dt
1
el
1
fi
1
3»
1
k» a
2
1
u
3
2
1
mr 2
1
nr
2
1
op
1
pp
4
2
17 derivB.
12 derivB.
49—2
388
A TENTH MEMOIR ON QUANTICS.
[693
Table No. 99 (continued).
Ord.
16
0
2
4
6
8
Col. 8 concl.
9*
</V
by
afo
..abgy
..X,
ff'q
9Jo
b'yq
agt
abjq
gti
ji
6'm
acq
ahv
. . emo
9'
qr
bg^m
. . b'gr
..adf
. . 9'h'
hgf
¥jo
adgq
■ .gir
bmq
. . bdgo
. . adu
. .gkp
. . bo'
. . bdt
ngjm
. .glo
dg^
bfn
..af
• ■ gn?
djq
hgjk
. . aor
..h-q
..dv
bjs
by
. .Itko
g'h
. . bmr
b*q
..hw?
ff^
. . bnq
b^gni
. . ijo
■ . gko
. .dg^k
6'/
. .jkn
. . gm?
. . dgs
Vdgj
..khn
..hu
. . djr
byh
. . It
../m
..dkq
bViq
. .ps
..kt
. . dmo
. . b-'ko
. . OS
• • efj
..bhn"
•
..r'
■ ■ egj
. . ev
. . ghr
9n
■ • 9JP
. . gmn
. . hjo
. . iu
. .jkm
bcf
bcgq
bcu
. . hd^g'
. . bd?q
. . bdjm
. . bego
. . bet
. . bghm
..bgk'
. . bhf
. . bks
. . bnr
. . bop
cg'^m
cmq
. . cd'
. . d^gm
■ . (Pf
. . dghj
. . dkr
. . dno
..efk
. . egs
■ ■ ^jf
1
do
1
ev
1
>
1
Jt
1
kt
1
U
1
op
1
0$
1
PP
4
2
V
3
2
1
13 derivs.
693]
A TENTH MEMOIR ON QUANTICS.
389
Table No. 99 (continued).
Ord.
17
18
19
1
3
5
0 2
4
6
1
3
^j
bfo
ag*
w bg*
bg'r
• • a^j
g'o
W
m
hgt
ag"-q
bg^q
. . bgjo
mi
gH
bgjq
gv
boq
agu
bgu
bjt
agv
gqo
bgv
ju
g^k
aq-
bf
bqr
aju
ou
bju
gh
WJ
g'm
. . dg^o
by
qt
dg*
■ -m
h'k
ff
..dgt
Wgq
dg\
9kq
¥v
gmq
. . doq
b^u
. . djt
. .gmo
bdg^
..90^
g^n
hym
dq^
..fo
bdgq
/?
gpjk
bw-
fjni
ku
bdu
. .jv
m
Vmq
• -gf
. . mt
bgjm
mu
. . gmr
.bV
■ ■ gor
qs
..bf
. . bor
. .dghn
■■dsP
. . dmq
. . at
gnq
■fr
jkq
. .jmo
. . kv
bdgv
hdjq
. . bdv
bfh
hgJuj
jmq
. .jo-
. . mv
. . rt
..do"
nu
. bgko
mv 1
fhj
. . bgm-
nv
1
. . gkr
bhu
rt 1
. . gno
. bpm,
% i
. bkt
. . hv '
. . bos
■ -jko
.br'
• • 311^
eg*
. . nt
cg^q
. . rs
cgu
cq-
hv
1
.(P^
iv
1
■ (Pgq
ml
1
. d'u
nt
1
■ dgjm
rs
2
1
. dor
■ eg^o
• egi
. eoq
. g'^hm
■ ghf
■ .9*«
■ gnr
■ gop
. hmq
.M
■ jhr
.jno
.k\
. kmo
.m'
.pt
. s^
jv 1
kv
1
Iv
1
at 1
pt
1
88
2
6 derivs.
6 derivs.
3 derivs.
390
A TENTH MEMOIB ON QUANTIC8.
Table No. 99 (concluded).
[693
Deg.
20
21
22
24
Old.
0
2
4
1
0 2
0
^
but
i'g*
S^
gw btf
^
9'q
gh-
bVq
9^9
bg>q
A
ff'u
ff^o
b^gn
gh,
bfu,
9^
9f
■ ■sat
by
gju
bgq'
ff
qu
gqr
bj^m
jf
bqu
gqu
■ -joq
bgmq
qv
ghn
<t
. . ov
ffT
M^
ru
. . bgo'
TO 1
fmq
bfq
■ . fo-
tv 1
■ ■ bjv
fffq
bmu
• -SJv
. . hot
ymu
rfA
. . got
dyjq
A
. .dgv
mq''
dju
..o'q
g*h
. . e
fhq
..fko
sv 1
..fm'
ghu
■ -apm
..gkt
. . go8
..gr^
w
■ ■r
■ .jor
. . koq
. . m?q
. . mo'
. . St
ov
1
pv
1
sv
1
3 derivs.
1 deriv.
1 deriv.
1 deriv.
693] A TENTH MEMOIR ON QUANTICS. 391
384. The Canonical form (using the divided expressions, Table No. 98) is peculiarly
convenient for the calculation of the derivatives. Some attention is required in regard
to the numerical determination: it will be observed that A is given in the standard form
(Ao, Aj, A^, Aj, A^, A,^x, yY, while the other covariants are given in the denumerate
forms B = (B„, £,, Bi^x, yY &c. : these must be converted into the other form
S = (5o, ^B„ B,-^x, y)\ C={C„ JC„ i^G,, i^C„ i^G,, ^G„ G.^x, y)', &c., the numerical
coefficients being of course the reciprocals of the binomial coefficients. We thus have,
for instance, the leading coefficients,
Lc. of AG2 = A^ . -igG, - 2 . A,. jtC, + A, . G„
but
„ „ BG2= B„.^C,-2.^B,.i.G, + B,.C,.
Moreover, as regards the covariants AA2, AA4-, &c., we take what are properly the
half-values,
I.e. of AA2 = A„Ai— Aj" (instead of A,^A3 — 2A,Ai + A.iAo),
„ „ AA4i = AaAt-4iAiA3 + SAi^ (instesid of AaAi-4'AiAt + 6AiA.2-4!A,Aj-AiAo),
&c., ^
and similarly
l.c. of BB2 = B^,-{^B,y,
„ „ CG2 = G,.^G,-{^G,r,
&c.
Any one of these leading coefficients, for instance l.c. of AG2, is equal to the
corresponding covariant derivative, multiplied, it may be, by a power of a: the index
of this power being at once found by comparing the deg-orders, these in fact differing
by a multiple of 1 . 5 the deg-order of a. Thus
aa2, A^Ai — 4,', deg-orders are 2.6, 2.6: or aa2 = A^A.^ - .d,',
oaA, il,.d4-4.4,.4,-H 34,', deg-orders are 2.2, 4.12: or aa4 = - (il„4«-4.4,.4j-|-34,'');
we have in fact
A^i- Ai^=\.c—(P = c : and aa2 = c,
AAt - 4-4,-4, + 34,' = 1 . (a'6 - :k') -4.0 ./+ 3 . c-, = ct'b : and aa4 = b.
An another instance, and for the purpose of showing how the calculation is actually
effected, consider the derivative ch2, which is to be calculated from the leading
coefficient of GH2, = (7„ . ^F, - 2 . ^C, . ^/T, + t^O, . /T, : this is
= c (j^rt'i/ — 2ahd — ch)
-2.^fi^be-l)
+ {jta'b-d')h
392
A TENTH MEMOIR ON QUANTICS.
[693
= column next written down ; but this column contains congregate terms which have
to be replaced by their segregate values (see Table No. 96, deg-order 8.16); and we
thus obtain
o^' a'6' a*bh a'cy abed b^c' c'h
ia%A
+ i
+ J«V
+ i
-2abcd
-2
-\hef
-i
+ 3
+ 2
-2c'h
-2
+/1
h
1
~ 5
+ §
1
-1
_ 2
+ 2
11
1
viz. the terms other than those divisible by a' all disappear: we may either abbreviate
the calculation by omitting them ab initio, or retain them for the sake of the
verification afforded by their disappearance. The factor a^ divides out, and the final
result is
which is the proper segiegate expression of the derivative ch2 : of course, we have
deg-order CH2 = 8. 16, deg-oi-der c/i2 = 6.6, and the difference is 2.10, the double of
1 . 0, so that the factor a^ is as it ought to be.
Table No. 100 (The Derivatives up to the Sixth Order).
Degree 2.
2.2
aa4
+ 1
2.6
+ 1
Degree 3.
3.1
ao5
4.0
9
ae5
- 2
bb2
-i
cc6
1
3.3
d
ab2
- 3
ac 4
+ ¥
4.2
adZ
aei
0
0
3.5
t
ab 1
+
i
ac3
+
1
ST!
Degree 4.
3.7 \ ab
4.4
6''
h
ad2
1
+ i
ae3
4
a/5
+ u
S3
be 2
+ i
-h
cc4
+A
1
3.9 I /
acl \ + i
4. 6
i
adl
+ i
ae2
+ 1
»/4
-^
6c 1
+ i
693]
A TENTH MEMOIR ON QUANTICS.
393
4.8
ad
be
ae 1
a
- 2
a/3
+ n
5
cc2
+ i
1
~ ins
Table No. 100 (continued).
4.10 ae 4.12 I a=6
a/2
+ 1
a/1
Degree 5.
5. 1
.;■
aA4
+ 2
at 5
1
bd2
-*
ce 5
8
5. 3
k
aA 3
+ *
ai4
+ i
bdl
1
be 2
cd3
-t- •■'
cei
+ 11^
c/6
1. 19
5. 5
ag
bd
ah 2
+ \
— 2
aiS
+ 0
- 2
he\
-i
+ Y
cd2
0
2
~ TT
ce3
X
48
df5
1
j. 8
5.7
6e
«
aAl
- 2
- 4
ai2
0
+ ^
6/2
c<^l
ce 2
(^/4
7
0
1
1
+ 1
+ i
43
5TT
5.9
ab^
oA
erf
atl
1
+ ^
+ 3
6/1
+ f
- i
+ 3
ce 1
0
- J
+ 1
d/3
+ ^
1 1
120
4.sg
TiSTJ
5. 11
rf/2
TT
+ TSJ5
5 . 13 I a^d dbc
dfl\+\ -I
Degree 6.
6.0
ci 6
6.2
ig
m
aA3
0
- 4
alb
0
+ ^
bh2
_ I
5
- 2
ch4
A
2
T
ci5
0
+ §
dd2
0
+ \
deZ
0
_ 4
Tf
ee4
- 1
_ 48
ITS
//8
s
68
867
6 .4
n
ajl
- 1
ak2
+ J
ali
-3^
bh\
+ i
bi2
+ 1
ch3
+ A
ci 4
+ A
de2
1
«/5
«4
6. 6
al
aZ3
6tl
ch2
ci3
del
d/3
ee2
e/4
//6
aj
bh
eg
~ t
C. X.
le
FT
1
Ti
1
2
~ TT
+ ST?
4
~ ^T
, 236
+ 3TT
_ 1529
T»8S
+
+
+
1
~ a'
1
+ TT
_ 143
+ ^
4
TTTT
, 2873
^ 7638
- 3
5
T
1 1
-3'
_i
IT
425
TT7
38
^T
10
+
+
+
+
. 3 6 3 3
+ TT5TF
+
+
3
T
0
+ BIT
1
T
139
TTT
-4- »
^ IfT
71
~ TTT
_ 5691
'STTTsf
50
394
A TENTH MEMOIR ON QUANTICS.
[693
Table No. 100 (concluded).
6.8
ok
bi
al2
-A
+ A
chl
+ i
+ i
ci2
+ I
+ ^
d/2
-^
-^
e/S
+ ih
+ /^
. 10
aV
abd
b*C
cA
ail
0
_ 8
- 1
+ 1
oil
0
- i
+ i
I
~ TT
df\
0
1
+ 1
_ I
71
e/2
+
1
+ 1
19
_ 4
T
//4
+
Tk
S3
4. 89
8
6. 12
^/'l
a6« o^ ct
6. U
a'fc^
aPh
acd
ic"
//2
_ 4
ST
+ ^
+ f
2
7
which is complete to the sixth degree. I had calculated the derivatives up to the
tenth degree, but the results were not in the segregate form.
On the form of the Numerical Generating Functions: the N.O.F. of a Bextic.
Art. Nos. 385, .386.
385. It is to be remarked that the R.G.F. is derived not from the fraction in
its least terms, which is algebraically the most simple form of the N.G.F., but from
a form which contains common factors in the numerator and denominator : thus for
the quadric, the cubic, and the quartic, writing down the two forms (identical in the
case of the quadric) these are —
Quadric
N.G.F. =
l-cud'.l-a-
Cubic
N.G.F. =
Quartic
N.G.F. =
\ — ax-\- a?a?
\-a*af
1 — a* . 1 — aa^ .\ — ax
1 - CM^ + g V
l-aM-a».l-aa^.l-aa;»
\ - a* .1 - CO? .1 -a^a? A - a*a?'
\-a^
l-a'.l-a".l-aar*.l- a'^.l-a*afi '
693] A TENTH MEMOIR ON QUANTICS.
For the quintic the two forms are, N.G.F. =
395
( 1
- a«
+ a^) a;»
+ (-1 :
+ a*
+ ■20."
- a"') oaf
+ ( ] +a-
-a'
+ a'") x'
+ (-1 !
+ a*
+ a'
+ a»
-a'"
- a") an?
+ (+1
+ «'
-a*
- a«
-a»
+ a"^) oV
H
-a'
+ a*
-d">
)aV
+ (+1
-2a«
-a»
+ o'=) oV
+ (-1
+ a'
- o'^) « V
divided by
and
1 — a'' . 1 — a" . 1 — rt' . 1 — «a:° . 1 - aa? A - ax;
( 1
+ «i«) x"
a<
+ o«
+ a'»
+ o«
)ax
o*
+ a«
+ a»
+ «"
+ «»
- a") aW
( 1
+ o»
+ a*
+ a»
)a?a?
( i
+ a»
+ «*
+ a«
+ a"
-a"
( 1
+ a*
+ «•
-n"
a"
-a"
-a"
)aW
««
-o"
-a"
-a"
-a"
- a") aa^
.
-o"
-a«
-a"
-a"
- a») aW
-««
-o»
-«'•
-«>»
-«»
-a*
-a»
-oF
-a"
(-1
- «>») aV
divided by
1-aM-aM -a'M-aa^.l -a'af.l -a'af:
this last being in fact equivalent to that used for the iletermination of the R.G.F.
50—2
396 A TENTH MEMOIR ON QUANTIC8.
386. For the sextic the forms are, N.G.F.=
[693
+ a
- a'
- a*
- a'
+ a'
+ ««)«'
(-1
-a
+ a*
+ 2a»
+ 2o«
+ cf
-a'
- a*) aa?
(-1
+ a«
+ a»
+ a*
+ o»
-o'
- a*)aa^
■¥a
- o»
- «*
- a»
-a«
+ a") aV
+ a
- a»
-2a'
-2a»
-a«
+ a^
+ a«)aV
(-1
-a
+ a'
+ a*
+ a»
-a'
- a») oV
divided by
and
1+a.l^a^.l -a^A -a*.l-aKl-aafiA -aa^.l-aa^:
{ 1
+ «")«»
+( 1
+ a«
+ a*
+ o»
+ «'
+ a«
)a»aJ'
+(
+ a''
+ a»
+ a*
+ «»
+ a»
+ «'
+ a«
+ a»
+ a"
)aV
+( 1
+ o
+ 2a»
+ »•
+ o»
+ a»
-a"
)a»iB«
+ ( +a
J
■t-a
iji
+ a
4^
-a
W.6
— o
12 Ji
— a
14.5 ) „2..a*
+ (
+ a'
-a"
-a»
-«'«
-2o"
-o»
-a" )aV
+ (
-a*
-o«
-a'
-a'
-a»
-a'«
-a"
- a"
-a"
)a»x"
+ (
-o*
-a»
-a'»
-a"
-a"
- a" ) a'aJ*
+ (-1
'
- a» ) aV
divided by
1 - aM - aM - aM - o'M - rta;M - aV . 1 - aW,
where observe that in the middle term, although for symmetry a' (= Va) has been
introduced into the expression, the coefficient is really rational, viz. the term is
(a' + a''+a'- a" - a" - a") a^.
The second form or one equivalent to it is due to Sylvester: I do not know whether
he divided out the common factors so as to obtain the first form. I assume that
it would be possible from this second form to obtain a R.G.F., and thence to establish
for the 26 covariants of the sextic a theory such as has been given for the 23 covariants
of the quintic : but I have not entered upon this question.
693]
A TENTH MEMOIR ON QUANTICS.
397
Table No. 93 bis (The covariant S, adopted form = — (D, M)).
In this Table, a, b, c, d, e, f denote, as in the tables of former memoirs, the
coeflScients of the quintic form (a, b, c, d, e, f\x, y)^.
S={
aWc>f' - 2
a»6Vt^ - 3
o'6»a/y^ + 3
a'ft'rfy^ + 2
c'def + 15
cV/^ + 3
c<fey^ - 6
t/^ey^ - 6
cV/ - 9
ctPeP^ 24
cey + 3
cfey + 6
ccPf - 9
cefey - 42
d?ep - 3
e« - 2
ccPelf - 6
ce» + 18
dP-eJ + 6
a'ftcrfy - 15
cde* + 9
f^y^ - 18
cfc' - 3
cdeP+ 30
d>e/ + 9
d'ej + 33
a^i^c^!/'^ - 3
cey - 15
cPe' - 7
dPe' - 15
dep + 6
d^eP + 15
a'b^c'r + 6
a^J^crf/^ + 6
ey - 3
tf'ey - 30
cdef - 30
c^P - 6
..ic^f/r-- - 24
rfe" + 15
c^f + 18
t^'e/^ - 24
c^e-p + 24
„ 6wy» + 9
(Pf + 9
rfey + 42
c(?'^/2+ 78
c'ey-^ - 9
cPey + 6
e» - 18
cd^f - 108
c'd'eP- 21
de* - 9
„ 6 cy» + 3
ce* + 30
c»c^ey+ 15
„ b ^ep - 15
i?deP~ 78
d*p - 24
cV + 6
<?^ + 21
cVy + 69
d'e-f + 24
crfy^ + 3
<?d^ - 6
cd^P + 93
„6«cy» + 18
cd^ej 4- 21
<?i^ + 18
crf^ey- 51
(fdep- 93
ccPe^ - 24
cd^ef + 30
cde* - 33
c'ey + 21
rfV - 9
cd?(? - 51
rfV - 57
chPp + 36
d*e' + 9
d?f - 36
d»e' + 54
(r'(f'ey+ 123
a 6'e£y» + 9
dV + 39
„6Vey + 24
<?d>^ - 51
deP - 18
„6Vrf/' - 3
c'rfy - 36
erf's/ - 111
ey + 9
<*ej + 45
(?d^f - 9
cdPe' + 39
„6Vrf/» + 6
c'rf'e/- 84
c'e^ - 54
rfy + 27
c'ey^ - 6
<?d^ - 63
c^d?ef -v 24
rfV - 9
cd'ep + 6
c^rfy + 45
c^f^'e' + 129
a6'c((/-» + 42
cd^f - 24
<?d?^ + 150
crfy + 9
c^P - 42
ce» + 18
cd^e - 117
crfV - 114
dPep - 69
d*p - 45
f/' + 27
d"« + 27
flfey + 96
cPey + 96
a'6V - 6
a'6^#^ - 3
e» - 27
d'e* - 51
de/» + 15
ey + 3
„6V/' - 33
„b<^P - 9
«y - 9
„6'cy» - 6
d'deP+ 51
tr'cfey'- 30
..i'cV' + 30
cdeP + 108
cVy + 48
<?e>f + 66
cd'P - 15
c«y - 96
c<Pp + 9
A/y= + 84
cd«y + 24
cPT - 21
cc^Vy - 147
c'd'ej- 36
ce* - 45
d?e?f - 48
crfe* + 39
cV«^ - 102
d'ef - 66
<fe* + 63
d*ef + 78
cd'ef - 174
rf'e' + 72
„6Ve/' - 24
ffe' - 45
ccfe» + 210
„6Vrf/' - 21
<?d'f - 123
„hc*ep + 57
rfy + 63
c»ey - 96
c^d^f + 147
c-Wy- 24
cPe' - 72
ed^ef + 36
c'e* + 66
e>dey - 78
„ iVe/'' + 36
<?d^ + 213
ct/vy + 78
c»e< - 60
c*dp - 45
cd*/ +120
C6P«» - 186
c^d^ef + 36
(^de'f - 120
cd?e? -303
rfy + 51
cr'rfV + 108
cV - 6
t^e + 51
d^t^ - 9
cdY - 24
<rWV/ + 204
5a;, y)»
{contimied on next page.)
398
A TENTH MEMOIR ON QUANTIC8.
(continued from last page.)
[693
a^h&'r + 9
a'bc'df +111
« 6 cd^e' - 6
« 6Vd»e» + 120
c*(fc/ + 174
cV/ - 78
rf'e - 9
<^dy - 66
c<e» - 36
e'd'ef- 36
„6W/^ - 9
i^d'e' - 240
c'cPf - 204
c'cfe* - 54
c»ey - 51
ccfe +144
^iPe^ - 174
c'rfy - 96
cVc/+ 96
d« - 27
<?<Pe +330
c^d>^ + 150
c^(ie» + 111
ao^cd/^ - 9
cd* - 99
cd»e + 30
c't/y - 27
cey^ + 9
„h'>(*ef - 63
d' - 27
c^«P«r' - 234
d'er - 18
(^(Pf + 66
„/,V/^ - 27
c=d»e + 141
dey + 45
<fd,? + 99
(^def + 24
cd' - -It
e» - 27
c'iPe - 147
cV + 54
a^bhlf' - 18
„6V/» + 7
c'lP + 45
c*dy + 27
e'f + 18
c>'c/e/'+ 51
a'fty" + 2
c^cPe' - 93
„6vy» + 15
rV/- - 72
,,6'ce/' - 15
cW^e + 6
c*/" + 33
c(^> + 63
<PP - l&
cW + 9
ce>f - 63
cPey - 213
dej - 18
o»6»c/' + 3
(Zy^ + 54
c<Ze^ +171
e' + 27
deP - 30
d'ey - 66
d'ef + 36
„h*<?dp + 24
«-y + 27
de* + 27
dV - 43
<?^f + 51
..iVrfy^" + 51
..fi^.r'ey^ - 54
„6V«r - 39
etf-'e/' + 102
cck:y - 39
c^dy - 129
c'tZy^ - 150
c<fc» - 171
<•«< - 27
c''<fey+ 186
<?d^f + 303
t^/ + 6
d?ef + 60
cV + 45
cV - 18
ePe" + 18
rfV - 45
ct^V + 51
cW«/ + 174
.,6»cy» - 9
„6Vrf/-^ - 39
ccP^ - 96
c'd'f? - 345
<?def - 210
cV/ + 45
dy - 54
ctZy - 99
c'e" + 43
cWe/ - 108
rfV + 48
crfV +192
<?d?f - 120
c'd^ + 96
„ l)'c*d/' +114
<fe - 18
cWe'' + 345
rd^f - 111
c^ey + 9
„6cV/^ +117
ce;«e - 87
«rfV +147
c»(fV-150
c=ey - 51
d« - 2
d^e - 30
c»tfe^ - 147
c\Pef - 330
„6Ve/ + 72
„ 6V/^ + 9
e'dy + 93
e'de' + 87
«*«P/ + 240
<^def + 6
corf's' + 150
f^rfy + 147
c^de'^ - 192
c^e' - 48
cePe - 87
<?d?e^ + 186
<?d?e - 186
t-c/y + 234
(P + 18
cWe - 201
cW + 96
e'Pe' - 150
„6cy^ - 27
erf' + 45
„h(»df -144
oWe - 108
d>de/ - 30
„6Vr - 27
c»«' + 18
ed^ + 57
c»«» + 30
<;«*/ + 99
(fd^e + 201
„ 6 cV + 9
c*dy - 6
cV + 2
c*ci* - 87
cW/ - 141
c*dV + 108
Azy - 45
„6V/ + 27
(T'de^ + 87
c»d«e - 96
rVe^ - 96
c^de - 45
c*cPe + 96
c^d' + 21
c*d*e + 87
c'c^^" + 20
f^rf" - 51
„6»cV/ + 27
c'rf' - 20
„6Vrf/ + 27
My - 9
cV - 18
<^de' - 57
c»rf»e - 21
c'rf'e + 51
<fd* + 12
r*fP - 12
I remark that I calculated the first two coefficients (S„, /S,, and deduced the
other two S^ from Si, and S3 from So, by reversing the order of the letters (or
which is the same thing, interchanging a and f, b and e, c and d) and reversing
also the signs of the numerical coefficients. This process for /Sj, S, is to a very
great extent a verification of the values of S^, S,. For, as presently mentioned, the
693] A TENTH MEMOIR ON QUANTICS. 399
terms of S^ form subdivisions such that in each subdivision the sum of the numerical
coeflBcients is =0: in passing by the reversal process to the value of S-j, the terms
are distributed into an entirely new set of subdivisions, and then in each of these
subdivisions the sum of the numerical coefficients is found to be = 0 ; and the like
as regards /S, and S^.
If in the expressions for S^, S^, S,, S3 we first write d = e = f=l, thus in effect
combining the numerical coefficients for the terms which contain the same powers in
a, b, c, we find
80= a»(-2c='+6c»-6c + 2)
+ a' {b' (Gc* - 12c - 6) + 6 (- Ibd' + SSc- - 21c + 3)
+ b" (42c* - 147c^ + 105c=- 117c + 27)j
+ a {b^.O + y (30c- - 36c + 6) 4- 6- (- 117c» + 2490=^ - 183c + 51 )
+ b(9<^ + 138c* - 378c» + 330c= - 99c) + bf (- 63c« + 165c» - 147c* + 45c')}
+ a» {6' . 2 + 6» (- 15c + 3) + 6* (75c^ - 69c + 24) + ft' (- 9c* - 167^" + 225c= - 87c - 2)
+ 6=(72c» + 48c* - 186c» + 96c=) + b{- 126c'' + 201c^ - 87c*)
+ 6''(27c«-45c'+20c«)J;
which for c = 1 becomes
= 26»- 126° + 306* -406' +306- -126 + 2, that is, 2(6-l)«,
and for 6=1 becomes = 0.
8, = a' (Oc= + Oc + 0)
+ a' {6' (Oc + 0) + 6 (3c' - 9c' + 9c - 3) + 6" (24c* - 99c' + 153c= - 105c + 27))
+ 0 {6*.0 +6'(-6c» + 12c-6) + 6'(-24c' + 90c=-108c + 42)
+ 6 (33c* - 90c' + 54c^ + 30c - 27) + 6" (- 27c« + 78c' - 66c* + 6c' + 9c0j
+ a» {6» (3c - 3) + 6* (- 15c + 15) + 6» (6^" - 12c' + 36c - 30)
+ 6» (9c» - 42c* + 84^" - 108c' + 57c) + 6 (9c« - 54c» + 96c* - 51c')
+ 6" (9c' -90*);:
which for c = 1 becomes = 0.
8,= a»(0c + 0)
+ a* {6». 0 + 6(0c' + Oc + 0) + 6«(18c* - 72c»+ 108c»- 72c + 18)j
+ a {f (Oc + 0) + fr" (- 33c» + 99c» - 99c + 33) + 6 (57c* - 162c' + 144c' - 30c - 9)
+ 6° (- 60c» + 207c* - 261c» + 141c' - 27c)j
+ a" {6» . 0 + 6* (15c» - 30c + 15) + 6» (- 54c» + 102c' - 42c - 6)
+ 6' (123c* - 297c' + 243c' - 87c + 18) + 6 (- 27c' + 102c* - 96c' + 21c')
+ 6" (27c' - 66c» + olc" - 12c*)! :
which for c = 1 becomes = 0.
400 A TENTH MEMOIR ON QUANTICS. [693
S,= (I'.O
+ a^{b (0c + 0) + 6<'(0c' + 0c" + 0c+0)}
+ a {6".0 +6»(0c» + Oc + O) + 6(-9c*+36c»-54c> + 36c-9)
+ 6" (36c» - 171c* + 324c' - 306c> + 144c - 27)}
+ a* [b* (Oc + 0) + ft" (7c> - 21c' + 21c - 7) + ¥ (- SQc* + 135c» - 171c' + 93c - 18)
+ b (66c° - 243c* + 333c' - 201c^ + 45c)
+ 6" (- 27c' + 101c* - 1 41c» + 87c* - 20c»)} :
which for c = 1 becomes = 0.
It follows that, for c = d = e=/=l, the value of the covariant S is =2(6— 1)V,
which might be easily verified.
694] 401
694.
DESIDERATA AND SUGGESTIONS.
No. 1. The theory of groups.
[From the American Journal of Mathematics, t. i. (1878), pp. 50 — 52.]
SuB.STiTUTiONS, and (in connexion therewith) groups, have been a good deal
studied ; but only a little has been done towards the solution of the general problem
of groups. I give the theory so far as is necessary for the purpose of pointing out
what appears to me to be wanting.
Let a, /8, ... be functional symbols, each operating upon one and the same number
of letters and producing as its result the same number of functions of these letters ;
for instance, a (x, y, z) = {X, Y, Z), where the capitals denote each of them a given
function of {x, y, z).
Such symbols are susceptible of repetition and of combination ;
a'(x, y, z) = cl{X, Y, Z), or ^a{x, y, z) = ^{X, Y, Z),
= in each case three given functions of (x, y, z) ; and similarly for a', a'/3, &c.
The symbols are not in general commutative, ayS not =/8a; but they are as-
sociative, a/3 . 7 = a . ^7, each = 0/87, which has thus a determinate signification.
The associativeness of such symbols arises from the circumstance that the
definitions of a, yS, 7, . . . determine the meanings of a/3, a7, &c. : if a, /3, 7, . . . were
quasi-quantitative symbols such as the quaternion imaginaries i, j, k, then a^ and /37
might have by definition values S and e such that a/3 . 7 and a . /37 (=87 and ae
respectively) have unequal values.
Unity as a functional symbol denotes that the letters are unaltered, \{x, y, z)=(x, y, z);
whence la = al = a.
C. X. 51
402
DESIDERATA AND SUGGESTIONS.
[694
The functional symbols may be substitutions; a (a;, y, z) = {y, z, x), the same letters
in a diflFerent order : substitutions can be represented by the notation a = — , the
xyz
substitution which changes xyz into yzx, or as products of cyclical substitutions,
a = ^ , = (ityz) (uw), the product of the cyclical interchanges x into y, y into z,
xyz uw
and z into x ; and u into w, w into u.
A set of symbols a, /9, 7, . . . , such that the product a/8 of each two of them (in
each order, a^ or /3a), is a symbol of the set, is a group. It is easily seen that 1
is a symbol of every group, and we may therefore give the definition in the form
that a set of symbols, 1, a, /S, 7, ... satisfying the foregoing condition is a group.
When the number of the symbols (or terms) is =n, then the group is of the nth
order ; and each symbol a is such that a" = 1 , so that a group of the order n is,
in fact, a group of symbolical nth roots of unity.
A group is defined by means of the laws of combination of its symbols: for the
statement of these we may either (by the introduction of powers and products)
diminish as much as may be the number of independent functional symbols, or else,
using distinct letters for the several terms of the group, employ a square diagram
as presently mentioned.
Thus, in the first mode, a group is 1, yS, /3», a, a/9, a/8» (a='=l, ^=\, a/9 = ;ff'a);
where observe that these conditions imply also a^ = /8a
Or, in the second mode, calling the same group (1, a, /8, 7, h, e), the laws of
combination are given by the square diagram
1
a
y3
y
8
c
1
1
a
a
J8
y
8
<
a
1
y
P
(
8
y
P
y
c
8
a
1
y
8
<
1
a
^
s
8
y
1
c
)8
a
1
t
P
a
8
y
for the symbols (1, a, /3, 7, B, e) are in fact = (1, a, /3, a/3, /3^ a^).
The general problem is to find all the groups of a given order n ; thus if n = 2,
the only group is 1, a (a' = l); if n = .3, the only group is 1, a, a' (a' = l); if n=4, the
groups are 1, a, a", a» (a<=l), and 1. a, 0, a/3 (a»=l, ^ = 1, a/3 = /8a); if n = 6, there
are three groups, a group 1, a, a.\ a', a*, a" (a«=l), and two groups 1, /3, /3', a, a/3.
694] DESIDERATA AND SUGGESTIONS. 403
a/S* (a' = l, /8' = 1); viz. in the first of these a^=:^a, while in the other of them (that
mentioned above) we have aff = ^a, a^ — ^a.
But although the theory as above stated is a general one, including as a
particular case the theorj- of substitutions, yet the general problem of finding all the
groups of a given order n, is really identical with the apparently less general problem
of finding all the groups of the same order n, which can be formed with the substitu-
tions upon 71 letters; in fact, referring to the diagram, it appears that 1, a, /8, y, 8, e
may be regarded as substitutions performed upon the six letters 1, a, 0, y, B, e,
viz. 1 is the substitution unity which leaves the order unaltered, a the substitution
which changes lo/SySe into aly^eh, arid so for /8, 7, S, e. This, however, does not in
any wise show that the best or the easiest mode of treating the general problem is thus
to regard it as a problem of substitutions : and it seems clear that the better course
is to consider the general problem in itself, and to deduce from it the theory of
groups of substitutions.
Cambridge, 2&th November, 1877.
\
No. 2. The theory of groups; graphical representation.
[From the American Journal of Mathematics, t. i. (1878), pp. 174 — 176.]
In regard to a substitution-group of the order n upon the same number of letters,
I omitted to mention the important theorem that every substitution is regular (that
is, either cyclical or composed of a number of cycles each of them of the same order).
Thus, in the group of 6 given in No. 1, writing a, b, c, d, e, f in place of 1, a,
^8, 7, Z, e, the substitutions of the group are 1, ace.bfd, aec.bdf, ab.cd.ef, ad. be. of,
of. be . de.
Let the letters be represented by points; a change a into b will be represented
by a directed line (line with an arrow) joining the two points; and therefore a cycle
abc, that is, a into b, b into c, c into a, by the three sides of the trilateral aba,
with the three arrows pointing accordingly, and similarly for the cycles abed, &c. :
the cycle ab means a into b, b into a, and we have here the line ab with a two-
headed arrow pointing both waj's; such a line may be regarded as a bilateral. A
substitution is thus represented by a multilateral or system of multilateral, each side
with its arrow ; and in the case of a regular substitution the multilaterals (if more
than one) have each of them the same number of sides. To represent two or more
substitutions we require different colours, the multilaterals belonging to any one
substitution being of the same colour.
In order to represent a group we need to represent only independent substitutions
thereof; that is, substitutions such that no one of them can be obtained from the
others by compounding them together in any manner. I take as an example a group
51—2
404
DESIDERATA AND SUGGESTIONS.
[694
of the order 12 upon 12 lettere, where the number of independent substitutions is
= 2. See the diagram, wherein the continuous lines represent black lines, and the
dotted lines, red lines.
The diagram is drawn, in the first instance, with the arrows but without the
letters, which are then affixed at pleasure; viz. the form of the group is quite indepen-
dent of the way in which this is done, though the group itself is of course dependent
upon it. The diagram shows two substitutions, each of them of the third order : one is
represented by the black triangles, and the other by the dotted triangles. It will be
observed that there is from each point of the diagram (that is, in the direction of
the arrow) one and only one black line, and one and only one dotted line; hence a
symbol B, " move along a black line," B', " move successively along two black lines,"
BR (read always from right to left), " move first along a dotted line and then along a
black line," has in every case a perfectly definite meaning and determines the path
when the initial point is given ; any such symbol may be spoken of as a " route."
1
abc . def . ghi . jkl {=■ B)
acb . dfe . gVi . jlk
ad . bl . ch . eg .fj . ik
aeh . bjd . cil . fkg
of I . bkJi . cgd . eij
agj . bfi . eke . dlh
ahe . bdj . cli . fgk
at . be . cj .dk.fh. gl
ajg . bif . cek . dhl (= R)
ak . bg . rf . di . el . Jij
alf . bhk . cdg . eji
a
b
c
d
e
f
9
h
i
j
k
I
h
c
a
e
f
d
h
i
9
k
I
J
c
a
b
f
d
e
i
9
h
I
j
k
d
I
h
a
9
j
e
c
k
f
i
b
e
J
i
b
h
k
f
a
I
d
9
c
f
k
9
e
i
I
d
b
)
e
h
a
y
f
k
I
c
i
3
d
b
a
e
h
h
d
I
j
a
9
k
e
c
b
f
i
i
e
j
k
b
h
I
f
a
c
d
9
J
i
e
h
k
b
a
I
f
9
c
d
k
9
f
i
I
c
b
j
d
h
a
e
I
h
d
9
j
a
c
k
e
i
b
f
694] DESIDERATA AND SUGGESTIONS. 405
The diagram has a remarkable property, in viHue whereof it in fact represents a
group. It may be seen that any route leading from some one point a to itself, leads
also from every other point to itself, or say from b to h, from c to c,..., and from
I to I. We hence see that a route, applied in succession to the whole series of
initial points or letters abcdefghijkl, gives a new arrangement of these letters, wherein
no one of them occupies its original place ; a route is thus, in effect, a substitution.
Moreover, we may regard as distinct routes, those which lead from a to a, to b, to
c,...,to I, respectively. We have thus 12 substitutions (the fii-st of them, which leaves
the arrangement unaltered, being the substitution unity), and these 12 substitutions
foi-m a group. I omit the details of the proof; it will be sufficient to give the
square obtained by means of the several routes, or substitutions, performed upon the
primitive arrangement abcdefghijkl, and the cyclical expressions of the substitutions
themselves : it will be observed that the substitutions are unity, 3 substitutions of
the order (or index) 2, and 8 substitutions of the order (or index) 3.
It may be remarked that the group of 12 is really the group of the 12 positive
substitutions upon 4 letters abed, viz. these are 1, abc, acb, abd, adb, acd, adc, bed,
bdc, ah . cd, ac . bd, ad . be.
Cambridge, I6th May, 1878.
I
No. 3. The Newton-Fourier imaginary problem.
[From the American Journal of Mathematics, t. Ii. (1879), p. 97.]
The Newtonian method as completed by Fourier, or say the Newton-Fourier
method, for the solution of a numerical equation by successive approximations, relates
to an equation f{x) — 0, with real coefficients, and to the determination of a certain
real root thereof a by means of an assumed approximate real value f satisfying
prescribed conditions: we then, from f, derive a nearer approximate value ^i by the
formula ^\ = ^— frri^^ and thence, in like manner, fj, fs, fa. ••• approximatmg more
and more nearly to the required root a.
In connexion herewith, throwing aside the restrictions as to reality, we have what
I call the Newton-Fourier Imaginary Problem, as follows.
Take /(«), a given rational and integral function of u, with real or imaginary
coefficients ; ^, a given real or imaginary value, and from this derive fi by the formula
^1 = ^—^fL< and thence f,, f,, fs, ... each from the preceding one by the like
formula.
A given imaginary quantity x + iy may be represented by a point the coordinates
of which are {x, y): the roots of the equation are thus represented by given points
406 DESIDERATA AND SUGGESTIONS. [694
A, B, C, .... aud the values f, f„ f,, ... by points P, P,, P the first of which is
assumed at pleasure, and the others each from the preceding one by the like given
geometrical construction. The problem is to determine the regions of the plane such
that, P being taken at pleasure anywhere within one region, we arrive ultimately at
the point A; anywhere within another region at the point B; and so for the several
points representing the roots of the equation.
The solution is easy and elegant in the case of a quadric equation: but the next
succeeding case of the cubic equation appears to present considerable diflGculty.
Cambridge, March 3rd, 1879.
No. 4. The mechanical construction of conformable figures.
[From the American Journal of Mat/ie7natics, t. ii. (1879), p. 186.]
Is it possible to devise an apparatus for the mechanical construction of conformable
figures ; that is, figures which are similar as regards corresponding infinitesimal areas ?
The problem is to connect mechanically two points P, and Pj in such wise that P,
(1) shall have two degrees of freedom (or be capable of moving over a plane area)
its position always determining that of P,: (2) that if Pj, Pj describe the infinitesimal
lengths PiQi, PjQj, then the ratio of these lengths, and their mutual inclination, shall
depend upon the position of Pj, but be independent of the direction of PiQ,: or
what is the same thing, that if Pj describe uniformly an indefinitely small circle,
then Pj shall also describe uniformly an indefinitely small circle, the ratio of the
radii, and the relative position of the starting points in the two circles respectively,
depending on the position of Pj.
Of course a pentagraph is a solution, but the two figures are in this case
similar; and this is not what is wanted. Any unadjustable apparatus would give one
solution only: the complete solution would be by an apparatus containing, suppose, a
flexible lamina, so that P, moving in a given right line, the path of P^ could be
made to be any given curve whatever.
Cambridge, July dtk, 1879.
695]
407
695.
A LINK-WORK FOR a^: EXTRACT FROM A LETTER TO
MR. SYLVESTER.
[From the American Journal of Mathematics, t. I. (1878), p. 386.]
I SUPPOSE the following is substantially your link-work for of. I use a slot to
make D move in the line OA ; but this could be replaced by proper link-work.
Supposing 0 and A fixed ; the line OB is movable, and I wanted to have the
distance OB measured in a fixed direction. This can be done by a hexagon OABQB'A'
with equal sides, and two other equal links B'R, BR: then of course, if 0, R, Q
are in line4, the hexagon will be symmetrical as to OQ, and OB' will be equal to OB,
and B" may be made to move in the fixed line OB'. If
then
or
BOA=^0, OA = AB = a, AC=CD = ^a,
OB = 2acos^0, OD = a (1 + cos 0) = 2a cos' ^5,
2a.0D = {0By.
November 30, 1877.
408 [696
696.
CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY
SEVENTHIC.
[From the Amei-ican Journal of Mathematics, t. Ii. (1879), pp. 71 — 84.]
For the binary seventhic (a, ...Jjr, yf the number of the asyzygetic covariants
(a, ...)*(«, yY, or say of the deg-order {6 . fi), is given as the coefficient of a^af- in
the function
1-a;-'
1 — aaF . 1 — aaf . 1 — oaf . 1 — cue. 1 — aar^ . 1 — ax'" . 1 — oar* . 1 — ax~''
developed in ascending powers of a. See my "Ninth Memoir on Quantics," Fhil.
Tram., t. CLXi. (1871), pp. 17—50, [462].
This function is in fact
where, developing in ascending powers of a, the second term —-^-A. (-) contains only
negative powers of x, and it may consequently be disregai-ded : the number of
asyzygetic covariants of the deg-order (^.^) is thus equal to the coefficient of a*af- in
the function A {x), which function is for this reason called the Numerical Generating
Function (N.G.F.) of the binary seventhic ; and the function A {x) expressed as a
fraction in its least terms is said to be the minimum N.G.F.
According to a theorem of Professor Sylvester's {Proc. Royal Soc, t. XXVIII.
(1878), pp. 11—13), this minimum N.G.F. is of the form
Z„+aZ, + a^Z,+ ...+a''Z„
1 - ax . 1 - oaf . 1 - axi' .1 - ax' . I - a* . 1 - a' . I - a\ 1 - a^' .1 - a^'
696] CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. 409
where Z^, Z^, ..., Z^ are rational and integral functions of x of degrees not exceeding
14 ; and where, as will presently be seen, there is a symmetry in regard to the
terms Z^, Z^; Zi, Z3,; &c., equidistant from the middle term Z^^, such that the
terms ^o. •••.^w being known, the remaining terms Zi,,...,Z3g can be at once written
down.
Using only the foregoing properties, I obtained for the N.G.F. an expression
which I communicated to Professor Sylvester, and which is published, Comptes Rendus,
t. Lxxxvii. (1878), p. .505, but with an erroneous value for the coefficient of a' and
for that of the corresponding term a^.* The correct value is
Numerator of Minimum N.G.F. is =
1
{—x — a^ — afi)
{x' + X* + 2af + a^ + x">)
(- a? - a? - aP - a^'^)
(2ar* + a? + a^*)
{x + 'ia^-a?- a;")
(- 1 + 2a^ - a^ - a? - a^" + a^-)
{'kc + of + ^af - a? ■\- «")
(2-a?-^af-^a?-a^''- «")
{x-lrZ3f + a?-a^ + 2af>-ir2a^)
' (- 1 + 4ar' - a^ - 2a;« - ^x"" - a^*)
(ojT + SaH" + 2«» - «' - 2a* - a;" + a^O
1(5 + ar" - 4a^- Qa?- 4a^»-a^' + 2a;'*)
' (a; - 4ia;» - 4af - a:^ + a!" + 4a;")
' (2 + Sar" + a;* + a^ - 2a;» + 3a;" - a;"*)
' (3a; - a;» - aH* - 7a;' - 5a;» - a;" - a;")
' (6 + 3a;> + 3a;* - 4a.-« - 3a;« - a;" + oa;"*)
' (- a; - 2a;^ - Qa;" - 8a;' - 4a;» - 3a;" + 4a;")
' (2 + 6a;» + a;* + 2a;» + 2a;« + a;"" + 6a;" + 2a;'*)
' (4a; - 3a;» - 4a;» - 8a;' - 9a;» - 2a;" - a;")
' (5 - ar' - 3a;« - 4a;« + Sa;" + 3a;" + Ga;"*)
' (- a; - a;* - 5a;» - 7a;' - a;» - a;" + 3a;")
I (- 1 + 3a;= - 2a;* + a;«+ a;'»+ 5a?» + ^a^*)
'(4a;+a;»-ar'-4af-4a;' + a.")
+ a
+ a>
+ a'
+ a*
+ a»
+ a«
+ a'
+ a»
+ a»
+ a"'
+ a"'
+ a"'
+ a"'
+ a»'
+ a»'
+ a"'
+ rt"(
+ a'»'
+ a""
+ a*"
+ a"'
+ a"'
+ 0"
* The existence of these errors was pointed out to me by Professor Sylvester in a letter dated 13th
November, 1878.
c. X. 52
410 CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY 8EVENTHIC. [696
+ 0*'
+ 0»
+ o"
+ 0"
+ 0"
+ a"
+ 0*
+ a"
+ ««'
+ a''
+ a**
+ a»
+ 0,"
(«-(r»-2«»-a?+ar» + ar'' + 5a;")
{-l-2x*-2ai'-afi + 4^'>-x"')
(2a; + 2aH' - a? + <r» 4- &c" + «")
(_ aJ - a;* - 3a;« - 3a* - a;" + 23;"*)
(a;» - a;» + Sa;* + a;" + 4a;")
(- af - af> + 2x'' + sd^')
{l+af+ 2a;"')
(—x — a^ — af — x')
(a;* + a;* + 2a;» + a;'" + a;")
(-a;»-a;''-a;")
.a;"*.
Denominator (as mentioned before) is
= 1 - cue .1 - oaf .1 - aaf> .1 - ax' .1 - a* .1 - a" .1 -a^ .1 - a'" .1 - a}\
The method of calculation is as follows : write for a moment
1 - a;-»
*^^"' '^^^l-aa^.l-a^.l-aa^.l-ax.l-ax-Kl-ax-'.l-ax-'.l-ax-''
then </)(a, a;), developed in ascending powers of a, and rejecting from the result all
negative powers of a;, is
^ Zo + aZ,+ ...+a''Z„
~ 1 -aa;. 1 -aa;». 1 -aa;». 1 -oar'. 1 -aM -aM -oM -a". 1 -a"'
developed in like manner in ascending powers of a; for the determination of the Z'a
up to Zis we require only the development of <f>(a, x) up to a"; and, assuming that
each Z is at most of the degree 14 in x, we require the coefficients of the different
powers of o in <f>{a, x) only up to a;". Assuming then that </>(«, x) developed in
ascending powers of a, up to a'', rejecting all negative powers of x, and all positive
powers greater than a^\ is
= Zo + aXi + . . . + a"Z,8,
we have
Y ^ Y A. u. isr Zo + aZr + ... + a'%s
or say
Zo+aZ, + ... + o"^,s= 1 -aM -a«. 1 -aM -a".l -a".
l-ax.l-aa^.l-a^.l-aa!'.(X, + aX, + ... + a'»X,,);
viz. developing here the right-hand side as far as a", but in each term rejecting
the powers of x above a.-", the coefficients of the several powers a", a', . . . , a" give the
696] CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. 411
required values Z„, 2,,..., Zjs. We require, therefore, only to know the values of these
functions Xq, Xi, ..., Xig.
To make a break in the calculation, it is convenient to write
l-aa;.l-a^.l-aa^A-ax^{Xo+aXt+ ...+ a'^X.^) = Y<, + a7i+ ... + a"F,8;
putting then
1 — ax.l — asfi . 1 — ax^ . 1 — aai' = 1 —ap+ a^q — a?r,
where (up to ai^*)
p = x + a?-\-af + a?,
gr = ar* + a^ + 2a;' + a^" + a;"-,
r = a:" + a;" + a;",
we have
F„ + aF, + a'Fs + . . . + a" F,8 = (1 - op -t a»9 - aV) (Z„ + aX, + a^X^ +...+ a'^X^).
The values of Y^, Fj, ..., Fjg then are
Jo •■1 ^2 'S . . • • 1^18
= JLo Xi Jl, Jlj Xya
-pXo -pXi —pX^ —pX,-,
+ qXo + qXi + qXu
- rXo - rX^i
the values being taken to cc^* only; and we then have
^<, + a-Zr, + a% + ...+a'«^,8=l-aM-aM-aM-a'M-tt"'(F„ + aF,+ ...+ffi"F8);
viz. the values of Zo, ^j, ..., ^is are
Zf, Zi Z« Z3 Zi Zj Zf Z-j Zg Zf
•— Fo Ii Fj F3 F4 Yi li !■! Fg F9
-F, -F, -F, -F3 -F, -F,
— F — F — F — F
-•o -"i ■'2 ■'3
— F — F
Zn •^12 ■^1.1 -^n ■^15 ■^18 ■^ir ■^i
18
*= Iw In -'^ij •'u -l^M •« IB ■'^le -«I7 '18
— •« 6 ^7 — ^8 ~ ■J 9 — ^10 ~ •'11 ~ i 12 ~ ■'IS "~ ■« 14
'l •''5 -" « ~ ' 7 ~ ■» 8 ■« 9 ~ •• 10 ~ ■'11 ~ ■« 12
~ Fj ~ i s — F4 — Fj — Fn — 1^7 — Fg ~ Fj — XiQ
+ 2F. +2F. +2F, H-2F +2F,
+ 2Fo +2F +2F,
+ F„.
The rule of symmetry, before referred to, is that the coefficient Z^^^p of a'*"^ is
obtained from the coefficient Zp of a^ by changing each power afl into x^*~t, the
coefficients being unaltered; in particular Z^, the coefficient of a", must remain
52—2
412 CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY 8EVENTHIC. [696
unaltered when each power sfl is changed into a^*~^ ; and the verification thus obtained
of the value
2r„ = 2 + ear" + a^ + 2a* + 2x« + a^» + 6ir" + 2«"
is in fact almost a complete verification of the whole work. Some other verifications,
which present themselves in the course of the work, will be referred to further on.
We have, therefore, to calculate the coefficients X^, Xi, ..., Xi^; the function <f>{a,x)
regarded as a function of a is at once decomposed into simple fractions ; viz. we have
l-a;-»
<^(o, x) =
1 — aa? . 1 — oaf . 1 — aa^ .l—ax.l— aar^ . 1 — ax~* . 1 — aar' . 1 — aar^
of* 1
ar" 1
l-a?.l-a^.l-««.l-««.l-a;'M-a;" X-am?
^« 1
■l-ar'.(l-a:'>'.l-a:«.I-a^.l-a;'» Y'-^a^
^8 1
^» 1
ar* 1
l-a?.(l-ar«)».l-a;«.l-a!».l-a?« l-oo;-'
1 1
\-a?.\-a*.\-ofi.\-s^.\-oi}^.\-a^^ X-aar
l-a;*.l-a!«.l-a;».l-a;'M-a^M-a^* 1-aar''
In order to obtain the expansion of ^ (a, x) in the assumed form of an expansion
in ascending powers of a, we must of course expand the simple fractions -^, &c.,
in ascending powers of a, but it requires a little consideration to see that we must
also expand the a;-coefficients of these simple fractions in ascending powers of x. For
instance, as regards the term independent of a, here developing the several coefficients
as &r as a:", the last five terms give (see post)
- «•«
+ 0^"+ a;'^+ S^^H- 5a^'+ 9a^»
- «*- af-'&a?-^"'- 8ic'>-lla;'*-l&r«'-24a:'»
1 + ar" + 2ar« + 3a;« + 5a:» + ra;"" + lla;"' + 14a;'* + 20a;'« + 26a:"
-or* .-a?- a;«-2a:»-2a^-4a;'»- 4a;"- Gar"*- 7a;"-10a;'«
= -ar^ + l 00000 0 0 0 0
viz. the sum is = 1 - a;~" as it should be*.
* To give the last degree of perfection to the beautiful method of Professor Cayley it would seem
desirable that a proof should be given of the principle illustrated by the example in the text, and the
nature of the mischief resulting from its neglect clearly pointed out. — Eds. of the A. J. M.
696] CALCULATION OF THE MINIMUM N.G.P. OF THE BINARY SEVENTHIC. 413
The expansion is required only as far as x'*: the first four terms are therefore
to be disregarded, and, writing for shortness
1
F =
G =
H==
l-af.(l-a*y{l-afy.l-a^'
1
l-a^.il-x'y.l-af .1-0^.1 -x">'
1
1 - xK 1 - x" .1 -af .1 - af .1 - ie"> .1 - x"'
1
1 -ir«. 1 -a;". 1 -a;". 1 -a^M-a^. 1 -a;» '
we have
which is
^ _ x">E a*F G x-^H
9(P"^)-l_a^, l-aa^»"^l-aa^« \-ax-
= '^"^ (1 + aar'^ + a^ar^ + ...)
- ai'F {l+aar^ + a'x-^ + ...)
+ Gil+axT'^-^-a^x-^"^ ...)
-oT'H^l + aar'' + a?x-^*+ ...),
where tte several series are to be continued up to a'*, and, after substituting for
E, F, G, H their expansions in ascending powers of x, we are to reject negative
powers of x, and also powers higher than x^*. The functions E, F, G, H contain
each of them only even powers of x, and it is easy to see that we require the
expansions up to x^, u^, «'*• and a^*" respectively. For the sake of a verification, I in
fact calculated E, ^ up to a;" and G, H up to x^*': viz. we have
(l-af)E = (l-x''>)F,
from the coefficients of E we have those of (1 — of) E, and in the process of calculating
F we have at the last step but one the coefficients of (1 — x^") F, the agreement
of the two sets being the verification ; similarly,
(l-a^)G=(l-a^*)H
gives a verification. The process for the calculation of E,
1
1 - a;'. (1 - ar«)' (1 - a^)'.! - «" '
414 CALCULATION OF THE MINIMXJM N.G.F. OF THE BINARY 8BVENTHIC. [696
is as follows:
Ind. X
0 2 4 6 8 10 12 14 16 18 20 22
-a?)-'
-^)-'
-a*)-'
-a*)-^
-a^r
E
_ /l _
-a^)-'
1
1
1
1
1
1
1
1
1
1
1
1
2
2
3
3
4
4
5
5
2
2
3
3
4
4
5
5
6
6
1
1
3
3
6
6
10
10
15
15
3
3
6
6
10
10
15
15
21
21
1
1
3
4
7
9
14
17
24
3
4
7
9
14
17
24
29
38
45
1
1
3
5
8
12
19
25
36
3
5
8
12
19
25
36
48
63
81
1
1
3
5
9
13
22
30
3
5
9
13
22
30
45
61
85
111
the alternate lines giving the developments of the functions
(1 - af)-\ (1 - af)-^ (1 - x*)-\ (1 - a;")-' (1 - *-*)-^ . . .,
which are the products of the a;-functions down to any particular line. And in like
manner we have the expansions of the other functions F, G, H respectively. I give
first the expansions of E, F, G, H ; next the calculation of the X's; then the cal-
culation of the F's: and from these the Z's up to Z,8, or coefficients of the powers
a", a', . . . , o" in the numerator of the N.G.F. are at once found ; and the coefficients
of the remaining powers a" a" are then deduced, as already mentioned.
Writing in the formula x — 0, we have, for the numerator of the N.G.F. of the
invariants, the expression
1 - a« + 2a« - a"" + 5a" + 2a" + 6a" + 2a" + 5a» - a'" + 2a" - a* + a«
agreeing with a result in my " Second Memoir on Quantics," Phil. Trans., t. CXLVi.
(1856), [Number 141, vol. II. in this Collection, p. 266]; this, then, was a known result,
and it affords a verification, not only of the terms in «", but also of those in a;". Thus,
in calculating the foregoing expression of the numerator, we obtain Z^= {2a^ + afi + a^*),
viz. the term is
o*(2a;* + a?+a;"),
and we thence have the corresponding term a** (1 + a:° + 2ie"'), which, when x = 0,
becomes = a", a term of the numerator for the invariants : and the term la^* of Z^
696] CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. 415
is thus verified, viz. so soon as, in the Calculation, we arrive at this term, we know
that it is right, and the calculation up to this point is, to a considerable extent,
verified. And similarly, in continuing the calculation, we arrive at other terms which
are verified in the like manner.
Expansions of the Functions E, F, G, H.
Ind. X
E
F
G
H
Ind. X
E
F
G
H
0
1
1
1
1
16
45
36
20
6
2
1
1
1
0
18
61
47
26
7
4
3
3
2
1
20
85
66
35
10
6
5
4
3
1
22
111
84
44
11
8
9
8
5
2
24
113
58
16
10
13
11
7
2
26
141
71
17
12
22
18
11
4
28
183
90
23
14
30
24
14
4
30
225
110
26
Ind. X F
G
H
Ind. X
G
H
Ind. X
H
32
284
136
33
70
2172
419
108
2265
34
344
163
37
72
2432
472
110
2426
36
425
199
47
74
2702
515
112
2623
38
508
235
52
76
3009
576
114
2807
,40
617
282
64
78
3331
629
116
3026
42
729
331
72
80
3692
699
118
3232
44
872
391
86
82
4070
760
120
3479
46
1020
454
96
84
4494
843
122
3708
48
1205
532
115
86
4935
913
124
3981
50
1397
612
127
88
5427
1007
126
4240
52
1632
709
149
90
5942
1091
128
4541
54
1877
811
166
92
6510
1197
130
4828
56
2172
931
192
94
7104
1293
132
5164
58
2480
1057
212
96
7760
1416
134
5481
60
2846
1206
245
98
8442
1525
136
5850
Qi
3228
1360
269
100
9192
1663
138
6204
64
3677
1540
307
102
9975
1790
140
6609
66
1729
338
104
10829
1945
142
6998
68
1945
382
106
2088
416 CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY 8EVENTHIC. [696
Calculation of the X's.
Ind. X even or odd according as suffix X is even or odd.
Ol
23
h
67
«9
lOu
12x3
14
1
1
3
- 1
—
1
- 3
- 4
- 8
- 11
1
1
2
3
5
7
11
14
- 1
- 1
-
2
- 2
- 4
- 4
- 6
x.=
1
0
0
0
0
0
0
0
1
1
3
-1
^ 1
- 3
—
4
- 8
- 11
- 18
3
5
7
11
14
20
26
-2
- 4
- 4
-
6
— 7
- 10 ■
- 11
x,=
0
0
0
+
1
0
0
0
1
1
3
5
- 1
- 3
- 4
-
8
- 11
- 18
- 24
- 36
7
11
14
20
26
35
44
58
- 6
- 7
- 10
_
11
- 16
- 17
- 23
- 26
X,= 0 + 1 0 + 1 0 + 1 0 + 1
1 1 3 5
- 4 - 8 - 11 -18-24 - 36 - 47
20 26 35 44 58 71 90
- 16 - 17 - 23 - 26 - 33 - 37 - 47
J:,= 0 + 1+1+1+2 + 1+1
113 5 9
- 8 - 11 - 18 - 24 - 36 - 47 - 66 - 84
35 44 ■ 58 71 90 110 136 163
- 26 - 33 - 37 - 47 - 52 - 64 - 72 - 86
X^= 1 0+3 + 1
1 1
- 18 - 24 - 36 - 47
71 90 110 136
-52 - 64 - 72 - 86 - 96 - 115 - 127
3
5
9
66
- 84
- 113
63
199
235
2:.= 1 + 2
696] CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. 417
Ol h H «7 «9 1^11 12^3 14
J« =
Xr =
X,=
1
1
3
5
9
13
- 24
—
36
—
47
—
66
—
84
—
113
—
141
- 183
110
136
163
199
235
282
331
391
- 86
-
96
-
115
-
127
-
149
-
166
-
191
- 212
0
+
4
+
2
+
7
+
5
+
8
+
8
+ 9
1
1
3
5
9
13
- 47
—
66
—
84
—
113
—
141
—
183
—
225
199
235
282
331
391
454
532
- 149
-
166
-
192
-
212
-
245
-
269
-
307
3
+
4
4-
7
+
9
+
10
+
11
+
13
1
1
3
5
9
13
22
- 66
—
84
-
113
—
141
—
183
—
225
—
284
- 344
282
331
391
454
532
612
709
811
- 212
-
245
-
269
-
307
-
338
-
382
-
419
- 472
4
+
3
+'-
10
+
9
+
16
+
14
+
19
+ 17
1
1
3
5
9
13
22
- 113
—
141
-
183
—
225
—
284
—
344
—
425
454
532
612
709
811
931
1057
- 338
-
382
-
419
-
472
-
515
-
576
-
629
4
+
10
+
13
+
17
+
21
+
24
+
25
x,=
1 1 3 5 9 13 22 30
- 141 - 183 - 225 - 284 - 344 - 425 - 508 - 617
612 709 811 931 1057 1206 1360 1540
- 472 - 515 - 576 - 629 - 699 - 760 - 843 - 913
2:,,= 0 + 12 + 13 + 23 + 23 + 34 + 31 + 40
1 3 5 9 13 22 30
- 225 - 284 - 344 - 425 - 508 - 617 - 729
931 1057 1206 1360 1540 1729 1945
- 699 - 760 - 843 - 913 - 1007 - 1091 - 1197
Z„= 8 + 16 + 24 + 31 + 38 + 43 + 49
1 3 5 9 13 22 30 45
- 284 - 344 - 425 - 508 - 617 - 729 - 872 - 1020
1206 1360 1540 1729 1945 2172 2432 2702
- 913 - 1007 - 1091 - 1197 - 1293 - 1416 - 1525 - 1663
2:,,= 10 + 12 + 29 + 33 + 48 + 49 + 65 + 64
C. X. 53
418 CALCULATION OF THE MINIMUM N.O.F. OF THE BINARY SEVENTHIC. [696
Oj -^3 ^5 ^7 «9 ^Qll ^^13 ^^
3 5 9 13 22 30 45
- 425 - 508 - 617 - 729 - 872 - 1020 - 1205
1729 1945 2172 2432 2702 3009 3331
- 1293 - 1416 - 1525 - 1663 - 1790 - 1945 - 2088
Jf.,=
14
+ 26
+ 39
+ 53
+ 62
+ 74
+ 83
3
5
9
13
22
30
45
61
- 508
- 617
- 729
- 872
- 1020
-1205
-1397
- 1632
2172
2432
2702
3009
3331
3692
4070
4494
-1663
- 1790
-1945
- 2088
- 2265
- 2426
- 2623
-2807
-^14 =
4
+ 30
+ 37
+ 62
+ 68
+ 91
+ 95
+ 116
5
9
13
22
30
45
61
- 729
- 872
- 1020
-1205
- 1397
-1632
-1877
3009
3331
3692
4070
4494
4935
5427
-2265
-2426
- 2623
-2807
-3026
-3232
-3479
x„=
20
+ 42
+ 62
+ 80
+ 101
+ 116
+ 132
5
9
13
22
30
45
61
85
- 872
-1020
- 1205
- 1397
- 1632
- 1877
- 2172
- 2480
3692
4070
4494
4935
5427
5942
6510
7104
-2807
- 3026
-3232
-3479
- 3708
-3981
-4240
- 4541
-*:,.=
18
+ 33
+ 70
+ 81
+ 117
+ 129
+ 159
+ 168
9
13
22
30
45
61
85
-1205
-1397
- 1632
- 1877
-2172
- 2480
-2846
4935
5427
5942
6510
7104
7760
8442
-3708
-3981
-4240
- 4541
- 4828
- 5164
- 5481
jr„=
31
+ 62
+ 92
+ 122
+ 149
+ 177
+ 200
9
13
22
30
45
61
85
111
- 1397
- 1632
- 1877
-2172
-2480
- 2846
- 3228
- 3677
5942
6510
7104
7760
8442
9192
9975
10829
-4541
-4828
-5164
-5481
- 5850
-6204
- 6609
- 6998
jr„=
13
+ 63
+ 85
+ 137
+ 157
+ 203
+ 223
+ 265
696] CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. 419
Calculation of the Y's.
Tnd. X even or odd as suffix X is even or odd.
0
8n
10
11
12
13
14
i;=
- 1
-- 1
i'i=
- 1
1^,=
0
1
0
1
0
0
1
—
1
— 1
—
1
- 1
1
1
2
1
0
1
I
2
1
0
0
1
1
1
2
1
-1
- 1
- 2
—
2
— 2
—
2
1
•
-
1
- 1
-
1
0
0
0
- 1
1
\
1
3
- 1
1
-2
1
2
- 5
3
3
- 5
2
2
- 5
4
Y* =
+ 2
0
1
0
1
- 1
3
-4
4
-5
1
4
- 7
•>
5
9
4
1
4
- 9
6
- 1
y>=
+ 1
- 1
0
4
-1
2
-3
1
5
10
5
13
5
7 9
16 - 17
11 10
1 - 2
Y.=
+ 3
+ 1
0
53—2
420 CALCULATION OF THE MINIMUM N.Q.F, OF THE BINARY 8EVENTHIC. [696
^1
23
^5
«7
«9
10„
12l3
14
3
4
7
9
10
11
13
- 4
- 6
-13
- 18
- 22
- 27
1
3
7
- 1
12
- 1
17
- 4
y,=
3
0
+ 2
- 1
- 2
0
- 1
4
3
10
9
16
14
19
17
- 3
- 7
-14
- 23
- 30
- 37
- 43
4
6
17
- 1
20
- 3
33
- 6
Ys =
4
0
+ 3
- 1
- 1
0
- 1
+ 1
4
10
13
17
21
24
25
- 4
- 7
-17
-26
-38
- 49
- 58
3
7
17
27
- 4
40
- 6
Y> =
0
+ 3
- 1
— 2
0
_ 2
+ 1
12
13
23
23
34
31
40
- 4
- 14
- 27
- 44
- 61
- 75
- 87
4
7
21
29
- 3
52
- 7
61
- 14
y.o=
0
+ 8
+ 3
+ 3
0
- 1
+ 1
0
8
16
24
31
38
43
49
-12
- 25
- 48
-71
- 93
- Ill
4
14
31
- 4
54
- 7
78
- 17
!'„ =
8
+ 4
+ 3
- 3
- 6
- 3
- 1
10
12
29
33
48
49
65
64
- 8
-24
-48
- 79
-109
-136
-161
12
25
60
- 4
84
- 14
128
- 27
Y„ =
10
+ 4
+ 5
- 3
- 6
- 4
- 1
* 4
696] CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. 421
Oj 23 4g 67 89 10„ 12,3 1^
U
26
39
53
62
74
83
-10
-22
-
51
- 84
- 122
- 159
- 195
8
24
56
95
- 12
141
- 25
1^:3=
4
+ 4
-
4
- 7
- 4
_ 2
+ 4
4
30
37
62
68
91
95
116
- 14
-
40
- 79
-132
- 180
- 228
-272
10
22
61
96
- 8
161
- 24
204
- 48
Yu =
4
+ 16
+
7
+ 5
- 3
- 1
+ 4
0
20
42
62
80
101
116
132
- 4
- 34
-
71
-133
-197
-258
-316
14
40
93
158
233
I
- 10
- 22
- 51
Y^ =
16
+ 8
+
5
- 13
- 13
- 6
- 2
18
33
70
81
117
129
159
168
-20
-
62
- 124
-204
- 285
-359
-429
4
34
75
163
- 14
238
- 40
350
- 79
r^.=
18
+ 13
+
12
- 9
- 12
- 7
- 2
+ 10
.
31
62
92
122
149
177
200
-18
-51
-
121
-202
-301
-397
-486
20
62
144
- 4
246
- 34
367
- 71
T„ =
13
+ 11
-
9
- 18
- 12
- 8
+ 10
13
63
85
137
157
203
223
265
-31
-
93
- 185
-307
- 425
- 540
- 648
18
51
139
235
- 20
389
- 62
511
- 124
Y,,^
13
+ 32
+
10
+ 3
- 11
- 7
+ 10
+ 4
Cambridge, December 7th, 1878.
422
[697
697.
ON THE DOUBLE ^-FUNCTIONS.
[From the Jcmmal fiir die reine und angewandte Matliematik (Crelle), t. LXXXVII. (1878),
pp. 74—81.]
I HAVE sought to obtain, in forms which may be useful in regard to the theory
of the double ^-functions, the integral of the elliptic differential equation
^
dx
+~
= 0:
^/a — x.b — x.c—x.d — x \/a — y.h — y.c — y.d — y
the present paper has immediate reference only to this differential equation; but, on
account of the design of the investigation, I have entitled it as above.
We may for the general integral of the above equation take a particular integral
of the equation
dx dy 4-—,,^=..=^-^ =0-
'Ja — x.h — x.c — x.d — x 'Ja — y.h — y.c — y.d — y~ "^a — z.b — z.c — z.d — z
viz. this particular integral, regarding therein ^ as an arbitrary constant, will be the
general integral of the first mentioned equation. And we may further assume that z
is the value of y corresponding to the value o of x.
I write for shortness
a — X, b — X, c — X, d — x = a,, b, c, d,
a-y. b-y, c-y, d-y = &i, bj, Cj, dj;
and I write also (xy, be, ad), or more shortly (be, ad) to denote the determinant
I, x + y, xy
1, b + c, be
1, a + d, ad
we have of course (ad, bc)=-(bc, ad), and there are thus the three distinct determinants
(ad, be), (bd, ac) and (cd, at).
697] ON THE DOUBLE ^-FUNCTIONS. 423
We have then for each of the functions
la — z h — z Ic — z
y cT^z' y d^z' V~d^z
a set of four equivalent expressions, the whole system being
la — z _ '^a — b.a — c {VadbiCi + Vaidibc} _ »Ja — h.a — c{x — y)
y d-z~ (be, ad) ~ Vadb^ - Vaidibc
_ Va — 6 . a - c {Vabcidi + Vaibicd} _ ^a — b.a — c {Vacb;di + VaiCibd} _
(a — c) Vbdbjdi —(b—d) Vaca,Ci (a - b) ^/cdc^d^ —(c—d) Vaba,bi
/T a/ — J {(a - c) Vbdbjd, + {b-d) VacaiC,) a / "^ — ^ [ Vabcjdi - Vajbicd}
V d-z (be, ad) VadbA - Va^dTbc
a/— ^,(cd, ah) a/ — — -^ {(a -d) VbcbjCi + (6 - c) Vadaid,}
(a — c) Vbdbidj — (6 — d) VacajCi (a — b) Vcdcid, — (c — d) Vabajbi
i—^-- a/ ——, {(a - b) Vcdcidi +(c — d) v'abaibi} */ -— j jVacbidj — VajCibd)
V d-z (6c, ad) VaHbA - '^ajdjbc
a/ -— j {(a - d) VbcbiC, -(b-c) Vada^d,} \/~~Z^ i^' "^)
(a — c) Vbdbidi -(b — d) Vaca,c, (a — 6) Vcdcjd, — (c — d) Vaba,b,
The expressions in the like fourfold form for the functions sn (u + v), en (u + v), dn (u + v)
are given p. 63 of my Treatise on Elliptic Functions.
It is easy to verify first that the four expressions for the same function of z are
identical, and next that the expressions for the three several functions
la — z lb — z Ic — z
V d^z' V d^z' y d^z'
are consistent with each other. For instance, comparing the first and second expressions
of fu j-^- > the equation to be verified is
adbjCi — a,d,bc = (x — y) (be, ad),
which is at once shown to be true. Again comparing the first and second expressions
Ib^z
for */ j-^- , we ought to have
{(a - c) \^bdbidi + (6 - d) VacaiCi} {\^adbiCi - Vaid,bc) = (6c, ad) {Vabcjdi - Vaibicdj.
424 ox THE DOUBLE ^-FUNCTIONS. [697
Here the product on the left-hand side is
= (a-c) |b,d Vabc^d, - bd, Va,b,cd} +(b-d) {- a,c Vabc,d, + ac, Vaibicdj,
viz. this is
= VabcA {(a - c) b,d - (6 - d) a,cl - Va,b,cd {(a - c) bd, - (6 - d) acj},
and in this last expression the two tei-ms in { j are at once shown to be each
= (6c, ad) ; whence the identity in question.
Comparing in like manner the first expressions for lu -r^^ and V^T^T" ^'
spectively, we have
(6 - d) {be, ady ., ^ = (« - 6) (a - c) (b - d) (adbjCi + ajdibc + 2 VabcdaibiC,d,},
(d — a)(bc, ady-j — =
(t ^ z
- (o - 6) {(a - cf bdb,di 4- (6 - df acajCi + 2(a-c)(b-d) VabcdaibiCidi},
whence, adding, the radical ou the right-hand side disappears; the whole equation
divides by —(a—b), and omitting this factor, the relation to be verified is
(be, adf = (a — c)' bdbid, ■\- {b - df acajCi — {a-e){b~ d) (adbjCi -f- aid,bc) ;
the right-hand side is here
= [{a - c) b,d — (b — d) a,c} {(a — c) bd, — (6 — d) aci},
and each of the two factors being = (be, ad), the identity is verified. It thus appears
that the twelve equations are in fact equivalent to a single equation in x, y, z.
Writing in the several formulae x = a, b, e, d successively, they become
x = a, X = b, x = c, x = d,
a — z _&i c — abi 6 — ac, a — 6.a — cd,
d — z di' d — b'ci' d— e'hj' d — b.d — e'sii'
b — z _ht c — b &i b — a.b — c di a — bci
d — z di' d—a'ci' d — a. d — c' hi' d — c'a,'
e — z _ Ci c — a . c — b d, 6 — ca, a — cbj
d-z~di' d-o.d-6'c,' ~ d-a' bj' ~ d - 6 ' a, '
viz. for x = a, the relation is z = y, but in the other three cases respectively the
relation is a linear one, z = ~ — ?.
7y + S
Rationalising the first equation for a/ j , we have
(6c, ady (a-z) = {a — 6) (a — c) (d - z) {adbjCi + aidibc + 2 VabcdaibiC,dj},
and thence
{{be, ady {a-z)-{a- b) {a -c){d- z) (adb.c, -l- aidibc)}'
= (a - 6)= (a - c)' (d - zy . 4abcda,b,Cidi.
697J ON THE DOUBLE ^-FUNCTIONS. 425
Expanding, and observing that
(adbiCi + aidibc)" = (adbjCi - ajdibc)^ + 4abcdaibiCidi = {he, ad)- {x-yf + 4abcdaibiCidi,
the whole equation becomes divisible by {he, adf, and omitting this factor, the
equation is
(be, adf {a - z)- — 2{a — h) {a — c){a — z) {d — z) (adbjCi + ajdibc)
-\-{a-hy{a-c)'{d-zy{x-yy = 0,
or, as this may also be written,
2»{(6c, adf -2(a-6)(o-c)(adbiCi + aid,bc) +{a-hf{a-c)-{x-yf ]
- 2z [{he, ad) a- {a-h){a-c) (adbjCj + ajdibc) {a + d)+{a- by {a - cf {x -yfd}
+ \{bc, ad)a' -2{a-b){a-c) (adbjCi + ajdibc) ad +{a- hf {a - cf {x - yf d»} = 0.
This is really a symmetrical equation in x, y, z of the form
A
-ir2B{x + y + z)
+ G{afi + y^ + z')
+ 2D {yz •{■ ZX+ xy)
+ 2E {y^z + yz^ + z^x + za^ + oc'y + any')
+ 4!Fxyz
+ 2G {x'yz + xy^z + ooyz^)
+ H {y^z'' + z'^x'' Jf a?f)
+ 27 {xfz'^ + oe^yz'^ + x^z)
+ Jsii'y''z'' = 0 ;
the several coefficients being symmetrical as regards b, c, d, but the a entering un-
symmetrically : the actual values are
A= a* {6V + l^d^ ■hd'd:'- 2hcd {h + c + d)} + 2a'bcd {be -\-bd + cd) - Sa'h'c'd\
B = 2a*bcd - a' {b''<f + M' + c'd^) + ab^&d\
C = - 4!a'hcd + a" {be + bd+ cdf - 2ahcd {be + hd+ cd) + h'e^d",
D = - a*{be + bd + ed) + a^{b'c + bc!' + b'd + hd' + c'd + cd:'-2bcd)
+ a' {b'c' + b'd' + d'd'- bed {b + e + d)}- b'd'd'',
E = a>{bc+bd + cd) - a' {b'c + be' + b'd+bd^ + i^d + cd^) + abed {b + c + d),
F = a*{b + c + d)-a'{b' + (^ + d' + hc + hd + ed)+ Gashed
- a [b'c' + b''d' + c'd- + bcd{b + e + d)} + bed {be + bd + cd),
G=- a* + a'{¥ + c'' + d'-bc-bd- cd) + a {b'e + he' + h'd + hd' + c'd + cd' - 2bcd)
-bcd{b+c + d),
H= a*-2a^{b + c + d) + a^{b + c + dy-iabcd,
/= a^-a{B'+c^+d') + 2hed,
J =~3a'' + 2a{b + c + d) + h' + d' + d'-2{he + hd + ed).
c. X. 54
426 ON THE IX)UBLE ^-FUNCTIOKS. [697
It may be remarked by way of verification that the equation remains unaltered
on substituting for x, y, z, a, b, c, d their reciprocals and multiplying the whole by
I further remark that, writing a = 0, we have
^ = 0, B = 0, C = ¥(?d\ D = -b'c'd', E = 0, F = bed (be + bd + cd),
Q = -bcdib + c + d), 11 = 0, I = 2bcd, J=b' + d' + d'-2(bc+bd + ed);
and writing also
6=1, -B = (b + c + d), y = be + bd + cd, -^ = bcd,
(whence
a — x.b — x.c — x.d—x = ^a;-\- ya^ + Bx' + ear*),
we have the formula
^{a^ + y' + z'- 2yz - 2zx - 2xy)
— 4/97 ^^
— 2/98 xyz (x + y + z)
— 4y9e xyz (yz -^ zx + xy)
+ {p'-iye)a?y^z^ = 0,
given p. 348 of my Elliptic Functions as a particular integral of the dififerential
equation when the radical is V/Sa; + yx' + Za? + ea^.
Let the equation in {x, y, z) bo called it = 0 ; u has been given in the form
M = g2» — 253^ + ?I, and we thence have ^ -t- = S^: — S3 which, in virtue of the equation
a = 0 itself, becomes h'j~— ^®' ~ ^'^ ! ^^ ^^^ easily
93> - 816 = (a - hf (a - c)» (a - d)» {(adb.c, + a,d.bc)' - (Jbc, adf {x - y)%
or, attending to the relation
(adbjCi + aid,bcy = (adbiCi — ajdibc)' + 4abcdaibiCid,
= (6c, ad)' (jr — y)' + 4abcdaibiCidi,
this is
»» - Sie = 4 (o - 6)" (o - ef (a - df abcdaibAd, ,
or we have
J J- = (a — 6) (a - c) (a — d) Vabcd VaibiCjd,.
Writing
a — z, b — z, c — z, d — z = &3, bj, Cj, d,,
we have of course the like formulae
i ^ = (a - i») (a - c) (a - d) VaAcA VaAcS,
i T- = (a - 6) (a - c) (ft - d) Vabcd Vaab,Cadj ;
697]
ON THE DOUBLE ^-FUNCTIONS.
427
and the equation dii = 0 then gives
dx
dy
+ -
dz
= 0,
Vabcd VajbiCid, 'Ja^^„
as it should do. The diflferential equation might also have been verified directly from
any one of the expressions for
fa — z Ih — z Ic—z
d^z' Vd3-^ ""' Vd^z-
Writing for shortness
X=a — x.b — x.c — x.d — X, etc.,
then the general integral of the differential equation
dx dy dz _
by Abel's theorem is
!^, X, 1, V^
>t, y.
1, ^/Y
z\ z.
1, 'JZ
V?, w,
1, VTT
= 0,
where w is the constant of integration : and it is to be shown that the value of w
which corresponds to the integral given in the present paper is w = a. Observe that
writing in the determinant w = a, the determinant on putting therein x = a, would
vanish whether z were or were not =y; but this is on account of an extraneous
factor a — w, so that we do not thus prove the required theorem that (w being =a)
we have y = z when x = a.
An equivalent form of Abel's integral is that there exist values A, B, G such
that
Aa? Jr Bx + C = ^X,
Af + By +G = ^fY,
Az^ +Bz +G = sIZ,
A'u?^Bw-\-G=^W,
or, what is the same thing, that we have identically
{AGf -It Be -^^ Gf -% ^{A" -\) . e - X .e -y .e - z .e - w.
We have therefore
or say
C - ahcd = {A'' — 1) ocyzfw,
C" - ahcd
icyzw = ■
A'-l
54—2
428 ON THE DOUBLE ^-FUNCTIONS. [697
which equation, regarding therein A, B, C &a determined by the three equations
Ax' +Bx +C = s/X,
Af+By+C = ^Y,
Attf' + Bw + C = >^W,
is a form of Abel's integral, giving z rationally in terms of x, y, w.
Supposing that, when x = a, z='y. then the last-mentioned integral gives
C - ahcd
where A, C are now determined by the equations
Aa''+Ba+C = 0,
Af+By+C = ^7,
Aw^ + Bw + C = >/W,
and, imagining these values actually substituted, it is to be shown that the equation
C'-abcd
is satisfied by the value w = a.
We have
A.a — y.a — w.w — y= (a — w)\/Y— {a — y)'JW,
B.a — y.a — w.w — y = (a — w)(a+w)'/Y — (a-y)(a + y)^W,
C.a — y.a-w.w — y = {a — w)aw 'fY—{a—y)ay '/W,
or writing as before
a-y, b-y, c-y, rf-y =ai, b,, Cj, d,,
and also
a—w, h — w, c — w, d—w = &3, bj, Cj, d,,
then y = aib,Cidi, Tr = ajb3C8d3, and the formulae become
-4 = 7 ■ , — {VajbiCidi - VaibjCsda},
(w - y) Vaia,
B = — T=- {- (a + w) VajbiCidi + (a + y) VaibjCsd,},
(w-y)Va,a3
C = . — {aw VajbiCid, — ay Va,bjcA}-
(w - y) Va,a,
697] ON THE DOUBLE ^-FUNCTIONS. 429
If in these formulae w is indefinitely nearly = a, then as is indefinitely small, so that
VajbiCidi may be neglected in comparison with VaibsCadj : also w — y may be put = a^ ;
the formulae thus become
. VbsCjds J, . , VbsCads _, VbsCsds
ai va, ai Vaj aiVaa
where the values of A, B, C are each of them indefinitely large on account of the
factor Va^ in the denominator; the value of G is C=ayA, and substituting this value
in the equation
G^-abcd
and then considering A as indefinitely large, the equation becomes ayhu = a^y-, that is,
w = a; so that w = a is a value of w satisfying this equation.
Cambridge, 3 July, 1878.
430 [698
698.
ON A THEOREM RELATING TO COVARIANTS.
[From the Journal fur die reine und angewandte Mathematik (Crelle), t. Lxxxvii. (1878),
pp. 82, 83.]
The theorem given by Prof. Sylvester, Crelle, vol. Lxxxv., p. 109, may be stated as
follows: If for a binary quantic of the order i in the variables, we consider the
whole system of covariants of the degree j in the coeflBcients, then
Wn(j)
where 6 denotes the number of asyzygetic covariants of the order 6 in the vaiiables,
the values of 6 being ij, ij - 2, t}'— 4, ..., 1 or 0, according as ij is odd or even.
In the case of the binary quintic (a, . . .$«, yf, (i = 5), we have a series of
verifications in the Table 88 of my " Ninth Memoir on Quantics," Phil. Trans, vol. CLXI.
(1871), [462]: viz. writing the small letters a, b, c, ..., u, v, w (instead of the capitals
A, B, etc.) to denote the covariants of the quintic, a, the quintic itself, degree 1,
order 5, or as I express it, deg-order 1.5: b, the covariant deg-order 2 . 2, etc., the
whole series of deg-orders being
a, 6, c, d, e, f, g, h, i, j, k, I,
1.5, 2.2, 2.6, 3.3, 3.5, 3.9, 4.0, 4.4, 4.6, 5.1, 5.3, 5.7,
m, n, 0, p, q, r, s, t, u, v, w,
6.2, 6.4, 7.1, 7.5, 8.0, 8.2, 9.3, 11.1, 12.0, 13.1, 18.0,
then the table shows for each deg-order, the several covariants of that deg-order, and
698]
ON A THEOREM RELATING TO COVARIANTS.
431
the number of them which are asyzygetic ; for instance, i = 5 as above, j = 6, an
extract from the table is
30
1
28
0
26
1
24
1
22
2
20
2
18
3
16
2
14
4
12
3
10
4
8
2
6
4
4
1
2
2
0
0
a*e
«'/
a*b, oV
a\ acf
aH, a^bc, c?, P
aH, ahf, ace
a^h^, a%, acd, be', ef
abe, al, ce, df
a^^, abd, 6'c, ch, e'
ak, bi, de
aj, W, bh, eg, cp
n
bg, m
{k + \)e
31
0
27
26
46
42
87
34
60
39
44
18
28
5
6
0
462 = -"111)-
n (5) n (6) '
where, for instance deg-order 6 . 14, the covariants are a'6', a?h, acd, be', ef, but the
number against these in the third column being (not 5 but) 4, the meaning is that
there exists between these five terms one syzygy, making the number of asyzygetic
covariants of the deg-order 6 . 14 to be 4. The second column thus in fact contains
the several values of k, and the third column the corresponding values of 0 ; whence,
forming the several products (k + l) as shown, the sum of these is as it should be
= 462..
Cambridge, 13 July, 1878.
432
[699
699.
ON THE TRIPLE ^-FUNCTIONS.
[From the Jourmil fur die reine und angewandte Mathematik (Crelle), t. Lxxxvii. (1878),
pp. 134—138.]
There should be in all 64 functions proportional to irrational algebraical functions
of three independent variables x, y, z; there is no difficulty in obtaining the expression
of these 64 functions in the case of the system of differential equations connected
with the integral
\dx : 'Ja — x.b — x.c — x.d — x.e — x.f — x.g — x.h — x;
but this is not the general form of the system for the deficiency (Geschlecht) p = Z;
and I do not know how to deal with the general form : the present note relates
therefore exclusively to the above-mentioned hyper-elliptic form.
I.
If in the Memoir, Weierstrass, "Theorie der Abel'schen Functionen," Crelle, t. Lii.
(1856), pp. 285 — 380, we take p = 3, and write x, y, z; u, v, w; a, b, c, d, e, f g
instead of .t,, x^, a;,; w„ it,, w,; Ui, cu,, a,, at, a^, a,, a^; then, neglecting throughout
mere constant factors, we have
X = a — x.b — x.c — x.d — x.e — x.f—x.g — x,
with the like values for Y and Z: the differential equations are
, _ b — x.c — x.dx b — y.c — y.dy b — z.c — z.dz
'*'*" v^ "*■ vF ^ :jz '
J _'c — x.a — x.dx c — y.a — y.dy c—z.a — z.dz
**" IJX "*" V7 "^ Tz '
, a—x.b — x.dx a — y.b — y.dy a — z.b — z.dz
""' V^ + VF ^ V2 '
699] ON THE TRIPLE ^-FUNCTIONS. 433
and if we ^vrite the single letters A, B, C, B, E, F, G for al (it, v, w)i, al(it, v, w\,
al(M, V, w\, &\{u, V, w\, a\(u, v, w\, &\{u, v, w\, a,l(u, v, w\ respectively, each of the
capital letters thus denoting a function of (m, v, w), the expressions of these functions
in terms of {x, y, z) are
A —'^a — x.b — x.c — w, (seven equations).
Similarly, instead of the 21 functions al(M, v, w^a al (u, v, w)g, writing AB, ...,F0,
each of these binary sjonbols denoting in like manner a function of (u, v, w), the
definition of AB is
AB = AVB-BSJA,
where
_ d d d
du dv dw '
we have
, , dx a — y.a — z.j ,, b — y.b — z, ,, c — y.c — z, ,.,
0 — c.c — a.a — o.-p^ = (o — c)du-\ (c — a)dv-\ (a — b)dw,
s/X x — y.x — z x — y.x — z ' x — y.x — z^
dy a — z.a — x,. ,, b — z.b — x, ., c — z.c — x, ,. ,
„ -ttt = \b — c)du-{ (c — a)dv-\ (a — 6) dw,
tjY y — z.y—x y — z.y-x y — z.y — x
dz a — x.a — yi,, , , b — x.b — y, , , c — x.c — y, ,, ,
-T-_ = ^ {b - c) du + ^ (c -a)dv-^- ^ (a -b)dw;
>jZ z — X .z — y z — x.z — y z — x.z — y^
hence
b-c.c-a.a-b^ a-y.a—z, . b-y.b-z .c-y.c-z, ,.
-_ Vx= ^ {b-c) + "- (c-a) + ^ {a-b),
i/X x — y.x — z x—y.x — z x — y.x — z
_ b — c.c — a.a — b
x — y.x — z '
that is, '
Vx =
x — y.x — z
and similarly
V V = , w z= .
■' y — x.y — z z — x.z — y
Hence from the equation
A = '^a — x.a — y.a — z
we have
VA=-^a(^—Vx + -^ Vy+ — V^),
\a — x a — y^a — z I
that is,
y — z.z — x.x — y\a — X d — y a — z J
and similarly
^B i^ l^^VX + f— Vf + P^VII;
y — z.z — x.x — y{b—x b- y b — z )
consequently
AB = hici-b)AB |(y-z)VZ {z-x)^Y ^ {x-y)^Z)
y — z.z — x.x — y\a — x.b — x a — y.b — y a — z .b — z) '
C. X. 55
434 ON THE TRIPLE ^-FUNCTIONS. [699
or substituting for A and B their values, and disregarding the constant factor ^(a — 6),
this is
AB= \{y—z)'Ja — y.h — y.a—z.h — z.c — x.d — x.e — x.f—x.g—x
■>r{z —x)*Ja — z.h—z.a — a.h — x.c — y.d — y.e—y.f—y.g — y
+ {x — y)^a — x.b — x.a — y.b — y.c—z.d — z.e— z.f—z.g— z\.
We have thus in all 21 equations, which exhibit the form of the Weierstrassian
functions al (m, v, w)jj, ..., al (m, v, w)^.
To complete the system, there should it is clear be 35 new functions al(«, v, w)ib,
..., al(«, V, w)„, represented by ABC, ..., EFG, viz. the whole number of functions would
then be
7 + [^+[^|(=7 + 21 + 35) = 63, =64-1,
since the functions represent ratios of the ^-functions.
n.
Starting now with the radical
'Ja — x.h — x.c — x.d — x.e — x .f— x.g — x.h — x
composed of eight linear factors, and writing, as in my " Memoir on the double
^-functions," t. Lxxxv. (1878), pp. 214 — 245, [665]; a, b, c, d, e, f, g, h to denote
these factors, and similarly a,, b,, Cj, dj, Cj, fi, gi, hj and aj, \, c^, da, e,, fj, gj, h, to
denote a — y, b-y, etc., and a — z, b — z, etc., so that X = abcdefgh, F=aibiCidieif,gihi,
Z = ajbjCjdjejfjgjhj ; then, instead of the Weierstrassian form, the diflferential equations
may be taken to be
"^''^ s/^ ^ ^/¥^ ^/Z'
dx dy dz
, xdx y dy zdz
dw =
a?dx y'dy z^ dz
^JX^ ^fY >JZ-
We then have 64 S^-functions and an w-function, viz. writing
d = y — z.z — x.x — y,
and then
'Ja = Vaa,a, (8 equations)
r-T- 1
•^/ahc =g{(y- z) VaibiCjajbjC^efgh -ir{z — x) VaabjCaabcdieif^ihi -^{x — y) VabcaibiCidjeafj^jhs
• : (56 equations)
699] ON THE TRIPLE ^-FUNCTIONS. 435
the equations, which define the ^-functions A, B,...,H, ABC,..., FOH, and the
o)-function fl, are
A=^^s/a (8 equations)
ABC = il's/abc (56 equations)
and one other relation which I have not as yet investigated.
As regards the algebraical relations between the 64 ^-functions, it is to be
remarked that, selecting in a proper manner 8 of the functions, the square of any
one of the other functions can be expressed as a linear function of the squares of
the 8 selected functions. To explain this somewhat further, observe that, taking any
5 squares such as {ABCf, we can with these 5 squares form a linear combination
which is rational in x, y, z. We have for instance, writing down the irrational part
only,
(^ BCy = ~ {abc (^ - a;) (a; - y) VT^ + a,b,Ci (a; - y) (y - ^r) Viz + a,b,Ca (y - 2) (0 - a;) Vl" 7} ,
and forming in all five such equations, then inasmuch as the coefficients abc, ... of
(z —x)(x— y) \YZ are each of them a cubic function containing terms in aP, x^, a?,
a?, we have a determinate set of constant factors such that the resulting term in
{z —x){x — y)'^YZ will be = 0 ; but the coefficients ajbiCj, ... of (x-y)(y- z) 'JZX only
differ from the first set of coefficients by containing y instead of x, and the same
set of constant factors will thus make the resulting term in {x—y){y — z)*JZX to
be = 0 ; and similarly the same set of constant factors will make the resulting term in
(y — z) (z — x) vXY to be =0; viz. we have thus a set of constant factors, such that
the whole irrational part will disappear. It seems to be in general true that the same
set of constant factors will make the rational part integral; viz. the rational part
is a function of the form ^ multiplied by a rational and integral function of x, y, z,
and if this rational and integral function divide by &^, then the final result will be a
rational and integral function, which, being symmetrical in x, y, z, is at once seen to be
a linear function of the symmetrical combinations 1, x + y + z, yz + zx + xy, xyz. Such
a function is obviously a linear function of any four squares A^, B', C\ B^; or the
form is, linear function of five squares (ABCy = linear function of four squares A',
that is, any one of the five squares is a linear function of 8 squares.
As an instance, consider the three squares (ABC)-, {ABD)\ (ABEf, which are
such that we have a linear combination which is rational : in fact, we have here in
each function the pair of factors ab, which unites itself with {z — x){x — y)'JXY,
viz. it is only the coefficient of &h{z — x)(x — y)*JXY which has to be made =0:
the required combination is obviously
(d - e) (,ABGf + (e - c) {ABDf + (c - d) {ABEf.
55—2
436 ON THE TRIPLE ^-FUNCTIONS. [699
Here the irrational part vanishes and the rational part is found to be
{(d — e) CjCade
+ (e-c)d,d^e ■
+ (c — d) e,ejdc
(d — e)cjcd,e,^
+ a-jbaabfigih, (z -xf- + (e — c) djdc,e, ,
+ {c -d) ejcdiC, j
)(d — e) cCidjCj^
+ (e -c)dd,CjejV].
+ (c — d) eeidjCaj
The three terms in { } are here = — {c — d){d-e){e — c) multiplied by (z — x){x — y),
ix—y)(y — z), {y — z){z — x) respectively; hence the term in [ ] divides by d and the
result is
(c — d){d- e) (e — c)
e
[ajbia-jb/gh (y-z)
or finally this is
multiplied by
+ a^bja b figibj {z — x)
+ abaibif2g2hj(a;-y)],
= -{c — d){d-e){e — c)
{{a? + ab + ¥)fgh - {a^b + ab') (fg +fh + gh) + a»6' (/+ g + h)]
+ (x + y + z){ -(a + b)/gh+ ah {fg + fh + gh) - a!¥ }
■\-{yz-\-zx + xy){ fgh- ab{f+g + h) +a^b + a¥ }
+ ccyz{-{fg+fh+gh)+ {a + b){f+g + h)-{a?-^ab-\-h') },
that is, we have (d-e){ABGf + {e-c){ABDf + {c-d){ABEf = ai. sum of four squares,
viz. we have here a linear relation between 7 squares.
I have not as yet investigated the forms of the relations between the products
of pairs of ^-functions.
Cambridge, 30 September, 1878.
700] 437
700.
ON THE TETRAHEDROID AS A PARTICULAR CASE OF THE
16-NODAL QUARTIC SURFACE.
[From the Journal fur die reine und angeivandte Mathematifc (Crelle), t. Lxxxvii. (1878),
pp. 161—164.]
In the paper "Sur un cas particulier de la surface du quatrifeme ordre avec seize
points singuliers," Crelle, t. Lxv. (1866), pp. 284 — 290, [356], I showed how the surface
called the Tetrahedroid could be identified as a special form of Kummer's 16-nodal
quartic surface ; but I was not then in possession of the simplified form of the
equation of the 16-nodal surface given in my paper "Note sur la surface du quatrieme
ordre dou^e de seize points singuliers et de seize plans singuliers," Crelle, t. LXXIII.
(1871), pp. 292, 293, [442] ; see also my paper, " A third memoir on Quartic surfaces,"
Proc. Lcmd. Math. Soc. t. ill. (1871), p. 250, [454, this Collection, t. vii., p. 281]. Using
the equation last referred to, I resume therefore the consideration of the question.
Taking the constants a, ^, y, a.', ff, y', a", fi", y", such that
a+y9 + 7=0, a' + /9' + 7'=0, a" + /3" + 7" = 0,
and writing also
M = aa"(^ -7 ) + y3'y8"(7 -a ) + 7'7"(a -^ )
= «"« 08' -7') + i8"/3 (7' -c^) + y"y («' -^)
^aa'{0'- y") + /3 /3' (7" - a") + 77' («" - /8")
i {(^ - 7) (/8' - 7) {^' - 7") + (7 - «) (7 - «') (y - «") + (a - /8) (a' - /8') («" - /3")),
(the equivalence of which different expressions for M is verified without difficulty) :
writing also X, Y, Z, TT as current coordinates, the equation of the 16-nodal surface is
■ W^{X^+Y'+Z^-2YZ-2ZX-2XY)
0 = ■ +2W [aa' a!' {Y^Z- YZ^) + /3/9'/3" (Z»Z - ZX') + 777" (X'Y- XY') + MXYZ]
+ iaa'a"YZ+^^^"ZX + yy^XYf,
438
ON THE TETRAHEDROID AS A PARTICULAB
[700
where, a, ^, y, a, ff, 7', a", ^8", 7" being connected aa above, the number of constants
is =6.
The equations of the 16 singular planes are
Z = 0, F=0.
a (7V'F- /3'y3"Z) - Tf = 0, /3 (aV^- 7'7"Z) - TF = 0,
d (7"7 F- /9")8 -?) - Tr= 0, ^ (a"aZ- y"y X)-W = 0,
a" iry' Y-^ff Z)-W= 0, ff' (aa' Z- 77' Z) - IT = 0,
Z = 0, Tr = o,
7 {^'^"X - a'a" F) - TT = 0, y37X + 7a F + a^Z = 0,
7' (y3"/3 X - a"o F) - F = 0, /3'7'Z + 7'a'F+ a'yS'Z = 0,
7" (/ays' X - aa' F) - >f = 0, ^'i'X + 7"a" F + al'^'Z = 0.
Writing x, y, z, w as current coordiDates, the equation of the Tetrahedroid is
v)Wpa* + w%y + Pm?h^z' +/yhh(j*
+ (Pp - my - nVtO (lyz^ +f'a^') + (- Pf^ + my - n^¥) (m^z^af + g^yhu')
+ (_ pp - my + n*) (jiWy" + h^zHi/') = 0,
where, inasmuch as /, g, h, I, m, n enter homogeneously, the number of constants
is = 5.
The equations of the 16 singular planes, written in an order corresponding to that
used for the 16-nodal surface, are
• ny—mz-\-fw= 0
fx —gy—hz * = 0
—mx—ly • +hw=0
nx * +lz +gw=0
— nx * +lz +gw=0
■nuc+ly * +hw=0
~fx -^gy—hz * =0
* —ny—mz+fw=0
mx—ly m +Aw=0
—7ix * —Iz +gw=0
* ny+mz+Jw=0
-fa -gy+hz * =0
—fx —gy—hz * =0
* ny—mz—fw=0
—nx * +lz —gw—0
mx—ly m — /iw = 0.
These equations can be made to agree each to each with those of the 16 singular
planes of the 16-nodal surface, provided that we have
m _n n _l
y~'$' e?~7'
8" ^7" /=-''^«''*"' g = -mPS'^'. h = -iviy'y",
where observe that the first three equations give a^'y = d'^y', which is the relation
between the constants when the 16-nodal surface reduces itself to a tetrahedroid in
the above manner. And if we then assume
X =ny — mz-¥fw, Y= -nx + lz+gw, Z — vix — ly + luv, W=—fx— gy — hz,
the 16 linear functions of X, F, Z, W will become mere constant multiples of the
coiTesponding 16 linear functions of x, y, z, w; the constants, by which the several
700] CASE OF THE 16-NODAL QUARTIC SURFACE. 439
functions of x, y, z, w have to be multiplied in order to reduce them each to the
corresponding linear function of X, Y, Z, W, being given by the table
1,
^ {la -m/9),
1,
1,
(la
n ^
- m^),
1,
|j.(^'7"-r7').
a a
^(m^-ny').
~^, irn^ - ni),
-?o.^'
- ny'),
|^(7"«-7«").
.r/ (^" _;„"),
t ("-y" - '"">•
^- <"^"
-la").
^(«/3' -«W.
For instance, we have
a(7V'F-/8'/3"Z)- W=2^(la-m^)(fx-gy-hz),
viz. substituting for Y, Z, W their values, the relation is
ma/S . y'y" {—no: * +lz + gw) \
- ma0 . /3'/3" ( mx-ly * +hw)\- = (la - ni/9) (fx -gy- hz).
m^{-fx -gy-hz * ))
As regards the terms in y, z, and w, the identity is at once verified. As regards
the term in x, we should have
mays (- ny'y" - m/3'/3") - {la - 2m0)f= 0,
viz. substituting for / its value, — laa'a" = — maa'/3", the equation divides by ma and
we then have
0 {- ny'y" - to/3'/3") + a'jS" {la - 2m0) = 0,
that is,
laa'ff"-m0^'{0' + 2a')-n0y'y" = O,
or writing herein m/S" = la", wy' — la', and /3' 4- 2a' = a' — y, the equation becomes
o'a/3"-a"/3(a'-7')-a'/37''=0, that is, a' (a/3" - a"/3) = aW - a'/Sy' ; or writing herein
a"y87' = a'^'7, the equation divided by a' becomes a^" — a";8 = 0y" — 0"y, which is true
in virtue of o + /3 + 7 = 0 and a" + yS" + 7" = 0. And in like manner the several other
identities may be verified.
The equation o.'0"y = d'^y might have been obtained as the condition of the
intersection, in a common point, of four of the singular planes of the 16-nodal
surface; and when this equation is satisfied, there are in fact four systems each of
four planes, such that the four planes of a system meet in a common point: viz. we
have
Planes
^■=0, /37Z+7ay+a/3Z=0, 7'(y3"^X-o"ar)-lf=0, 0" {aa'Z -yy'X)-W=0,
F = 0, 7 (y9'/3"Z - o'a" F) - TF = 0, ^y'X + 7'a' F + a'^Z = 0, a" (77' F - 00' Z) -W = Q,
Z =0, 0 {a'a"Z - y'i'X) - TT = 0, a! {i'y Y - 0"0Z) - TT = 0, 0"y"X + 7"a" F+ <^'0"Z = 0,
TT = 0, a (7'7" Y - 010!' Z) - Tf = 0, 0' {a"aZ - y^Z) - F = 0, 7" {00'X - aa'F) - TF = 0,
440 ON THE TETRAHEDROID. [700
meeting in points
0, -A 7, a^y'.a.
a'. 0, -7'. a'W-)8'.
-a", r, 0, a"/37'.7".
j8".aa'a", a" . ^^^'. 7" • «' W- 0.
the four points being in fact the vertices of the tetrahedron formed by the four planes
of the tetrahedroid. Observe that, if the singular planes of the 16-nodal surface in
their original order &re
1. 2, 3, 4,
5, 6, 7. 8,
9, 10, 11, 12,
13, 14, 15, 16,
then the planes forming the last-mentioned four systems of planes are
(1, 8, 11, 14),
(2. 7, 12, 13).
(3, 6. 9, 16).
(4, 5. 10, 15),
viz. they correspond each of them to a term which in the determinant formed with
the 16 symbols would have the sign +.
The equation a'^'y = a"/37' is evidently not unique. The triads (a, 0, 7), (a', ^, 7'),
(a", y9", 7") enter symmetrically into the equation of the 16-nodal surface; by taking
the singular planes of one of the surfaces in a different order, the equation would
present itself under one or other of the different forms
a'/3"7 = a'/Sy, a'W = a^'7". a^y" = a'l3"y,
a'/97" = a"/3'7, a"/S'7 = a/9'V. a0"y' = a'0y".
Cambridge, 9 December, 1878.
701]
441
701.
ALGORITHM FOR THE CHARACTERISTICS OF THE TRIPLE
^-FUNCTIONS.
[From the Journal fii/r die reine iind angewandte Mathematik (Crelle), t. Lxxxvii. (1878),
pp. 165—169.]
The characteristics of the triple ^-functions may be represented, the 28 odd
characteristics by the binary symbols or duads, 12, ..., 78, and the even ones
f other than , = 0 ) , say the 35 even characteristics, by the ternary symbols or
triads 123 567: which triads may be regarded as abbreviations for the double tetrads
1238.4567, ..., 5678.1234, the 8 being always attached to the expressed triad. The
correspondence of the symbols is given by the diagram :
npper line of characteristic
000
100
010
110 ! 001
101
Oil
111
o
(KK)
0
236
345
137
467
156
124
257
1
100
237
67
136
12
157
48
256
35
<e
010
245
127
23
68
134
357
15
47
110
126
13
78
146
356
26
46
234
1
001
667
146
125
247
45
17
38
26
1
101
147
68
246
34
16
123
27
367
oil
136
347
14
57
28
36
167
456
111
.S46
24
56
235
37
267
457
18
C. X.
56
442 ALGORITHM FOR THE CHARACTERISTICS OF THE TRIPLE ^FUNCTIONS. [701
Or, what is the same thing, it is
apper line of characterUtio
000 100 {
010 110
001
101
oil
111
12
100
18
110
14
Oil
16
010
16
101
17
001
18
111
23
010
24
111
25
110
26
001
27
101
28
Oil
34
101
35
100
36
oil
37
111
38
001
45
001
46
110
47
010
48
100
66
111
67
Oil
58
101
67
100
68
010
78
110
3.
701] ALGORITHM FOR THE CHARACTERISTICS OF THE TRIPLE ^-FUNCTIONS. 443
000
100
010
110
001
101
oil
lU
123
101
124
000
125
001
126
110
127
010
134
010
135
oil
136
100
137
000
145
110
146
001
147
101
156
000
157
100
167
i_
oil
234
110
235
111
236
000
237
100
245
010
246
101
247
001
256
100
257
000
267
111
345
000
346
847
111
oil
356
110
357
010
367
101
456
oil
457
111
467
000
667
001
g-
56—2
444 ALGORITHM FOR THE CHARACTERISTICS OF THE TRIPLE ^-FUNCTIONS. [701
by means of which the two-line-characteristic is at once found when the duad or
triad is given.
The new algorithm renders unnecessary the Table I. of Weber's memoir "Theorie
der Abel'schen Functionen vom Geschlecht 3" (Berlin, 1876). In fact, the system of
six pairs corresponding to an odd characteristic such as 12 is
13.23, 14.24, 15.25, 16.26, 17.27, 18.28,
and that corresponding to an even characteristic such as 123 (=1238.4567) is
12.38, 13.28, 18.23, 45.67, 46.57, 47.56:
80 that all the (28 + 35 =) 63 systems can be at once formed.
The odd characteristics correspond to the bitangents of a quartic curve, and as
regards these bitangents the notation is, in fact, the notation arising out of Hesse's
investigations and explained Salmon's Higher Plane Curves (2nd Ed. 1873), pp. 222 — 225.
It may be noticed that the geometrical symbols corresponding to the before-mentioned
two .systems are:
2 and
Hence, selecting out of the first system any two pairs, we have a symbol D : but
selecting out of the second system any two pairs, we have a symbol which is either
D or nil; so that in each case (Salmon, p. 224) the four bitangents are such that
the eight points of contact lie on a conic.
The 28 bitangents of the general quartic curve
ViTif 1 -I- Vajjfj -f- Va^j^s = 0,
represented by the equations given by Weber, I.e., pp. 100, 101, and taken in the order
in which they are there written down, have for their duad-characteristics
18, 28, 38, 23, 13, 12, 48, 14, 58, 15, 68, 16, 78, 17, 24, 34, 25, 35,
26, 36, 27, 37, 67, 57, 56, 45, 46, 47
respectively. Taking out of any one of the 63 systems three pairs of bitangents at
pleasure, these give rise to an equation of the curve of a form such as
701] ALGORITHM FOR THE CHARACTERISTICS OF THE TRIPLE ^-FUNCTIONS. 445
and the whole number of the forms of equation is thus = 1260. The triads of pairs
which enter into the same equation may be
triads such as 12 . 34, 13 . 42, 14 . 23
„ „ 12 . 34, 13 . 42, 56 . 78
2 5 7
No. = 70,
= 630,
„ = 560,
making the whole number = 1260, as already mentioned.
Cambridge, 7 December, 1878.
I
446 [702
702.
ON THE TRIPLE ^-FUNCTIONS.
[From the Journal fiir die reine und angewandte Mathematik (Crelle), t. Lxxxvii. (1878),
pp. 190—198.]
A QUARTIC curve has the deficiency 3, and depends therefore on the triple
^-functions: and these, as functions of 3 arguments, should be connected with functions
of 3 points on the curve ; but it is easy to understand that it is possible, and may
be convenient, to introduce a fourth point, and so regard them as fimctions of 4
points on the cui-ve : thus in the circle, the functions cos u, sin u may be regarded
as functions of one point cos u = x, sin u = y, or as functions of two points,
cos u = xx-^ + yy, , sin u = xy^ — x^y.
And accordingly in Weber's memoir "Theorie der Abel'schen Functionen vom Geschlecht
3," (1876), see p. 156, the triple ^-functions are regarded as functions of 4 points
on the curve: viz. it is in effect shown that (disregarding constant factors) each of
the 64 functions is proportional to a determinant, the four lines of which are
algebraical functions of the coordinates of the four points respectively: the form of
this determinant being different according as the characteristic of the ^-function is
odd or even, or say according as the ^-function is odd or even. But the geometrical
signification of these formulae requires to be developed.
A quartic curve may be touched in six points by a cubic curve : but (Hesse,
1855*) there are two kinds of such tangent cubics, according as the six points of
contact are on a conic, or are not on a conic ; say we have a conic hexad of points
on the quartic, and a cubic hexad of points on the quartic. In either case, three
points of the hexad may be assumed at pleasure ; we can then in 28 different
ways determine the remaining three points of the conic hexad, and in 36 different
• See the two memoirs "Ueber Determinanten und ilire Anwendung in der Geometrie" and " Ueber die
Doppeltangenteu der Curven vierter Ordnung," Crelle, t. xlix. (1855).
702] OK THE TRIPLE ^-FUNCTIONS. 447
ways the remaining three points of the cubic hexad : or what is the same thing,
there are 28 systems of cubics touching in a conic hexad, and 36 systems of cubics
touching in a cubic hexad. The condition in order that four points of the quartic
curve may belong to a hexad (conic or cubic) is given by an equation 0 = 0, where
fl is a determinant the four lines of which are algebraical functions of the coordinates
of the four points respectively: but the form of such determinant is different according
as the condition belongs to a conic hexad, or to a cubic hexad : we have thus 28
conic determinants and 36 cubic determinants, fl ; and the 64 ^-functions are pro-
portional to constant multiples of these determinants; viz. the odd functions correspond
to the conic determinants, and the even functions to the cubic determinants.
First, as to the conic hexads : the points of a conic hexad lie in a conic with
the two points of contact of some one of the bitangents of the quartic curve : so
that, given any three points of the hexad, these together with the two points of
contact of the bitangent determine a conic which meets the quartic in the remaining
three points of the hexad. Suppose that a, b, c, f, g, h are linear functions of the
coordinates such that the equation of the quartic curve is
'^af+ ^bg + ^fch = 0 ;
then a = 0, 6 = 0, c = 0, f=0, y = 0, h=0 are six of the bitangents of the curve,
and the bitangent a=0 touches the curve at the two points of intersection of this
line with the conic bg — ch = 0. The general equation of a conic through these two
points a = 0, bg — ch = 0, may be written
bg-ch + a (Ax + By + Cz) = 0,
where for x, y, z we may if we please substitute any three of the six linear functions
o. b, c, f, g, h, or any other linear functions of the coordinates (*•, y, z): and the
equation may also be written
a/± Q)g -ch) + a (Ax + By + Cz) = 0.
Adopting this latter form, and considering the intersections of the conic with the
qtiartic, that is, considering the relation
-^qf+'Jbg + \/ch = 0
as holding good, we have
af+ bg — ch = — 2 ^afbg,
of— bg + ch = — 2 'Jafch,
and we thus have at pleasure one or other of the two equations
- 2 sfajbg + a(Ax + By+Cz) = 0,
- 2 -/a/ch + a (Ax + By + Cz) = 0,
that is,
-2-^/bg + ^a(Ax + By + Cz) = 0,
-2V7cA +Va(Ax + By + Cz) = 0.
448
ON THE TRIPLE ^-FUNCTIONS.
[702
Hence the condition in order that the four points (a;,, y,, 2,), {x„ y^, ?,), {xt, y„ ^,),
(^4. y*y ^*), assumed to be points of the quartic, may belong to the conic hexad, may
be written
"JfAgu a^Vo,, y,\^, ZiVoi
^/J>i9l, !Ci'J(h, yt'^ch, •^jVoj
V/Ajf,, a^Vaj, y,Va„ ^•.Va,
V/Afl'i, «Wa4. 3/4^^04, 04Va4
= 0, or
^/jCjAi, a!,Va,, y,Va,, ^jVa,
V/^A. ^s^^Os, ys^> ^W"i
V/,cA„ a!,Va,. jr,Va„ ^jVo,
^f*cA, a;4^^. y4^^4, -24 ^a4
= 0.
where, as before, the a;, y, z may be replaced by any three of the letters a, b, c,
f, g, h, or by any other linear functions of («, y, z): and, moreover, although in
obtaining the condition we have used for the quartic the equation
^af-\-'Jbg + '^ch = 0,
depending upon six bitangents, yet from the process itself it is clear that the condition
can only depend upon the paiticuiar bitangent a = 0 : calling the condition £1—0, all
the forms of condition which belong to the same bitangent a = 0, will be essentially
identical, that is, the several determinants fl will differ only by constant factors ; or
disregarding these constant factors, we have for the bitangent a = 0, a single determinant
n, which may be taken to be any one of the determinants in question. And we
have thus 28 determinants fi, corresponding to the 28 bitangents respectively.
Coming now to the cubic hexads, Hesse showed that the equation of a quartic
curve could be (and that in 36 diflferent ways) expressed in the form, symmetrical
determinant = 0, or say
a, h, g, I =0,
h, b, f, m
9, f, c, n
I, m, n, d
where (a, b, c, d, f, g, h, I, m, ?i) are linear functions of the coordinates; and from
each of these forms he obtains the equation of a cubic
a, h, g, I, a = 0,
h, b, f, in, /3
9, /, c, n, y
I, m, n, d, B
a, 0, y, S
containing the four constants a, /3, y, S, or say the 3 ratios of these constants,
touching the quartic in a cubic hexad of points : that the cubic does touch the
quartic in six points appears, in fact, from Hesse's identity
702]
a, h, g, I, a
h, h, f, m, ^
g, f, c, n, 7
I, m, n, d, S
a, A 7> S
ON THE TRIPLE ^-FUNCTIONS.
a, h, g, I, a'
h, b, f, m, ^
g, f, 0, n, i
I, m, n, d, 8'
a', ff, 7. 5'
a, h, g, I, a
h, b, f, 7)1, /S
g, f, c, n, y
I, m, n, d, B
a', 13', y, S'
a,
h,
g.
I
til
h,
b.
f.
m
g'
/.
c,
n
I,
m,
n,
d
where Z7 is an easily calculated function of the second order in a, b, c, d, f, g, h,
I, m, n, and also of the second order in the determinants ayS' — a'/9, etc.
We can obtain such a form of the equation of the quartic, from the before-
mentioned equation
viz. this equation gives
•, h, g, a =0,
h, *, /, b
g, /, *, c
a, b, c, «
which is of the required form, symmetrical determinant = 0 ; the equation is, in fact,
alf" + by + c* - 2bcgh - 2cahf- 2abfg = 0,
which is the rationalised form of
Va/+ V6^ + Vc^ = 0,
and we hence have the cubic
*, It, g, a, a
h, », /, b, fi
g, f, *, 0, y
a, b, c, », S
«, ^. 7. ^. *
= 0,
the developed form of which is
a:'bcf+ ff'cag + y'abh + S'fgh
- (a^y + faS) (- af+ bg + ch)
- (670 + £f/8S) ( af- bg + ch)
- (cay9 + hr/i) ( af+bg- ch) = 0.
Considering the intersections with the quai-tic
Vo^+ Vbg + 'Jch = 0,
we have
-af+bg + ch, af -bg + ch, af+bg — ch = — 2 '^bcgh, — 2 'Jcahf, — 2 'Jabfg,
and the equation thus becomes
{a'Jb^+^'^cag + y'/aM, + S'^/ghy = 0-,
c. X. 57
450 ON THE TRIPLE ^-FUNCTIONS. [702
viz. for the points of the cubic hexad we have
a Vfcc/ + /8 Vca^ + 7 VoM + 8 V/gA = 0,
and hence the condition in order that the four points {xi, y^, z^j, (asj, yj, ^i), («», y%, 2^3).
("'ii y*. ^i) naay belong to the cubic hexad is
V6,Ci/i, VciO.flf,, VoiJiA,, V/,5rA
V6a/!, Vc/Is<7j, Vojtj/ij, V/,5r^
= 0,
VijCs/,, VcA5'». "^oJhK, '■^fsgA
^bfijt, 'Jctaigt, Vutbthi, "^AgA
viz. we have thus the form of the determinant H which belongs to a cubic hexad.
It is to be observed that the equation
\/af+s/bg + 'Jch = 0
remains unaltered by any of the interchanges a and f, h and g, c and h; but we
thus obtain only two cubic hexads; those answering to the equations
o \'bcf + /8 Vc^ + 7 Va6A + 8 '^fgh = 0,
and
a 's/agh + yS Vbhf + 7 Vc^ + S \/abc = 0,
which give distinct hexads. The whole number of ways in which the equation of the
quartic can be expressed in a form such as
^af+ "Jbg + VcA = 0,
attending only to the pairs of bitangents, and disregarding the interchanges of the
two bitangents of a pair, is = 1260, and hence the number of forms for the determ-
inant fi of a cubic hexad is the double of this, = 2520, which is = 36 x 70 : but
the number of distinct hexads is = 36, and thus there must be for each hexad,
70 equivalent forms.
To explain this, observe that every even characteristic except . .. , and every odd
characteristic, can be (and that in 6 ways) expressed as a sum of two different odd
characteristics ; we have thus (see Weber's Table I.) a system of (35 + 28 =) 63
hexpairs; and selecting at pleasure any three pairs out of the same hexpair, we have
a system of (63x20=) 1260 tripairs; giving the 1260 representations of the quartic
in a form such as
Vo/"-!- V6^ + VcA = 0.
Each even characteristic (not excluding j can be in 56 different ways (Weber,
p. 23) expressed as a sum of three different odd characteristics, and these are such
that no two of them belong to the same pair, in any tripair; or we may say that
each even characteristic gives rise to 56 hemi-tripairs. But a hemi-tripair can be in
5 different ways completed into a tripair; and we have thus, belonging to the same
702]
ON THE TRIPLE ^-FUNCTIONS.
451
even characteristic (56 x 5 =) 280 tripairs, which are however 70 tripairs each taken
4 times. A tripair contains in all (2' =) 8 hemi-tripairs, but these divide themselves
into two sets each of 4 hemi-tripairs such that for each hemi-tripair of the first set
the three characteristics have a given sum, and for each hemi-tripair of the second
set the three characteristics have a dififerent given sum. Hence considering the 70
tripairs corresponding as above to a given even characteristic, in any one of the 70
tripairs, there is a set of 4 hemi-tripairs such that in each of them the sum of the
three characteristics is equal to the given even characteristic ; and taking the bitangents
/, g, h to correspond to any one of these hemi-tripairs, the bitangents which corre-
spond to the other three hemi-tripairs will be b, c, f; c, a, g and a, h, h respectively ;
and we thus obtain from any one of these one and the same representation
a V6c7+ /8 "Jmg + 7 •Jabh + B "^fgh = 0
of the cubic hexad. And the 70 tripairs give thus the 70 representations of the
same cubic hexad.
The whole number of hemi-tripairs is 36x56=2016: it may be remarked that
there exists a system of 288 heptads, each of 7 odd characteristics such that selecting
at pleasure any 3 characteristics out of the heptad, we obtain always a hemi-tripair :
we have thus in all 288 x 35 =< 10080 hemi-tripairs: this is =2016x5, or we have
the 2016 hemi-tripairs each taken 5 times. Weber's Table II. exhibits 36 out of the
288 heptads.
I recall that in the algorithm derived from Hesse's theory the bitangents are
represented by the duads 12, 13, ..., 78 formed with the eight figures 1, 2, 3, 4, 5,
6, 7, 8; these duads correspond to the odd characteristics as shown in the Table,
and the table shows also triads corresponding to all the even characteristics except
000
000"
Top line of characteristic.
•C
000
100
010
no
001
101
on
111
000
236
345
137
467
156
124
257
100
237
67
136
12
157
48
256
35
010
245
127
23
68
134
357
15
47
no
126
13
78
145
356
25
46
234
001
567
146
125
247
45
17
38
26
101
147
58
246
34
16
123
27
367
on
135
347
14
57
17
36
167
456
111
346
24
56
235
37
267
457
18
57—2
452
ON THE TRIPLE ^-FUNCTIONS.
[702
See my "Algorithm of the triple ^-functions," Crelle, t. Lxxxvii. p. 165, [701],
The (35 + 28 =) 63 hexpaire then are
35 hexpairs such as
5 _6
, say this is 1234.5678 or for
and
shortness 567 (the 8 going always with the expressed triad) : that is, 567
denotes the hexpair
12.34; 13.24; 14.23; 56.78; 57.68; 58.67:
28 hexpairs such as i
2, say this is 12; that is, 12 denotes the
hexpair
13.32; 14.42; 15.52; 16.62; 17.72; 18.82.
It is to be noticed that the odd characteristics, as represented by their duad
symbols, can be added by the formulae
or, what is the same thing,
and
12+23 = 13, etc..
12 + 13 + 23 = 0, =J^^, etc.,
1 2 + 34 = 13 + 24 = 14 + 23 = 56 + 78 = 57 + 68 = 58 + 67 = 567, etc.
Thus, referring to the table,
and
which are right.
The 288 heptads are
8 heptads such as
i9_loq io 110^010 100
12 + 23= 13 means ioo + 010 = lIO'
TO O^ rc^ 110 110 000
12 + 34 = 567 means ioo + 101=001'
, say this is the heptad 1, denoting
2 3 4 5 6 7 8
the seven duads 12, 13, 14, 15, 16, 17, 18:
702]
and
ON THE TRIPLE ^-FUNCTIONS.
1
453
280 heptads such as // \\ / \ , say this is the heptad 1.678,
2 3 4 5 7 8
denoting the seven duads 12, 13, 14, 15, 67, 68, 78.
We hence see that the 2016 hemi-tripairs are :
V
280 hemi-tripairs
12, 13, U:
1680 hemi-tripairs
(I.), say this is 1 . 234, denoting the three duads
1 6
(II.), say this is 12 (6 . 78), denoting the three duads
2 7 8
12, 67, 68: i
1
56 hemi-tripairs
13, 23;
(III.), say this is 123, denoting the three duads 12,
2016.
We further see how each hemi-tripair may be completed into a tripair in 5
1 I
different ways: thus (I.) gives the 5 tripairs 2^.
-.-'4 2<. 3
•4 ; (III) gives the
5 tripairs
2^
; while (II.) gives the 3 tripairs
5,6,7or8
34
35
or (7 )8 and the 2 tripairs
4a
4,5,6,7or8
1 6 2 6
/.....7\ -v.-/
"8 8
r.,4or3
454 ON THE TRIPLE ^FUNCTIONS. [702
To each even characteristic there belongs a system of 56 hemi-tripairs ; thus for
000
the characteristic -^^ , the 56 hemi-tripairs are 123, that is, 12, 13, 23, etc. : whence
the 70 tripairs are 1234, that is, 12.34; 13.24; 14.23, etc.; and in any such
tripair, say in 1234, we have the set of four hemi-tripairs 123, 124, 134, 234, for
each of which the sum of the three characteristics is
(l2+23-H3 = JJJ,etc.):
000 /,„ .„„.,„ 000
000
and the other set 1.234, 2.134, 3.124, 4.123, for each of which the sum of the
three characteristics is
= 567(12-1-13 + 14, =23 + 14, =567, = qqJ) •
To find the hemi-tripairs that belong to any other even characteristic ; for instance,
^.^ , corresponds to 567: we have 4 such as 1.234; 24 such as (5.12)34; 4 such as
5.678; and 24 such as (1.56)78; in all 4+24 + 4 + 24, =56. The tripairs are the
2, 1234, 5678; 16 such as 54(123); 16 such as 15(678); 36 such as (5162)34.78;
in all 2 + 16 + 16 + 36, =70; and in each of these it is easy to select the hemi-
tripairs for which the sum of the 3 duads is = 567.
Cambridge. 27 December, 1878.
703]
455
703.
ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS.
[From the Journal fur die reine und angewandte Mathematik (Crelle), t. Lxxxviii. (1879),
pp. 74—81.]
I ASSUME in general
%==a-&.b-e.c-e.d-e.e-d.f-e,
and I consider the variables x, y, z, w, p, q, connected by the equations
1, 1, 1, 1, 1, 1 =0,
X, y, z, w, p, q
of, y\ z\ vf, ]^, q^
a?, y', z*, vfi, p', (f
vr, \^, vz, v¥, vp, vq
equivalent to two independent equations, which in feet serve to determine p, q, or say
the symmetrical functions p + q and pq, in terms of x, y, z, w.
These equations, it is well known, constitute a particular integral of the differential
equations
dx dy dz dw dp dq
VZ vT ^Z Vf
vp^vt ^'
xdx ydy zdz wdw pdp qdq _
or what is the same thing, regarding ^, 5' as arbitrary constants, they constitute the
general integral of the differential equations
dx dy dz dw _
xdx ydy zdz wdw _
456 ON THE ADDITION OF THE DOUBLE ^FUNCTIONS. [703
I attach the numbers 1, 2, 3, 4, 5, 6 to the variables x, y, z, w, p, q, respectively:
and write
Aj, —"Ja — x.a — y, An = 'Ja — z.a — w, A„ — 'Ja—p.a — q;
(six equations),
ABa = (Vo — x.b — x ./— X . c —y.d—y.e—y — 'Ja — y.h — y ./— y .c — x.d — x.e — x]; etc.
X — y
(ten equations),
where it is to be borne in mind that AB is an abbreviation for ABF.CDE, and
so in other cases, the letter F belonging always to the expressed duad: there are
thus in all the sixteen functions A, B, C, D, E, F, AB, AC, AD, AE, BG, BD,
BE, CD, CE, DE, these being functions of x and y, of z and w, and of p and q,
according as the suffix is 12, 34, or 56.
It is to be shown that the 16 functions Ai^, AB^ of p and q can be by means
of the given equations expressed as proportional to rational and integral functions of
the 16 functions -d,,, AB^, A^, AB^ of x and y, and of z and w respectively: and
it is clear that in so expressing them we have in effect the solution of the problem
of the addition of the double S^-functions.
I use when convenient the abbreviated notations
a — a; = ai, a — ^ = 8,, etc.,
6 — a; = b,, etc.,
6it = x-y, e3i=z-io, 6K=p-q;
we have of course
X = aibiCjd,eifi,
-4-^12 = z~ KaibifiCjdjej — VaJMAdlej}, etc.
17,2
Proceeding to the investigation, the equations between the variables are obviously
those obtained by the elimination of the arbitrary multipliers a, /8, 7, 8, e from the
six equations obtained from
by writing therein for 0 the values x, y, z, w, p, q successively; we may consider
the four equations
aa^+^x'+yx+B = e y/X,
af +0y^ +yy +S = e '/T,
az" +^z' +yz +B = e VZ,
aiu' + ^w" + yw + S = e •sTW,
703]
ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS.
457
as serving to determine the ratios a : 0 : y : S : e in terms of x, y, z, w; and we
have then for the determination of p, q the remaining two equations
op" + ySp- + 7p + 8 = e VP,
which two equations may be replaced by the identity
{a0' + $0' + ye+Sf-e^ = a''-e\0-x.e-y.0-z.e-w.0-p.0-q.
Writing herein 0 = any one of the values a, b, c, d, e, f, for instance 0 = a, and
taking the square root of each side, we have
oa' + fia^ + ya + 8 = Va' — ^'Ja — x.a — y'^a — z.a — w»Ja — p.a — q,
or as this may be written
aa» + ^a» + 7a + S = Va'"^^ ^18 . -4 S4 . 4„ ,
which equation when reduced gives the proportional value of A^.
For the reduction we require the value of oa' + y3a^ + ya + h: calling this for the
moment H, we join to the four equations a fifth equation
aa» + )8a' + 7a + 8 = n.
Eliminating a, /3, 7, 8, we find
a^, a?, X, 1, e^X
f, y-, y, 1, e-JY
!?, ^^ z, 1, eVZ
ttV", w=, w, 1, eVW
a', a', a, 1, X2
= 0,
or, what is the same thing.
11
X, 1
y*. f, y, 1
^. z\ z. 1
W*. W-, IV, 1
+ e
^^. 2/", y\ y, 1
V^, r", Z-, z, 1
VW, m;», vfl, w, 1
a', a^ a, 1
= 0;
viz. this is
H.x—y.x— z.x— w.y—z.y—w.z—w = — e {VX .y — z.y—w.y — a.z—w.z — a .w — a
+ VF .z—w.z—a.z—x.w — a.w — x.a — x
+ VZ .w—a.w — x.w — y.a —x .a — y.x —y
+ ^W .a — x.a —y .a—z.x—y.x — z .y —z\,
c. X. 58
458 ON THE ADDITION OP THE DOUBLE ^-FUNCTIONS. [703
or as it may be written
fl.x — z .x — w.y — z.y — w^-- '- —\y — z.y — w.a — y. \/X — x — z.x — w.a — x.'JY]
+ — ^ '-- — -[w — x.w — y.a — w.'^Z—z — x.z — y.a — z.^fW],
an equation for the determination of H.
Consider first the expression which multiplies e.a — z.a—w; this is
= 3-{y — «.y— w.a, ^X — x — z.x — w.a,i */¥} ;
we have
BEii = 2- {Vb,e,fia,Cjd. - Vbjejfja,c,dj},
and multiplying this by
^,5 . C,j . Da, = Va,Cid,ajCsd„
we derive
BEn . C„ . Du . ^„ = ^ (cjdA V^ - Cidia, VF},
and similarly two other equations; the system may be written
BE.G.D.A = ^ [c4^ VZ - Cid,a, V Y],
GE.D.B.A= „ {dA„ „ -d,b,„ „ },
DE.B.C.A= „ {bjCj,, „ -bic, „ „ },
the suflBxes on the left-hand side being always 12. The letters b, c, d which enter
cyclically into these equations are any three of the five letters other than a; the
remaining two letters e and / enter symmetrically, for BE is a mere abbreviation for
the double triad BEF.ACD; and the like for GE, and DE.
Multiplying these equations by
b — z.h — w c — z.c — w d — z .d — w
b — c.b-d' c — d.c-b' d — b.d — c'
respectively, and then adding, the right-hand side becomes
= 3- {y —z .y —w .Si, 'JX —x — z.x — w.s^ vT}.
Writing
b-z.b-w -1
b-c.b-d c-d.d-b.b-c" *•■"«'«»«•'
703] ON THE ADDITION OF THE DOUBLE ^FUNCTIONS. 459
the left-hand side becomes
= — r/''l ^— {c-d.B^^ . BE,, .C^.D,, + d-b. (7„» . CE^ .D,,.B,,+ b-c. i)„> . DE,, . B^ . (?„},
c—d.d-b.h—c^
which for shortness may be written
— A
~ — j~^j — ir^ — 1[c — d. B^ . BEa . Cu . Aj}.
c — a.a — o.o — c
the summation referring to the three terms obtained by the cyclical interchange of
the letters b, c, d. The result thus is
3- {y — z.y — w. a^VX —x — z.x — w. ai VF}
— A
= j-j — r— T li {c — d. Btt' . BEa . C12 . Aj)-
c — a.a — o.o — c
Interchanging x, y with z, w respectively, we have of course to interchange the suffixes
1, 2 and 3, 4 ; we thus find
^{w-x.w-y.&t '/Z — z — x.z — y .&t VTT}
= j^~j — T^T S {c — d . Bii" . BE,^ . Csi . Dm},
c—a.a—o.o — c
and we hence find the value of il.x— z.x — w.y — z.y — w. But fl, = aa' + /3a' + ya+B,
is =^0.^- e* . Ai.2.Aii.Ax: the resulting equation divides by A^^.A^t- throwing out this
factor, we have
Va» — e*
{x — z.x — w.y — z.y — w){c — d.d — b.b — c) A^
= 4„2 {c - rf . £«' . BE» . C„ . D„] +^„ t{c-d. B,,' . BE^ . C„ . D^],
where, as before, the summations refer to the three terms obtained by the cyclical
interchange of the letters b, c, d; these being any three of the five letters other
than a; and the remaining two letters e, / enter into the formula symmetrically.
The formula gives thus for A„ ten values which are of course equal to each other.
Writing for a each letter in succession, we obtain formulaj for each of the six
single-letter functions .4j« of p and q ; and the factor
Vo'- e*
{x — z.x — w.y — z.y — w)
is the same in all the formulae.
We require further the expressions for the double-letter functions of p, q. Con-
sidering for example the function BEa, which is
= JT {^^d5e5f5a,b,C8 - Vdee,f,^5b,c,).
58—2
460 ON THE ADDITION OP THE DOUBLE ^-FUNCTIONS. [703
then multiplying by
we have
DE„ .A„.B„.C„ = ^ {a,b^s '^ - a^b^c, VQ},
tfft
= {a-q.b-q.c — q. s/P — a—p.b-p. c-p . VQ},
or recollecting that e Vp, e VQ are = op* + /Sp' + yp + B and o^ + /Sj' + yq + S respectively,
this is
e . DEk .Ax.Bx.Cn
= — - {a-q.b-q.c-q. (op* + ^p' + yp + B)- a-p .b- p .c - p .(a^ + ^^ + yq + S)].
Using the well-known identity
op' + /8p» + 7P + S = aa" + /3a^ + 7a + S . 5 :iP<'-P-^-P
0 (t . C '~~ CL * Ct — Of
c-b.d — b.a — b
a — c. a — c.b — c
a — d.b — d.c-d
and the like expression for ac^ ■\- ^q^ + yq + B, there will be on the right-hand side
terms involving
aa' + /3a» -f- 7a + 8, at' + /36= -t- 76 + S, ac^ -H/Sc^ + 7c -l- S :
but the term in ad? 4- ^d" + 7^-1-8 will disappear of itself.
The term in oa' -f /3a' + 7a + S is
1 oo' -H iSa' -I- 7a -H S ,
where the expression in () is =d — a.p — q: hence the term is
b-a.c-a b'q.o-q.b-p.c-p.
which is
_aa'-h^a'-t-7aH-8
6-a.c-a ^'^ •^"•
Forming the two other like terms, the equation is
o — a.c — a
^+^bf + yb + B ,
^aC-h/3c'+7C + a
a-c.b-c
703]
ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS.
461
But the expressions
are
= ^f^Zr^A,,.A^.A,„ Va»-6*5,,.5«.5„, ^ce-eG„.C^.C„,
respectively : the whole equation thus divides by J^^ . fij, . C„ ; throwing out this factor,
Va' — 6"
and then multiplying each side by — , we find
Va'-e"
DE„
'"^b-c.c-a.a-bi T") ^ b-c.A,,.A^.B^.C„
in which formula if we imagine
\/a' - e' „ Vtt« - e" ^
e € e
each replaced by its value in terms of the an/- and zw-functions, we have an equation
of the form ^
{x — z.x — w.y — z.y — w) DE^ =
M,
X — z.x — w.y— z.y — w
where Jlf is a given rational and integral function of the 16 and 16 functions
^,2, ABa and A^,, AB-u of x and y and of z and w respectively. The factor
{x— z.x — w.y— z.y — tu)
is retained on the left-hand side as being the same factor which enters into the
equations for A„, etc.: but on the right-hand side x — z.x — w.y — z.y — w should be
expressed in terms of the xy- and ^^w-functions. This can be done by means of the
identity
\, x+ y, xy \\\, x+ y, xy
X — z.x — w.y — z.y — w = z
1, z -^w, zw
^\ 1, a+ b, ah
7« = Z. :
1, Z +W, ZW
1, a+ c, ac
a — b.a — c
where the summation refers to the three tenns obtained by the cyclical interchange
of the letters a, b, c. The first determinant, multiplied by a — b, is in fact
a-z.a-w, a — x.a — y
b — z .b —w, b — X .b — y
and the second determinant, multiplied by a — c, is
= 1 a-z.a-w, a — x.a — y
I c — z .c —w, c — X .c — y
462 ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS. [703
SO that the formula may also be written
a — z.a — w, a — x.a — y
h —z.b —w, b —x.h —y
a — z.a — w, a — x.a — y
c — z .c —w, c —x.c — y
x-z.x-w.y-z.y-w^l (^-6)»(a-c>'
or, what is the same thing, it is
<c-z.x-w.y-z.y-w = ^ ______ ,
which is the required expression for x-z.x — w.y — z.y — tu\ the lettere a, h, c, which
enter into the formula, are anj' three of the six letters.
As regards the verification of the identity, observe that it may be written
^\L + M(a + b) + Ifab]{L+M(a + c) + Nac}
x-z.x-w.y-z.y-w^^^- a-b.a-c '
where L, M, N are
= {x-\-y)zw — {z + w) xy, xy — zw, and z-\-tu —x — y:
this is readily reduced to
x-z.x — tv.y — z.y — w = M^ — NL,
which can be at once verified.
Cambridge, I2th March, 1879.
I take the opportunity of remarking that, in the double-letter formulae, the sign
of the second term is, not as I have in general written it — , but is +,
AB — {Vabfcidiei + Vaibificdel, etc.
X — y
In fact, introducing a factor to which is a function of x and y, the odd and even
^-functions are =ft)Vaai, etc., and
{VabfCidiC, + Vajbificdej, etc.,
respectively; w is a function which on the interchange of x, y changes only its sign;
and this being so, then when x and y are interchanged, each single-letter function
changes its sign, and each double-letter function remains unaltered.
Cambridge, 29t/t July, 1879.
704] 463
704.
A MEMOIR ON THE SINGLE AND DOUBLE THETA-
FUNCTIONS.
[From the Philosophical Transactions of the Royal Society of London, vol. 171, Part III.,
(1880), pp. 897—1002. Received November 14,— Read November 28, 1879.]
The Theta-Functions, although arising historically from the Elliptic Functions,
may be considered as in order of simplicity preceding these, and connecting themselves
directly with the exponential function (e* or) exp. x\ viz. they may be defined each
of them as a sum of a series of exponentials, singly infinite in the case of the
single functions, doubly infinite in the case of the double functions ; and so on. The
number of the single functions is = 4 ; and the quotients of these, or say three of
them each divided by the fourth, are the elliptic functions sn, en, dn ; the number
of the double functions is (4''=) 16; and the quotients of these, or say fifteen of
them each divided by the sixteenth, are the hyper-elliptic functions of two arguments
depending on the square root of a sextic function. Generally, the number of the
j:>-tuple theta-functions is = 4'' ; and the quotients of these, or say all but one of
them each divided by the remaining function, are the Abelian functions of p arguments
depending on the irrational function y defined by the equation F{x, y) = 0 of a curve
of deficiency p. If, instead of connecting the ratios of the functions with a plane
curve, we consider the functions themselves as coordinates of a point in a space of
(4p— 1) dimensions, then we have the .single functions as the four coordinates of a
point on a quadri-quadric curve (one-fold locus) in ordinary space; and the double
functions as the sixteen coordinates of a point on a quadri-quadric two-fold locus in
15-dimen.sional space, the deficiency of this two-fold locus being of course = 2.
The investigations contained in the First Part of the present Memoir, although
for simplicity of notation exhibited only in regard to the double functions are, in
fact, applicable to the general case of the /j-tuple functions; but in the main the
464 A MEMOIR ON THE SINGLE AND DOUBLE THETA- FUNCTIONS. [704
Memoir relates only to the single and double functions, and the title has been given
to it accordingly. The investigations just referred to extend to the single functions ;
and there is, it seems to me, an advantage in carrying on the two theories simul-
taneously up to and inclusive of the establishment of what I call the Product-
theorem : this is a natural point of separation for the theories of the single and the
double functions respectively. The ulterior developments of the two theories are indeed
closely analogous to each other; but on the one hand the course of the single theory
would be only with difficulty perceptible in the greater complexity of the double
theory ; and on the other hand we need the single theory as a guide for the course
of the double theory.
I accordingly stop to point out in a general manner the course of the single
theory, and, in connexion with it but more briefly, that of the double theory; and
I then, in the Second and the Third Parts respectively, consider in detail the two
theories separately; first, that of the single functions, and then that of the double
functions. The paragraphs of the Memoir are numbered consecutively.
The definition adopted for the theta-functions differs somewhat from that which
is ordinarily used.
The earlier memoirs on the double theta-functions are the well-known ones : —
Rosenhain, " M^moire sur les fonctions de deux variables et a quatre periodes, qui
sont les inverses des int^grales ultra-elliptiques de la premiere classe." [1846.] Paris:
M6m. Savans Strang., t. xi. (1851), pp. 361 — 468.
Gopel, "Theoriae transcendentium Abelianarum primi oi-dinis adumbratio levis,"
Crelle, t. xxxv. (1847), pp. 277—312.
My first paper — Cayley, "On the Double ^-Functions in connexion with a 16-nodal
Surface," Crelle-Bmxhardt, t. LXXXiii. (1877), pp. 210—219, [662]— was founded directly
upon these, and was immediately followed by Dr Borchardt's paper,
Borchardt, "Ueber die Darstellung der Kummersehe Flache vierter Ordnung mit
sechzehn Knotenpunkten durch die Gopelsche biquadratische Relation zwischen vier
Thetafunctionen mit zwei Variabeln," Ditto, pp. 234 — 244.
My other later papers, [663, 664, 665, 697, 703], are contained in the same Journal.
FIRST PART.— INTRODUCTORY.
Definition of the theta-functions.
1. The p-tuple functions depend upon i^p (p — 1) parameters which are the co-
efficients of a quadric function of p ultimately disappearing integers, upon p arguments,
and upon 2p charactei-s, each =0 or 1, which form the characteristic of the 4^ functions;
but it will be sufficient to write down the formulae in the case p = 2.
As already mentioned, the adopted definition differs somewhat from that which
is ordinarily used. I use, as will be seen, a quadric function J (a, /(, &$?», n)" with
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 465
even integer values of m, n, instead of (a, /(, b'^m, nf with even or odd values; and
I write the other term ^vi {mu + nv), instead of mu + nv ; this comes to affecting the
arguments u, v with a factor -rri, so that the quarter-periods (instead of being iri)
are made to be = 1.
2. We write
and in like manner
( ' j=:J(«, h, b'^m, n)- + ^iri (mu + nv),
Z+'^'l+f)^^^"' '*' ^^'" + «■'* + ^^" + i'"" K'« + a) (« + 7) + (n + ^){v + 8)},
and prefixing to either of these the functional symbol exp. we have the exponential
of the function in question, that is, e with the function as an exponent.
We then write, aa the definition of the double theta-functions,
.(;f)(..,=s»p. (■;:-; ::f
where the summation extends to all positive and negative even integer values (zero
included) of m and n respectively; a, /8, y, B might denote any quantities whatever,
but for the theta-functions they are regarded as denoting positive or negative integers ;
this being so, it will appear that the only effect of altering each or any of them by
an even integer is to reverse (it may be) the sign of the function ; and the distinct
functions are consequently the (4^=) 16 functions obtained by giving to each of the
quantities a, /9, y, B the two values 0 and 1 successively.
3. We thus have the double theta-functions, depending on the parameters (a, h, b)
which determine the qnadric function (a, /(, b^ni, nf of the disappearing even integers
(m, n), and on the two arguments {u, v): in the symbol I ' ^j , which is called
the characteristic, the characters a, /3, y, B are each of them =0 or 1 ; and we thus
have the 16 functions.
The parameters («, h, b) may be real or imaginary, but they must be such that
reducing each of them to its real part the resulting function ( * ][«t, nf is invariable
in its sign, and negative for all real values of vi and n : this is, in fact, the condition
for the convergency of the series which give the values of the theta-functions.
\y,
ay + /3S is even or odd.
' 5^ j is said to be even or odd according as the sum
Allied functions.
5. As already remarked, the definition of
'<.,e
C: ?]*"'•'
)
c. X. 59
466 A MEMOIR ON THE SINGLE AND DODBLE THETA-FUNCTI0N8. [704
is not restricted to the case where the a, y9, 7, S represent integers, and there is
actually occasion to consider functions of this form where they are not integers : in
particular, a, y3 may be either or each of them of the form, integer + ^. But the
functions thus obtained are not regarded as theta-ftmctuyiis, and the expression theta-
function vdll consequently not extend to include them.
Properties of the Theta-Functions : Various sub-headings.
Even-integer altei'ation of characters.
6. If X, y be integers, then m, n having the several even integer values from
— 00 to + C30 respectively, it is obvious that r/t + a + 2x, n + /3 + 2y will have the same
series of values with m + a, n + ^ respectively ; and it thence follows that
^la+2x,^ + 2y\,^^ .^_a/«. ^^
(--;f-^^)(..,..(;f)<«,*
M:+2.:f+j<»'')
V7
Similarly if z, w are integers, then in the function
Kit- "
the argument of the exponential fimction contains the term
i^iri {ni + a . It + 7 + 22 + n + /3 . y + S + 2w} ;
this differs from its original value by
\iti (m + a .2z + n + ^ . 2w),
= iri (mz + nw) + iri {az + /8w),
and then, m and n being even integers, mz ■+ mo is also an even integer, and the
term iri (mz + nw) does not affect the value of the exponential : we thus introduce
into each term of the series the factor exp. iri (az + fiw), which is, in fact, = (— )«+^«' ;
and we consequently have
K7+2.;«^2j<"'^>=(->"'^'""^(;"f)(«'^)'
or, uniting the two results,
K::":f:rj(-)-(-)"-K:;f)<-*
This sustains the before-mentioned conclusion that the only distinct functions are the
16 functions obtained by giving to the characters a, 0, 7, S the values 0 and 1
respectively.
Odd-integer alteration of characters.
7. The effect is obviously to interchange the different functions.
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 467
Uven and odd functions.
8. It is clear that — m — a, — «— /3 have precisely the same series of values
with m + a, ?i + /9 respectively : hence considering the function
K:f) <-«■->
the linear term in the argument of the exponential may be taken to be
^Tri {— m — a . — u + y + — 11 — ^ . — V + B],
which is
= ^TTt {m + a.u+y+ n + ^ .v+ B} -Tti [m + a.y + n + ^ .8};
the second term is here
= — TH (my + «S) — Tn (ay + ^S),
where, my + nB being an even integer, the part — wi (my + nB) does not alter the value
of the exponential : the effect of the remaining part — tti (ay + /SS) is to affect each
term of the series with the factor exp. —^(ay+ffB), or what is the same thing,
exp. iri (ay + _8B), each of these being, in fact, = (— )«v+?'.
VIZ.
^("' x)^"' ^^ ^^ ^" even or odd function of the two arguments (w, v) conjointly,
according as the characteristic ( ' jj ) is even or odd.
The quarter-periods unity.
9. Taking z and w integers, we have from the definition
.(;.?)<.+.. +..)=^(;^,;f,J(.,.).
viz. the effect of altering the arguments u, v into u + z, v + w is simply to interchange
the functions as shown by this formula.
If z and w are each of them even, then replacing them by 2z, 2w respectively,
we have
^(;;f)(.-.2.,.+2.)=^(;_^^^>f_^j(.,.),
which by a preceding formula is
=(-r*^"'^(";f)(w. ^x
59—2
468 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
or the function is altered at most in its sign. And again writing 2z, 2w for z, w,
we have
^("* f)<" + *^' «' + 4«') = ^("' f){u, V).
In reference to the foregoing results we say that the theta-functions have the
<luarter-period.s (1, 1), the half-periods (2, 2), and the whole periods (4, 4).
The conjoint quarter quasi-periods.
10. Taking x, y integers, we consider the effect of the change of u, v into
M H . (ax + hy), v-\ — . Que + hy\
■in ^ in •^
It is convenient to start from the function
^(""'';f~^)(« + i-(«^+%). v + ^ihx + by));
the argument of the exponential is here
i (a, h, 6$m + a—x, n + ^ — yf
+ i7n\m + a — x.u + y-\ — .(ax + hy) + n + ^ - y . v + S + —.(hx+by)l ,
which is
= i (a, h, b'^vi + a, n+ ^y + ^m, (m + a .u + y + n + ^ .v + 8)
+ other terms which are as follows : viz. they are
- i (a, h, b^m + a, n + ^'^x, y) + ^ (m + a .ax + hy + n+^ .hx + by)
+ i (a. k b'$x, yy -^iriix.u + y + y.v + B)
-l(x.ax + hy + y.hx + by),
where the terms of the right-hand column are, in fact,
= + i («> /*. ^5™ + a. « + /S $*•, y)
. —^Triix.u + y +y .v+S)
-i(«. K b~$_x, yf,
and the other terms in question thus reduce themselves to
- i (a, h, b^x, yf -^Tri(x.u + y + y .v + B),
which aie independent of m, n, and they thus affect each term of the series with
the same exponential factor. The result is
^(""'';f"^)(« + ^.(a- + %), v + lihx + by))
= exp.{-i(a, h, b-^x, yy-^',n{x.u+y + y.v + S)}'^(^' ^](«,t));
704] A MEMOIR ON THE SIN^GLE AND DOUBLE THETA-FUNCTIONS. 4(59
or (what is the same thing) for a, /3, writing a + x, /3 + y respectively, we have
^ (7. 8 ) {'' + ^i ^"^ + ''^^' ' + ^- ^''^ + ^^^)
= exp.{-J(a, A, 6$«, y)r -^iriix . u+ y + 1/ . v + B)}"^ (^^'^'^'' g '■')(«. i')-
Taking «, y even, or writing 2a:, 2y for a;, y, then on the right-hand side we have
fa + 2x, fi + 2y
which is
^(""^""r •"'^)(ii,v),
=m;;^) (...):
but there is still the exponential factor.
11. The formulas show that the effect of the change u, v into wH .(ctoc + hu),
■m ^
f H — ; (hx + by), where x, y are integers, is to interchange the functions, affecting them
however with an exponential factor ; and we hence say that — ; (a, h), — ; (h, b) are
conjoint quarter quasi-periods.
The product-theorem.
12. We multiply two theta-functions
^ ("; f) (« + «'. ^ + ^'), ^ ("'/ f,') (« -n',v- v') ;
it is found that the result is a sum of four products
^f^(a + a')+p. i(^ + ^;) + Y^(2,, 2.).0(^(«-"?+^' *<^-f + '^) (2«', 2v),
\ 7+7 , 6+6 / \ 7-7 , 0-6 J
where p, q have in the four products respectively the values (0, 0), (1, 0), (0, 1), and
(1, 1); B is written in place of ^ to denote that the parameters (a, h, b) are to
be changed into (2a, 2A, 26). It is to be noticed that, if a, a' are both even or
both odd, then ^ (a + a'), J (a — a') are integers ; and so, if y8, /3' are both even or
both odd, then ^(/8 + /3'), i(y3— /3') are integers; and these conditions being satisfied
(and in particular they are so if a = a', /8 = /8') then the functions on the right-hand
side of the equation are theta-functions (with new jjarameters as already mentioned) ;
but if the conditions are not satisfied, then the functions on the right-hand side are
only allied functions. In the applications of the theorem the functions on the right-
hand side are eliminated between the different equations, as will appear.
13. The proof is immediate : in the first of the theta-functions, the argument
of the exjwnential i.s
'm+a , «+/3 \
\«-hw' + 7, w-t-y' + S/'
470 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
and in the second, writing vi, n' instead of m, n, the argument is
/m' + a , n' + ^ \
I « - u' + 7', V -v' + B'J '
hence in the product, the argument of the exponential is the sum of these two
functions, viz.
= 1 (a, h, 6$«i +0, n + ^y + ^m(m +a .u + u' + y +n +^ .v + v' + S)
+ J(a, h, 65m' + o', n' -i- ^y + i7ri{m' + a' .u-u' + y +n' + 0' .v -v' + 8').
Comparing herewith the sum of the two functions
V2M + 7 + 7' ,2t> + 8+8' /' 12m' + 7-7' ,2v' + 8-B' /'
= i (2a, 2h, 26$/i + i (a + «')- " + i (/3 + /3'))^
+ ^Tn {/x + i (a + a') . 2tt + 7 + 7' + 1/ + i (/3 + /9') • 2i; + S + S'j
+ i(2a, 2h, 26$/ + i(a-a'), ,/' + ^ (/3 - /3'))^
+ i Tri {/ + i (a - «') • 2m' + 7 - 7' + 1'' + H/3 - /8') • St)' + 8 - S'i ,
the two sums are identical if only
m + m' = 2fi, n + n = 2v,
m — m' = 2/a', n — m' = 2/,
as may easily be verified by comparing the quadric and the linear terms separately.
The product of the two theta-functions is thus
~-^''P-V2m + 7 + 7' ,2v + B + B' J-^®''P-U«' + 7-7' ,2i;'+S-8' j'
with the proper conditions as to the values of fj,, v and of ;tt', v in the two sums
respectively. As to this, observe that m, m! are even integers ; say for a moment
that they are similar when they are both =0 or both = 2 (mod 4), but dissimilar
when they are one of them = 0 and the other of them = 2 (mod 4) ; and the like
as regards n, «'. Hence if m, nt' are similar, /t, yu are both of them even ; but if
TO, m' are dissimilar, then /i, y! are both of them odd. And so if n, n' are similar,
V, v are both of them even ; but if n, n are dissimilar, then v, v are both odd.
14. There are four cases:
m, m' similar, n, n' similar,
TO, to' dissimilar, n, n similar,
TO, to' similar, n, n' dissimilar,
TO, to' dissimilar, n, n' dissimilar.
In the first of these, /*, v, fi, v are all of them even, and the product is
-e<;:;> *i:f)<^»- ^')-«e<::?; *i:f )<-'■ -■•■
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 471
In the second case, writing fi + l, ^' + 1 for /*, fj.', the new values of /a, fi' will
be both even, and we have the like expression with only the characters ^ (a + a'),
^(a — a') each increased by 1 ; so in the third case we obtain the like expression
vdth only the characters ^ (/3 + /3'). k (^ ~ ^') ^ach increased by 1 ; and in the fourth
case the like expression with the four upper characters each increased by 1. The
product of the two theta-functions is thus equal to the sura of the four products,
according to the theorem.
Resume of the ulterior theory of the single functions.
15. For the single theta-functions the Product-theorem comprises 16 equations,
and for the double theta-functions, 256 equations : these systems will be given in
full in the sequel. But attending at present to the single functions, I write down
here the first four of the 16 equations, viz. these are
0.0 ^ Q (m -I- w')-^(o )(«-«')= XX'+YY',
1.0 ^ J „ ^ J „ = YX' + Xr,
0.1 ^ ? „ ^ ? „ = XX' -YY',
0
ji
^ 0
1
»
-l
1
1
a
-\
1.1 ^ J „ ^ J „ =- YX' + XY';
where X, Y denote 0| j(2u), Bf j(2m) respectively, and X', Y' the same functions
of 2u' respectively. In the other equations we have on the left-hand the product of
different theta-functions o{ u+ u, u — u' respectively, and on the right-hand expressions
involving other functions, X,, F,, X/, F/, &c., of 2m and 2u' respectively.
16. By writing w' = 0, we have on the left-hand, squares or products of theta-
functions of u, and on the right-hand expressions containing functions of 2m : in
particular, the above equations show that the squares of the four theta-functions are
equal to linear functions of X, F; that is, there exist between the squared functions
two linear relations : or again, introducing a variable argument x, the four squared
functions may be taken to be proportional to linear functions
^{a-x), ^{h-x), i§,ic-x), 2)(d-a;),
where 21, 33, 6, 3), a, b, c, d, are constants. This suggests a new notation for the
four functions, viz. we write
^(o)(">- ^Q<">-
= Au, Bu,
and the result just mentioned then is
A-'u : B'u
= 2l(a-x) : «(6-.c)
^Qiux
k;)(.o
Cu,
Du;
Ou :
D'u
{5ic-x):
3) (d - .>:),
472 A MEaiOIK ON THE SINGLE AND DOUBLE THETA-rUNCTIONS. [704
which expresses that the four functions are the coordinates of a point on a quadri-
quadric curve in ordinary space.
17. The remaining 12 of the 16 equations then contain on the left-hand products
such as
A(u + u').B(u-u');
and by suitably combining them we obtain equations such as
u+u'u-«' u+u'«-u'
^-p-^^-p-^ = function OO,
where for brevity the arguments are written above; viz. the numerator of the
fraction is
B(u + u') A(u-u')-A {u + u') B(u- u),
and its denominator is
G{u + u')D {u - ?0 + D {u + «') G {u - «').
Admitting the form of the equation, the value of the function of u' is at ouce found
by writing in the equation u=0; it is, as it ought to be, a function vanishing for
u' = 0.
18. Take in this equation u indefinitely small ; each side divides by u', and
the resulting equation is
AuRu - BuA'u
„ ,-, = const.,
UuDu
where A'u, Bit are the derived functions, or differential coefficients in regard to v.
It thus appears that the combination AuBu — BuA'u is a constant multiple of
CuDu : or, what is the same thing, that the differential coefficient of the quotient-
D p
function ^— is a constant multiple of the product of the two quotient-functions -j—
Au Au
. Du
and r- •
Au
19. And then substituting for the several quotient- functions their values in terms
of X, we obtain a differential relation between x, u ; viz. the form hereof is
J Mda;
au ■■
'^a — x.h — x.c — m.d—x
and it thus appears that the quotient-functions are in fact elliptic-functions : the
actual values as obtained in the sequel are
sn Ku = — - = Du -=- Cu,
Ik'
en Kn '= sj -r Bu -r- Cu,
dnZM= ^/FAu^Cu;
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 473
and we thus of coui-se identify the functions An, Bu, Cu, Du with the H and the 0
functions of Jacobi.
20. If in the above-mentioned four equations we write first u = 0, and then i(' = 0,
and by means of the results eliminate from the original equations the quantities
X, Y, X' , Y' which occur therein, we obtain expressions for the four products such
as A{%i + u') A(u — u'). One of these equations is
0'0.G{u + u)C (i( - u') = Ou C'u' - DhiD'u.
Taking herein m' indefinitely small, we obtain
CuC"u-{C'uf _C^_ (^V D^u
Chi ~ CO [col 'Ou'
where the left-hand side is in fact t ^ log Cu, or this second derived function of the
theta-function Cu is given in terms of the quotient-function ^i hence, integrating
twice and taking the exponential of each side, we obtain Cu as an exponential the
argument of which contains the double integral I ( >♦, (duy, of a squared quotient-
function. This, in fact, corresponds to Jacobi's equation
21. From the same equation
CK).C{u + u') C {u - u') = CuC^u' - B^uD'u',
differentiating logarithmically in regard to u' and integrating in regard to u, we obtain
an equation containing on the left-hand side a term log ^-, k , and on the right-
hand an integral in regard to u ; this, in fact, conesponds to Jacobi's equation
B'a , . , 0 (w - a) „ . .
^ea+*^''^0(^^:^ a)- "("■")
Itfsnacnadaa sn' u du
-I
1 — h? sxi' a sn' u
22. It may further be noticed that if, in the equation in question and in the
three other equations of the system, we introduce into the integral the variable x
in place of u, and the corresponding quantity f in place of u', then the integral is
that of an expression such as
dx
T'Ja — x.b — x.c — x.d — x'
where T is = a; — f, or is = any one of three forms such as
] , a; -I- f , x^
1, a+ h, ab
1 , c +d, cd
c. X. 60
474 A MEMOIR OK THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
B^nU of the ulterior theory of the double functions.
23. The ulterior theory of the double functions is intended to be cai-ried out
on the like plan. As regards these, it is to be observed here that we have not only
the 16 equations leading to linear relations between the squared functions, but that
the remaining 240 equations lead also to linear relations between binary products of
different functions. We have thus between the 16 functions a system of quadric
relations, which in fact determine the ratios of the 16 functions in terms of two
variable parameters x, y. (The 16 functions are thus the coordinates of a point on
a quadri-quadric two-fold locus in 15-dimensional space.) The forms depend upon six
constants, a, h, c, d, e, f: writing for shortness
Va = 'Ja — x.a—y,
>Jah = {Va — x.h — x ./— x.c — y.d — y.e — y + ^a — y. b — y.f—y.c — x.d — x.e — x},
X— y
(observe that in the symbols Va6 it is always / that accompanies the two expressed
letters a, b — or, what is the same thing, the duad ab is really an abbreviation for
the double triad ahf.cde): then the 16 functions are proportional to properly determined
constant multiples of
Vo, Vft, Vc, Vd, Vc, V/, ^/ab, ^ac, 'J ad, Vae, Vic, VW, Vje, \fcd, '^ce, "Jde:
and this suggests that the functions should be represented by the single and double
letter notation A{u, v),..., AB(u, v),...; viz. if for shortness the arguments are omitted,
then we have
A. B, C, D, E, F, AB, AC, AD, AE, BG, BD, BE, CD, CE, DE,
proportional to determinate constant multiples of the before-mentioned functions
Va, ..., Va6, ..., of a; and y.
24. It is interesting to notice why in the expressions for 'Jab, &a, the sign
connecting the two radicals is -t-; the effect of the interchange of x, y is, in fact, to
change (u, v) into (— w, —v); consequently to change the sign of the odd functions,
and to leave unaltered those of the even functions: the interchange does in fact leave
Va, &c., unaltered, while it changes Va6, &c., into - Va6, &c. ; and thus, since only
the ratios are attended to, there is a change of sign as there should be.
25. The equations of the product-theorem lead to expressions for
»+«' «-»' «+«' «-«'
A.B - B.A,
where the arguments, written above, are used to denote the two arguments, viz. u + u'
to denote (u+it', v + v) and u-u' to denote {u-u', v-v); and where the letters
A, B denote each or either of them a single or double letter. These expressions
704] A MEMOIK ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 475
are found in terms of the functions of («, v) and of («', *'): in any such expression
taking u', v' each of them indefinitely small, but with their ratio arbitrary, we obtain
the value of
u u u u
A.dB-B.dA,
(viz. w here stands for the two arguments (u, v), and 9 denotes total differentiation
dA = du-j- A (u, v) + dv-j- A (u, v) ),
as a quadric function of the functions of (u, v) : or dividing by A", the form is 9 -j- equal
A
to a function of the quotient-functions -j, &c., that is, we have the differentials of
the quotient-functions in terms of the quotient-functions themselves. Substituting for
the quotient-functions their values in terms of x, y, we should obtain the differential
relations between dx, dy, du, dv, viz. putting for shortness
X = a — x.h — x.c — x.d — x.e — x.f—x,
and
Y=a-y.b-y.c-y.d-y.e-y.f-y,
these are of the form
dx dy xdx ydy
'/l~Vf' vT~7f'
each of them equal to a linear function of du and dv : so that the quotient-functions
are in fact the 15 hyperelliptic functions belonging to the integrals \-j=y \~7=]
and there is thus an addition -theorem for them, in accordance with the theory of
these integrals.
26. The first 16 equations of the product-theorem, putting therein first m=0,
D = 0, and then u =0, v' = 0, and using the results to eliminate the functions on the
right-hand side, give expressions for
A . B, &c.,
that is, they give A{u + u', v + v') .B(u — u', v — v'), &c., in terms of the functions of (u, v)
and {u', i/) : and we have thus an addition-with-subtraction theorem for the double
theta-functions. And we have thence also consequences analogous to those which present
themselves in the theory of the single functions.
Remark as to notation.
27. I remark, as regards the single theta-functions, that the characteristics
Q. ©. ©■ G).
might for shortness be represented by a series of cunent numbers
0, 1, 2, 3:
60—2
476 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
and the functions be accordingly called ^o". ^i". ^2W> ^s"; but that, instead of this,
I prefer to use throughout the before-mentioned functional symbols
A, B, G, D.
As regards the double functions, I do, however, denote the characteristics
00 10 01 11 I 00 10 01 11 I 00 10 01 11 00 10 01 11 j
00' 00' 00" 00 10' 10' 10' 10 or or or oi ir ir ii' ii I
by a series of current numbers
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. 14, 15,
and write the functions as ^„, ^i ^u accordingly; and I use also, as and when it
is convenient, the foregoing single and double letter notation A, AB,..., which
correspond to them in the order
BD, GE, CD, BE, AG, C, AB, B, BG, DE, F, A, AD, D, E, AE.
Moreover, I write down for the most part a single argument only : thus, A(u + u')
stands for A(u + u', v + v'), A (0) for A (0, 0) : and so in other cases.
SECOND PART.— THE SINGLE THETA-FUNCTIONS.
Notation, Jkc.
28. Writing exp. a = g, and converting the exponentials into circular functions,
we have, directly from the definition,
^ (m) = ^w = .4u = 1 + 2^- cos TTU + 2q* cos ^iru + 2q^ cos S-iru + ... ,
^^(u) = %U = Bu= 2}* cos ^TTW-l- 2}* cos f TTU + 2^^008 fTTM-l-...,
^ 1 (w) = ^j" = Cm = 1 — 2^ cos TTU + 2q* cos 27ru — 2q' cos Sttu + ..,(= 0 (Ku), Jacobi),
% - (w) = %u = Du = — 2g* sin jTTit + 2q^ sin f Trit — 2^" cos f ttw +...(= — H (Ku), Jacobi),
where o is of the fonn a = — a + /3t, a being non-evanescent and positive : hence
5 = exp. (— a -f /8i) = e~» (cos /3 + i sin jS), where e~", the modulus of q, is positive and
less than 1 ; cos/9 may be either positive or negative, and q^ is written to denote
exp. i (— a + /3i), viz. this is =e~l* {cos JjS + isin J/8). But usually /3 = 0, viz. 5' is a
real positive quantity less than 1, and g* denotes the real fourth root of q.
1 have given above the three notations but, as already mentioned, I propose to
employ for the four functions the notation Au, Bu, Gu, Du: it will be observed that
Du is an odd function, but that Au, Bu, Gu are even functions, of u.
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 477
The constants of the theory.
29. We have
^0 = 1 + 2^ +25* +22» +...,
C0=l-2q +2q*-2^ +...,
D0 = 0,
1/0= -■:r{q^-Sq^+5q''^ -...}.
If, as definitions of k, k', K, we assume
*~^H)' AH)' BO 'CO'
then we have
„ 7r(l+2g + 2g«+...)(l.-3g' + 5g«--) 1 n .A ^A." ■ f^ , . X
where I have added the first few terms of the expansions of these quantities. We
have identically
k'+k"- = l.
It will be convenient to write also, as the definition of B,
we have then
moreover,
._E_1 (TO _2-ir^ g- 47*+ %»-...
K~K^' CO ' ~ K^' l-2q-<r2q*+ ...'
giving
^= l-8g + 48g»-2243» + ...,
and thence
£ = iir {1-4^ + 202=- 64g» +...}.
30. Other formulae are
"-"""iXl+qA + f...]'
_ a-q.i-(f...y
tl+9.1+g'...['
. [l+g.l+g^-l-gM-9^-)'
478 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
31. Jacobi's definition of q is from a different point of view altogether, viz. we
riT
K
d<f>
have q = exp. „ , where
H
oVl -h'Bm^<f>'
ttK'
and K' is the like function of k' ; the equation gives log 3 = — ^, viz. we have
K' = -^\ogq,
IT
and, regarding herein ^ as a given function of q, this equation gives £"' as a function
of q.
The product-theorem.
32. The product-theorem is
Vt+Y/ \7-7/ \ 7+7 / \ 7-7 /
a a'
Here giving to , , their different values, and introducing unaccented and accented
capitals to denote the functions of 2m and 2m' respectively, the 16 equations are
A. A ^jM + tt'&jM-u'= XX' + YY', (square-set)
B.B ^J „ ^J „ = YX+ XY',
C.C ^J „ ^J „ = XX'- YY',
D.D ^J „ ^J .. =- YX'+ XY';
C.A ^^u + u'^^u-u'= X,X; + F, F/, (first product-set)
A.c ^0 „ ^J „ = x,x:-yj:.
D.B ^J „ &J „ = y,x;+x,y:,
B.D ^J „ ^J „ = y^:-x,y:;
704] A MEMOIR OX THE SINGLE AND DOUBLE THETA-FUNCTIONS. 479
B.A '^ u + u"^ u-u' = PP' + QQ', (second product-set)
A.B ^^ „ ^J „ = PQ' + QP',
D.c ^J „ a J „ = iPP' - iqq,
CD ^J „ ^J „ = iPq - iQP';
D.A ^ J M + m' S> J M - w' = P,P; + Q,Q;, (third product-set)
A.D ^^ „ ^J „ = iP,q:-iQ,P:,
B.C ^J „ ^J „ =-iP,P: + iQ,Q;,
c.B ^J „ ^J „ = p,q;+ q,p;.
33. Here, and subsequently, we have
®0' ®0' ®1' ®J(2iO=Z, Y,X„ Y, I ej. 0^.0*, 0|(2«)=P, Q,P., Q„
„ „(2«') = ^', F', z;, f;' Ij „ „ „ „(2«') = P', Q', p;, q;,
, » (0) =a , /3, a„ ^, i! „ „ „ „ (0) = p, q, p^, q^;
viz. we use also a, /3, a,, ;S, and p, q, p,, q, to denote the zero-functions; yS, is =0,
but we use B,' to denote the zero-value of -;- Y.
' du '
34. In order to obtain the foregoing relations, it is necessary to observe that
0"+2^0«;
7 7 -
by which the upper character is always reduced to 0, 1, ^ or f; and that, for re-
ducing the lower character, we have
0 „=0 : 0 „ =- 0 ;
7-t-2 7' 7-1-2 7'
0* ^iB* fM)i „ = -i0*; 0^ „ = -10^,0^ „=^0^;
7-1-2 7 7-2 7' 7-1-2 7 7-2 7'
by means of which the lower character is always reduced to 0 or 1 : in all these
formulae the argument is arbitrary, and it is thus = 2m, or 2ti' as the case requires.
The formulae are obtained without difficulty directly from the definition of the
functions 0.
t9 Jt
it 1}
480
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8.
[704
35. As an instance, taking , ' = i » -i > *^6 product-equation is
^J(« + u').^J(M-w') = 0j(2»).eJ(2«') + 0|(2M).eJ(2M'),
= {& J (2m) . 0 J (2u) - i& J (2m) . 0 ^ (2m'),
which agrees with the before-given value.
-iQ-Qf,
36. The following values are not actually required : but I give them to fix the
ideas and to show the meaning of the quantities with which we work.
M = 0
a =1 +2q^ + 2f + ...,
13 = 2qi + 2q^+...,
a, =1 - 2q^ +2q^- ...,
^: = 2T(-qi + Sqi-...)
= ^r, for «=0.
du '
Z = 0 (2ii) = \-\-2f cos 2irit -1- 29* cos 47rM + ... ,
F = 0 „ (2m) = 25* cos TTM -1- 2(^ cos Sttm + . . . ,
Z, = 0 (2m) = 1-25-'' cos 27rM -I- 2f cos 4irM - ... ,
F, = 0.(2m)= -25* sin TTM -t-29*sin37nt- ...,
P = 0 ^ (2m) = 9* (cos \itu + i sin ^7nt) -f- (^ (cos f ttm — % sin Ittm)
+ q^ (cos \tru + i sin ^ttm) + ... ,
Q = 0 ^ (2m) = 7* (cos ^TTit - i sin ^ttu) + (^ (cos Ittm -J- i sin Ittm)
+ q^ (cos ^TTM — I sin ^ttm) 4- . . . ,
P, = 0 I (2m) = -~j^ \(^ (cos ^TTM + i sin ^ttm) - g* (cos f mt - 1 sin \inji)
— q^ (cos f TTM -I- i sin ^ttm) +...[,
Q, = 0 * (2m) = -~jJ^ jj* (cos \iru - i sin ^7n<) - 5* (cos |7rw + i sin f ttm)
— g^(cos§TrM — isinf7rM)-f ...{■;
and therefore also
P = q = 5'* + 9* + 9^ +
P,=
V2
|gi_5l_5¥+3¥ + 5V__..j, q^=L__?|Do.}: p, = iq,,
704] A MEMOIR OS THE SINGLE AND DOUBLE THETA-FDNCTIONS. 481
The square set, ii = 0 ; and x-formulcB.
37. We use the square-set, in the first instance by writing therein ?t' = 0 ; the
equations become
A-'u = olX +^Y, =0)^21 {a-x),
Bhi=^X+ aY, = «^ (6 - x),
Ou=aX -^Y, = coHK (c -x),
DHi, = ^X - aY, = 0)^2) (d - x),
viz. the equations without their last members show that there exist functions «" and
xm', linear functions of X and Y, such that 21, S, g, 5), 2la, 336, Sc, !Drf, being
constants, the squared functions may be assumed equal to 2la . w'' — 21 . to^x, &c., that
is, w*2l (a — x), &c., respectively : the squared functions are then proportional to the
values 21 (a - x), &c.
To show the meaning of the factor w", observe that, from any two of the equations,
for instance from the first and second, we have an equation without to,
A'u - R-u = 21 (a -x)^^{b-x);
and using this to detennine x, and then substituting in w" = A^u -=- 21 (a — x), we find
, ^A"-a - ^{Bhi
•" ' {a-b)Wd '
where the numerator is a function not in anywise more important than any other
linear function of A^i and Bhi.
38. The function Du vanishes for !( = 0, and we may assume that the corresponding
value of a; is = d. Writing in the other equations u = 0, they become
AH) = («'-■ + /30 = '»o'2l (a - d),
BH) = 2a/3 = ft)„=33 {b - d),
CH)= a--^ = a)„'g (c - d),
where &)„' is what tS' becomes on writing therein x = d. It is convenient to omit
altogether these factoi-s &)' and to^; it being understood that, without them, the
equations denote not absolute equalities but only equalities of ratios: thus, without
the Wo'i the last-mentioned equations would denote
AH} : BH) : CH) = a:'+^- : 2a^ : a^-/3^ =2l(«-d) : 3J(6-d) : (£(c-d).
The quantities 21, 33, 6, 2) only present themselves in the products ^m", &c., and
their absolute magnitudes are therefore essentially indeterminate : but regarding to' as
containing a constant factor of properly determined value, the absolute values of
21, ©, (S, 2) may be regarded as determinate, and this is accordingly done in the
formulae 2l' = — agh, &c., which follow,
ex. 61
482 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
Relations between the constants.
39. The formulae contain the differences of the quantities a, b, c, d; denoting
these differences
b — c, c — a, a — b, a — d, b — d, c — d,
in the usual manner by
a, b, c, f, g, h,
so that
. -h +g -a = 0,
h . _f _b = 0,
-g+f . -0 = 0,
a +b +c . =0,
and also
af+bg + ch =0,
and then assuming the absolute value of one of the quantities 21, S3, S, I), we have
the system of relations
2l» = - agh, JBda = 2l2)f, Slbcf = - 3363), SlSgD = abefgh,
33= = bhf, SSlb = - JBDg, 33cag = 6212),
&= cfg, 2lS3c = -e2)h, 6abh= 21332),
3)= = -abc, 2)fgh = - 21336,
0=33' + b»e= - f =2)= = bcf (af + bg + ch), = 0.
- c»8l» . + a=e» - g=D» = cag ( „ ), = 0,
- b=2l= + a==33= . -h=2)= = abh( „ ), =0,
-f=2l= + g»33'+h=g» . =fgh( „ ), =0.
It is to be remarked that, taking c, a, b, d in the order of decreasing magnitude,
we have — a, b, c, f, g, h all positive ; hence 21°, 33^ S', 2)" all real ; and taking as
we may do, 2) negative, then 21, 33, 6 may be taken positive; that is, we have
— a, b, c, f, g, h, 21, S3, 6, - 2) all of them positive.
40. We have
The foregoing equations
give
^=0 =
o= + y3^ = 2lf.
R0 =
2a/9 =33g,
CH) =
a»-/3' = eh.
"'AH)'
2lf'
,, 6h
]^=
satisfying
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 483
and we thence have
La „
-af -af
41. Observe further that, substituting for a, b, c, f, g, h their values, we have
^^ =c - b.b -d.c -d, = c —d.d—h.h - c,
^■=c— a.c— d.a — d, = d—a.a—c.c—d,
©2 =a-h .a — d .h —d, = — a— b.b—d.d — a,
^^ = c — b.c— a.a— b, = — b—c.c—a.a—b,
where in the first set of values all the differences are positive, but in the second set
of values, we take the triads of abed, in the cyclical order bed, cda, dab, abc. There
is in this last form an apparent want of symmetry as to the signs (viz. the order
which might have been expected is H 1 — ), but taking the order of the letters to
be (5, 21, 35, 2) and c, a, b, d, then the cyclical arrangement is
6- = — b—d.d—a.a—b,
!H^ =-d-c.c -b.b-d,
33''= — c —a.a — d.d — c,
^' = -a-b.b-c.c-a,
where the four outside signs are all — . Observe that the triads of abed, and abdc, are
bed, eda, dab, abc,
bdc, dca, cab, abd.
and
where in the first and second columns the terms of the same column correspond to
each other with a reversal of sign, whereas in the third and fourth columns the lower
term of either column corresponds to the upper term of the other column, but without
a reversal of sign.
The product-sets, u ± u' : and u' indefinitely small, differential formulce.
42. Coming now to the product-sets, these may be written
\[G.A+A.C\ = X,X;,
„{D.B + B.D}=Y,X:,
ii{B.A + A.B]= {P + Q)(P' + Q'),
„[D.C + C.D]= i(P - Q){P' + Q'),
^{D.A + A.D}= (P, - iQ,) (P; + iQ,'),
„{B.C + C.B} = -i {P, - iQ,) (P; + iQf),
u+u' u-u' u-hu' u~u'
^{c.a-a.C} = y,y;,
„{d.b-b.d} = x,y;,
^{B.A-A.B}= (i^ - Q)(P' - Q'l
„{D.C -C.D]= i(P + Q)(F - Q-),
^{D.A-A.D}= {P, + iQ,) (P; - iQ,'),
.,{B.C-C.B} = -i{P,- iQ,) (P; - iQ,').
61—2
484 A MEMOIR ON THE SENGLE AND DOUBLE THETA-FUNCTJONS. [704
43. We can from each set form two fractions (each of them a function of m + m'
and M — m'), which are equal to one and the same function of u' only : for instance,
Y' . .
from the first set we have two fractions, each J., : putting in such equation u = 0,
we obtain a new expression for the function of u' involving only the theta-functions
A it', &c., which new expression we may then substitute in the equations first obtained :
we thus arrive at the six equations
C^A ^A^ _ D.B-B.D _ Du'.Bu'
D.B + B.D~C.A + A.C~Gu'.Au"
_ S.A-A.B _ D.G-G.D _ Du' . Cu
I).C + C.D~B.A+A.B~Bu'.Au"
_ B^-C^ B _ D.A-A.D _ Du'^v[
D.A+A.'D~ B.C+C.B~ Bu'.Cu"
■where observe that the expressions all vanish for u' = 0.
44. Taking herein u' indefinitely small, we obtain
Au.C'u- Cu.A'u _ Bu.D'u-Du.B'u _ D'O . BO __„B'0
Bu.bu ~ Cu.Au ~ CX).AO ~ A'O '
Au.B'u-Bu.A'u _ Gu .B'u-Du. Cu _ D'O . CO _ _ ^ C=0
Cu.Du ~ Au.Bu ~AO.BO~ X=0'
Cu.Ru-Bu.C'u _Au.iyu-Du.A'u_D'O.AO_ „
Aa.Du ~ 'BuTOu £OTCO ~ '
where the last column is added in order to introduce K in place of UO.
45. These formulae in effect give the derivatives of the quotient-functions in terms
of quotient-functions : for instance, one of the equations is
d Du _ Bu Gu
du Au Au ' Au'
substituting herein for the quotient-fractions their values in terms of x, this becomes
dliV a-x~ WS12) a-x '"~^Va 7^^ '
or the left-hand being
-jf dx
(a — x)i Vd — « du'
this is
i Vaf . da
Kdu =
Va — x.b — x.c — x.d—x
where on the right-hand side it would be better to write V— af in the numerator
and a; — d in place of d — x in the denominator.
sn.
en.
704] A MEMOIR ON THJE SINGLE AND DOUBLE THETA-FUNCTIONS. 485
Comparison with Jacobi's fornudw.
46. The comparison of the formulae with Jacobi's formulae gives
dn Ku= VFAu^Cu, =\/j\/"^'
where it will be recollected that
*^~-af' * ~-af
It may be remarked that we seek to determine everything in terms of a, b, c, d.
The formula just written down, ^ = bg -h — af, gives k in terms of these quantities ;
and k, K being each given in terms of q, we have virtually .ff' as a function of k,
that is, of a, b, c, d: but it would not be easy from the expressions of k, K, each
in terms of q, to deduce the actual expression
Jo's/l-k^
of K as a function of k.
sin' <!> '
The square-set, u ± u'.
47. Reverting to the square-set
A{u + iOA{u-u')= XX'+YY',
B(u + u')B{u-u')= YX' + XY',
C(u + u') C (m - u,') = XX' - YY',
D{u + u')D{u-u') = - YX' + XY',
if we first write herein u' = 0, and then m = 0, using the results to determine the
values of X, Y, X', Y' we find
aCHi; - ^I>u' = (a= - ^') X',
/SOm' - al>u' = „ Y',
^Ou - aUhi = „ Y,
and thence
(a» - ^y XX' = a' . ChiOu' + ^ . I^uD'u' - a0 {O^uBr-ii + IfuOu'),
YY'=^' „ +0C' „ -a^
whence
{a' - 0'f {XX' + YY') = (a» + ^) {G^CH' + IfiaD'^u) - 2a^ {Ghi]>u' + DhiChi'),
(a' - ^) {XX' - YY') = {CHiC'ii' - D'uDhi'),
486
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8.
[704
where observe that, in taking the difference, the right-hand side becomes divisible by
o'-zS", and therefore in the final result we have on the left-hand side the simple
factor a* - /S* instead of (o» - yS*)*.
Similarly
(a» - yS") YX' = a/3 {ChiChi,' + iyul>u') - a^D'uCHi,' - ^-CHD^u',
„ XY'=a$ „ -^ „ -a= „ .
and thence
(a? -^y{ YX' + X Y') = 2aj8 (C^Chi' + DhiD^u) - (a" + /3=) {C'ulhi' + LhiC^u),
(a. _ ^) (_ YX' + XY')= DHtCHi' - Chil>u'.
48. Hence recollecting that
^''O = a^ + /3=,
BK) = 2a/3,
the original equations become
O^O .A(u + u')A (u - u') = AH) {GhiChi; + DhiDhi') - R-O {ChilT-u' + l>uG^ii'),
C*O.B(u + u') B(u-u')=BH) {G^uGHi' + D'-uDhi') - AH) (Ghil>u' + DhiG"-u'),
C«0 .G{u + u')G{u- u') = GhiGhi' - rr-aiyu',
CH).D{u + u') D{u- u') = DhiGHi' - G'uDhi'.
49. It will be observed that the four products A(u + v!) A{u — u), &c., are each
of them expressed in terms of C^u, Ifu, Ghif, Lht. Since each of the squared functions
AHi, Bht, C'u, Dhi is a linear function of any two of them, and the like as regards
A'u, J5*u', Ghi', D'u', it is clear that in each equation we can on the right-iiand
side introduce any two at pleasure of the squared functions of u, and any two at
pleasure of the squared functions of u. But all the forms so obtained are of course
identical if, taking x the same function of u that a; is of it, we introduce on the
right-hand side x, x instead of u, u' ; and the values of A (m -|- m') . ^ (m — m'), &c.,
are found to be equal to multiples of V, V,, Vj, V^, where
S7 = x-x', V,=
V,=
Xj 3/ *^ U/ f 3/Su
1, a + d, ad
1, h + c, be
50. In fact, from the equations
AHi = ^{a-x), ^V = 2l(fi-a;').
1, x + x', ocx'
1, h + d, bd
1, c + a, ca
V,=
1, x + x' , xx'
1, c + d, cd
1, a + b, ab
we have
V = ^g {R-uGHi' - G^uBhi'), = ^ {G-uAhi' - AhiChi), = ^„,^ {AHi,Br-ii' - B'uAHi'),
= Y^^ (AHiB^' - DhiA'u'), = -^ {BhiL^u' - I^uB'a'), = ^ {G-'uD^u' - IhiG'u).
704] A MEMOIR OX THE SINGLE AND DOUBLE THETA-FUNCTION8.
where it will be recollected that
f2lS)=a33e, -g33S) = bg2l, -hg2) = c2l33.
487
Moreover
(6 — c)V, = — h — x.b — x', c — x.c — x'
I b — a.b—d, c — a.c — d
, (a-d)V,=
a — x.a — x', d — x.d — x'
a—b.a — c, d—b.d — c
(c — rt)V, = — j c—x.c — x', a — x.a — x' j, (b — d)Vs = \ b—x.b — x', d — x.d — x'
\ c — b .c — d, a — b.a — d\ \ b — c .b - a, d— c.d — a
(a — 6)Vg = — a — x.a — x', b—x.b—x' , (c — d)^3=\ c—x.c — x', d — x.d — x' I,
a— c.a — d, b — c .b — d i c —a.c — b, d—a.d—b\
or as these may be written
V 1 = {bh .b — x.b—x'. + cg.c—x.c— x'}, = j [gh. . a — x . a — x . + he . d — x . d — x'},
V „ = — r {c{ . c — X . c — x' . + ah . a — X . a — x'}, = - {ht . b — x .b — x . + ca. . d — x . d - x'],
O
that is,
or finally
V.= - ■'■ ( B^uB'io' + C'uC'u'), =-^(A'uA^u' + B'uDhO,
af
V , = - ^ ( 6'=mC V - A'uA'u'), = ^ (5=it£-^2t' - I>uD'u'),
V , = - -^ (- A"-uA^u' + BhiRu'), = \ (C-uCW - BhilPio').
eh
61. Hence V, V,, V,, Vj denoting these functions of x, x or of u, u', we have
A{u + u')A{u-u')^^V„
B{u-^u')B(u-u') = ~V„
(7(« + w')C(m-m') = -| V,,
D{u + u')D{u-u')=1)V.
488 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
The square-set u ± u', u' indefinitely small : differential formxdw of the second order.
52. I consider the original form
C'OC(u + u') C (« - u') = CuCht' - I>uir-u',
which is of course included in the last-mentioned equations.
Writing this in the form
f,,^C(u + u')G(u-u')_ DhiDhi'
and taking «' indefinitely small, whence
(7 (m + m') = Cm + m'C7'« 4^ u'C'u, Gu' = CO,
C{u-u') = Cu-u'C'u + ^u'^C"u, Du' = u'iyO,
C(u + u')G (u - u') =GHi + u'^ {CuC'u - (O'w)'},
the equation becomes
that is,
C"u _ /G'uV _G^_ /D'Oy mi
Gu \Cu) CO Kca] G^'
f d\^ TP-u
viz. we have ( ,1 log (7« expressed in terms of the quotient-function -^ - , and conse-
quently Gu given as an exponential, the argument of which depends on the double
integral \du \du j^— .
53. To complete the result, I write the equation in the form
d' , „ C"0 1 fB'Oy 1 fD'Oy /, , D-'u\
I/O - G"0
-gcj is = — ^kK, and -^ is =K (K -E); hence the equation is
or integrating twice, and observing that -r~ log Gu and log Gu, for u = 0, become = 0
and log CO respectively, we have
log C« = log CO -t- J ^1 - J) K^^ - k' j duj du K^ sn= Ku,
which is in fact
log 0 {Ku) = log CO -I- J (l - ^) Khfi -k>j duf du K' sn» Ku,
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
489
agreeing with Jacobi's formula
log 0w = log 00 + ^ ( 1 - -^j M= - A;- 1 duj du sn" u.
Elliptic integrals of the third kind.
54. We may write
V:
AJji + v!) A {u -u') _ 1
l^uAhi Igh a-x.a-oo"
B{u+u')B(u-u') 1 V,
BhiBhi' ~mi b-x.b-x"
C{u + u') C{u-u') _ J^ V,
C^uC^u' ~ (Sfg c-x.c-(c"
D(u + u')D{u — u') _ 1 x — x
DhiDht! ~ "^ d-x.d-x"
We differentiate logarithmically in regard to u'. Observing that
Kdu'=^
, Wafda;' Waf , ,
-7 -^ , , = ^—=^ dx,
\a — af .h — x .c — x .d — x' v X'
suppose, the first equation gives
A'u A'{u-u') .A'{u + u')^ K-JX' d ^^ _V,_
Au A {u- u') A {u + u') Vaf dx a-x'
and if for a moment
v., =
IS put
then
d
dx'
1, x + x', xo^
1, a-\-d, ad
1, b +c, be
= P(a-x') + Q(d-x),
:''°«a-a;" da/'^^V +^a-W '^ {a-x')V,' (a-x')V,-
But, writing a;' = a, we have
Q(rf-a), =-Qf =
that is,
or
1, a+x, ax \, —{a — b){a — c){d — x), =-hc{d — x),
1, a+d, od I
Qf=-bc(d-a;),
d , V, bc(d — «)
Hence the equation is
a — x' (a — a;')V,'
2 il'(M') ^'(m-m') _ A'iuJrv!) _ 2A'^c ,y d-x
4(m') il(w-M') Z7m + m')~ Vaf (a-a;')'^i'
O. X,
62
490 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTION8. [704
and similarly
^Rju') R (u - u') F{u + u') ^ 2Kca ^,j, d-x
B(u')'^ B{u-u') B(u + u') Vif {b-x')V,'
C'(m') C'(u-u') _ C'ju + u') ^ 2K^^Y' ^'"^
Ciu') Ciu-u) C(u + u') \/af (c-a;')V3'
■^(m') !)'(«-«') iy(u + u')_ 2^ ^^, d-x
D(u') D{u-u') l){ji + u') Vaf {d-x'){x-x')'
V 3,f (l-W
55. Multiply each of these equations by du, = | -— --^ , and integrate. We
have equations such as
2u ■ ^ ■ + log — ^^ = const. + -7= / 5^ f-r- ,
A{u) ^A{u + u') Vaf (a -a;')-' V,VZ
showing how the integrals of the third kind
f{d — x)dx f(d — x)dx t{d — x)dx f{d — x)dx
J V,VZ~' i~V,VZ ' i V,VZ ' J (^-^')'^X
depend on the theta-functions.
If, instead, we work with the original equation
Ciu + u')C{u-u') _-. _Dhi, D^u'
Cu.C'u' ~ C^uC'u"
we find in the same way
C'ju') C'ju-u') C'(u + u') d^ A.^D^mWx
C{u')'^ G{u-u') C(u + u') ~ du' ^^ [ CHiC'u) '
= - ^, log (1 - A= sn» Ku sn'' Ku'),
. • _ 2/c'K sn Ku' en Ku' dn Ku' sn' Ku .
l-A»sn»^M'Tn^^M '
or, multiplying by ^du and integrating, we have
£^u') C(u-u')_rk^sn Ku' en Ku' dn -gw' sn' ^m . Kdu
" (7 («') "^ * ''^ C (it + m') j 1-k' sn= /^-m' sn' i^-^
which is in fact Jacobi's equation
8'a , 0 (m — a) /"sn a en a dn a sn' udu „ . .
^eH+^^^geo7+^=j l-ifc'sn'asn'^t ' ="^"' ">'
I do not effect the operation but consider the forms first obtained,
A (if + u') A {u - w') = -^ V„ &c.,
as the' equivalent of Jacobi's last-mentioned equation.
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
491
Addition-formuloi.
56. The addition-theorem for the quotient-functions is of course given by means
of the theorem for the elliptic functions : but more elegantly by the formulae relating
to the form dx -5- Va — x.h — x.c — x.d — x obtained in my paper " On the Double
^-Functions" (Crelle, t. Lxxxvii. (1879), pp. 74—81, [697]); viz. for the differential
equation
dx dy dz ^
1 — — = = 0
VZ VF VZ '
to obtain the particular integral which for y = d reduces itself to z = x, we must,
in the formulae of the paper just referred to, interchange a and d : and writing for
shortness a, b, c, d = a—x, b—x, c — x, d — x, and similarly a,, b,, c,, d^ = a — y,
b — y, c — y, d — y, then when the interchange is made, the formulae become
Id-z
^d-b.d-c (Vadb,c, + Va,d,bc}
(6c, ad) .
-Jd-h.d-c^ .X — y
Vadb^ - Va,d,bc
Vd - 6 . d - c {Vbdc,a, + VbAca}
" (d^ c) Vaba,b, - (6 - a) v'cdc,d, '
-Jd-b.d-c ( VcdaX + '^abc,d j
{d-b) Vaca,c, -{c-a) Vbdb^d,
/h-z
(6c, ad)
sl i^a i^^^ - ^b,d,ac{
Vadb^c, — Va,d,bc
y/^(ac, 6d)
(d — c) Vaba^b, — (6 — a) Vcdc,d, '
(J TZr K^ ~ ") ^bct,c, + (6 - c) Vaba,bJ
{d - 6) Vacate, -{c-a) Vbdb^
d^^ {(d - i) ^^cac,a, -I- (c - a) Vbdb^dJ
(6c, ad)
y d^a f^^^*'^' ~ '^abc,d j
^/:
Vadb,c, — Va,d,bc
J—- {(d - a) Vbcb,c, - (6 - c) Vada d j
(d — c) VabaF, - (6 - a) Vcdc^d,
^^^^(^6, cd)
(rf — 6) Vacate, — {c — a) v'bdb,d.
62—2
492
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8.
[704
57. In the foregoing formulae, (6c, ad), (ac, bd), and {ab, cd) denote respectively
1, x+1/, xy , \, x + y, xy
1, c + a, ca 1, a + h, ah
\, h+d, hd 1, c + d, cd
and substituting for 21, ©, S, i) their values, and for a, b, &c., writing again a-x,
b — x, &c, we have moreover
1,
^ + y, xy
y
1,
6 + c, he
1,
a-\rd, ad
A*u = ^c — b.h — d.c — d (a — x),
Bhi = 's/c—a.c — d.a — d (b — x),
C*u = va — b.a — d.b — d (c —x),
D^ = Vc - 6 . c — o . a — 6 {d-x),
^»(m + «) = V
B' (u + v) = V
(7«(m + i;) = V
D»(u + u) = V
(a-y),
(b-y).
(c - y).
(d-y),
(a - z),
ih-z),
{o-z\
(d-z),
the constant multipliers being of course the same in the three columns respectively.
According to what precedes, the radical of the fourth line should be taken with the
sign -. The functions {be, ctd), &c., contained in the formulae, require a transformation
euch as
(6 - c) (be, ad) = b — x.b - y, c — x.c — y
b — a.b — d, e — a.c — d
in order to make them separately homogeneous in the differences a — x, b — x, c — x,
d — x, and a — y, b — y, c — y, d— y, and therefore to make them expressible as linear
functions of the squared functions AH, &c., and Ahi, &c., respectively : the formulae then
give the quotient-functions ^ (m + u) ^ 2) (m + v), &c., in terms of the quotient-functions of
M and V respectively.
Doubly infinite prodioct-fonns.
58. The functions Au, Bu, Cu, Du may be expressed each as a doubly infinite
product Writing for shortness
TO -I- n . — ; = (m, n),
m
TO -t- 1 + n . . = (to, re),
TO + (n + 1) — . = (m, n),
TM
m -I- 1 + (ft + 1) — . = (ni, n),
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 493
then the foiinulse are
Au = Ao. nn-ii +
(m, n)J '
Bu = Bo. nnli + Tj^l,
(to, n)\
Cu = co. nnli + -^l,
Du = D'O . uun \l + ^-^l ,
( (m, n)]
where in all the formulae, m and n denote even integers having all values whatever,
zero included, from — x to + oc ; only in the formula for Du, the term for which m
and n are simultaneously =0, is to be omitted.
59. But a further definition in regard to the limits is required : first, we assume
that m has the corresponding positive and negative values, and similarly that /; has
corresponding positive and negative values*; or say, in the four formulae respectively,
we consider m, n as extending
m from —if,to/i+2, n from - v to v + 2,
„ -fi ., fj. + 2, „ „ -V „ V,
„ -fi „ fi , „ „ -V „ v + 2,
„ -/J. ., fl , „ „ —v„ V,
where /* and v are each of them ultimately infinite. But, secondly, it is necessary
that n should be indefinitely larger than v, or say that ultimately - = 0.
/*
60. In fact, transforming the g'-series into products as in the Fundamenta Nova,
and neglecting for the moment mere constant factors, we have
Au= (1 + 25 cos TTM + 5') (1 + 2(f cos TTtt + j"). . .,
Bu = cos i^-n-ii {I + 2q-coa'7ru + q*){l + 2gr*cos7rM + 5^)...,
Cu = (1 - 2? cos TTM + q") (1 - 25^ cos TTU + q^)...,
Du = sin Jttm (1 — 29' cos wu + 9*) (1 — 2q* cos iru + (f). ..,
and writing for a moment 0= — ; and therefore qi+q~i, = ei"' 4- e~i'"', =2cosi7ro, &c.,
each of these expressions is readily converted into a singly infinite product of sines
or cosines
Au = n . cos ^ (« + no),
Bu = Tl . cos ^TT (m + wo).
Cm = n . sin ^ (« + no),
Du = n . sin ^TT (m + no),
* This is more than is necessai-y ; it would be enough if the ultimate values of m were - fi, ix', /j. and fi'
being in a ratio of equality ; and the like as regards n. But it is convenient that the numbers should
be absolutely eqaal.
494 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8. [704
where n is written to denote n + 1, and n has all positive or negative even values
(zero included) from - « to + oo , or more accurately from — v to i/, if v is ultimately
infinite.
61. The sines and cosines are converted into infinite products by the ordinary
formulae, which neglecting constant factors may conveniently be written
sin ^u = n (« + m), cos ^vu = !!(« + fh),
where m is written to denote m + 1, and ?)i has all positive or negative even values
(zero included) from — oo to + x , or more accurately from — /x to /*, if /* be ultimately
infinite. But in applying these formulae to the case of a function such as
sin ^ir (u + na),
it is assumed that the limiting values fi, — fi of m are indefinitely large in regard to
u + na; and therefore, since n may approach to its limiting value + v, it is necessary
that ft should be indefinitely large in comparison with v, or that - = 0.
62. It is on account of this unsymmetrj' as to the limits - = 0, - = x , that
we have 1 as a quarter-period, — . only as a quarter-quasi-period of the single theta-
functions.
The transformatio7i q to r, log q log r = tt-.
68. In general, if we consider the ratio of two such infinite products, where for
the first the limits are (+ /*, ± v), and for the second they are (± /a', + v), and
where for convenience we take ij.> ijf, v> v, then the quotient, say [/tt, v\ -h [/*', r'] is
= exp. {Mu*), where
M
=-»//
dm dn
(m, ny
taken over the area included between the two rectangles. We have
(m, n)=m -!-—.»;, =m + i6n
TTl
suppose, where (a being negative) 6 = is positive : the integral is
IT
[fjmdn_ _[ 1/ 1
J J (m + idnr • -i'^'^-'id \^r+^
idn).
id j \m - i0v m + idv) '
— -Li "* ~ *^''
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 495
or finally between the proper limits the value is
where the logarithms are
log (fi - idv) = log slfi? +v' — i tan-' — , &c.,
A'
and the tan"' denotes always an arc between the limits — ^tt, + ^tt. Hence, if
- = 00 , ^ = 0, the value is
V V
_(_0^-0^ + i,^^ + i^)=-g-, =- — ; or^ = i-.
Hence finally
]ji^v, = X ] -;- [/i, H- 1/, =0] = exp. ^i^M'J.
64. We have a, =log5', negative; hence taking r such that log g' log ?• = tt, we
have a' = log r, also negative ; and r, like q, is positive and less than 1. We consider
the theta-functions which depend on r in the same manner that the original functions
did on q, say these are
A{u, r) = A (0, /•) nn<^
B{u, r) = B (0, r) UUr^
m + n
a
u
y
ffi + n
a'
u
y
a'
VI + n
Tri
u
m+n
a'
■ni,
C(u, r) = C' (0, r) UH '^ "^
j){u, r) = z)'(o, r)Mnnr ■•■
limits as before, and in particular - = x; it is at once seen that if in the original
functions, which I now call A (m, q), B (u, q), C (u, a), D (u, q), we write — . for u, we
obtain the same infinite products which present themselves in the expressions of the
new functions A{u, r), &c., only with a different condition as to the limits; we have
for instance
au
nn| 1+ -^^ i=nn
7/1 + n —
TTt
I n — m — :/ I n + m — :/
\ -Tl/ \ TTl/
496 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
which, interchanging m, n, and of course also fi, v, is
= nn
with the condition - = 0 instead of - = ac . Hence disregai-ding for the moment
V V
constant factors, and observing that a is replaced by a', we have
Z)(«, r)-D(g, q) = [,i^v, =x]^[f.^v, =0]
= exp. [\ ^ M") . = exp. (i«' log q).
65. We have equations of this form for the four functions, but \vith a proper
constant multiplier in each equation : the equations, in fact, are
A (u, r) = {A (0, r)^A (0, ,7)} exp. {\u^ log 9) J (^ , ?) ,
B («, r) = {B (0, r) - B (0, q)] „ B (^. , 9) ,
C(u.r)={CiO,r)^C(0.q)] „ <^{~i'9)>
D (u, r) = {ly (0, r)^iy (0, q)} ^' „ i) g , q
It is to be observed that ?• is the same function of k' that q is of A". This
from
Trfi"
would at once follow from Jacobi's equation log q = v^ , for then log q log r = tt^
and therefore logr = ~ (only we are not at liberty to use the relation in question
log g = j^\ : assummg it to be true, we have
£»(o,_£) w.o^coj) ^(0. g)iy(o, g)
4'(o, 5)' "- ^no, ?)' 5(0, 9)^(o> 9) '
.^C'(0. r) y^B'jO, r) A (0, r) D' (0, r)
A^{0, r)' ^'(0, r)' B(0, r)C(0. r) '
, ttK' , irK
log? -j^> logr = -^,
where, if the identity of the two values of k or of the two values of k' were proved
independently (as might doubtless be done), the required theorem, viz. that r is the
same function of y that q is of k, would follow conversely: and thence the other
equations of the system.
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 497
66. We have, in the Fundamenta Nova, p. 175, [Jacobi's Ges. Werke, t. i., p. 227],
the equation
Hjiu, k) _ . A ."^ H(u, k')
6(0, k)~'^V k' 0(0, k')'
writing here K'u instead of u the equation becomes
HjiK'u.k) fk /, Tir X HjK'u. k')
6(0, k) -^V k'^'^^-V K "")• @{0,k') '
or, what is the same thing,
^{■Tri' V . /k , 1 n ^ ^(». ^)
which can be readily identified with the foregoing equation between D( — ., q) and
J){u, r). But the real meaning of the transformation is best seen by means of the
double-product formulae.
THIRD PART.— THE DOUBLE THETA-FUNCTIONS.
Notations, &c.
67. We have here 16 functions ^( ^{u, v): this notation by characteristics,
containing each of them four numbers, is too cumbrous for ordinary use, and I
therefore replace it by the current-number notation, in which the characteristics are
denoted by the series of numbers 0, 1, 2,..., 15: we cannot in place of this introduce
the single-and-double-letter notation A, B,..., AB, &c., for there is not here any cor-
respondence of the two notations, nor is there anything in the definition of the
functions which in anywise suggests the single-and-double-letter notation : this first
presents itself in connexion with the relations between the functions given by the
product-theorem : and as the product-theorem is based upon the notation by charact-
eristics, it is proper to present the theorem in this notation, or in the equivalent
cvirrent-number notation: and then to show how by the relations thus obtained
between the functions we are led to the single-and-double-letter notation.
68. There are some other notations which may be referred to: and for showing
the correspondence between them I annex the following table : —
c. X. 63
498
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
The double th eta-functions.
1.
2.
8.
4.
s.
6.
7.
8.
Asterisk
denotes the
odd functions.
Current
number.
Character.
Single-and-
double-letter,
Cayley.
(JSpel.
G6pel.
Cayley.
Bosenhain.
Weier-
strass.
Kommer.
a,
^00
BD
p,,,
A
Sn
5.
12
10
00
CE
R'"
R,
n
4
8
01
00
CD
Q'"
Q.
23
01
10
11
00
BE
S"'
s.
33
23
6
4
00
10
AC
F
A
02
34
4
*
10
10
C
iK
R,
12
3
16
01
10
AB
Q'
Qx
03
2
2
*
11
10
B
iS
s.
13
24
14
00
01
BC
P"
A
20
12
9
10
01
DE
B"
R.,
30
03
5
♦
10
01
01
F
iQ"
Q.
21
02
11
«
u
11
01
A
1
iS"
s.
31
13
7
11
00
11
AD
P
p
00
0
1
*
u
10
11
D
iR
R
10
04
13
♦
14
01
11
E
iQ
Q
01
1
3
U
11
11
AE
S
s
11
14
15
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 499
69. These are the notations: —
1. By current-numbers. It may be remarked that the series was taken 0, 1,..., 15,
instead of 1, 2,..., 16, in order that 0 might correspond to the characteristic ;
2. By characteristics;
3. By single-and-double letters;
4. Gopel's, in his paper above referred to, and
5. The same as used in my paper above referred to;
6. Rosenhain's, in his paper above referred to ;
7. Weierstrass', as quoted by Konigsberger in his paper "Ueber die Transfor-
mation der Abelschen Functionen erster Ordnung," Crelle-Borchardt, t. LXIV. (1865),
p. 17, and by Borchardt in his paper above referred to;
8. Not a theta-notation, but the series of current-numbers used in Kummer's
Memoir "Ueber die algebraischen Strahlen-systeme," Berl. Abh. 1866, for the nodes
of his 16-nodaI quartic surface, and connected with the double theta-functions in my
paper above referred to.
But in the present memoir only the first three columns of the table need be
attended to.
70. It will be convenient to introduce here some other remarks as to notation, &c.
The letter c is used in connexion with the zero values m = 0, v = 0 of the
arguments, viz. : —
•*0> '''ll ^3i ^3» ^4> •'■«) ^8> ^9) ^n> •'IS
are even functions, and the corresponding zero-functions are denoted by
Co, Ci, Cj, C3, C4, C(j, Cj, Cj), Cij, Cii'.
there are thus 10 constants c.
When (m, v) are indefinitely small each of these functions is of course equal to
its zero-value plus a quadric term in (w, v), and we may write in general
^ = c-|-^(c"', c^ c^^w, vY:
there are thus 30 constants c'", c'", c'.
The remaining functions
•J 5) -J?) ^101 ^n> "Jiai ''^n
are odd functions vanishing for u = 0, v = Q; when these arguments are indefinitely
small, we may write in general
^ = (c', c"$M, v):
there are thus 12 constants c', c".
63—2
I
500 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
71. All these constants are in the first instance given as transcendental functions
of the parameters, or say rather of exp. a, exp. h, exp. h, which exponentials cor-
respond to the q of the single theory: viz., in a notation which will be readily
understood, the constants c, c'", c'", c' of the even functions are
2, exp. I
m + a, n + /3
7 i
- i7r»2 (m + af, 2 (w + a) {n + 0), (n + /3y, exp. ('" '*'"''' "^ ^ ;
and the constants c', c" of the odd functions are
i mZ (m + a), (n + p), exp. f g "^ j .
72. It is convenient for the verification of results and otherwise, to have the
values of the functions, belonging to small values of exp. (— a), exp. (— b) ; if to
avoid fractional exponents wo regard these and exp. (— h) as fourth powers, and write
exp. (- a) = Q*, exp. (- h) = i^^ exp. (- b) = S*,
also
QiP/S = A, QR-^S = A', and therefore AA' = Q'S^
then attending only to the lowest powers of Q, S we find without difficulty
^0 (u) = 1, and therefore also Co = 1,
^1 = 2Qcos^7ru, c, =2Q,
% = 2S COS Is ITU, Co =2S,
X = 2Acosi7r(M + i)) + 2A'cosi7r(M-'y), c-, =2A + 2A',
^4 =1. C4 = 1,
^» =—2Q sin ^TTM,
&. = 2/Scosi7n>, c, =2S,
^7 = — 2A sin Jtt (m + v) - 2A' sin ^ir {u - v),
% =1, Cs = 1,
&, = 2Qcosi7m, c, =2Q,
^10 = — 2S sin ^TTV,
^,1 = - 2A sin ^TT (m + 1)) + 2A' sin ^tt (m - v),
X. = 1, c, = l,
^13 = — 2Q sin ^TTW,
^14 = — 2/Ssin ^TTj;,
^„ = - 2A cos J^TT {u + v) + 2A' cos iTT (it - V), c,5 = - 2A + 2A'.
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 501
73. The expansions might be carried further; we have for instance
&„ («) = 1 + 2^ cos TTM + 2S* cos TTV, Co = 1 + 2Q* + 2S^
^4 =1-2Q' „ +2S* „ , c,=l-2Q* + 2S*,
&, =1 + 20' „ -2-S* „ , 0^=1 + 2(^-28*,
^„ =1-20* .. -2-S* „ . C, = 1-2Q*-2S^
^1 = 2Q cos ^TTM + 2Q» cos f TTM + 2.4 cos ^ TT (u + 2t;) + 2 J.' cos J tt (m - 2v),
C=2Q + 2Q> + 2A + 2A',
^5 = -2Qsin^7ru + 2Q»sinf7TO- 24siniTr(M+2i;)-2i4'sin j7r(M-2v),
&, = 2Q cos ^TTU + 2Q» cos firM - 2A cos ^ir {u + 2v) — 2A' cos ^tt (m — 2i)),
c, =2Q+2Q»-2^-24',
^is = -2Qsin^7n( + 2Q»sinf7rw + 24sin|7r(« + 2«) + 2-4'sin|7r(M-2t)),
in which last formulae
A30! A'^jS*"
A = Qi^*s^, = ^ ; ^' = QR-'S*. = -^.
74. In the single-and-double-letter notation we have six letters A, B, C, D, E, F;
and the duads AB, AC, ..., BE are used as abbreviations for the double triads ABF,
CLE, &c., the letter F always accompanying the expressed duad ; there are thus in
all six single-letter symbols, and 10 double-letter symbols, in all 16 symbols, cor-
responding to the double-theta functions, as already mentioned in the order
^0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
BD, CE, CD, BE, AC, C, AB, B, EC, DE, F, A, AD, D, E, AE,
where observe that the single letters C, B, F, A, D, E correspond to the odd functions
5, 7, 10, 11, 13, 14 respectively.
75. The ground of the notation is as follows : —
The algebraical relations between the double theta-fiinctions are such that, intro-
ducing six constant quantities o, h, c, d, e, f and two variable quantities x, y, it
is allowable to express the 16 functions as proportional to given functions of these
quantities (a, h, c, d, e, /; x, y) ; viz. there are six functions the notations of which
depend on the single letters a, b, c, d, e, f respectively, and 10 functions the notations
of which depend on the pairs ab, ac, ad, ae, be, bd, be, cd, ce, de respectively: each of
the 16 functions is, in fact, proportional to the product of two factors, viz. a constant
factor depending only on (a, b, c, d, e, /), and a variable factor containing also x
and y. Attending in the first instance to the variable factors, and writing for
shortness
a — x, b — x, c — x, d — x, e — x, f— x = a, h , c, d, e, f; x — y = 6;
a-y, b-y, c-y, d-y, e-y,f-y = &^, b,, c„ d,, e„ f,;
502 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
these are of the forms
»/a = Vaa], Vat = ^ {Vabfc,d,e, + Va^bXcde}.
I remark that to avoid ambiguity the squares of these expressions are throughout
written as (Vo)' and (Va6)* respectively.
76. There is, for the constant factors, a like single-and-double-letter notation which
will be mentioned presently; but in the first instance I use for the even functions
the before-mentioned 10 letters c, and for the odd ones six letters k. I assume
that the values x, y = oo , oo (ratio not determined) correspond to the values w = 0,
0 = 0 of the arguments; and that w is a function of (x, y) which, when (x, y) are
interchanged, changes only its sign, and which for indefinitely large values of {x, y)
becomes =-. — r|. This being so, we write (adding a second column which will be
afterwards explained)
0=BD = toe, \^bd,
l = CE = „Ci v/ce,
2= CD==„Ca v^cd,
S = BE = „c, Vhe,
4t = AG — „Ci v^ac,
&=AB = „c, Vab,
7= B =„hVh,
8 = BG = „c Vbc,
9 = DE = „c>y/de,
10= i' =„K^,
11=4= ,,^11 Va,
12 = .4Z) = „c,j\/ad,
13= i) =„k,yd.
14= E =„h.^e, k, =„</\
15 = AE=„c^t\/ae, Cij=„v^;
viz. here, on writing x, y = so , oo , each of the functions Vtd, &c. becomes = 2 ^^ ■
_ a;-y'
and each of the functions Va, &c., becomes = Va;y ; hence by reason of the assumed
Co
=x-yU,
Ci
= ,, v^ce,
Cj
= „ v^^,
C3
=„<^re,
C4
= „ v^oc,
k.
= „ v^c,
Ce
= .,^^^,
k.
= .,^^,
Cs
= „ v^Tc,
C9
= „v^.
"-10
1
=.n
h
I
= „v/a,
Cm
-,.</^.
k:
!
= ,M
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 503
form of ft), the odd functions each vanish (their evanescent values being proportional
to ki, b,, kio, kii, ki3, ku respectively), while the even functions become equal to Co, Ci,
Cj, C3, C4, Cg, Cs, c„ C12, c,5 respectively.
Observe further that on interchanging x, y, the even functions remain unaltered,
while the odd functions change theii' sign; that is, the interchange of x, y corresponds
to the change u, v into —ii, —v.
77. As to the values of the 10 c's and the six k's in terms of a, b, c, d, e, f,
these are proportional to fourth roots, v a, &c., v oft, &c. ; in v a, a is in the first
instance regarded as standing for the pentad bcdef, and then this is used to denote
a product of differences hc.hd .he. hf .cd.ce.cf.de .df.ef; similarly ab is in the first
instance regarded as standing for the double triad abf. cde, and then each of these
triads is used to denote a product of differences, ab.af.bf and cd.ce.de respectively.
The order of the letters is always the alphabetical one, viz. the single letters and
duads denote pentads and double triads, thus:
a = bcdef, ab = abf . cde,
b — acdef ac = iwf . bde,
c = abdef, \. ad = adf. bee,
d = abcef ae = aef . bed,
e = abcdf be = bcf . ade,
f = abcde, bd = bdf. ace,
be = bef . acd,
cd = cdf. abe,
ce = cef . abd,
de —def.abc.
There is no fear of ambiguity in writing (and we accordingly write) the squares of
Va and Vab as v^a and Vab respectively ; the fourth powers are written (v^a)=
and (v/a6)=; the double stroke of the radical symbol v' is in every case perfectly
distinctive.
This being so we have as above Co = \vbd, &c., ki = X\^c, &c. : it is, however,
important to notice that the fourth roots in question do not denote positive values,
but they are fourth roots each taken with its proper sign (+, — , +i, —{, as the
case may be) so as to satisfy the identical relations which exist between the sixteen
constants; and it is not easy to detei-mine the signs.
The variables w, y are connected with u, v by the differential relations
adu + Tdv = -^&-^\.
■ du+pdv = — i \ -7= — ^
^ Wx \
dy
V7
504
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTION8.
[704
where A'=abcdef, F = a,b,c^d,e,f, ; which equations contain the constants vr, p, a, t, the
values of which will be afterwards connected with the other constants.
78. The c's are expressed as functions of four quantities o, /8, 7, S, and connected
with each other, and with the constants o, b, c, d, e, f, by the formulae
0 = a» + /?■ + 7= + a' = a>„' v/td,
1 = 2(0/3 + 78) = „ v/^,
2 = 2(a7+/35) = „ v^cd,
3 = 2(oS + y87) = „ v/6i,
4 = a=-/y + 7'-8»= „ ^aic,
6=2{eeY-^B) = „ -^ab,
8 = o» + /3»-7'-S»= „ v^tc,
9 = 2(ay3-7S) = „ V^,
16 = 2(a«-^7) = „ \/^.
It hence appears that we can form an arrangement
Cn 1
cx^
c,'
<h\
-c.»,
c.'
<h\
-c>,^
-Cs'
a ,
h.
c
a'.
V,
c'
a",
h",
c"
a system of coefficients in the transformation between two sets of rectangular
coordinates.
We have, between the squares of these coefficients of transformation, a system of
6 + 9 equations
a= +b' +c» =1,
a'» + 6'" + c'» = 1,
a"= + 6"» + c"» = l,
a» +a'» +a"==l,
6" + 6'» + 6"= = 1,
c- +c'» + c"»=l,
fr" + c» = a'» + a"*, b'^ + c'» = a"" + a», 6"" + c"" = a' + a'=,
c> + a» = 6'»+6"», c'» + a''' = 6"» + 6% c" + a"' = b' + b'\
a» + 6' = c'' + c"» , a'" + b'^ = c"' + c\ a"» + 6"» = c- + c'» :
704]
that is,
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
= 0;
605
c*
c*
c*
c*
12
+ 1
+ 6
- 0
9
+ 4
+ 3
- 0
2
+ 15
+ 8
- 0
12
+ 9
+ 2
- 0
1
+ 4
+ 15
- 0
6
+ 3
+ 8
- 0
1
+ 6
- 9
- 2
6
+ 12
- 4
- 15
12
+ 1
- 3
- 8
4
+ 3
_ 2
- 12
3
+ 9
- 15
- 1
9
+ 4
- 8
- 6
15
+ 8
- 12
- 9
8
+ 2
- 1
- 4
2
+ 15
- 6
- 3
and we have, between their products, a system of 6 + 9 equations
a'a" + b'h"-\-c'c" =0,
a'a + h"b + c"c = 0,
aa + hb' + cc' = 0,
be +b'c' +b"c"=0,
ca + c'a' + c"a" = 0,
ab +ab' +a"b" = 0,
a, b, c =b'c"-b"c', c'a"-c"a', a'b"-a"b',
a', b', c' b"c-bc", c"a-ca", ab"-a"b,
a", b", c" be' -b'c , ca' -c'a , ab' -a'b :
C. X.
64
506
that is,
A MEMOIR ON THE SINGLE AND DOUBLE TH ETA -FUNCTIONS.
= 0;
[704
c»
c"
c»
c»
c»
<?
9
2
+ 4
15
- 3
8
2
12
- 15
1
- 8
6
12
9
- 1
4
+ 6
3
1
6
- 4
3
+ 15
8
6
12
+ 3
9
- 8
2
12
1
- 9
4
- 2
15
- 0
12
+ 4
8
+ 3
15
- 0
1
+ 3
2
+ 8
9
- 0
6
- 9
15
+ 2
4
- 0
9
-15
6
+ 8
1
+ 0
4
- 8
12
- 6
2
- 0
3
- 12
15
- 1
2
- 0
2
+ 1
3
+ 4
6
+ 0
15
+ 6
9
- 3
12
+ 0
8
- 12
4
- 9
1
each of the first set of 15 giving a homogeneous linear relation between four terms
c*; and each of the second set giving a homogeneous linear relation between three
terms c'.c', formed with the 10 constants c. Thus the ni-st equation is
c,./ + c,* + C6^-Co* = 0;
and so for the other lines of the two diagrams.
79. I form in the two notations the following tables: —
Table of the 16 Kummeu hexads.
A
A
A
A
A
B
B
B
B
C
C
C
D
D
E
A
B
C
D
B
F
C
D
E
F
D
E
F
E
F
F
B
AB
AC
AD
AE
AB
BC
BD
BE
AB
CD
CE
AC
DE
AD
AE
C
CD
BD
BC
BC
AC
AD
AC
AC
BC
AB
AB
BC
AB
BD
BE
D
CE
BE
BE
BD
AD
AE
AE
AD
BD
AE
AD
CD
AC
CD
CE
E
DE
DE
CE
CD
AE
DE
CE
CD
BE
BE
BD
CE
BC
DE
DE
F
11
11
11
11
11
7
7
7
7
5
5
5
13
13
14
11
7
5
13
14
10
5
13
14
10
13
14
10
14
10
10
7
6
4
14
12
6
8
0
3
6
2
1
4
9
12
15
5
2
0
8
8
4
12
4
4
8
6
6
8
6
0
3
13
1
3
3
3
12
15
15
12
0
15
12
2
4
2
1
14
9
9
1
2
15
9
1
2
3
3
0
1
8
9
9
10
704]
A MEMOIE ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
507
SS
o
g
O
H
n
e5
I
fe,
0
§
0
R5
ft)
ft!
aq
ft!
0
ft)
0
fe^
^5
"=5
C
"n
C3
-t;
a;
0
C)
!<.
^
0
«;
1^
9
0
ftl
c:^
=5
0
^
^
'^
a;
0
=^
fe)
0
1 fe5
0
a^
!;i
^
aq
<^
(5|
«5
0
ft,
'^
a;
0
!«,
S
5S)
=5
sq
0
^
^
§
ft)
0
C)
«3
ft)
-^
a^
c;
ft)
fel
ft)
S5
ft!
^
0
a^
'=5
a;
?;:>
0
=5
ft,
-<^
a;
q
c>
0
«;
S
0
0
ft)
a;
ft)
a;
!;3
0
0
^
0
ft,
'^
.a^
i
65
&,
g
C5
0
a?
^
§
^
05
0
ft)
^
0
<^
^
fe3
C«3
0
ft)
ft!
«5
0
«5
«5
0
«;
ft,'
"^
0
q
C5
§
0
ft)
0
fti
«?
03
^
^
ft.
^
0
ft)
C)
§
§ i
0
«;
ft)
0
=q
=5
^ 1
ft.
^
«^
f^
*i
S
S
§
0
C5
ft)
'"?
§
§
^
a?
0
<^
ft)
fej
05
0
ftl
ft)
0
a;
0
a?
"«!
«5
0
"^
"^
ft,
a;
d
0
c^
§
C5
0
^
ft)
a;
^
'<«
«5
''J
0
^
ft)
ft,
a?
0
ft)
0
^
C5
>3
fti
0
ft)
a;
§
^
55
^
'^
ft,
a?
c>
ft)
OQ
^
0;
ft)
09
§
s
5
■^
0
''1
ft)
&;
0
=5
ft)
0
(M
0
00
in
CO
I-H
05
I— 1
to
-*
T—,
I—,
t~
in
CO
t-H
0
1—,
n
00
c,
0
CI
OJ
I— ,
to
■*
in
)-H
t^
in
':(<
-,*
1—1
CO
l-H
1
j a>
00
■*
to
CO
T— 1
1—,
0
CI
' d
l-H
I-H
1:~
in
0
f— «
CJ
a
0
-*
00
Cl
l-H
m
«o
1—i
in
f-H
t^
CO
I-H
l-H
t— 1
«
CO
O)
1
CO
0
Cl
to
10
■*
00
Cl
0
f— f
1
^
l:~
CO
(M
0
OJ
CI
CO
in
l-H
to
>o
■«*<
00
I— 1
0
I— t
l-H
l-H
Ir-
l-H
0
05
I— t
01
to
00
0
CO
t~
■*
in
1—t
in
CO
l-H
l-H
-<),
in
1—4
,_,
o»
CO
CI
Cl
r-H
-*
t^
<D
00
0
0
1— 1
l-H
l-H
m
CO
l-H
(M
(N
OS
0
l-H
in
■*
t^
(O
00
CO
0
l-H
l-H
l-H
in
l-H
in
-*
CI
1— t
00
OS
■n
CJ
l-H
t~
to
0
CO
©
f-H
f-H
I— t
CO
l-H
l-H
0
r-t
04
I—,
CI
(O
•«c
Cl
l-H
m
l-H
^
00
0
eo
t-
in
CO
l-H
l-H
»— 1
<n
r-H
OS
10
l-H
Cl
0
00
f-H
I— 1
«o
-*
CI
l-H
0
f-H
t-
in
eo
l-H
I— 1
0
<M
05
CI
l-H
CO
00
to
-*
in
r— 1
0
f-H
l:~
in
l-H
10
00
0<>
f— 1
•*
o>
CO
0
1—t
to
f-H
in
l-H
0
l-H
t~
CO
■*
1-
00
0
CO 1
to
03
f— 1
Cl
-*
CI
m '
0
f-H
in
eo
f-H
r-H
G4— 2
508
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTION8, [704
81. Table of the 60 GOpel tetrads.
A.B.AE.BE
A.B.AD.BD
A.B.AC .BC
C .D.CE .DE
CE. CD.DE
CF .AB.DE
E .F .AB .CD
D.F .AB .CE
D.E.CD.CE
AC .BD.AD.BC
AC .BE .AE .BC
AD.BE .AE . BD
A.C.AE.CE
A.C .AD.CD
A.C.AB.BC
B.D.BE.DE
B.E.BD DE
B.F.AC .DE
E .F.AC .BD
D.F.AC .BE
D.E.BD.BE
AB.GD.AD.BC
AB .CE .AE .BC
AD.CE .AE.CD
A.D.AE.DE
A. D.AC. CD
A.D.AB.BD
B.C .BE .CE
B.E.BC .CE
B.F.AD.CE
E .F.AD.BC
C .F.AD.BE
C .E. CD.DE
AB.CD.AG . BD
AB .DE .AE.BD
AC .DE. AE.CD
A.E.AD.DE
A.E.AC .CE
A.E.AB.BE
B.C.BD.CD
B.D.BC .CD
B.F.AE.CD
D.F.AE.BG
C .F.AE.BD
C .D.BC .BD
AB.CE.AC . BE
AB .DE .AD.BE
AC .DE. AD.CE
A.F .BC .DE
A.F .BD. CE
A.F. BE. CD
B. CAB. AC
B.D.AB.AD
B.E .AB .AE
D.E .AD.AE
C .E .AC .AE
G .D .AC .AD
BD .CE .BE . CD
BC .DE.BE.GD
BC .DE.BD.CE
11 7
15
3
5
13
1
9
14
10
6
2
4
0
12
8
11 7
12
0
5
14
2
9
13
10
6
1
4
3
15
8
11 7
4
8
5
10
6
9
13
14
2
1
12
3
15
0
11 5
\5
1
7
13
3
9
14
10
4
0
6
2
12
8
11 5
12
2
7
14
0
9
13
10
4
3
6
1
15
8
11 . 5
6
8
7
10
4
9
13
14
0
3
12
1
15
2
11 13
15
9
7
5
3
1
14
10
12
8
6
2
4
0
11 13
4
2
7
14
8
1
5
10
12
3
6
9
15
0
11 13
6
0
7
10
12
1
5
14
2
9
4
9
15
2
11 14
12
9
7
5
0
2
13
10
15
8
6
1
4
3
11 14
4
1
7
13
8
2
5
10
15
0
6
9
12
3
11 14
6
3
7
10
16
2
5
13
8
0
4
9
12
1
11 10
8
9
7
5
6
4
13
14
12
15
0
1
3
2
11 10
0
1
7
13
6
12
5
14
4
15
8
9
3
2
11 10
3
2
7
14
6
15
5
13
4
12
8
9
0
1
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 509
The prodvct-theorem, and its results.
82. The product-theorem was
K;:f)<"+"'>-*(;: ?)<«-«'>
7+7 , 6+0 7—7 . 0—6
where only one argument is exhibited, viz. m + it', u — ti, 2m, 2m' are written in place
of (m + m', v + v'), (it — u', v — v'), (2m, 2v), (2m', 2v') respectively. The expression on the
right-hand side is always a sum of four terms, corresponding to the values (0, 0),
(1, 0), (0, 1), and (1, 1) of (p, q). For the development of the results it was found
convenient to use the following auxiliary diagram.
Upper half of characteristic.
A
I-H
l^H
J_,
.— »
f-H
1— <
I.H
+
+
+
+
+
+
+
+
'd
20.
'a
So.
=??«.■
'a
«a.
~8 ca.
~«
«a.
a
5a.
'a
^
+
+
1
1
+ +
1
1
+ +
1
1
+
+
1
1
C5 OX
*B
JQ.
^
s
a
s
a «i
a
s
^S
^
S
^
s
^
s
•HN
-Ot
-*»
-^M
; HN -*>
-4S
1
0
0 1
-*«
H«
•^1
'HN
HN
-♦m
0 0
0
0
i 0
0
0
0
1
•< 1 0
0
1
1
1
1
1
1 0
Q
0
i
0
i
0
i# 0
0
i 1
i
1
1 3
1 '
1
3
1
0 1
0
0
0
h
0
h
1 i
1
h
« 1
0
3
1
1
3
1
3
If
1 1
0
0
i
h
J
i
if h
3
h
1 3
i
3
IT
3
2
3
2
3
2
3
0 0
1
0
h
0
0
i
0
i 1
.•J
1
3
1
h
1
1 0
1
0
1
0
0
0
j 0 0
1
0
1 1
0
1
0
1
1
1
0 1
1
0
i
J
3
h
1 h
i
J
J f
3
IT
3
2
3
1
i
3
1 1
1
0
1
h
0
i
0 i
I
1
1 f
0
3
0
1
1
3
2"
0 0
0
^ !
0
h
0
1
ll *
1
3
IT
0 f
0
h
1
3
1
J
1 0
0
i
h
i
3
If i
3
3
■J
h f
i
h
t
3
2
3
i
0 1
0
0
1
0
0
1 1 1
1
0
0 0
0
1
1
0
1
1
1 1
0
i
1
i
0
if 1
3
5
0
i 0
i
1
3
If
0
3
■J"
1
0 0
1
h
i
3
1
If i
i
3
h 1
t
i
3
3
2
h
h
1 0
1
1
h
0
3
0 h
1
3
5^
1 t
0
h
0
3
2
1
h
0 1
1
1 1
i
1
3
5
0
1 1
i
0
h 0
3
1
3
0
i
1
1 1
1
1
1
0
0
0 1
1
0
1 0
0
1
0
0
1
1
510
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
Lower half of characteristic.
^••o
Vto
+ +
1
1
+ +
1
1
£0
V2o
+ +
1
to
1
eo
+ +
1
1
0 0
0 0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
1 0
0 0
1 0
1
0
1 0
1
0
1 0
1
0
1 0
1
0
0 1
0 0
0 1
0
1
0 1
0
1
0 1
0
I
0 1
0
1
0 0
1 1
1
1
1 1
1
1
1 1
1
1
1 1
1
1
1 0
1 0
- 1
0
1 0
- 1
0
1 0
- 1
0
1 0
-1
0
1 0
2 0
0
ol
2 0
0
0
2 0
0
0
2 0
0
0
1 0
1 1
- 1
1
1 1
- 1
1
1 1
- 1
1
1 1
- 1
1
1 0
2 1
0
1
1
2 1
0
1
2 1
0
1
2 1
0
1
0 0
0 1
0 1
0
-1
0 1
0
- 1
; 0 1
0
- 1
'. 0 1
0
- 1
0 1
1 1
1
- 1
1 1
1
- 1
1 1
1
- 1
' 1 1
1
- 1
0 1
0 2
a
0 j
0 2
0
0
0 2
0
0
10 2
0
0
0 1
1 2
1
0
1 2
1
0
1 2
1
0
1 2
1
0
1 1
1 1
- 1
-1
1 1
- 1
- 1
1 1
- 1
- 1
1 1
- 1
- 1
1 1
2 1
0
- 1
2 1
0
- 1
2 1
0
- 1
2 1
0
- 1
0 1
1 1
1 2
- 1
0
0 2
- 1
0
0 2
- 1
0
' 1 2
- 1
0
1 1
2 2
0
0
2 2
0
0
2 2
0
0
r- '
0
0
83. The upper characters of the 0's have thus the values 0, 1, ^, f ; the lower
characters are originally 2, 1, 0, or —1, and these have when necessary to be, by
the addition or subtraction of 2, reduced to 0 or 1 ; the effect of this change is
either to leave the 0 unaltered, or to multiply it by — 1 or + i, as follows :
= -i0*,
7
3
= 10^,
7
where only the first column of characters is shown, but the same rule applies to the
second column ; and where we must of course combine the multipliers corresponding
to the first and second columns respectively : for instance
7+26+2 76 70
«0 -
®7±2-
7
0* -
*^7 + 2"
<■
•^7-2
0^ -
-<■
0t =.
7 + 2
-.^,
«' + 2
•04]
A MEMOIR OX THE SINGLE AND DOUBLE THETA-FUNCTIONS.
511
Thus taking the tenth line of the upper half, and the fifth line of the lower half,
we have
10
01
i 1 i f
tiff
i f i i
i f i i
00
10
1 0-1 0
1 0-1 0
1 0-1 0
1 0-1 0
giving the value of
^ll(u + u').^ll{u-a')-
viz. this is
2 i f \ —{©t^c \ lai i I
+ ®i o^")-®_?5(.. ) -i^l o("^-®i 0^")'
where the first column is the value given directly by the diagram: it is then reduced
to that given by the second colunin.
84. But instead of the 0's, we introduce single letters (X, Y, Z, W), {E, F, G, H),
{I, J, K, L), (M, N, P, Q), with the suffixes (0, 1, 2, 3), in all 64 symbols, thus
0 00 10 01 11 (2u) = X Y Z W
00
0
10
1
01
2
11
3
that is,
0^(2«) = X, 0j^=F, &c.,
that is,
05J(2«) = Z,,..
0 ^0 il fO |1 {2u) =
E F G H
00
0
10
1
01
2
11
3
0^(2.) = A'.
&c.,
512
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
e OJ li Of 1H2«)= I J K L
[704
00
10
01
11
The functions of (2«') are denoted in like manner by accented letters
e^(2«')=^'. &c..
« H J4_Ji_J_(2«)= M N P Q
0
00
10
01
11
85. To simplify the expression of the results, instead of in each case writing
down the suffixes, I have indicated them by means of the column headed " Suff."
Thus
Sufi.
I 8-0 I ^l]u-\-n'.'^^^u-u' = XX' +YY' + ZZ' + WW |2
01 ' 00
means that the equation is to be read
= X,X; + KF/ + ZX' + W,W,'.
It is hardly necessary to mention that the | 8 - 0 of the left-hand column shows
the current numbers of the theta-functions ; viz. the left-hand side of the equation is
^8(m-|-m').^„(m-m').
And by a preceding remark the single arguments u + u' and u-u' are written in
place of (m -I- m', v + v') and (m - u', v - v') respectivel} .
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
The 256 equations now are
86. First set, 64 equations.
513
I
0-0
nOO
„oo
= XX'
+ rr
+ ZZ'
+ WW
Snffixes.
0
i-O
00
10
00
00
= XX
+ Yl"
+ ZZ'
+ WW
1
8-0
00
01
00
00
= XX'
+ YY'
+ ZZ'
+ WW
2
12-0
00
11
00
00
= XX'
+ YY'
+ ZZ'
+ WW
3
0-4
„oo
„oo
•^10"-"
= XX'
-YY
+ ZZ'
- WW
1
4-4
00
10
00
10
= XX
-YY'
^^ZZ'
- WW
0
8-4
00
01
00
10
= XX-
-YY'
+ ZZ'
-WW
3
12-4
00
11
00
10
= XX'
-YY'
^ZZ'
- WW
2
0-8
S^f^U + U
nOO
= XX'
+ YY
-ZZ!
- WW
2
4-8
00
10
00
01
= XX'
+ YY'
-ZZ
- WW
3
8-8
00
01
00
01
= XX'
+ YY
-ZZ'
-WW
0
12-8
00
11
00
01
= XX
+ YY
-ZZ
- WW
1
0-12
„oo
SqqU + U
„oo
•^11"-"
= XX'
~YY'
-ZZ'
+ WW
3
4-12
00
10
00
11
= XX'
- YY'
-ZZ'
+ WW
2
8- 12
00
01
00
11
= XX'
- YY'
-ZZ'
+ WW
1
12-12
00
11
00
11
= XX'
- YY'
-ZZ'
+ WW
0
C. X.
65
514
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
Fii-st set, 64 equations (continit^d).
1-1
'J2»-'
•»:»-«■
= rx" +xr
+ WZ'
+ ZW
Suffixes.
0
6-1
10
10
10
00
= YX' +zr
+ WZ
+ ZW
1
9-1
10
01
10
00
= YX" +XY'
+ WZ'
+ ZW
2
13-1
10
11
10
00
= YX' +XY'
+ WZ'
+ ZW
3
1-5
a 10
olO
•^10"-"
= YX' -XY'
+ WZ'
-ZW
1
5-5
10
10
10
10
= -YX! ^XY'
- WZ'
+ ZW'
0
9-5
10
01
10
10
= YX' -XY'
+ WZ
-ZW'
3
13-5
10
11
10
10
=-yx' +zr
- WZ'
+ ZW'
2
1-9
^s— ■
qIO
= YX' -vXY'
- WZ'
-ZW'
2
5-9
10
10
10
01
= YX' +zr
- WZ'
-ZW'
3
9-9
10
01
10
01
= YX +XY'
- WZ'
-ZW'
0
13-9
10
11
10
01
= YX +zr
- WZ"
-ZW
1
1-13
„io
3qqU + u
„io
^l"-«
= YX' -XY'
-WZ'
+ ZW'
3
6-13
10
10
10
11
= -YX +xr
+ WZ'
-ZW'
2
9-13
10
01
10
11
= YX -XY'
- WZ"
+ ZW
1
13-13
10
11
10
11
=-Yx +zr
+ WZ"
-ZW'
0
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
515
First set, 64 equations (continued).
2-2
a 01
a 01
= ZX'
+ wv
+ XZ' + YW'
Suffixes.
0
6-2
01 *
10
01
00
= ZJT
+ Wl"
+ XZ' + YW'
1
10-2
01
01
01
00
= ZX'
+ WY'
+ XZ' + YW'
2
14-2
01
11
01
00
= ZX'
+ WY'
+ XZ' + YW'
3
2-6
oOl
SqqU + U
oOl
= ZX
- WY'
+ XZ' -YW'
1
6-6
01
10
01
10
= ZX
- WY'
+ XZ' - YW'
0
10-6
01
01
01
10
= ZX'
- WY'
+ XZ' - YW'
3
14-6
01
11
01
10
= ZX'
-WY'
+ XZ' -YW'
2
2-10
a 01
■»:!-«•
= ZX'
+ WY'
-XZ' -YW'
2
6-10
01
10
01
01
= ZX'
+ WY'
- XZ' - YW'
3
10-10
01
01
01
01
= -ZX'
- WY'
+ XZ' + YW'
0
14-10
01
11
01
01
= -ZX'
- WY'
+ XZ' + YW'
1
2-14
':;»-■
a 01
= ZX
- WY'
- XZ' + YW'
3
6-14
01
10
01
11
= ZX'
- WY'
-XZ' +YW'
2
10-14
01
01
01
11
= -ZX
+ WY'
+ XZ' - YW'
1
14-14
01
11
01
11
= -ZX'
+ WY'
+ XZ' -YW'
0
65—2
516
A MEMOIR ON THE SINGLE AND DODBLE THETA-FUNCTIONS.
[704
First set, 64 equations (concluded).
3-3
^^^»-
all
^00"-"
= WX'
+ ZY'
+ YZ'
+ XW'
Saffizes.
0
7-3
11
10
11
00
= WX'
+ ZY'
+ YZ'
+ XW'
1
11-3
11
01
11
00
= WX'
+ ZY'
+ YZ'
+ XW'
2
15-3
11
11
11
00
= WX'
+ ZY'
+ YZ'
+ XW'
3
3-7
all
all
= WX'
-ZY'
+ YZ'
-XW'
1
7-7
11
10
11
10
= -WX'
+ ZY'
-YZ'
+ XW'
0
11-7
11
01
11
10
= WX'
-ZY'
+ YZ'
-XW'
3
15-7
11
11
11
10
= -WX'
+ ZY'
-YZ'
+ XW'
2
3-11
Sllu..'
<"-"'
= WX'
+ ZY'
-YZ'
-XW'
2
7-11
11
10
11
01
= WX'
+ ZY'
-YZ'
-XW'
3
11-11
11
01
11
01
= -WX'
-ZY'
+ YZ'
+ XW'
0
15-11
11
11
11
01
= - WX'
-ZY'
+ YZ'
+ XW'
1
3-15
a'J^u^w
.,;;„-.
= WX'
-ZY'
-YZ'
+ XW'
3
7-15
11
10
11
11
= -WX''
+ ZY'
+ YZ'
-XW'
2
11-15
11
01
11
11
= -WX'
+ ZY'
+ YZ'
-XW'
1
15-16
11
11
11
11
= WX'
-ZY'
- YZ'
+ XW'
0
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
517
87. Second set, G4 equations.
1-0
a 10
.^Z"-"'
= EE'
+ GG'
+ FF'
+ HH'
Suffixes.
0
5-0
10
10
00
00
= EE'
+ GG'
+ FF'
+ HH'
1
9-0
10
01
00
00
= EE'
+ GG'
+ FF'
+ HH'
2
13-0
10
11
00
00
= EE'
+ GG'
+ FF
+ HH'
3
1-4
a 10
nOO
= - iEE'
+ iGG'
-iFF'
+ iHH'
1
5-4
10
10
00
10
= iEE'
-iGG'
+ iFF'
-iHH'
0
9-4
10
01
00
10
^-iEE'
+ iGG'
-iFF'
+ iHH'
3
13-4
10
11
00
10
= iEF
-iGG'
+ iFF'
- iHH'
2
1-8
.J2-.'
•^2?«-'
= EE"
+ GG'
- FF'
- HH'
2
5-8
10
10
00
01
= EE'
+ GG'
- FF'
- HH
3
9-8
10
01
00
01
= EE'
+ GG'
- FF'
- HH'
0
13-8
10
11
00
01
= EE
+ GG'
- FF'
- HH'
1
1-12
^Z'*"'
.».-w
= -iEE'
+ iGG'
+ iFF'
-iHH
3
5-12
10
10
00
11
= iEE
- iGG'
-iFF'
+ iHH'
2
9-12
10
01
00
11
= -iEE'
+ iGG'
+ iFF'
-iHH'
1
13-12
10
11
00
11
= iEE"
-iGG'
-iFF'
■viHH'
0
518
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
Second set, 64 equations {continued).
0-1
a 00
■^:»-"'
= EG'
+ GE'
+ FH'
+ HF'
Saffizen.
0
4-1
00
10
10
00
= EG'
+ GE"
+ Fir
+ HF'
1
8-1
00
01
10
00
= EG'
+ GE'
+ FW
+ HF'
2
12-1
00
11
10
00
= EG'
+ GE'
+ Fir
+ HF'
3
0-5
.»»....
„io
= iEG'
-iGE'
+ iFH'
-iHF'
1
4-5
00
10
10
10
= iEG'
-iGE'
+ iFH'
- iHF'
0
8-5
00
01
10
10
= iEG'
-iGE'
+ iFU'
-iHF'
3
12-5
00
11
10
10
= iEG'
-iGE
+ iFH'
-iHF'
2
0-9
»22— •
«io
^oi"-"
= EG'
^.GE'
- FH'
- HF'
2
4-9
00
10
10
01
= EG'
+ GE
- FH'
- HF'
3
8-9
00
01
10
01
= EG'
+ GE
- FH'
- HF'
0
12-9
00
11
10
01
= EG'
+ GE
- FH'
- HF'
1
0-13
„oo
.-„-..
= iEG'
-iGE
-iFir
+ iHF'
3
4-13
00
10
10
11
= iEG'
-iGE
-iFH'
+ iHF'
2
8-13
00
01
10
11
= iEG'
-iGE'
-iFH'
+ iHF'
1
12-13
00
11
10
11
= iEG'
-iGE
-iFir
+ iHF'
0
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
519
Second set, 64 equations {continued).
3-2
Sllu.u'
.01
= FH'
+ IIG'
+ EF'
+ GH'
Suffixes.
0
7-2
11
10
01
00
= FE'
+ HG'
+ EF'
+ GH'
1
11-2
11
01
01
00
= FE'
+ HG'
+ EF'
+ GH'
2
15-2
11
11
01
00
= FE
+ HG'
+ EF'
+ GH'
3
3-6
„ 11
nOl
= -iFE
+ iHG'
-iEF'
+ iGH'
1
7-6
11
10
01 ^
10
= iFE
-iHG'
■viEF'
-iGH'
0
11-6
11
01
01
10
= -iFE
+ iHG'
-iEF'
+ iGH'
3
15-6
11
11
01
10
= iFE'
- iHG'
+ iEF'
-iGH'
2
3-10
^llu.u'
„01
= FE'
+ HG'
- EF'
- GH'
2
7-10
11
10
01
01
= FE
+ HG'
- EF'
- GH'
3
11-10
11
01
01
01
= - FE'
- HG'
+ EF'
+ GH'
0
15-10
11
11
01
01
= - FE
- HG'
+ EF'
+ GH'
1
3-14
sllu.W
lOl
■'ll"-"
= -iFE
+ iHG'
+ iEF'
-iGH'
3
7-14
11
10
01
11
= iFE
-iHG'
-iEF'
+ iGH'
2
11-14
11
01
01
11
= iFE
-iHG'
-iEF'
+ iGH'
1
15-14
11
11
01
11
= -iFE
+ inG'
+ iEF'
-iGH'
0
520
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8.
[704
Second set, 64 equatioiis (concluded).
„ 01
« 11
2-3
■» A«
« + u'
. ^ -„ M - m'
00
00
01
11
6-3
10
00
01
11
0-3
01
00
01
11
4 — 3
11
00
= FG' + HE' + EW + GF'
= FG' + HE + EH' + GF'
= FG' + HE + EH' + GF'
= FG' + HE' + EH' + GF'
2-7
6-7
10-7
U-7
a 01 . .n
00
01
10
01
01
01
11
10
11
10
11
10
11
10
iFG' -iHE' +iEH' -iGF'
= iFG' -iHE' +iEH' -iGF'
= iFG' -iHE' +iEH' -iGF'
= iFG' -iHE' +iEH' -iGF'
2-11
6-11
10-11
14-11
qOI , „11
00
01
10
01
01
01
11
01
11
01
11
01
11
01
= FG' + HE' - EH - GF'
= FG' + HE' - EH' - GF'
= - FG' - HE' + EH' + GF'
= - FG' -HE' + EH' + GF'
01
11
2-15 ^„„m + m'.5,, m-m' = iFG' -iHE' -iEH' +iGF'
00
11
6-15
10-15
14-15
01
10
01
01
01
11
= iFG' -iHE' -iEH' +iGF'
= -iFG' +iHE' +iEH' -iGF'
= -iFG' +iHE' +iEH' -iGF'
Suffixes.
0
1
2
3
1
0
3
2
2
3
0
1
3
2
1
0
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
521
88. Third set, 64 equations.
2-0
^qqU + U
nOO
= //'
+ jj'
+ KK'
+ LL'
Sn£Bxes.
0
6-0
01
10
00
00
= //
+ JJ'
+ KK'
+ LL'
1
10-0
01
01
00
00
= //'
+ JJ'
+ KK'
+ LL'
2
14-0
01
11
00
00
= //'
+ JJ'
+ KK'
+ LL'
3
2-4
„oi
„oo
= //'
- JJ'
+ KK'
- LL
1
6-4
01
10
00 '
10
= //'
- J J'
+ KK'
- LL'
0
10-4
01
01
00
10
= //■
- JJ'
+ KK'
- LL'
3 '
14-4
01
11
00
10
= //'
- J J'
+ KK'
- LL'
2
2-8
a 01
.»«-„■
=-i//'
-UJ'
+ iKK'
+ iLL'
2
6-8
01
10
00
01
= -ijr
- ijj'
+ iKK'
+ iLL'
3
10-8
01
01
00
01
= iir
+ ijj'
-iKK'
-iLL'
0
14-8
01
11
00
01
= iir
+ ijj'
- iKK'
-iLL'
1
2-12
nOl
„oo
= -iir
+ ijj'
+ iKK'
-iLL'
3
6-12
01
10
00
11
=-iir
+ ijj'
+ iKK'
-iLL'
2
10-12
01
01
00
11
= iir
-ijj'
- iKK'
+ iLL'
1
14-12
01
11
00
11
= iir
-iJj'
- iKK'
+ iLL'
0
C. X.
66
522
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
Third set, 64 equations (continued).
3-1
^5J« + "'
= //'
+ iJ'
+ LK'
+ KL'
Saffixes.
0
7-1
10
00
= jr
+ IJ'
+ LK'
+ KL'
1
11-1
10
00
= jj'
+ IJ'
+ LK'
+ KL'
2
15-1
10
00
= jr
+ //'
+ LK'
+ KL'
3
3-5
3 u + u'
= jr
- I J'
+ LK'
- KL'
1
7-5
10
10
=- jr
+ IJ'
- LK'
+ KL'
0
11-5
10
10
= jr
- IJ'
+ LK'
- KL'
3
15-5
10
10
= - Ji'
+ I J'
- LK'
+ KL'
2
3-9
^ M + U'
a 10
= -ur
-iij'
+ iLK'
+ iKL'
2
7-9
10
01
=^-ijr
-iij'
+ iLK'
+ iKL'
3
11-9
10
01
= ur
+ iij'
-ilK'
- iKL'
0
15-9
10
01
= ijr
+ iij'
-iLK'
-iKL'
1
3-13
& u + u'
„io
= -ijr
+ iij'
+ iLK'
-iKL'
3
7-13
10
11
= ur
-iij'
-iLK'
+ iKL'
2
11-13
10
11
= ijr
- iij'
- iLK'
+ iKL'
1
16-13
10
11
= -ijr
+ iij'
+ iLK'
-iKL'
0
ro4]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
523
Third set, 64 equations (continued).
0-2
SqqU + u
a 01
= IK'
+ JU
+ KI'
+ LJ
Suffixes.
0
4-2
00
10
01
00
= IK'
+ JL'
+ KI'
+ LJ'
1
8-2
00
01
01
00
= IK'
+ JL'
+ KI'
+ LJ'
2
12-2
00
11
01
00
= IK'
+ JL'
+ KI
+ LJ
3
0-6
„oo
^qqM + W
„oi
•^10"-"
= IK'
- JL'
+ KI'
- LJ'
1
4-6
00
10
01
10
= IK'
- JL'
+ KI
- LJ'
0
8-6
00
01
01
10
= IK'
- JL'
+ KI'
- LJ'
3
12-6
00
11
01
10
= IK'
- JL'
+ KI'
- LJ'
2
0-10
„oo
..»;.-„■
= UK'
+ UL'
- iKI'
-iLJ'
2
4-10
00
10
01
01
= UK'
+ iJL'
- iKI'
-iLJ'
3
8-10
00
01
01
01
= UK'
+ iJL'
- iKI'
-iLJ'
0
12-10
00
11
01
01
= UK'
+ UL'
- iKI'
-iW
1
0-14
.-»..■
nOl
S u—u
= UK
-UL'
- iKI'
+ iLJ'
3
4-14
00
10
01
11
= UK'
-UL'
-iKI'
+ iLJ'
2
8-14
00
01
01
11
= UK'
-UL'
- iKI'
+ iLJ'
1
12-14
00
11
01
11
= UK'
-UL'
-iKI'
+ iLJ'
0
66—2
^24
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
Third set, 64 equations (concluded).
1-3
»J2"»'
»;;-"■
= JK-
+ IL'
+ LI' + KJ'
Soffixes.
0
5-3
10
10
11
00
= JK'
+ IL
+ LI' + KJ"
1
9-3
10
01
11
00
= JK'
+ IL'
+ LI' + KJ'
2
13-3
10
11
11
00
= JK'
+ IL
+ LI' + KJ'
3
1-7
„ 10
oil
= JK'
- IL
+ LI' - KJ'
1
5-7
10
10
11
10
= - JK'
+ IL
- LI' + KJ'
0
9-7
10
01
11
10
= JK'
- IL
+ LI' - KJ'
3
13-7
10
11
11
10
= - JK'
+ IL
- LI' + KJ'
2
1-11
a 10
all
•^oi"-"
= UK'
+ iIL
-ill' -xKJ'
2
5-11
10
10
11
01
= UK'
+ iIL
-iLI' -iKJ'
3
9-11
10
01
11
01
= UK'
+ iIL
-ill' -iKJ'
0
13-11
10
11
11
01
= UK'
+ iIL
-iLI' -iKJ'
1
1-15
„ 10
.S\\u-u'
= UK'
-ilL
- iLI' + iKJ'
3
5-15
10
10
11
11
= -UK'
+ iIL
+ iLr -iKJ'
2
9-15
10
01
11
11
= UK'
-ilL
- iLI' + iKJ'
1
13-15
10
11
11
11
= - UK'
+ iIL
+ iLr -iKJ'
0
704]
A MKMOIR OX THE SINGLE AND DOUBLE THETA-FUNCTIONS.
525
89. Fourth set, 64 equations.
t
3-0
all
= MM'
+ NN'
+ PF
+ QQ'
Suffixes.
0
7-0
11
10
00
00
= MM'
+ NN'
+ PF
+ QQ'
1
11-0
11
01
00
00
= MM'
+ NN'
+ PF
+ QQ'
2
15-0
11
11
00
00
= MM'
+ NN'
+ PF
+ QQ'
3
3-4
»llu-^u'
„oo
= - iMM'
+ iNN'
-iPF
+ iQQ'
1
7-4
11
10
00
10
= + iMM'
-iNN'
+ iPF
-iQQ'
0
11-4
11
01
00
10
= -iMM'
+ iNN'
-iPF
+ iQQ'
3
15-4
11
11
00
10
= + iMM'
-iNN
+ iPF
-iQQ'
2
3-8
»llu + u'
nOO
•^oi"-"
= -iMM'
-iNN
+ iPF
+ iQQ'
2
7-8
11
10
•00
01
= -iMM'
-iNN
+ iPF
+ iQQ'
3
11-8
11
01
00
01
= iMM'
+ iNN'
-iPF
-iQQ'
0
15-8
11
11
00
01
= IMM'
+ iNN'
-iPF
-iQQ'
1
3-12
»lln + u'
a 00
^ u-u
= - MM'
+ NN
+ PF
- QQ'
3
7-12
11
10
00
11
= + MJf
- NN'
- PF
+ QQ'
2
11-12
11
01
00
11
= + MM'
- NN
- PF
+ QQ'
1
15-12
11
11
00
11
= - MM'
+ NN'
+ PF
- QQ'
0
526
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
Fourth set, 64 equations {corUintted).
2-1
a 01
-j:-.-
= JOT'
+ NM'
+ PQ'
+ QP
Saffixes.
0
6-1
01
10
10
00
= MN'
+ NM'
+ PQ'
+ QP
1
10-1
01
01
10
00
= MJ\r'
+ NM'
+ PQ'
+ QP"
2
U-1
01
11
10
00
= MN'
+ NM'
+ PQ'
+ QP
3
2-5
„«1
„io
•^10"-"
= iMN'
-iNM'
+ iPQ'
-iQP
1
6-5
01
. 10
10
10
= iMN'
- iNM'
+ iPQ'
-iQP
0
10-5
01
01
10
10
= iMN'
-iNM'
+ iPQ'
-iQF
3
U-5
01
11
10
10
= iMN'
-iNM'
+ iPQ'
-iQP'
2
2-9
oOl
nlO
= -iMN'
-iNM'
+ iPQ'
+ iQF
2
6-9
01
10
10
01
= - iMN'
-iNM'
+ iPQ
+ iQF
3
10-9
01
01
10
01
= iMN'
+ iNM'
-iPQ'
-iQP'
0
14-9
01
11
10
01
= iMN'
+ iNM'
-iPQ'
-iQP
1
2-13
nOl
olO
= MN'
- NM'
- PQ'
+ QF
3
6-13
01
10
10
11
= MN'
- NM'
- PQ'
+ QP
2
10-13
01
01
10
11
= - MN'
+ NM'
+ PQ'
- QF
1
14-13
01
11
10
11
= - MN'
+ NM'
+ PQ'
-QF
0
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
527
Fourth set, 64 equations (continued).
1-2
a 10
Ky-
= MF
+ NQ' + PM'
+ QN'
Suffixes.
0
5-2
10
10
01
00
= MF
+ NQ' + PM'
+ QN'
1
9-2
10
01
01
00
= MF
+ NQ' + PM'
+ QN'
2
13-2
10
11
01
00
= MF
+ NQ' + PM'
+ QN'
3
1-6
n 10
„oi
= -iMF
+ iNQ' - iPM'
+ iQN'
1
5-6
10
10
01 i.
10
= IMF
-iNQ' +iPM'
- iQN
0
9-6
10
01
01
10
= -iMF
+ iNQ' -iPM'
+ iQN'
3
13-6
10
11
01
10
= iMF
-iNQ' +iPM'
-iQN'
2
1-10
a 10
a 01
= iMF
+ iNQ' -iPM'
-iQN'
2
5-10
10
10
01
01
= iMF
+ iNQ' -iPM'
-iQN'
3
9-10
10
01
01
01
= iMF
+ iNQ' -iPM'
-iQN'
0
13-10
10
11
01
01
= iMF
+ iNQ' -iPM'
-iQN'
1
1-14
a 10
^qqU + U
..«.-„.
= MF
- NQ' - PM'
+ QN
3
5-14
10
10
01
11
= - MF
+ NQ' + PM'
- QN'
2
9-14
10
01
01
11
= MF
- NQ' - PM'
+ QN'
1
13-14
10
11
01
11
= - MF
+ NQ' + PM'
- QN'
0
528
A MEMOIB ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
Fourth set, 64 equations (concluded).
0-3
„oo
<-'■
= MQ'
+ NF
+ FN'
+ QM'
Saffixes.
0
4-3
00
10
11
00
= MQ'
+ NF
+ FN"
+ QM'
1
8-3
00
01
11
00
= MQ'
+ NF
+ FN'
+ QM'
2
12-3
00
11
11
00
= MQ'
+ NF
+ FN'
+ QM'
3
0-7
':»-•
= iMQ'
-iNF
+ iPN'
-iQM'
1
4-7
00
10
11
10
= iMQ'
-iNF
+ iPN'
-iQM'
0
8-7
00
01
11
10
= iMQ'
- iNF
+ iPN'
-iQM'
3
12-7
00
11
11
10
= iMQ
- iNF
+ iPN'
- iQM'
2
0-11
oOO
.^l\u.u'
= iMQ'
+ iNF
- iPN'
-iQM'
2
4-11
00
10
11
01
= iMQ
+ iNF
- iPN'
-iQM'
3
8-11
00
01
11
01
= iMQ'
+ iNF'
-iPN'
-iQM'
0
12-11
00
11
11
01
= iMQ'
+ iNF
- iPN'
-iQM'
1
0-15
a 00
„11
= - MQ'
+ NF
+ PN'
- QM'
3
4-15
00
10
11
11
= - MQ
+ NF
+ PN'
- QM'
2
8-15
00
01
11
11
= - MQ
+ NF
+ FN'
- QM'
1
12-15
00
11
11
11
= - MQ
+ NF
+ FN'
- QM'
0
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
529
90. I re-arrange these in sets of 16 equations, the equations of the first or
square-set of 16 being taken as they stand, but those of the other sets being
combined in pairs by addition and subtraction as will be seen. And I now drop
altogether the characteristics, retaining only the current numbers : thus, in the set of
equations next written down, the first equation is
^0 (u + m') ^„ (u - u') = XX' + YY' + ZZ'+WW':
in the second set, the first equation is
h {% (w -1- u) % (u - u') + \{u + u) ^4 (it - u')] = X,X,' + Z,Z,',
and so in other cases.
First or square-set of 16.
»+«'
«— u'
3
»
(Suffi
xes 0.)
0
0
=
XX'
+ YY'
+ ZZ'
+ WW'
i
4
=
XX'
- YY'
+ ZZ'
- WW'
8
8
rL:
XX'
+ YY'
- ZZ'
- WW'
12
12
=
XX'
- YY'
- ZZ'
+ WW'
1
I
=
YX'
+ XY'
+ WZ'
+ ZW'
5
5
=
- YX'
+ XY'
- WZ'
+ ZW
9
9
=
YX'
+ XY'
- WZ'
- ZW'
13
13
= ■
- YX'
+ XY'
+ WZ'
-ZW
2
2
=
ZX'
+ WY'
+ XZ'
+ YW
6
6
=
ZX'
- WY'
+ XZ'
- YW
10
10
zz .
- ZX'
- WY'
+ XZ'
+ YW
14
14
=
- ZX'
+ WY'
+ XZ'
- YW
3
3
z=
WX'
+ ZY'
+ YZ'
+ xw
7
7
= -
- WX'
+ ZY'
- YZ'
+ xw
11
11
= .
- WX'
- ZY'
+ YZ'
+ xw
15
15
=
WX'
- ZY'
- YZ'
+ xw
91
. Second set of 16.
u+u'
u— u'
u+u'
«— tt'
ii^
. »
+
9 .
^ }
(Suffixes
1-)
4
0
0
4 =
XX' +
ZZ'
12
8
8
12
XX' -
ZZ'
5
1
1
5
YX' +
WZ'
13
9
9
13
YX' -
WZ'
6
2
2
6
ZX' +
XZ'
14
10
10
14
ZX' +
XZ'
7
3
3
7
WX' +
YZ'
15
11
11
15
WX' +
YZ'
«+tt'
tt-tt'
U+u'
It— u'
h{^
. S
-
» .
^}
(Suffixes
1.)
4
0
0
4 =
YY- +
WW
12
8
8
12
YY' -
WW
5
1
1
5
XY' +
ZW
13
9
9
13
XY' -
ZW
6
2
2
6
WY' +
YW
14
10
10
14
WY' +
YW
7
3
3
7
ZY' +
XW
16
11
11
15
ZY' +
XW
C. X.
67
530 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8. [704
92. Third set of 16.
ti-U'
. a +
u+tf
9
u-u'
. J}
(Suffixes 2.)
8
0
0
8 =
XX'
+ YY'
12
4
4
12
XX'
- YY'
9
1
1
9
rx'
+ XY'
13
5
5
13
- YX'
+ zr
10
2
2
10
ZX'
+ WY'
14
6
6
14
ZX'
- WY'
11
3
3
11
wx
+ ZY'
15
7
7
15
- WX'
+ ZY'
u+u'
u— u'
, 3 -
u+u'
u-u'
(Suffixes 2.)
8
0
0
8 =
ZZ'
+ WW'
12
4
4
12
ZZ'
- WW'
9
1
1
9
WZ'
+ ZW'
13
5
5
13
- WZ'
+ ZW'
10
2
2
10
XZ'
+ YW'
14
6
6
14
XZ'
- YW'
11
3
3
11
YZ'
+ XW'
15
7
7
15
- YZ'
+ XW'
93.
Fourth set of 16.
u+u'
«— «'
» +
«+«'
9 .
u— u'
9\
(Suffixes 3.)
12
0
0
12 =
XX'
+ WW'
8
4
4
8
XX'
- WW'
13
1
1
13
YX'
+ ZW'
9
5
5
9
YX'
-ZW'
14
2
2
14
ZX'
+ YW
10
6
6
10
ZX'
- YW'
16
3
3
15
WX'
+ XW
11
7
7
11
WX'
-XW'
«+«'
tt-U'
» -
«+«'
u— u'
9}
(Suffixes 3.)
12
0
0
12 =
YT'
+ ZZ'
8
4
4
8
- YY'
+ ZZ'
13
1
1
13
XY'
+ WZ'
9
5
5
9
-XY'
+ WZ'
14
2
2
14
WY'
+ XZ'
10
6
6
10
- WY'
+ XZ'
15
3
3
15
ZY'
+ YZ'
11
7
7
11
- ZY'
+ YZ'
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA- FUNCTIONS.
94. Fifth set of 16.
531
u+u'
5
I
13
9
7
3
15
11
5
tl+W
+ »
»}
(Suffixes 0.)
1
0
0
1 =
E+ G .
E' + G'
+
F + n.F' -i
■H'
5
4
4
5
i.E-G
JJ
+ i
.F-H
9
8
8
9
E+ G
J>
—
.F+H
13
12
12
13
i.E-G
J)
- i
.F-H
3
2
2
3
F+H
J»
+
E + G
7
6
6
7
i.F - H
it
-»- i
.E-G
11
10
10
11
- .F + H
»>
+
E+G
15
14
14
15
-i.F-H
J>
+ i
.E-G
u+u'
u—u'
u+u'
- » .
«— u'
(Suffixes
0.)
1
0
0
1 =
E-G .
E' -G'
+
F-H. r-
■H'
5
4
4
5
i.E + G
}>
+ i
.F+H
9
8
8
9
E-G
))
—
.F-H
13
12
12
13
i.E+G
JI
— i
.F+H ,
3
2
2
3'
F-H
Jl
+
E-G
7
6
6
7
i.F+H
»J
+ i
.E + G
11
10
10
11
- .F-H
>J
+
E-G
15
14
14
15
-i.F+H
n
+ i
.E+G
95.
Sixth set of 16.
u+u'
u-u'
u+u'
+ 5
U—u'
(Suffixes 1.)
5
0
0
5 =
E - iG
E + iG
+
F-iH. F
• + in'
1
4
4
1
-i.E + iG
—
i.F + in
8
13
13
8
E- iG
—
.F-iH
9
12
12
9
— i.E + iG
+
i.F+iH
7
2
2
7
F-iff
+
E-iG
3
6
6
3
-i.F + in
—
i.E + iG
15
10
10
15
- .F -iH
+
E-iG
11
14
14
11
i.F + in
-
i.E + iG
u+u'
u—u'
u+u^
- ^
«-tt'
• 5}
(Suffixes
1-)
0
4
8
12
2
6
10
14
0
4
8
12
2
6
10
14
o
1
13
9
7
3
15
11
E + iG . E' - iG' + F + iH .r - iW
-i.E -iG
E + iG
-i.E -iG
F + iH
-i.F -in
- .F + iH
+ i.F-iH
i.F -iH
.F + iH
i.F -in
E + iG
i.E-iG
.E + iG
i.E - iG
67—2
582
A MEMOIR ON THE SINGLE AND DOUBLE THET A -FUNCTIONS.
[704
96. Seventh set of 16.
u— u'
. &
tt+U'
+ 3
u— u'
(Suffixes
2.)
9
0
0
9 =
E+ G .
E" +G'
+
F-H .
F-H'
13
4
4
13
i.E -G
«
+
i.F + H
1
8
8
1
E + G
»»
-
.F-H
6
IS
12
5
i.E-G
»>
—
i.F + H
11
2
2
11
F + H
>»
+
E - G
15
6
6
15
i.F-H
»»
+
i.E + G
3
10
10
3
F + U
)f
—
.E-G
7
14
14
7
i.F-H
}I
-
i.E + G
u+u'
u— u'
5
u+u'
- »
«— u'
(Suffixes
2-)
9
0
0
9 =
E -G .
E-G'
+
F + H.F' + H'
13
4
4
13
i.E + G
»»
+
i.F-H
)9
1
8
8
1
E-G
jj
—
.F + H
»
5
12
12
5
i.E + G
ji
—
i.F-H
l>
11
2
2
11
F -H
»
+
E +G
n
16
6
6
15
i.F + H
»i
+
i.E-G
a
3
10
10
3
F-H
»»
—
.E +G
)}
7
14
14
7
i.F + U
11
-
i.E-G
»i
97.
Eighth set of 16.
U+W
i{5
u— u'
9
u+u'
+ »
tt— u'
(Suffixes
3.)
13
0
0
13 =
E-iG
E -\-iG
+
F+iH
F- - iH'
9
4
4
9
-i.E + iG
)j
-
i.F-iH
l»
5
8
8
5
E~iG
)}
—
. F + iH
»»
1
12
12
1
~i.E + iG
»
+
i.F-iH
>»
15
2
2
15
F -iH
1)
+
E + iG
J»
11
6
6
11
-i.F + iH
i»
-
i.E-iG
»»
7
10
10
7
F -iH
»
-
.E + iG
)»
3
14
14
3
-i.F + iH
)}
+
i.E-iG
)»
i«+«'
u— u'
u+u'
- & .
u— u'
(Suffixes
3.)
13
0
0
13 =
E + iG .
F - iG'
+
F-iH
F' + iH'
9
4
4
9
-i.E-iG
i>
—
i.F+iH
I»
6
8
8
5
E+iG
II
—
.F-iH
»
1
12
12
1
-i.E-iG
II
+
i.F+ iH
f»
15
2
2
15
F+iU
II
+
E-iG
1>
11
6
6
11
-i.F-iH
II
—
i.E + iG
»»
7
10
10
7
F +iH
II
_
.E-iG
))
3
14
14
3
-i.F-iH
II
+
i.E +iG
»>
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
98. Ninth set of 16.
533
«+u'
U9
«— u' U+tl'
. » + 3
«— n'
• ^ }
(Suffixes 0.)
2
0
0
2 =
I + K.
r
+
K'
+
J + L .
J'
+ L'
6
4
4
6
I + K
))
-
.J + L
»»
10
8
8
10
i.I - K
j»
+
i.J- L
)>
14
12
12
14
i.I -K
It
—
i.J-L
)l
3
1
1
3
J + L
j»
+
I + K
f)
7
5
5
7 —
.J + L
n
—
.1 + K
))
11
9
9
11
i.J- L
)i
+
i.I -K
)l
15
13
13
15 -
i.J - L
i»
-
i.I - K
)»
U-Mt'
i{5
tt— u'
- 5 ,
U— tt'
(Suffixes 0.)
2
0
0
2 =
I - K .
r
—
A"
+
J-L .
J'
- L'
6
4
4
6
I -K
))
—
.J-L
9i
10
8
8
10
i.I + K
»
+
i.J + L
)»
14
12
12
14
i.I + K
)l
-
i.J + L
»
3
1
1
3
J ~L
j»
+
I - K
It
7
5
5
7 —
.J - L
)»
+
I - K
»
11
9
9
11
i.J + L
II
+
i.I + K
»
15
13
13
15 -
i.J + L
))
+
i.I + K
1)
99. Tenth set of 16.
tt+u'
5
U+u'
+ 9
u— u'
■ ^}
(Suffixes 1.)
6
0
0
6 =
I + K.
r
+
K'
+
J-L .
J'
-L
2
4
4
2
I + K
—
.J-L
»
14
8
8
14
i.I - K
+
i .J + L
J>
10
12
12
10
i.I - K
—
i.J + L
n
7
1
1
7
J+ L
+
I - K
)i
3
5
5
3
J + L
—
.1 -K
>»
15
9
9
15
i.J-L
+
i.I +K
»»
11
13
13
11
i.J - L
-
i.I + K
»»
u+u'
tt— u'
5
u+u'
- » ,
u-u'
(Suffixes 1.)
6
0
0
6 =
I - K .
1'
—
K'
+
J + L .
J'
+ L
2
4
4
2
I -K
—
.J+ L
J)
14
8
8
14
i.I + K
+
i.J - L
n
10
12
12
10
i.I + K
—
i.J-L
»»
7
1
1
7
J-L
+
I + K
»
3
5
5
3
J - L
—
.1 + K
11
15
9
9
15
i.J+ L
+
i.I - K
M
11
13
13
11
i.J + L
—
i.I -K
>>
534 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8. [704
100. Eleventh set of 16.
u+u'
J{5
«-u'
. 5
U+u'
+ 9
u— u'
(SufSxes 2.)
10
0
0
10 =
J -i£.
r
+
iK'
+
.J-iL.
J'
+ iL'
14
4
4
14
I-iK
1)
-
.J-iL
})
2
8
8
2 -
i.I + iK
»
-
i.J+iL
>»
6
12
12
6 -
i.I + iK
»
+
i.J + iL
»)
11
1
1
11
J-iL
)»
+
I -iK
))
15
5
5
15 -
.J-iL
l»
+
I-iK
1}
3
9
9
3 -
i.J+iL
»1
-
i. I + iK
J»
7
13
13
7 +
i.J+ iL
))
-
i.I + iK
}|
n+u'
tt-«'
u+u'
- 9 ,
u-u-
(Suffixes 2.)
^
10
0
0
10 =
I + iK.
1
' —
Jk^
+
J + iL .
~J
'-W
14
4
4
14
I + iK
»
—
J+iL
>J
2
8
8
2 _
i.I -iK
»
—
i.J — iL
)>
6
12
12
6 -
i.I -iK
>>
■+
i.J-iL
)>
11
1
1
n
J+iL
>>
+
I + iK
n
15
5
5
15 -
.J + iL
>l
+
I+iK
>»
3
9
9
3 -
i.J - iL
)»
-
i.I-iK
»>
7
13
13
7 +
i .J — iL
»
-
i.I -iK
it
101.
Twelfth set of 16.
«+u'
U-tt'
u+u'
+ » .
«— tt'
(Suffixes
,3.)
14
0
0
14 =
I-iK.
I
' +
iK'
+
J+iL .
Tj
' - iL'
10
4
4
10
I-iK
»
—
.J+iL
»)
6
8
8
6 -
i.I + iK
J»
—
i.J-iL
»»
2
12
12
2 -
i.I + iK
)J
+
i.J- iL
)>
15
1
1
15
J-iL
»
+
I + iK
J)
11
5
5
11
J-iL
»
-
.I + iK
»»
7
9
9
7 -
i.J + iL
»
—
i.I-iK
If
3
13
13
3 -
i.J+iL
JJ
J-
i.I-iK
)»
u+u'
i{5 .
tt-tt'
- 5 .
tt— u'
(Suffixes
.3.)
14
0
0
14 =
I + iK.
T'
—
iK~
+
J-iL .
~J'
+ iL'
10
4
4
10
I + iK
»
_
.J-iL
»»
6
8
8
6 -
i.I -iK
»
—
i.J+iL
»
2
12
12
2 _
i.I -iK
11
+
i.J+iL
?>
15
1
1
15
J+iL
»
+
I-iK
)»
11
6
5
11
J+iL
))
—
.I-iK
i»
7
9
9
7 -
i.J - iL
»)
—
i.I+iK
i»
3
13
13
3 -
i.J - iL
J>
+
i.I +iK
11
704]
A MEMOIR OX THE SINGLE AND DOUBLE THETA-FUNCTIONS.
535
102. Thirteenth set of 16.
u+u'
. 5
u+u'
+ s
U—tt
• » }
(Su
ffixes 0.)
3
0
0
3 =
= M+Q.
M' + Q'
+ N + P.N' + P'
7
4
4
7
i.M-Q
- i.N -P
11
8
8
11
i.M-Q
+ i.N-P
15
12
12
15
- .M+Q
+ N + P
2
1
1
2
N + P
+ M+Q
6
5
5
6
-i.K -P
+ i.M-Q
10
9
9
10
i.N - P
+ i.M-Q
14
13
13
14
N + P
- .M+Q
u+u'
u+u'
- »
«— tt'
• 5 }
(Suffixes 0.)
3
0
0
3 =
-- M-Q.
M'-
-Q'
+ N -P.N'-
-F
7
4
4
7
i.M+Q
- i.N + P
i
11
8
8
11
i.M+Q
+ i.N- + P
J
15
12
12
15
- .M-Q
+ N-P
)
2
1
1
2 .
N -P
+ M-Q
»
6
5
5
6
-i.N +P
+ i.M+Q
i
10
9
9
10
i.N + P
+ i.M+Q
J
14
13
13
14
N -P
- .M-Q
»
103.
Fourteenth set
of 1
6.
«— u'
U+tt'
+ 5
tt-«'
(Suffixes 1.)
' 7
0
0
7 =
M-iQ
M'
+ iQ
+ N + iP . N' -iP'
3
4
4
3
-i.M + iQ
»»
+ i.N-iP
15
8
8
15
i.M+iQ
»
+ i.N-iP
11
12
12
11
M-iQ
»>
- .N + iP
6
1
1
6
N - iP
l>
+ M+ iQ
2
5
5
2
-i.N -iP
»)
+ i.M-iQ
14
9
9
14
+ i.N + iP
)1
+ i.M-iQ
10
13
13
10
N - iP
JJ
- .M+iQ
u+u'
u—it'
. 5
u+u'
- »
u— u'
(Suffixes 1.)
7
0
0
7 =
-. M + iQ .
M'
-iQ
+ N -iP. A
'' + iP
3
4
4
3
-i.M-iQ
»i
+ i.N + iP
15
8
8
15
-i.M-iQ
»j
+ i.N + iP
11
12
12
11
M+iQ
»f
- .N + iP
6
1
1
6
N + iP
If
+ M-iQ
2
5
5
2
-i.N -iP
»>
+ i. M + iQ
14
9
9
14
+ i . N - iP
»»
+ i . M + iQ
10
13
13
10
N + iP
>»
- .M-iQ
586
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
104. Fifteenth set of 16.
u+u'
u-«'
n+tt'
u— u'
i{5
. s
+ a
• » }
(Suffixes 2.)
11
0
0
11 =
M-iQ.
M'
+ iQ' + N-iP.
N' +iF
15
4
4
15
i.M+iQ
- i.N + iP
«
3
8
8
3
-x.M+iQ
- i.N+iP
)
7
12
12
7
M-iQ
- .N-iP
»
10
1
1
10
N -iP
+ M-iQ
»
14
5
5
14
-i.N + iP
„ + i.M+iQ
»
2
9
9
2
-i.N + iP
- i.M+iQ
»
6
13
13
6
- .N-iP
+ M-iQ
»
ii+«'
«-u'
u+«'
u-u"
i{5
• ■»
- »
. » )
(Suffixes 2.)
11
0
0
11 =
M+iQ.
M'
-iQ' + N + iP.
N'-iF
15
4
4
15
i.M-iQ
- i.N-iP
>>
3
8
8
3
-i.M-iQ
- i.N-iP
7
12
12
7
M+iQ
- .N + iP
10
1
1
10
N + iP
+ M + iQ
14
5
5
14
-i.N -iP
„ + i.M-iQ
2
9
9
2
-i.N -iP
- i.M-iQ
6
13
13
6
- .N +iP
+ M + iQ
105.
Sixteenth set of 16.
u-Mi'
«— u'
«+u'
u—u'
i{^
. 3
+ S
• ^}
(Suffixes 3.)
tf
15
0
0
15 =
M-Q.
M'
-Q' + N + P.N'
+ f
11
4
4
11
-i.M+Q
+ i.N-P
yj
7
8
8
7
-i.M+Q
- i.N-P
»»
3
12
12
3
- .M-Q
+ N + P
it
14
1
1
14
N -P
+ M+Q
>i
10
6
5
10
-i.N + P
+ i.M-Q
»
6
9
9
6
-i.N + P
- i.M-Q
»»
2
13
13
2
- .N-P
J
+ M+Q
>i
u+u'
U— tt'
u+u'
u— «'
u»
0
- »
0
15 =
(Suffixes 3.)
f
15
M+Q.
M'
+ Q' + N -P. N
'-/
11
4
4
11
-i.M -Q
+ i.N + P
»
7
8
8
7
-i.M-Q
- i.N + P
ft
3
12
12
3
- .M+Q
+ N-P
It
14
1
1
14
N + P
+ M-Q
11
10
6
5
10
-i.N -P
+ i.M+Q
If
6
9
9
6
-i.N - P
- i.M+Q
If
2
13
13
2
- .N + P
i
+ M-Q
If
ro4]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
537
106. In the square set, writing u' = v' = 0, and a, y3, 7, S for X', Y', Z', W ;
also slightly altering the arrangement,
the system becomes : and further writing herein u = 0, v = 0, it becomes
u X r Z W 0 c^
^
0
4
8
12
a /3 y 8
a - j3 y - 8
a /? -y -8
a -/3 -y 8
^
0
4
8
12
_
a» - j8= + y= - 8==
a^ - /S-- - y^ - 8^
a^ - /3^ - y^ + 8^
=
0
4
8
12
1
5
9
13
=
/3 0 8 y
P -a 8 -y
/3 a - 8 -y
/3 - 0 -8 y
1
5
9
13
=
2(ay8 + y8)
0
2(a|8-y8)
0
1
9
2
6
10
14
=
y 8 a y3
y -8 a - P
y B -a ~P
y -8 -a /8
i
2
6
10
14
=
2 (ay + ^8)
2 (ay -^8)
0
0
2
6
3
7
11
15
=
8 y /3 a
8 -y (3 -a
8 y -/3 -0
8 -y -^ a
3
7
11
15
2 (a8 + )3y)
0
0
2(a8-/3y)
3
15
viz. this last is the before-mentioned system of equations giving the values of the
10 zero-functions c in terms of the four constants o, /8, 7, S.
107. The system first obtained is a system of 16 equations
V(«, v) = aX + ^Y + yZ + SW, &c.,
showing that the squares of the theta-fimctions are each of them a linear function
of the four quantities X, Y, Z, W. If the functions on the right-hand side were
independent (a.syzygetic) linear functions of X, Y, Z, W, it would follow that any
four (selected at pleasure) of the squared theta-functions are linearly independent,
and that we could in terms of these four express linearly each of the remaining
12 squared functions. But this is not so ; the foi-m of the linear functions of
(X, Y, Z, W) is such that we can (and that in 16 different ways) select out of
the 16 linear functions six functions, such that any four of them are connected by a
linear equation ; and there are consequently 16 hexads of squared theta-functions, such
C. X. 68
5S8
A MEMOIE ON THE SINGLE AND DOUBLE THETA-FUN0TI0N8.
[704
that any four out of the same hexad are connected by a linear relation. The hexads
are shown by the foregoing "Table of the 16 Kummer hexads."
108. The d posteriori verification is immediately effected; taking for instance the
first column, the equations are
X Y Z W
A
B
AB
CD
CE
BE
11
7
6
2
1
9
= B
8
7
7
7
- 7
-B
a
a
/3
/8
a
a
B
8
-0.
-a.
-0,
7.
-7;
viz. it should thence follow that there is a linear relation between any four of the six
squared functions 11, 7, 6, 2, 1, 9: and it is accordingly seen that this is so. It
further appears that, in the several linear relations, the coefficients (obtained in the
first instance as functions of a, /3, 7, B) are in fact the 10 constants c : the 15
relations connecting the several systems of four out of the six squared functions are
given in the following table.
109.
9*
11
7
6
2
1
9
i?
6
- 2
1
- 9
6
+ 15
- 12
+ 4
- 2
-
15
+ 8
- 0
1
+
12
- 8
+ 3
- 9
—
4
+ 0
- 3
6
3
- 0
+ 8
- 2
—
3
+ 4
- 12
1
+
0
- 4
- 15
- 9
—
8
+ 12
+ 15
-16
+ 3
+
2
- 6
- 12
+ 0
+
1
- 6
- 4
+ 8
+
9
- 6
- 3
+ 15
+ 9
- 1
- 0
+ 12
-)■ 9
- 2
- 8
+ 4
+ 1
- 2
= 0.
Read
c.'V • + c,.» V - c„» V + c/V = 0, &c.
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
539
110. The first set of 16 equations is the square-set, which has been already
considered. If in each of the other sets of 16 equations we write in like manner m' = 0,
each set in fact reduces itself to eight equations ; sets 2, 3, 4 give thus 8 + 8-1-8,
= 24 equations; sets 5 to 8, 9 to 12, and 13 to 15, give each 8-1-8+8-1-8, =32
equations ; or we have sets of 24, 32, 32, 32, together 120 equations, the number
being of course one half of 256—16, the number of equations after deducting the
16 equations of the square set.
111. The first set, 24 equations.
This is derived from the second, third, and fourth sets, each of 16 equations, by
writing therein m' = 0. Taking a,, /3i, 71, S^ for the zero-functions corresponding to
X,, F,, Zj, Wi, then on writing u' = 0, X^, F/, Z^, W/ become ai, A. 7i. Si- In
the second set of 16 equations, the first equations thus are
&,M.Vf = a,Xi + 7A, 0 = AFi + 8,Tfi,
%,u . ^8« = o,Z, - 7,Zi, O = ;8, F - 81 TT,,
viz. the equations of the column require that, and are all satisfied if, /3i = 0, 8, = O :
hence the zero- functions are Oj, 0, 7,, 0; and this being so we have only the equations
of the first column. And similarly as regards the third and fourth sets ; the zero
values corresponding to
Z„ F,. Z„ W, X,, F, Z,, W, X„ F3, Z„ W,
are a, 0 7, 0 | a^ /9o 0 0 a, 0 0 83;
and we have in all 8 + 8+8, = 24 equations. These are
4
12
6
14
0 =
8 =
2 = y
(Suffixes 1.)
X z
a
a
y
■y
a
10 =
y -a
W
8
12
9
13
(Suffixes 2.)
X T
W
12
8
1.5
11
0
4
3
7
(Suffixes 3.)
X W
= a
= s
= -8
8
-8
a
a
5
1
=
a y
10
2
=
a H
13
1
=• a 8
13
9
=
a -y
u
6
=
a -P
9
5
= a -h
7
3
=
y a
11
3
=
H a
14
2
= 8 a
15
11
=
y -0
15
7
=
P -a
10
6
= -8 a
m
M
SO .
.90
50
50
4
0
=
a^ + Y'
8
0
=
€? + ^
12
0
= a-' + S'
12
8
=
d'-Y'
12
4
=
c?-^
8
4
= a^-S'
6
2
=
2ay
9
1
=
2ay3
15
3
= 2o8.
68—2
540
A MEMOIB ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
112. The second set, 32 equations.
To exhibit these in a convenient form, I alter the notation, viz. I write
E+Q, iiE-G), (F+H), i(F-H)
= X.
Z.
w
E, + iGu E,-iG,, F, + iHu F,-iH,
X,.
F„
Zu
w.
{E, + 0,), i(E,-G,), (F, + Hd, i(F,-H,) \ E, + iG„ E,-iG„ F, + iH„ F,-iH,
2lj,
F,.
Z„
w.
X,
Y„
Zj,
w„
so that as regards the present set of equations, X, Y, Z, W, signify as just mentioned.
And, this being so, the corresponding zero-values are
a, 0, 7, 0 I a„ 0, 7., 0 | a,, 0, 0. S, i a,, 0, 0, S,.
The equations then are
(Suffixes 0.)
(Suffixes 1.)
(Suffixes 2.)
(Suffixes 3.)
3u
»u X Z
3u . 3u
X Z
3u
3u
X W
3u
3u X W
1
0 = 0 y
1 4 =
— ia — iy
9
0 =
'T^
9
i = -ia -iS
9
8 = a — y
9 12 =
— ia + iy
1
8 =
s
1
12 = - ia + iS
3
2 = y 0
3 6 =
— iy — ia
15
6 =
8 a
15
2= 8 a
11
10 = y -0
11 U =
— iy + ia
7
14 =
-8 a
7
10 = - 8 a
Y W
4 = 0 y
5 0 =
Y W
a y
13
4 =
7 Z
■ a 8
13
Y Z
5
0= a 8
13
12 = a -y
13 8 =
a -y
5
12 =
a -8
5
8= a - 8
7
6 = y 0
7 2 =
y 0
11
2 =
-8 a
11
6 = - t8 - ia
15
14 = y — 0
15 10 =
y -a
3
10 =
8 a
3
14 = iS - ia
30
30
.50. .90
50
30
50
30
1
0 = a' + y
1 4 =
-t(a» + y)
9
0 =
a^-S'
9
4 = -t(a= + 8»)
9
8 = a' - y
9 12 =
-t(a=-/)
1
8 =
a- + 8^
1
12 = -i(a»-8')
3
2= 2ay
3 6 =
— 2iay
15
6 =
2a8
15
2= 2a8.
704]
A MEMOIB ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
541
113. Third set, 32 equations.
We again change the notation, writing
I + K, i(I-K), J+L, i(J-L)
= X, Y. Z, W
/a + iKi , I 2 — iKi , Ja + iLi , J2 — ii/3
X„ F„ Z„ W,
li + iKs, I^ — iKz, Jj + iL,, J^ — iL-i
X3, Y3, Z3, W3,
the zero values being
a, 0, 7, 0 I a,. 0, 0, S, | <x„ 0, 7,, 0 | a,, 0, 0, S,.
Then equations are
(Suffixes 0.)
.Sm . .iu X Z
2 0 = a 7
6 4 = a — y
3 1 = y "
7 5 = y -o
' r W
10 8 = a y
14 12 = a -y
11 9 = y a
15 13 = y -a
30 . .90
2 0 = a' + y
6 4 = o' - y
3 1 = 2ay
(Suffixes 1.)
3u . 3u X W
6 0= '~^^
2 4:= a 8
15 9 = 8 o
11 13 = - 8 o
SO. 90
6 0 =
2 4 =
15 9 =
o» + 8'
2a8
1
2
9t(,
8 =
(Suffixes 2.)
X Z
6
(Suffixes 3.)
9u X W
— ia
-ty
8 = -ia -iS
6
12 =
— to
+ iy
2
12 = - ia iS
3
9 =
-ty
— to
15
1=8 a
7
13 =
-iy
+ io
11
5 = - 8
10
0 =
Y
a
W
y
14
r ^
0 = ' a 8
14
4 =
a
-y
10
4= a - 8
11
1 =
y
a
7
9 = - iS - ia
15
5 =
y
— a
3
13 = iS - io
90
90
50
50
2
8-
-i(a»
+ 7=)
6
8 = -i(a^ + 8^)
6
12 =
-t(a»
-/)
2
12 = -i(a=-8')
3
9 =
- 2ioy
15
1 -. 2a8.
I
542
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8.
[704
114. Fourth set, 32 equations.
Again changing the notation, we write
= X. F, Z, W
Xu F„ Z,. W„
M+Q, i{M-Q). N+P, i{JS^-P) i M^ + iQ„ M.-iQ,, N, + iP„ K,-iP,
= Z.. F,. Z„ TT, I
the zero values being
a, 0, 7, 0 I a„ 0, 0, 8,
The equations then are
X„
F,,
Z^.
w„
«=, 0, 7„ 0 I 0, /Sj, 73. 0.
(Suffixes 0.)
(Suffixes
1.)
(Suffixes 2.)
(Suffixes 3.)
»u
3 =
X
Z
3
4 =
X
W
15
5m
4 =
X
z
5m
15
5m Y
.2'
0
a
y
— ia
Is
ia.
^
0 = -/3
y
15
12 =
— a.
y
15
8 =
ia
iS
3
8 =
— ia
-iy
3
12= p
y
2
1 =
y
a
6
1 =
8
a
14
5 =
iy
— ia
10
5= y
-/3
U
13 =
-y
a
10
13 =
- S
a
2
9 =
-iy
~ia
6
9 = -y
-/8
7 =
7
a
w
-y
7
0 =
Y
Z
11
0 =
Y
a
w
y
11
X
IT
4
a
~8
4 = r^
y
8
11 =
a
y
11
12 =
a —
8
7
12 =
a
-y
7
8 = -yS
-y
6
5 =
y
— a
2
5 =
iB -
ia
10
1 =
y
a
14
1= y
-/i
10
9 =
y
a
14
9 =
i8
i'a
6
13 =
y
— a
2
13= y
fi
.90
^0
^0
50
50
50
50
50
0
3 =
a?
^y"
3
4 =
-iio?-
8»)
15
4 =
i{a^
-f)
15
0=-(^
-f)
15
12 =
-{a?
-/)
15
8 =
i(a.^ +
8»)
3
8 =
-i(a»
+ /)
3
12= ^
+ /
2
1 =
2
ay
6
1 =
2aS
2
9 =
— 2ta7
6
9=- 2/3y.
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
543
115. It will be noticed that the pairs of theta-functions which present themselves
in these equations are the same as in the foregoing " Table of the 120 pairs." And
the equations show that the four products, each of a pair of theta-functions, belonging
to the upper half or to the lower half of any column of the table, are such that any
three of the four products are connected by a linear equation. The coefificients of
these linear relations are, in fact, functions such as the a- + 8", a" — B', 2aS written
down at the foot of the several systems of eight equations, and they are consequently
products each of two zero-functions c.
Thus (see
"The first set,
24
equations ")
we ha\
'e
(Suffixes
3.)
(Suffixes
3.)
(Suffixes 3.)
3u
. au X
W
8
5
9 =
7
Z
1
SO
4
.SO
4
8= a -
= a —
8 = a= - 8^
0
12= a
8
1
13 =
= a
8
0
12 = a^ + 8»
3
1.5= 8
a
2
14 =
= 8
a
15
3 = 2a8.
7
ll=-8
a
6
10 =
= -8
a
116. In the left-hand four o(f these, omitting successively the first, second, third,
and fourth equation, and from the remaining three eliminating the X3 and W3, we
write down, almost mechanically,
^u
^w
4
8
0
12
3
15
7
11
- 8- - a\ a'-Sf
-B- + a\ 0?+^
2aS
+ 2a8,
- 2aS,
- a» -»- SS S" + a', - 2aS
and thence derive the first of the next following system of equations ; read
CaCij^y^ij C0C12 J'a^ifl -r C^Cg ^7*T]i = U,
"~ CjjCij^T^ Jg -r C4C9 ^s-JiB ^O^lil^T^U ^ V,
Ctfii^i^S "^ C^Cq ^o-Jjo -f- C3Ci5-T7rJii ^ U,
where the theta-functions have the arguments «, v.
Observe that, on writing herein m = 0, i; = 0, the first three equations become each
of them identically 0 = 0; the fourth equation becomes
'~ C4 C^ "T Cq C12 "" C3 C15 ^ \)j
which is one of the relations between the c's and serves as a verification.
But in the right-hand system, on writing u = v = 0, each of the four equations
becomes identically 0 = 0.
544
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8.
[704
117. The equations are
9
4.8
0.12
3.15
7.11
e
3.15
-0.12
4.8
- 3.15
-4.8
-0.12
0.12
-4.8
3.15
-4.8
0.12
-3.15
5
6.8
2.12
1.15
5.11
e
1.15
-2.12
6.8
-1.15
6.8
-2.11
2.12
-6.8
1.15
-6.8
2.11
-1.15
•
0,
9
5.9
1.13
2.14
6.10
= 0,
e
3.15
-0.12
4.8
-3.15
4.8
-0.12
0.12
-4.8
3.15
-4.8
-0.12
-3.15
»
7.9
3.13
0.14
4.10
= 0,
c
1. 15
-2. 12
6.8
-1.15
6.8
-2. 12
2.12
-6.8
1.15
-6.8
2.12
-1.15
9
0.6
2.4
9.15
11.13
e
9.15
-2.4
0.6
-9.15
0.6
-2.4
2.4
-0.6
9. 15
-0.6
2.4
-9.16
9
1.7
3.5
8.14
10. 12
c
9.15
-2.4
0.6
-9. 15
0.6
-2.4
2.4
-0.6
9.15
-0.6
2.4
-9.15
3.6
1.4 9. 12 14. 11 =0,-
e
9.12
-1.4
3.6
-9. 12
3.6
-1.4
1.4
-3.6
9.12
-3.6
1.4
-9.12
9
2.7
0.5
8
13
10.15
c
9.12
- 1
4
3.6
-9,12
3
6
-1.4
1.4
-3.6
9.12
-3.6
1.4
-9
12
0,
9
8.9
0.1
2.3
10.11
e
-2.3
0.1
8.9
2.3
-8.9
-0.1
-0.1
8.9
2.3
-8.9
0.1
-2.3
0,
12.13
4.5
6.7 14.15 =0,
c
-2.3
0.1
8.9
2.3
-8.9
-0.1
-0.1
8.9
2.3
-8.9
0.1
-2.3
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 545
», 4.6 0.2 1.3 5.7 =0, S 9.11 13.15 12.14 8.10 =0,
c
-1.3
0.2
4.6
1.3
-4.6
-0.2
-0.2
4.6
1.3
-4.6
0. 2
- 1.3
1.3
-0.2
4.6
-1.3
-4.6
0.2
0.2 4.6
4.6 -0.2
1.3
1.3
5
6.12
2.8
3.9
7.13
c
3.9
-2.8
-6.12
-3.9
6. 12
2.8
2. 8
-6. 12
- 3.9
6.12
-2.8
3.9
3
1.11
5. 15
4. 14
0.10
c
3.9
-2.8
6.12
- 3.9
-6. 12
2.8
2.8
6. 12
-3.9
-6. 12
-2.8
3.9
6.15
0.9
7.14 =0,
c
0.9
-1.8
-6.15
-0.9
6.15
1.8
1.8
-6.15
-0.9
6.15
-1.8
0.9
3
2.11
5.12
4.13
3.10
c
0.9
-1.8
6.15
-0.9
-6.15
1.8
1.8
6.15
-0.9
-6.15
-1.8
0.9
9
4.9
1.12
2.15
7.10
c
2.15
-1.12
4.9
-2.15
4.9
-1.12
1.12
-4.9
2.15
-4.9
1.12
-2.15
»
0.13
5.8
6.11
3.14
a
2.15
1.12
-4.9
-2.15
-4.9
1.12
-1.12
4.9
2.15
4.9
-1.12
-2.15
= 0,
5
4.12
0.8
1.9
5.13
c
-1.9
0.8
4.12
1.9
-4.12
-0.8
-0.8
4.12
1.9
-4.12
0.8
-1.9
= 0,
9
3.11
7.15
6.14
2.10
c
1.9
-0.8
4.12
-1.9
-4.12
0.8
0.8
4.12
-1.9
-4.12
-0.8
1.9
C. X,
69
546
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
[704
»
4.16
3.8
2.9
5.14
e
-2.9
3.8
4.15
2.9
-4.15
-3.8
-3.8
4.15
2.9
-4.15
3.8
-2.9
5
0.11
7.12
6.13
1.10
c
-2.9
3.8
-4.15
2.9
4.15
-3.8
-3.8
-4.15
2.9
4.15
3.8
-2.9
a
6.9
3.12
0.15
5.10
c
-0.15
3.12
-6.9
0.15
-6.9
3.12
-3.12
6.9
-0.15
6.9
-3.12
0.15
»
2.13
7.8
4.11
1.14
c
0.15
3.12
-6.9
-0.15
-6.9
3.12
-3.12
6.9
0.15
6.9
-3.12
-0.15
3
12.15
0.3
1.2
13.14
c
1.2
-0.3
-12.15
-1.2
12.15
0.3
0.3
-12.15
-1.2
12.15
- 0.3
1.2
0,
.9
8.11
4.7
5.6
9.10
c
1.2
- 0.3
12.15
- 1.2
-12.15
0.3
0.3
12.15
-1.2
-12.15
- 0.3
1.2
= 0,
3
1.6
3.4
8.15
10.13
c
8.15
-3.4
1.6
-8.15
1.6
-3.4
3.4
-1.6
8.15
-1.6
3.4
-8.15
= 0,
.9
2.5
0.7
11.12
9.14
c
-8.15
-3.4
1.6
8. 15
1.6
-3.4
3.4
-1.6
-8.15
-1.6
3.4
8.15
9
2.6
0.4
8.12
10.14
c
-8.12
0.4
-2.6
8.12
-2.6
0.4
-0.4
2.6
-8.12
2.6
-0.4
8.12
0,
»
1.5
3.7
11.15
9.13
e
-8.12
-0.4
2.6
8. 12
2.6
-0.4
0.4
-2.6
-8.12
-2.6
0.4
8.12
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
547
118. The foregoing equations may be verified, and it is interesting to verify them,
by means of the approximate values of the functions : thus, for one of the equations,
we have
(2A + 2A')(-2A + 2A') .1.1
— 1 . 1 . 2A cos ^TT (m +v) + 2A' cos ^tt {u- v).
— 2A cos ^TT (m + u) + 2A' cos Itt (m — v)
+ 1 . 1 . — 2A sin Jtt (i« 4- 1)) — 2A' sin ^tt {u — v).
— 2 A sin ^TT (m + 1)) + 2 A' sin Jtt {u — v);
= 0.
viz. the equation to be verified is here
- 4A^ + 4A'»
+ 4A^ cos' ^TT (m + d) - 4 A'- cos= \tr (w - v)
+ 4A- sin^ \iT (m + ?)) — 4A'* sin= \it {u — v)
= 0,
= 0,
which is right.
119. In the equation
\
= 0,
2Q . 1 . 2Q cos ^TTM . 1
-2Q.1.2Qcosi7rM.l
= 0;
this is right, but there is no verification as to the term CsC^u'^n ', taking the more
approximate values, the term in question taken negatively, that is, —CsC^i^n is
= -(2A + 2A'). 28. -2Ssini7n). - 2A sin Jtt (m + 1;) + 2A' sin Jtt (m - d),
which is
= - 8«» (A + A'y cos ^TTU + 8<S» (A + A') A cos ^v (u + 2v) + 851* (A + A') A' cos ^tt (u - 2v),
and this ought therefore to be the value of the first two terms, that is, of
{2Q + 2Q> - 2A - 2A'){1 - 2Q' - 2S*) {2Q cos ^mi + 2Q> cos fv™
+ 2A cos Jtt (m + 2d) + 2A' cos ^tt (m - 2v)} {1 - 2Q* cos ttu + 2S* cos -rrv)
- (2Q + 2Q« + 2^ + 2A') (1 - 2Q« + 2S*) {2Q cos imi + 2<2» cos Ittm
- 2A cos ^TT (n + 2v) — 2A' cos Jtt (u — 2v)j (1 - 2Q* cos ttm - 2)S* cos ttv),
which to the proper degree of approximation is
= (2Q - 4Q« - 4,QS* + 2Q>-2A- 2A') [2Q cos ^mt - 4Q» cos ^ttm cos ttm
+ 4tQS* cos Jttm cos th) + 2Q» cos Ittm + 2A cos ^tt (m + 2v) + 2A' cos Jtt (m — 2t>)j
- (2Q - 4Q» + 4QS^ + 2Q» + 2^ + 24') {2Q cos ^ttm - 4Q^ cos l-mi cos ttw
— 4Q(S'* cos \nra cos ttw + 2Q» cos fTrw — 2-4 cos ^tt (m + 2«) — 2A' cos Jtt (m — 2d) j.
69—2
548 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
This is
(2iif„-2n,)(2itf+2n)
- (23f„ + 2a) (2M - 2n), = 8 (il/„n - Mil,),
if for a moment
M=Qcos ^TTU - 2Q» cos ^iru cos ttm + Q* cos f ttm, Jf„ = Q - 2Q» + Q",
n = 2QS'co8'7r«co8 7rt> + ilco8jTr(i* + 2t)) + -4'cos^7r(«-2i;), fl„=2QS* +-4+^1',
or substituting and reducing, the value of 8 (i¥„fl - J»m„) to the proper degree of
approximation is found to be
= - 8Q (2QS* + A + A')cos ^ttu
+ 8 (Q'S* + SQA) cos ^TT (m + 2i;) + 8 (Q--^^ + 8QA') cos ^tt (m - 2v),
which in virtue of the relations QA = A'S', QA' = A'''S°; Q'S^ = AA', is equal to the
foregoing value of CsCj^u^u. I have thought it worth while to give this somewhat
elaborate verification.
Risximi of the foregoing results.
120. In what precedes we have all the quadric relations between the 16 double
theta-functions : or say we have the linear relations between squares (squared functions)
and the linear relations between pairs (products of two functions) : the number of
the asyzygetic linear relations between squares is obviously = 12 ; and that of the
asyzygetic linear relations between paii-s is = 60 (since each of the 30 teti-ads of
pairs gives two asyzygetic relations) : there are thus in all 12 + 60, =72, asyzygetic
linear relations. But these constitute only a 13-fold relation between the functions,
viz. they are such as to give for the ratios of the 16 functions expressions depending
upon two arbitrary parameters, x, y. Or taking the 16 functions as the coordinates of
a point in 15-dimensional space, these coordinates are connected by a 13-fold relation
(expressed by means of the foregoing system of 72 quadric equations), and the locus
is thus a iS-fold, or two-dimensional, locus in 15-diraensional space.
Hence, taking any four of the functions, these are connected by a single equation ;
that is, regarding the four functions as the coordinates of a point in oitiinary space,
the locus of the point is a surface.
In particular, the four functions may be any four functions belonging to a hexad :
by what precedes there is then a linear relation between the squares of the four
functions: or the locus is a quadric surface. Each hexad gives 15 such surfaces, or
the number of quadric surfaces is (16 x 15 =) 240.
The 16-w odaZ quartic surfaces.
121. If the four functions are those contained in any two paire out of a tetrad
of pairs (see the foregoing "Table of the 120 pairs"), then the locus is a quartic
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 549
surface, which is, in fact, a Kummer's 16-nodal quartic surface. For if for a moment
x.y and z.iu are two pairs out of a tetrad, and r.s be either of the remaining
pairs of the tetrad; then we have rs a linear function of xy and zw. squaring, r^d'
is a linear function of ahj^, xyzw, z-w- ; but we then have t^ and s^ each of them
a lineal' function of a?, y'^, z-, w^; or substituting we have an equation of the fourth
order, containing terms of the second order in {a?, y^, z^, w-), and also a term in
xyzw. It is clear that, if instead of r . s we had taken the remaining pair of the
tetrad, we should have obtained the same quartic equation in {x, y, z, w). And
moreover it appears by inspection that, if acy and zw are pairs in a tetrad, then xz
and yw are pairs in a second tetrad, and new and yz are pairs in a third tetrad :
we obtain in each case the same quartic equation. We have from each tetrad of
pairs six sets of four functions {x, y, z, w): and the number of such sets is thus
(^6 . 30 =) 60 : these are shown in the foregoing " Table of the 60 Gopel tetrads," viz.
taking as coordinates of a point the four functions in any tetrad of this table, the
locus is a IG-nodal quartic surface.
122. To exhibit the process I take a tetrad 4, 7, 8, 11 containing two odd
functions; and representing these for convenience by x, y, z, w, viz. writing
%, %, %, ^„(M) = a;, y, z, w,
we have then X, T, Z, W linear functions of the four squares, viz. it is easy to
obtain
a (a,'» + z-)- B Of + ^if) = 2 (a- - B') X,
B( „ )-«( „ )=2( „ )W,
-^(x'-z^) + y(y'-'U^) = 2i0'-rf)Y,
-7( ,. ) + -8( „ ) = 2( „ )Z.
Also considering two other functions %{u) and ^i2(m), or as for shortness I wiite
them, ^0 and ^jj, we have
X' =aX + ^Y+yZ+BW,
%,' = aX-0Y-yZ+BW,
and substituting the foregoing values of X, Y, Z, W, we find
M^^" =Ax' + By' + Cz' + Diu\
Miii^ = Car" + Dy' + -42= + Bw'',
where, writing down the values first in terms of a, ^, y, B and then in terms of
the c's, we have
M= (a^ - B=) {^ - 'f) =i . c,*-c,\
A= ^-B^-a'i' =„ -c,V,
B = -aB{ff'-y') + 0y (a" -S')=„ c,V - CiW,
C= a-ff'-y'B' =„ CiV,
550 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8. [704
and we then have fiirther
that is,
whence equating the two values of ^0*^11' we have the required quartic equation in
«, y, «, tff.
123. But the reduction is effected more simply if instead of the c's we introduce
the rectangular coeflScients a, b, c, &c. We then have
if=(c"«-6'»), A=-a"c, C=a'b,
B = - b'c' - b"c", =bc, D = b'b" + c'c". = a'a" ;
and the equations become
(c"» - b'') V = - a"c^ + bey' + a'bz' - a'a"ii^,
(c"--6''')V= a'bx'-a'aY-a"c^+ bcv^,
V6'c"^„^„ -'Jaxz + 'J^y'dyw,
80 that the elimination gives
b'c" (- a:'c^ + bcy' + a'bz' - a'a"'uf) {a'ba? - a'a'y - a'cz" + bcw-)
= (c"' - b'-y [ax'-z- - b"c'yHv' + 2 'J^^'Yxyzw},
viz. this is
- a'a'Wcc" {x* + i/* + z* + w*)
+ a'b'cc" (a"» + 6») {a^' + zhu")
+ [6'c" (a''6' + a' V) - a (b'' - c"')'} ce'i'
+ {b'c" (a'a"' + b'c') + b"c' (b'' - c"'f} fw'
-a"bb'c"{a'' + (?)(a?w'-iry'z')
- 2 {V - c"')' -J-ab"c'xyzw = 0.
124. In this equation the coefficients of a?z' and yhu' are each = cl a"bc {b'- + c"'),
as at once appears &om the identities
(a'b.U -c".a"c=:a(b''-c"'),
\a'b.c"-b'.a"c= (b''-c"'),
la'a" .b'-c" .bc = - b" (b'' - c"'),
\a'a".c -b'.bc= c'{b''-c"%
by multiplying together in each pair the left-hand and the right-hand sides respec-
tively. Substituting and dividing by -a'a"bb'cc", we have
x' + y^ + z^ + iu*
a"* -f 6» , b'' + c"' a' -4- r'
^ 2(6'»-c"»)'V-a6"c'
a'a"bb'cd'
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
551
or, if we herein restore the c's in place of the rectangular coefficients, this is '
(v* + y* + z* + w*
which is the equation of the 16-nodal (juartic surface.
Substituting for x, y, z, w their values ^4, ^7, ^g, ^n («)> ^^ have the equation
connecting the four theta-functions 4, 7, 8, 11 of a Gopel tetrad. And there is an
equation of the like form between the four functions of any other Gopel tetrad: for
obtaining the actual equations some further investigation would be necessary.
The soy-expressions of the theta-functions.
12.5. The various quadric relations between the theta-functions, admitting that
they constitute a 13-fold relation, show that the theta-functions may be expressed as
proportional to functions of two arbitrary parameters x, y; and two of these functions
being assumed at pleasure the others of them would be determinate ; we have of
course (though it would not be easy to arrive at it in this manner) such a system
in the foregoing expressions of the 16 functions in terms of x, y; and conversely
these expressions must satisfy identically the quadric relations between the theta-
functions.
126. To show that this is so as to the general form of the equations, consider
first the ary-factors Vk, \lab, &c. As regards the squared functions {'Jaby, we have for
instance
(V^)» = 1 {abfcAe, + a>,f,cde + 2 ^TY],
{s/cdf = i {cdfa,b,e, + c,d,f,abe + 2 VXF) ;
2
each of these contains the same irrational part ^VXF, and the difference is therefore
rational : and it is moreover integral, for we have
CJahy - C^cdy = i (abc,d, - a,b,cd) (fe, - f,e),
where each factor divides by 0, and consequently the product by 6^; the value is in
fact
= (e-f)
1, x + y, xy
1, a + 6, ah
1, c -t-d, cd
552 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTI0N8. [704
a linear function of 1, x + y, xy. This \s the case as regards the difiference of any
two of the squares (Vo^)', {'Jacf, &c. ; hence selecting any one of these squares, for
instance (Vdc)*, any other of the squares is of the form
\ + n{x-iry) + vxy + p{'^de)\ (p = l);
and obviously, the other squares (Va)', &c., are of the like form, the last coefficient p
being =0. We hence have the theorem that each square can be expressed as a linear
function of any four (properly selected) squares.
127. But we have also the theorem of the 16 Kummer hexads.
Obviously the six squares
(Va)», {^h)\ (Vc)^ (\^)S (Ve)», (V/)'
are a hexad, viz. each of these is a linear function of 1, x + y, ooy: and therefore
selecting any three of them, each of the remaining three can be expressed as a linear
fiinction of these.
But farther the squares (Va)=, {'Jbf, {'Jabf, {'Jcdf, {-^ce)-, ("/d^f form a hexad.
For reverting to the expression obtained for {'Jaby — (\cdy, the determinant contained
therein is a linear function of aa, and bb,, that is, of (Va)* and (V6)»; we, in fact, have
(a-b)
= (b — c)(b — d) (a — x){a — y) — {a — c) (a — d){b — x) (b — y).
1, x + y, xy
1, a + b, ah
1, c + rf, cd
Hence {'■/aby — ("Jcdy is a linear function of (Va)^, (Vft)- ; and by a mere inter-
change of letters {^fcS>y — {*Jcey, ('/aby — ('/dey, are each of them also a linear function
of (Va)* and (Vt)*; whence the theorem. And we have thus all the remaining 15
hexads.
128. We have a like theory as regards the products of pairs of functions. A
tetrad of pairs is of one of the two forms
VoV6, s/acJbc, ^ad'/bd, VoiVte, and '^f'/ab, '/c'Jde, VdVce, 'Je'Jcd;
in the first case the terms are
Vaa^bb,,
^ {(ab, + a b) Vcdefc,d,e/, + (cfd,e,+ c/,de) V^^J,
^{ » .. + (dfc,e, + dXce) „ ),
^{ » „ + (efcA + e,f,cd) „ |,
and as regards the last three terms the difference of any two of them is a mere
constant multiple of Vaa,bb, ; for instance, the second term — the third term is
■■ ^ (cd, - c,d) (fe, - f,e) Vaa,bb, . = (c - d) (/- e) Vaa,bb, ;
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 553
we have thus a tetrad such that, selecting any two terms, each of the remaining
terms is a linear function of these.
In the second case, the terms are
3 {fVabc,d,e,t; + f V'a,bcdef},
^{c „ +c^ „ },
g{d .. +d, „ },
^ l^ " "^ ®/ " J'
whence clearly the four terms are a tetrad as above. And it may be added that
any linear function of the four terms is of the form
g |(\ + iix) -/abc^^, + (\ + fiy) Va,b,cdef}.
129. Considering next the actual equations between the squared theta-functions,
take as a specimen
that is,
C* {-Jabf - Cj* (Vcd)2 + c* (\/ce)= - c/ (Vde)= = 0,
where c«, Cj, c,, 0^=^ ah, V cd, vce, vde respectively. Since the functions {'Jaby,
&c., contain the same in-ational term ^ '/XV, it is clear that the equation can only
be true if
Cg Cg "T" Ci C(, = u ;
and, this being so, it will be true if
Cj< {(V^y - (Vcd)'} - Ca« [(\/a6)' - (V^)'j + c,* {(^aSy - {'^def] = 0,
where, by what precedes, each of the terms in { } is a linear function of (Va)" and
{^hf. Attending first to the term in {'^af, the coefficient hereof is
ef. bc.bd. c./ — df. bc.be. Ci" + cf. bd.be. c^*,
where for shortness be, bd, &c., are written to denote the differences b — c, b — d, &c. :
substituting for Cj* its value (vcdy, = cd . cf. df. ah . ae . be, and similarly for Ci'' and c/
their values, =ce.cf.ef.ab.ad.bd, and de.df.ef.ab.ac.be respectively, the whole ex-
pres-sion contains the factor ah.bc.hd.be. cf. df. ef, and throwing this out, the equation
to be verified becomes
cd.a£ — ce.ad + de.ac = 0,
C. X, 70
554 A MEMOIR ON THE SINGLE AND DOUBLE THETA- FUNCTIONS. [704
which is true identically. The verification is thus made to depend upon that of
Cf* — c* + Cj* — c,* ==0 ; and similarly for the other relations between the squared functions,
the verification depends upon relations containing the fourth powers, or the products
of squares, of the constants c and k.
130. Among these are included the before-mentioned system of equations involving
the fourth powers or the products of squares of only the constants c; and it is
interesting to show how these are satisfied identically by the values Co = vbd, &c.
Thus one of these equations is Cu* + c,* + c,* = Co^ ; substituting the values, there is
a factor ce which divides out, and the resulting equation is
ad. of. df. bc.be + cf.ef.ab.ad.bd + ab. af. b/.cd.de — ac.ae.bd.bf. d/= 0.
There are here terras in a', a, a' which should separately vanish ; for the terms
in a', the equation becomes
df. bc.be + bd. cf . e/+ bf .cd.de -bd. bf. df= 0,
which is easily verified; and the equations in a and a° may also be verified.
An equation involving products of the squares is Ci/c,' — d'c/- + CiC^ = 0. The
term c,j'c'^ is here 'Judf. bee *Jdef. abc which is = V(6c)^ (d/Y .ab.ac.ad. af. be.ce.de.ef,
which is taken =bc.df 'Jab. ac.ad.af.be. ce.de. ef; similarly the values of Ci%' and
Cj'c* ai-e =bd. cf and bf. cd each multiplied by the same radical, and the equation to be
verified is
be . df- bd . cf+ bf. cd = 0,
which is right: the other equations may be verified in a similar manner.
131. Coming next to the equations connecting the pairs of theta-functions, for
instance
this is
CsCuCoCij I V6d Vad - 'J be "Joe] + CiCj<>,kn .'Jb'Ja = 0,
the products ^bd '/ad and Vfce Voe contain besides a common term the terms
^ (dfc,e, + d,f,ce) Vaa,bb, , and ^ (efc^d, + e/,cd) Vaa^b^ ,
hence their difference contains ^(de, — d,e)(fc, -f,c) Vaa^bb, which is =de./c Vaa^bb,,
that is, de.fc'/a'Jb: hence the equation to be verified is
ae .JC . CgCiflCoCia t C^C^^Kh = U \
CaCijCoCu is =\/bef.acd\/arf.bcd\/bdf.ace\/adf.bce, where under the fourth root we
have 24 factors, which are, iu fact, 12 factors twice repeated; and if we write
n, =ab.ac.ad.a^.af .be. bd.be. bf. cd.ce.cf.de. df.ef for the product of all the 1.5
factors, then the 12 factors are in fact all those of IT, except ab, cf de; viz. we have
CaCCoCu = \/U^(aby(cfy(dey.
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 555
Again, cfijc^k^^, =\/acf .hde y/hcf .ade y/acdef y/hcdef, is a fourth root of a product of
32 factors, which are in fact 16 factors twice repeated, and in the 16 factors, ab does
not occur, cf and de occur each twice, and the other 12 factors each once : we thus
have
cfi^Ax = \/\\:'(cff{def-^(abf,
and the relation to be verified assumes the form
fc .de\/l^ (c/y {def + v/(c/)» {def = 0,
which, taking fc .de = — \/{cfY (deY, is right. And so for the other equations. It will
be observed that, in the equation de .fc . CaCijCoCi-j + ccsfc^kn = 0, and in the other equations
upon which the verifications depend, there is no ambiguity of sign : the signs of the
radicals have to be detennined consistently with all the equations which connect the
c's and the k'a : that this is possible appears evident d priori, but the actual verification
presents some difficulty. I do not here enter further into the question.
Further results of the product-theorem, the u ± u' forvmlcB.
132. Recurring now to the equations in u + u, u—u', by putting therein m' = 0,
we can express X, T, Z, W in' terms of four of the squared functions of u, and by
putting ?t = 0 we can express X', Y', Z', W in terms of four of the squared functions
of m' ; and, substituting in the original equations, we have the products
&( )« + »'. ^( )ih-u'
in terms of the squared functions of u and «'.
Selecting as in a former investigation the functions 4, 7, 8, 11, which were called
X, y, z, w, it is more convenient to use single lettei-s to represent the squared functions.
I write
Then
i + m')3
(u - u'}
^"-«
^V
a«o
4
4 = P,
4: = p,
4 = ]}',
4 = po{=c,%
7
7 = Q,
7 = q.
7 = q'.
7 = 0,
8
8 = R,
8 = r.
8 = »•',
8 = n{=cs''),
11
11 = S,
11 = s,
11 = s',
11 = 0.
X
Y Z
W
X
Y
Z W
X' Y' Z' W
Hence
P = X' -Y' + Z' -W, p=a-^+y-S, /=a-j8 + 7-S,
Q^W'-Z' +r-X', q = S-yi-^-a, q' = S-y+^-a,
R = X' +Y'-Z' -W, r = a + i8-7-S, r'=a + ^~y-S,
S=W'+Z' -V'-X', ,s=S+7-/3-a, s' = B + y-^-a.
a(p + r)-B(q + 8) = 2(a' - B')X, a (p' + r')-B (q' + s') = 2 {a? - B')X',
S „ -a „ =2 „ W, S „ -a „ =2 „ W,
-^{p-r) + y{q-8) = 2(^-y^)Y, - ^(p -r') + yiq' -s') = 2(^-rf) Y',
-y „ +/3 „ =2 „ Z. -y „ +^ „ =2 „ Z'.
70—2
556 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
By means of these values, we have
iia'-S'yX'X = iz^(p + r)(p' + r')+ S'iq + 8)(q' + s')- aS [ip + r){q' + 8') + (p' + r'){q + 8)l
4 „ W'W=S* „ +«' ., -aS[ » .. ].
4(/9»-y)»F'F =^ip-r)<j)' -r') + 'f(q-8)(q' -8')-fiy[(p-7-)(q' -s') + (p' -r'){q-8)l
4 „ Z'Z =7» „ +/3» „ -$y[ „ „ ].
Hence
4 (a» - S*) (X'X - TT'Tf ) = (p + r) (p' + r) -(q + s) {q' + s'),
i(^ ->/)(¥'¥ -Z' Z) = (p-r)ip' -r)-(q-s)(q-8'),
and substituting in the expressions for P and R,
4(a'-S«)(/S«-7»)P =
(^ - 7") [(P + r) (p' + r') - (? + s) (?' + s')] - (a^ - S') [(;> - r) (p - r') - (7 - s) (5-' - s)],
4 .. i2-
)) L " » J "r » L » »> J"
Similarly
i(ce -S'yW"X =
aS[{p + r){p' +r') + (q + 8)(q' + 8'y]-cc'(j) + r)(q' +s')-S'(q + s)(j)' + r),
4 „ XTr =
» L I) »i J ~ " " —a „ ,
4>(fi^-rfyz'Y =
^7 [(P - ^) (P' - r) + (q-s) {q' - «')] -^(p- r) (q -s')-rf(q- s) (p' - r'),
4 „ YZ =
»> [ »> » ] ~ T » ~ ^'' •• '
whence
i{<^ -S')(W'X-X'W) = -[(p + r){q' +s')-{p' + r')(q + 8)l
4(/9»-y)(Z'F - Y'Z) = -[(p-r)iq-s)-ip-r){q-s)l
and substituting in the expressions for Q and S
4(a«-S')(/9^-7^)<2 =
- (/S' - 7») [(p + r) (q + «') - (2J' + r) (q + «)] + (tf - 8^) [(p - r) (5' - 5') - (p - r') (q - s)].
~ » L » » ] ~ » L ■> » J*
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 557
133. Hence, collecting and reducing, ,
- (a^ - /3= + 7^ - S-*) {pp' - qq' + rr' - ss') + (a» + ^--y^- 8=) (p/ + p'r - qs' - q's),
4 „ R =
4 „ Q =
4 „ S =
we have
Po(= c/) = a=- /3^ + 7=- S', r„(= c/) = tf^ + /3^ - 7' - S^
and thence
ro^-i)o^=4(a=-&0(;8"--7');
the equations hence become
(n" - Po^) P = —po (pp' - qq' + rr'- ss') + r„ {pr' + p'r — qs' - q's),
R= n( , „ )-po{ .. )>
Q= Po(pq'-p'q + rs'-r's)-n( „ ),
&' = - r-o ( „ ) + Po ( >. )•
On writing in the equations u' = 0, then P, Q, R, S, p', q', r', s' become =p, q, r, s,
Pd, 0, To, 0; and the equations are (as they should be) true identically. The equations
may be written
u+tt' u—u' u u' u u' u u' u u' « «' « u' u u' u u'
(8-4) 4 4 = -4(4.4-7.7 + 8.8 -11.11) +8(4.8 + 8.4-7.11-11.7),
( „ ) 8 8 = +8( „ ) -4( „ ),
( „ ) 7 7 = +4(4.7-7.4 + 8.11-11.8 ) -8(4.11-11.4 + 8.7 - 7.8),
( „ ) 11 11 = -8( „ ) +4( „ ).
There is of course such a system for each of the 60 Gopel tetrads.
Differential relations connecting the theta-functions with the quotient-functions.
134. Imagine p, q, r, s, &c., changed into of, y', z^, vf, &c. ; that is, let x, y, z, w
represent the theta-functions 4, 7, 8, 11 of u, v\ and similarly x', y', £, w' those of
«', v', and Xa, 0, z^, 0 those of 0, 0. Let «', v be each of them indefinitely small;
and take 3, =u' ^- +v' -j- , as the sjonbol of total differentiation in regard to ;/, v,
au av
the infinitesimals u' and 1/ being arbitrary: then, as far as the second order, we have
in general
^(m + m', v+v')='it(u, v) + d^iu, v) + ^d'^(u, v),
558 A MEMOIR ON THE SINGLE AND DOUBLE TH ETA-FUNCTIONS. [704
and heDce
P^{x+dx+ ^x) (x-dx + ^d^x), =ar'+ {xd^x - (dx)*},
and similarly for Q, R, 8. Moreover, observing that x' and z' are even functions,
^ and v/ are odd functions, of u', v, we have
a/, y', z', w =Xo + ifi'x^, dy„, z„ + J3»^o. 9w,„
where 3'a;,, dy^, &c., are what d'^x, dy, &c., become on writing therein u = 0, v = 0 ;
3yo. Sw, are of course linear functions, d^x„, d'Zo quadric functions of u' and v. The
values of ar'», y'*, nl'', w'^ are thus x^^ + a;„9'a;„, (9^0^. V + ^o9%, (Swo)* ; and we have
x^Xo {dyof Zdd-Zi> {d^Unf
a^af* -y'y'* -k-zH"" -wHo"'- s^x^ + zW +«" -y* +z' -vf,
«*/* -fx'^ +zW^ —id'z'' =-y"-Xo- -iifzo" -y- +0^ -vf ■hz\
<c»/' -yW +z^x'- -vh)"' = ^•V +a;V +2' -'<^ +«° -y%
a^M,'* _ y2^'2 4. zHj- — wV = — vi^x^ — y%= — w^ +2^ — 3/^ + a?.
135. On substituting these values, the constant terms (or terms independent of
«', v) disappear of themselves; and the equations, transposing the second and third
of them, become
mS^x^ (dyoY ^o9% (^w,)'
(z,*-x,*){xd'x -idxy]= (-ar,V+V^O +( x,Y-z,V) +{-x,^z-' +z,^ci^) +( x,W-z„Y),
„ [ydhj -{dyy]= -( a;„y-Vw-) -{-x.^x? +z,^z^) -( XoHv^-ZoV) -(-Xo'z' +z,'x'),
„ {zd'z -{dzy}= (-a-„V+^„V) +( x,hv^-z,Y) +(-a-oV +^„=z= ) +( a;„y -z„V),
„ {wa«M;-(3w)'}= -( Xo'w^-Zoy) -(-aro^'Z-'+ZoV) -( x.y- -zM') -(-x,W+z,'z°-).
where it will be recollected that x, y, z, w mean ^4, %, %, %i (ii) ; .r,, is ^4 (0),
that is, C4, and z^ is ^9(0), that is, Cg. But the formulae contain also
3»aro = {Ct", C4", C4^$m', 2;')'''. ^2/0 = (c?', c," $m', v),
8'^o=(c,"', c,'\ Cs^Jm', t/)", &m;, = (c„', c„"$«', v').
The formulae may be written
C4a'C4
(Scr
{ ^.d^-{^y]
c».^ (f.^
{c,*-c,*){ 4 4 4 )= (-4 4 +8 8)
,,{77 7 )=-( 4 7-811)
,,{88 8 j= (-4 8+8 4)
„ {11 11 11 )=-( 4 11 ^8 7)
+(4 7-8 11)
-(-4 4 +8 8)
+( 4 11 -8 7)
-(-4 8 +8 4)
C^-Cs
c».^ c».y
+(-4 8+8 4)
-( 4 11 -8 7)
+(-4 4+8 8)
(ac„)--'
+( 4 11 -8 7),
-(-4 8 +8 4),
+ ( 4 7-8 11),
-( 4 7-8 11)1 -(-4 4+8 8),
■where d'Ct, 3«c», dcj, dcn are written in place of 8X. d'z,, dyo, 9wo- There is of course
a like system of equations for each of the Gopel tetrads.
704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
559
136. Observe that, dividing the first equation by ^4-(«)i or say by ^4-, the left-
hand side is a mere constant multiple of d" log ^j ; and the right-hand side depends
only on the quotient-functions ^7 -=- ^4, ^9 ^^4, ^,,-=-^4; each side is a quadric function
of ii, v'. Equating the terms in u', u'v, v'^ respectively, we have
log^..
dudv
log ^4. d^M^-
each of them expressed as a linear function of the squares of the quotient-functions
^7-H^i, ^8-f-^4, ^,1-7-^4. The formula is thus a second-derivative formula serving for
the expression jf a double theta-function by means of three quotient-functions.
Differential relations of the theta-functions.
137. In "The second set of 16," selecting the eight equations which contain F,
and Wi, these are
«+n' «-u' u+u' u—u' (bufuxes 1.)
^ . ^ ^ . ^ Y W
i { 4 0 - 0 4; = Y' + W",
12 8 - 8 12 = r - W,
6 2-2 6 =W' + Y',
14 10 - 10 U =W' - Y',
1 + 1
'i
X' +Z',
13 9+9 13 =X' -Z',
7 3+3 7 =Z' +X',
15 11 + 11 15 ^Z' -X'.
Then, con.sidering any line in the upper half and any two lines in the lower half,
we can from the three equations eliminate F, and W^, thus obtaining an equation
such as
^,^.-^0^4, Y', W =0,
J5 Ji + Ji J5 J Jl t Zi
viz. this is
-IX'Z' (^4 ^0-^0^4)
+ ( Z'>f'+FX)a^,+^,^„)
+ (- Z'F'+FX)(^,3S^„ + ^„%) = 0,
where the arguments of the theta-functions are as above, « + «', u—u', u+u', u — u'\
and the suffixes of the X', Y\ Z', W are all = 1.
560 A MEMOIR ON THE SINGLE AND DOUBLE THETA-KUNCTIONS, [704
188. Suppose that in this equation u' becomes indefinitely small. If u were =0,
the values of X', Y', Z', W would be a, 0, 7, 0 : hence u' being indefinitely small,
we take them to be a, 3y3, 7, 3S, where
'^' = («' i + ' i) '-' -^ ''• = («' i + ^' jJ ^ • ^" = ^ = ^>'
are, in fiict, linear functions of u' and v.
We have ^«^(, — ^r,,^, standing for
&, (u + w') ^0 («« - «') - ^0 (w + m') ^4 (« - «').
and here
^4 (w ± «') = ^4 + 5^4, ^0 (w ± w') = ^0 ± 3^0 ;
the function in question is thus
{X + 6^4) (^0 - S^o) - (^4 - S^4) {% + 5^o) = 2 [^^X - ^49^0),
where the arguments are w, i;, and the 9 denotes «' j~ + ^ j~ •
Also a-,^, + ^,^5, that is, S^,(if + m')^j(m — M') + ^,(M + it')^5(M -«'), becomes simply
= 2^,S-,, and similarly %^s, + ^(,^]3 becomes =2^13^9; and the equation thus is
- 2a,7, {^^X - V^„) + {a,dh, + 7.8/80 ^,^, + (- a,dh^ + T.^A) ^,3^, = 0,
where the proper suflSx 1 is restored to the a, 3/3, 7, and 3S.
139. The equation shows that the differential combination ^o9^4 — ^4f'^o is a linear
function of ^5^1 and ^,3^1,, the coefficients of these products being of course linear
functions of ii and v. Writing the equation
v^4 - ^49^0 = Ax% + m,^%,
we can if we please determine the coefficients in terms of the constants c , c", c'", c''', c";
viz. taking w, v indefinitely small, we have
^i,= Co, d^i = u' {ci"u-\-Ct''v) + v' {c^U + CiV),
^4 = C4, a^o = U {Co"'u + Co'>) + V' (Co'^M + Co^f),
^1 = C, , ^s = Ci'u + Ct'v,
or substituting, and equating the coeflBcients of u and v respectively, we have
Co(c;"u' + Ci''v') - C4 (Co"'lt' + C„''v') = ^CjC,' + Bc»c^^,
Co (C4'V + C4V) - C4 (Co'V + C(,V) = -AcjCj" + BciCii",
which equations give the values of id, B.
140. Disregarding the values of the coefficients, and attending only to the form
of the equation
'^^X - \d^o = A%% + £^,3^„
704] A MEMOIR ON THE SINGLE AND DOUBLE THET A- FUNCTIONS. 561
this is one of a system of 120 equations ; viz. referring to the foregoing table of
the 120 pairs, it in fact appears that taking any pair such as ^o^4 out of the upper
compartment or the lower compartment of any column of the table, the corresponding
differential combination ^o9^4 — ^^3^0 is a linear function of any two of the four pairs
in the other compartment of the same column.
Differential relation of u, v and x, y.
141. We have, as before, in the two notations, the paii's
A .B 11 .7
c
.DE
5
9
D
CE
13
1
E
CD
14
2
F
.AB
10
6
From the expressions given above for the four pairs below the line, it is clear that
any linear function of these four pairs may be represented by
(a - 6) 3 {(X + fiy) Vcdefa,b, + (X. + fue) v'c,d,e,f,ab},
where X, fj. are constant coefficients : the factor (a — b) has been introduced for con-
venience, as ^vill appear.
We have consequently a relation
Vaa^ 9 Vbb^ - Vbb, 9 Vaa, = — „- {(X + fiy) Vcdefa^b, + (X + /*«) Vc,d,e,f,abj,
where, as before, 9 is used to denote u -r- +v' -r , u and v' being arbitrary multipliers ;
considering it, v as functions of x, y, we have
d _dx d dy d
du du dx du dy '
d _dx d dy d
dv dv dx dv dy'
(L d . dso dx
and thence 9 = P -r- + Q ,- , if for shortness P and Q are written to denote u' -j- +v' t-
dx dy du dv
and w' j^ + ^ j^ respectively.
142. The left-hand side then is
= p(Vaa,^^Vbb,-Vbb.|V^) + Q (Va^,|Vbb,-Vbb,|Vaa,);
c. X. 71
562 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
the coefficients of P and Q are at once found to be
(a-bWaA (a,-b,)Vib
* Vab ' Va>,
respectively, or observing that a — b, =a^ — b,, =a — h, the equation becomes
P^' + Q ^ = - ? {(X + ^y) V^difkA + (^ + /^) ^/cAiX^} ;
vab va,b, "
or multiplying by \/aba,b, and writing for shortness abcdef = X, a^c.d^e/, = F, this
becomes
aA{i' + |a + /.y)VZ} + ablQ + |(\ + ^)VF} = 0.
143. There are, it is clear, the like equations
b,c, {P 4- 1 (V + M> ) VZ} + be {Q + I (\' + /x';c ) V Y) = 0,
CA {P + I (X" + m"2/) VZj + ca {Q + 1 (X" + ^"x) 'JY\ = 0,
and it is to be shown that X = X' = X" and iJi. = ft! = ij,". For this purpose, recurring
to the forms
Vaa,9Vbb, - Vbb,8Vaa^= — ^ ((X + fiy) Vcdefa,b^ + (X + fix) \/c,d,e/,ab},
Vbb, 9Vcc, - Vcc^ a Vbb,= -^ {(X' + fi'y) Vadefb,c, + (X' + filx) Va,d,eXbc],
v'cc^ 3 VaaT - Vaa^ d Vce^ = ^- {(X" + /'y) Vbdefc,a, + (X" + n"x) Vb,d,eXca),
multiply the first equation by Vcc,, the second by Vaa,, and the third by Vbb^, and
add : the left-hand side vanishes, and therefore the right-hand side must also vanish
identically.
144. But on the right-hand side we have the tenn ^ v'defa,b,c, multiplied by
(a - 6) c (X + fiy) + (6 _ c) a (X' + n'y) +{c-a)h (X" -H /i"y),
and the term — ^ v'd,e,f,abc multiplied by
(rt - b) c, (X + /tw;) -1- (6 - c) a, (X' + filx) + {c-a) b, (X" + fi"x),
and it is clear that the whole can vanish only if these two coefficients separately
vanish. This will be the ca.se if we have for X, X', X" the equations
(a - 6) X + (6 - c) X' + (c - a) X" = 0,
c „ +a „ +b „ =0,
704] A MEMOIR ON THE SINGLE AND DOUBLE TH ETA-FUNCTIONS. 563
and the like equations for fi, ti!, /i". The equations written down give
(a — 6) \ : (6 — c) X' : (c — ei) \" = a — 6 : h — c : c — a,
that is, \ = \' = \" : and similarly ft,= fx = n".
145. But this being so, the three equations in P, Q give
that is,
oje a« ^ — y
In these equations m' and v' are arbitrary; hence \ and fi must be linear
functions of u' and v' ; say their values are = xaru' + pv, au' + tv' respectively. We
have therefore
or, what is the same thing,
- i^ ^ = (w + o-y) dw + (p + ry) rft;,
- ^^ ^ = (ot + ax) du +{p + rx) dv,
whence also
7 7 , ^ dx dy
, , , /xdx ydy\
du + pdv^-l[^-^),
which are the required relations, depending on the square roots of the sextic functions
X = abcdef, and Y = a,b,c,d,e/, of x and y respectively ; but containing the constants
tr, p, a, T, the values of which are not as yet ascertained.
146. I commence the integration of these equations on the assumption that the
values ^< = 0, r = 0 correspond to indefinitely large values of x and y. We have
x.^(i-f....), .-=^(.-f....),
where S = a + b + c + d + e+f; and thence the equations are
,.„w.. i5(i4^..^)-i^j(..f....),
71—2
ffU
and thence
•BTM •
564 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
Hence integrating, we have
where the omitted terms depend on — , — &c.
Hence, neglecting these terms, we have
a-U + TV _ _ /'I 4. 1^
'efu + pv + \S((ru + Tv)~ \x y) '
an equation connecting the indefinitely small values of u, v, with the indefinitely
large values of x, y.
147. From the equations A = ^uCT Va, B = k-;ts Vj, taking (m, v) indefinitely small
and therefore {x, y) indefinitely large, we deduce
, 1 - Aa f- + - ]
Citt + Cu V ^ Aji \x yj
\x yJ
Cy'u + Cj"v h,
hence substituting for - + - the foregoing value, and introducing an indeterminate
X y
multiplier M, we obtain
Cii'« + Cii'v = Mku [btu + pv + ^S{(TU + Tv) + ^a (au + rv)],
which breaks up into the two equations
Similarly
C' =Mky {
b
C" =Mk, {
b
C' =Mh I
c
c," =Mk, {
c
Ca' = Mk;, {
d
Cis" = Mk3 {
d
C,' = Mku{
e
Cu" = Mku {
e
c,.' = m.o{
. /
Co" = Mk, {
f
which twelve equations determine the coefficients «r, a, p, t in terms of the c', c"
of the odd functions 5, 7, 10, 11, 13, 14; and moreover give rise to relations connecting
these c', c" with each other and with the constants a, h, c, d, e, f.
148. It is observed that if, as before,
~, , d , d r> d , ^ d
dx
dy'
704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 565
then, substituting for P and Q their values, we have
a = _|(W + ,.')(V^£WF|)-|(.u' + r.')(W^,l. + WF
= (wm' + pv') 9i + (<nt' + tV) dt,
if for shortness
then operating with 9 on the equations A = tsha sfah, &c., we have for instance
AdB- BdA = ■ur%,k, { (ont' + p«') (Va d,\/b- ^bd,-/a)
+ (au' + TV') (Va 9^6 - ^/bd,'^)},
which is one of a system of 120 equations, the A, B being in fact any two of the
16 functions.
These are in fact nothing else than the foregoing system of 120 equations giving
the values of the differential combinations ^o9^, — ^,9^u, &c., each as a sum of products
of pairs of functions, only on the right-hand sides we have expressions such as
Va9iV6— V69,Va, &c., which present themselves as perfectly determinate functions of
X, y: so that, regarding zru' + pv', au' + rv as given linear functions of the arbitrary
quantities n', v\ there is no longer anything indeterminate in the form of the equations.
566 [705
705.
PROBLEMS AND SOLUTIONS.
[From the Mathematical Questions rvith their Solutions frovi the Educational Times,
vols. XIV. to LXi. (1871—1894).]
[Vol. XIV., July to December, 1870, pp. 17—19.]
3002. (Proposed by Matthew Collins, B.A.) — If every two of five circles A, B,G,D,E
touch each other, except D and E, prove that the common tangent of D and E is just
twice as long as it would be if D and E touched each other.
Solution by Professor Cayley.
Consider the ellipse — +Ti = li foci R, S; the coordinates of a point U on the
ellipse may be taken to be (a cos u, b sin u), and then the distances of this point from
the foci will be
r = a (1 — e cos u), s = a (1 + e cos m).
Taking k arbitrarily, with centre R describe a circle radius a — k, with centre S
a circle radius a + k, and with centre U a circle radius k~ae cos u : saj' these are the
circles R, S, U respectively ; the circle U will touch each of the circles R, S (viz.
assuming ae<k<a, so that the foregoing radii are all positive, it will touch the circle
R externally and the circle S internally).
Considering next a point V, coordinates (a cos v, b sin v), and the circle described
about this point with the radius k — ae coav, say the circle F; this will touch in like
manner the circles R, S respectively. And the circles U, V may be made to touch
each other externally ; viz. this will be the case if squared sum of radii = squared
705]
PEOBLEMS AND SOLUTIONS.
567
distance of centres, or what is the same thing, squared difference of radii + 4 times
the product of radii = squared distance of centres ; that is,
a'e" (cos u — cos v)' + 4 (^• — ae cos u) (k — ae cos v) = a^ (cos u — cos vf + ¥ (sin u — sin v)*,
or
2(k — ae cos u) (k — ae cos v) = b'' [1 — cos (ft — v)}.
If for a moment we write tan^!t=a;, tan^« = y, and therefore
1 - ar' 1 - w2 . 2x' . 2y
cosM=:i 1, cos «; = :, — ^, sinM== -, sin t; = ,---„,
1+a?' i+f 1+0^ l+f
we have
, (l-a;»)(l -«n + 4aw , , , 2(x-yf
[ aeil-a?)] L ae(l-f)l^ b'(x-yf
r 1+ai' ]\ l+f ] (l+^Xl+y*)'
or
(A; - fte + (A; + ae) «=) jfc - ae + (k + ae) y'} =b^x- yf,
which is readily identified with the circular relation
, /k+ae\i ^ , /k+ae\i ^ jk^-a'^Xi
or, what is the same thing, in order that the circles U, V may touch, the relation
between the parameters u, v must be
Considering in like manner a circle, centre the point W, coordinates (acosw, isinw),
and radius A — a«cosw, say the circle W; this will, as before, touch the circles R, S;
and we may make W touch each of the circles U, V; viz. we must have
•""-{(li^y-H— -{(;
tan-'
{f^J *^" *"} - *^" {(1^3* *^""' M = *"""' fi
1.2 _ (i2e2N J
yk»
where, in the last equation, tan~'lr ) tan^«> must be considered as denoting its.
value in the first ec|uation increased by tt. Hence, adding the three equations, we have
that is,
fk' -a'e'\i . , /,
568 PROBLEMS AND SOLUTIONS. [705
or
^-s_a»g» = 3(a'-^='),
that is,
:3a''-4^•'^-aV = 0;
viz. this is the condition for the existence of the three circles U, V, W, each touching
the two others, and also the circles R, S.
The circle R lies inside the circle S, and the tangential distance is thus
imaginar}-; but defining it by the equation
squared tangential dist, = squaied dist. of centres — squared sum of radii,
the squared tangential distance is
= 4aV - 4al
But if the circles were brought into contact, the distance of the centres would be
2k, and the value of the squared tangential distance = 4i' — 4a= ; hence, if this be
=s one-fourth of the former value, we have
4(i-=-a») = aV-a=,
that is,
3a^ - 4i* + a^e^ = 0,
the same condition as above. The solution might easily be varied in such wise that
the circles R, S should be external to each other, and therefore the tangential distance
real ; but the case here considered, where the locus of the centres of the circles
17, F, TT is an ellipse, is the more convenient, and may be regarded as the standard
case.
[Vol. XIV., p. 19.]
3144. (Proposed by Professor Cayley.) — If the extremities A, A' of a given line
AA' describe given lines i-espectively, show that there is a point rigidly connected
with A A' which describes a circle.
[Vol. XIV., pp. 67, 68.]
3120. (Proposed by Professor Cayley,) — To find the equation of the Jacobian of
the quadric surfaces through the six points
(1, 0, 0, 0), (0, 1, 0. 0), (0, 0, 1, 0), (0, 0, 0, 1), (1, 1, 1, 1), (a, 0, y, B).
Solntion by the PROPOSER.
Writing for shortness
a=^-7, b = y-a, c = a-^, /=a-S, g = ^-B, h = y-B,
705] PROBLEMS AND SOLUTIONS. 569
(so that a+h—g = 0, &c., a+b + c = 0, af + bg + ch = 0), the six points lie in each of
the plane-pairs
X {hy —gz+ aw) = 0, y (— hx +/« + hw) = 0,
z igx — fy ■¥ cw) =0, w{—ax — by—cz) = 0.
We cannot take these as the four quadrics, on account of the identical equation
0 = 0, which is obtained by adding the four equations ; but we may take the first
three of them for three of the quadrics, and for the fourth quadric the cone, vertex
(0, 0, 0, 1), which passes through the other five points; viz. this is
aayz + b^zx + cyxy = 0.
We write therefore
P = X {hy — gz + aw), Q = y{-}uc +fz + bw),
R = z (gx —fy + cw), 8 = aayz + b^zx + cr^ayy ;
and we equate to zero the determinant formed with the derived functions of P, Q, R, 8
in regard to the coordinates {x, y, z, w) respectively. If, for a moment, we write
A, B, G to denote bg — ch, ch — af, af—bg respectively, it is easily found that the
term containing d^S is
{b^z + cyy) X {— agh, bhf, cfg, tibc, —af^, —gB, hC, a A, ¥g, —c-h\x, y, z, w)^:
the terms containing dyS and dzS are derived from this by a mere cyclical interchange
of the letters (», y, z), (A, B, C), (a, b, c), and (/, g, h). Collecting and reducing, it
is found that the whole equation divides by 2aic; and that, omitting this factor,
the result is
ayz {av? — Ba^) + fxw {^z^ — yy') \
+ bzx (^w- — By^) + gytu (yx" — az'') r = 0,
-I- cxy {yw- — Bz^) + hzw (ay" — ^a?)
which, substituting for a, b, c, f, g, h their values, is the required form.
If, in the equation, we write for instance x = 0, the equation becomes
ayzw (hy — gz + aw) = 0 ;
or, the section by the plane is made up of four lines. Calling the given points
1, 2, 3, 4, 5, 6, it thus appears that the surface contains the fifteen lines 12, 13, ..., 56,
and also the ten lines 123 . 456, &c. ; in all twenty-five lines. Moreover, since the
surface contains the lines 12, 13, 14, 15, 16, it is clear that the point 1 is a node
(conical point) on the surface ; and the like as to the points 2, 3, 4, 5, 6.
[Vol. XIV., pp. 104, 105.]
3249. (Proposed by Professor Caylet.) — Given on a given conic two quadrangles
PQRS and j)qr8, having the same centres, and such that P, p; Q, q; R, r; S, s
are the corresponding vertices (that is, the four lines PQ, RS, pq, rs all pass through
C. X. 72
570
PROBLEMS AND SOLUTIONS.
[705
the same point ; and similarly the lines PR, QS, pr, qs, and the lines PS, QR, ps. qr) :
it is required to show that a conic may be drawn, passing through the points p, q, r, «
and touched at these points by the lines pP, qQ, rR, sS, respectively.
Solution by the Proposer.
Taking the centres for the vertices of the fundamental triangle, the equation of
the given conic may be taken to be a^ + y' + z^ = 0; and then the coordinates of P,
Q. R, S to he {A, B, C), {A, - B, G), {A, B, -G), (A, -B, -C) respectively, where
.4' + .B* + C = 0 ; and those of p, q, r, s to be (a, /9, 7), (a, - /8, 7), (a, /3, - 7),
(o, — y9, —7) respectively, where a'^ + ;S' + 7'=0. The required conic, assuming it to
exist, will be given by an equation of the form ^ + my* ■\-m^ = 0. This must pass
through the point (a, /9, 7), and the tangent at this point must be
a; (£7 - C/8) + y (Ca - ^7) + ^ (.4/3 - 5a) = 0 ;
that is, we must have l<^ + vi^ + nrf = 0, and
la : m^ : ny=By-C^ : Ca-Ay : A^-Ba.
The first condition is obviously included in the second ; and the second condition
remains unaltered if we reverse the signs of B, yS, or of C, 7, or of B, /8 and C, 7.
Hence the conic passing through p, and touched at this point by pP, mil also pass
through the points q, r, s, and be touched at these points by the lines qQ, rR, sS,
respectively ; that is, the equation of the required conic is
By-l^a^+^-jzAiy^+^^^^l'^.^^o
13
or, what is the same thing,
^y^, ycty-, affz'
A , B , G
« , ^ . 7
= 0.
[Vol. XV., January to June, 1871, pp. 17 — 20.]
3206. (Proposed by Professor Cayley.) — In how many geometrically distinct ways
can nine points lie in nine lines, each through three points ?
3278. (Proposed by Professor Cayley.)— It is required, with nine numbers each
taken three times, to form nine triads containing twenty-seven distinct duads (or, what
is the same thing, no duad twice), and to find in how many essentially distinct ways
this can be done.
705] PROBLEMS AND SOLUTIONS. 571
Solution by the Proposer.
Let the numbers be 1, 2, 3, 4, 5, 6, 7, 8, 9. Any number, say 1, enters into three
triads, no two of which have any number in common. We may take these triads to
be 123, 145, 167. There remain the two numbers 8, 9; and these are, or are not, a
duad of the system.
First Case. — 8 and 9 a duad. In the triad which contains 89, the remaining
number cannot be 1 ; it must therefore be one of the numbers 2, 3 ; 4, 5 ; 6, 7 ; and
it is quite immaterial which ; the triad may therefore be taken to be 289. There is
one other triad containing 2, the remaining two numbers thereof being taken from the
numbers 4, 5 ; 6, 7. They cannot be 4, 5 or 6, 7 ; and it is indififerent whether they
are taken to be 4, 6 ; 4, 7 ; .5, 6, or 5, 7 : the triad is taken to be 247. We have
thus the triads
123, 14.5, 167, 289, 247;
and we require two triads containing 8 and two triads containing 9. These must be
made up with the numbers 3, 4, 5, 6, 7 : but as no one of them can contain 47, it
follows that, of the two pairs which contain 8 and 9 respectively, one pair must be
made up with 3, 5, 6, 7, and the other pair with 3, 5, 6, 4; say, the pairs which
contain 8 are made up with 3, 5, 6, 7, and those which contain 9 are made up with
3, 5, 6, 4 (since obviously no distinct case would arise by the interchange of the
numbers 8, 9). The triads which contain 8 must contain each of the numbers
3, 5, 6, 7, and they cannot be 83.5, 867, since we have 67 in the triad 167 ; similarly
the triads which contain 9 must contain each of the numbers 3, 5, 6, 4, and they
cannot be 845, 836, since we have 4.5 in 145. Hence the triads can only be
836, 857 I 934, 956,
837, 856 I 935, 946;
and clearly the top row of 8 must combine with the top row of 9, and the bottom
row of 8 with the bottom row of 9 ; that is, the .system of the nine triads is
123, 145, 167, 289, 247,
in combination with
836, 857, 934, 956,
or else in combination with
837, 856, 935, 946.
These are really systems of the same form, that is, each of them is of the form
BCa
0ya
bcG
CA^
yab
caA
A By
a/9c
abB;
viz. in the first and second systems respectively we have
ABCa^yabc
6 1 3287549 (First system).
5 13 2 9
4 6
7 8 (Second system).
72—2
572
PROBLEMS AND SOLUTIONS.
[705
as one out of maiiy ways of effecting the identification. Observe that there is not in
the system any triad of triads containing all the numbers. It thus appears that 8, 9,
a duad, gives only a single form of the system.
Cor. — It is possible to find in a plane nine points such that the points belonging
to the same triad lie in lined. The nine points are, in fact, on a cubic curve ; and
the figure is that belonging to a theorem of Prof Sylvester's, according to which it
is possible to find on a cubic curve a system of points 1, 2, 4, 5, 7, 8, &c., (a series of
numbers not divisible by 3), such that for any triad (such as 145) where the sum of
the numbers, one taken negatively, =0, the three points are in lined; and so also
that, if two of the points become identical, in the figure 13 = 14, then there is not
any new point, but the preceding points are indefinitely repeated; thus, 2, 14, 16 being
in lined, and 14 being =13, 16 must be =11, and so on.
Second and Third Cases. — 8 and 9 do not form a duad. There are thus three
triads composed of 8 with (2, 3 ; 4, 5; 6, 7), and three triads composed of 9 with
(2, 3 ; 4, 5 ; 6, 7). If with these numbers (2, 3 ; 4, 5 ; 6, 7) we form all the arrange-
ments of three duads other than those which contain all or any of the duads 23, 45, 67,
there are the eight arrangements
A = 24, 37, 56,
B = 24, 36, 57,
C = 25, 36, 47,
D = 25, 37, 46,
E = 26, 35, 47,
F = 26, 34, 57,
G = 27, 34, 56,
H = 27, 35, 46,
where A has a duad in common with B, with D, and with G:
in common with C, E, F, or H. We have thus the sixteen pairs
but it has no duad
AC,
AE,
AF, AH,
BD,
BE,
BG, BH,
CF,
CG,
GH,
DE,
DF,
DG,
EG,
FH,
where each pair contains six different duads.
705] PROBLEMS AND SOLUTIONS. 573
Combining AC with 8, 9, we have the triads 8 (24, 37, 56) and 9 (24, 36, 57),
that is, the triads
824, 837, 856 : 924, 936, 957 :
which, with the original three triads 123, 145, 167, form a system of nine triads;
8 and 9 might, of course, be interchanged, but no essentially distinct system would
arise thereby. Hence we have a system of nine triads by combining the original three
triads 123, 145, 167, with any one of the sixteen pairs AG, AE, &c. But it is
suflBcient to consider the combinations of the three triads with each of the pairs
AC, AE, AF, AH; in fact, these are the only systems which contain the triad 824;
and since there is no distinction between the two paii-s 4, 5 and 6, 7, or between
the two numbers of the same pair, it is allowable to take 824 as a triad of the system.
Hence —
Second Case. — The system consists of the three triads combined with AE; viz. it is
123, 145, 167: 824, 837, 856: 926, 935, 947:
which, it is to be observed, consists of three triads of triads, each triad of triads
containing all the nine numbers ; viz. the system is
123, 479, 568 : 145, 269, 378 : 167. 248, 359.
' 2-
3
Cor. — We may have nine points such that the points belonging to the stime triad
lie in lined, viz. the figure is that of Pascal's hexagon when the conic is a line-pair.
Third Case. — Combining the three triads with AC, AF, or AH, it is readily seen
that we obtain in each case a system of the form
Aaa', A^j, A^'y',
5/3/3', Bya, By'a' ,
Cyy' , Ca&, (7a'/3',
viz. in the case where the pair is AC; that is, the system is
123, 145, 167 : 824, 837, 856 : 925, 936, 947 ;
and in the cases where the pair is AF or AH, the identifications may be taken to be
ABC o /3 7 a' /3' 7'
9, 8, 1; 4, 5, 2; 7, 6, 3 (AC),
9, 8, 1; 2, 3, 4; 6, 77~5 (AF),
9, 8, 1; 5, 4, 6; 3, 2, 7 (AH).
574 PROBLEMS AND SOLUTIONS. [705
Observe that there is in the system a single triad of triads Aaa', B^ff, Cyy, con-
taining all the numbers; viz. for the system with AC, this is 123, 856, 947; for the
system with AF, it is 145, 837, 926 ; and for the system with AH, it is 167, 824, 935.
Cor.— It is possible to find a system of nine points such that the points belonging
to the same triad lie in lined. Such a figure is this:—
The solution shows that these are the only systems of nine points satisfying the
prescribed conditions.
[Vol. XV., pp. 66, 67.]
3329. (Proposed by Professor Cayley.) — It is required to show that every per-
mutation of 12345 can be produced by means of the cyclical .substitution (12345), and
the interchange (12).
Solution by the Proposer.
It is sufficient to show that the interchanges (13), (14), (15) can be so produced;
for then, with the interchanges (12), (13), (14), (15), we can, by at most two such inter-
changes, bring any number into any place.
Writing P = (12345), a = (12), we have
(12) = a,
(13) = a7^aP*a,
(14) = a Pa P*a P^a F'a Pa P*a,
(15) = P'a P,
as can be at once verified ; and the theorem is thus proved.
I remark that, starting with any two or more substitutions, and combining them
in every possible manner (each of them being repeatable an indefinite number of
times), we obtain a "group"; viz. this is either (as in the problem proposed) the
705] PROBLEMS AND SOLUTIONS. 575
system of all the substitutions (or say the entire group), or else it is a system the
number of whose terms is a submultiple of the whole number of substitutions. The
interesting question is, to determine those two or more substitutions, which, by their
combination as above, do not give the entire group; for in this way we should arrive
at all the forms of a submultiple group.
[Vol. XV., p. 80.]
3356. (Proposed by Professor Cayley.) — If the roots (a, /3, 7, B) of the equation
(a, b, c, d, e) («, ly = 0 are no two of them equal ; and if there exist unequal
magnitudes 9 and <f>, such that
(^ + a)' : (6 + 0)' : (0 + y)* : (6 + S)' = (<f> + <xy : {<}> + 0y : (.^ + 7)^ : (<f> + Sy;
show that the cubinvariant
ace — ad- — h^e — c'+ ibcd = 0 ;
anfl find the values of d, <f>.
[Vol. XVI., ifune to December, 1871, p. 6.5.]
3507. (Proposed by Professor Cayley.) — Show that, for the quadric cones which
pass through six given points, the locus of the vertices is a quartic surface having
upon it twenty-five right lines; and, thence or otherwise, that for the quadric cones
passing through seven given points the locus of the vertices is a sextic curve.
[Vol. XVL, ]). 90.]
3536. (Proposed by Professor Cayley.) — A particle describes an ellipse under the
simultaneous action of given central forces, each varying as (distance)"^, at the two
foci respectively : find the differential relation between the time and the excentric
anomaly.
[Vol. XVIL, January to June, 1872, p. 35.]
3591. (Proposed by Professor Cayley.) — If in a plane A, B, C, D are fixed points
and P a variable point, find the linear relation
a.PAB + ^.PBC + y.PGT) + i.PDA = 0,
which connects the area» of the triangles PAB, &c.
[Vol. XVIL, p. 49.]
2652. (Proposed by Professor Cayley.) — Find the differential equation of the
parallel surfaces of an ellipsoid.
576 PROBLEMS AND SOLUTIONS. [705
[Vol. XVII., p. 60.]
3677. (Proposed by Profe-ssor C.A.YLEY.) — Find at any point of a plane curve the
angle between the normal and the line drawn from the point to the centre of the
chord parallel and indefinitely near to the tangent at the point ; and examine whether
a like (juestion apj)lies to a point on a surface and the indicatrix section at such point.
[Vol. xvn., p. 72.]
3564. (Proposed by Professor Cayley.) — To determine the least circle enclosing
three given points.
[Vol. XVIII., July to December, 1872, p. 68.]
3875. (Proposed by Professor Cayley.) — Given the constant a and the variables
X, y, to construct mechanically ; or what is the same thing, given the fixed
y
points A, B, and the moving point P, to mechanically connect therewith a point P'
such that PP' shall be always at right angles to AB, and the point P' in the
circle APB.
[Vol. XX., July to December, 1873, pp. 106, 107.]
3430. (Proposed by W. J. C. Miller.)— Find the equation of the fii-st negative
focal pedal of (1) an ellipsoid, and (2) an ellipse.
Solution by Professor Cayley.
1. It is easily seen that if a sphere be drawn, passing through the centre of
the given quadric and touching it at any point {x', y', z'), then the point {x, y, z) on
the required surface, which corresponds to («', y', z), is the extremity of the diameter
of this sphere which passes through the centre of the quadric. We thus easily find
the expressions
--(2-:,). y = .'(2-|,). -^'(2-^);
where
Solving these equations for x, y', /, and substituting in the two equations
xuf + yy' + zz' = x'^ + y'^ + z\ ^" + ^V ^' = 1,
705] PBOBLEMS AND SOLUTIONS. 577
we get
" ■ '»■ +,-^=< <i),
(-^.) (-^) (-
«" 2/' ■^^ 1 /ox
a»(2--.V 6^f2-:^l c»(2
a-
--T
cV
Since (2) is the differential with respect to t of (1), the result of eliminating t
between these two equations is the discriminant of (1). Hence the equation of the
required surface is the discriminant of (1) with respect to t. Since (1) is only of
the fourth degree, this discriminant is easily formed. If (1) be written in the form
AP + iBt^ + 6Cf + 4i)« + £ = 0,
it will be found that A and B do not contain x, y, z, while C, D, E contain them,
each in the second degree. Now the discriminant is of the sixth degree in the
coefficients, and of the form A^ + B^'y^ (see Salmon's Higher Algebra, § 107); hence
it contains x, y, z only in the tenth degree. This is therefore the degree of the
required surface. \
The section of this derived surface by the principal plane z consists of the dis-
criminant of
-^+ J^-< w,
which is of the sixth degree, and is the first negative pedal of -;, + f^ = 1 ). together
with the conic (taken twice), which is obtained by putting t=2<? in (3).
This conic, which is a double curve on the surface, touches the curve of the
sixth degree in four points.
2. The formulae for the conic are quite analogous to those for the ellipsoid, viz.
we have
a. = z{2-^,(Z'+F»)}, y=F|2-i(Z=+F0},
leading to the equations
6 = 77 + -^
2-^ 2-^'
and its derived equation, from which to eliminate 6. The first is the cubic equation
{A, B, C, D){0, iy = 0, where
A = l, 5=-§(a'' + 6'), C = ^{aV + by + 4:d'b'), D = - 2w'b- (x" + y^).
c. X. 73
578 PROBLEMS AND SOLUTIONS. [705
Equating the discriminant to zero, this is
0 = ^= V = 4 (ilG - £»)' - (SABG - A^D - 1&)\
Or finally
(3a'a!» + 3&y - 4a* + 4a»6» - 46*)'
+ {9 ia? - 2¥) aV + 9 (6' - 2a'') fr'y' - 8a» + 12a*6« + 12a»6* - ft"}' = 0,
which is the required equation.
[Vol. XXI., January to June, 1874, pp. 29, 30.]
4298. (Proposed by J. W. L. Glaisher, B.A.) — With four given straight lines
to form a quadrilateral inscribable in a circle.
Solution hy Professor Cayley.
Let the sides of the quadrilateral taken in order be a, h, c, d; and let its
diagonals be x, y; viz. x the diagonal joining the intersection of the sides a, b
with that of the sides c, d; y the diagonal joining the intersection of the sides
a, d with that of the sides b, c ; then, the quadrilateral being inscribed in a circle,
the opposite angles are supplementary to each other. Suppose for a moment that
the angles subtended by the diagonal x are 6, tt — 0, we have
x' = b^ + c'' + 2bc cos 0, a^^a' + d^- 2ad cos 0 ;
and thence
(ad + be) X- = ad (6= + c=) + be (a^ + d') = (ac + bd) (ab + cd),
that is,
.^ = (ac + 6d)^--j^.
and similarly
, , ,j,ad + bc
v' = (ac + bd) — r -, ,
^ ^ 'ab + cd'
agreeing as they should do with the known relation xy = ac-\-bd: the quadrilateral
is thus determined by means of either of its diagonals. It is however interesting
to treat the question in a different manner.
Considering a, b, c, d, x, y as the sides and diagonals of a quadrilateral, we have
between these quantities a given relation, say
F(a, b, c, d, X, y) = 0,
and the quadrilateral being inscribed in a circle, we have also the relation xy = ac + bd;
which two equations determine x, y; and thus give the solution of the problem.
705] ■ PROBLEMS AND SOLUTIONS. 579
The expression of the function F is in effect given in my paper, "Note on the
value of certain determinants, &c.," Quarterly Mathematical Journal, t. iii. (1860),
pp. 275 — 277, [286] ; viz. a, b, c being the edges of any face, and /, g, h the remaining
edges of a tetrahedron, then
volume = Tii {b^'c' (g"- + h^ ) + d'a^h'' + p) + a'b" (p + g")
+ g-h? (¥ + c^) + h-p (c= + a^) +fy (a= + b^)
- ap {a? +p) - 6y (6' + g^) + c-'h? (c^ + h?)
- aYh'-b%Y--c'fy-a'b''d'},
where, when the tetrahedron becomes a quadrilateral, the volume is = 0.
In this formula, changing c, b, h, g, f, a into a, h, c, d, x, y, we have the required
equation F =0; viz. this is found to be
a'6V + b^c^d? + c'd'a' + dW6» - l^d'' {P + d^) - aV {a"" + c') + od^y^ {a^+b^ Jrc^ + d^ -a?- y^)
+ a? (aV + bH^ - a^d' - b^c") + y^ (aV + 6=d^ - a^"" - (fd^) = 0,
which, with xy = ac + bd, determines x, y. Substituting in the foregoing equation for
xy its value, the equation becomes
(ad + bcf *•= + (ab + cdf y^=1 {a^6V + b'^&d'' + &d^a? + dW6^ + ahcd {a^ + 6'' + c= + d^)],
or
{ad + bcY a? + {ab + cdf y^ = 2 (ad + be) (ab + cd) (ac + bd).
To show more clearly how this equation arises, I observe that we have identically
jP - (a' + &■ + c= + rf' - a^ - 2/=) (xy + ac + bd) (xy-ac — bd) — 2(ad + be) (ab + cd) (xy -ac- bd)
= {(ad + bc)x- (ab + cd) y}'.
The resulting equation (ad +bc)x — (ab + cd)y = 0, together with xy = ac + bd, gives
for X, y the foregoing values.
[Vol. XXI., pp. 81, 82.]
4392. (Proposed by S. Roberts, M.A.) — If Np denotes the number of terms in
a sjTnmetrical determinant of p rows and columns, show that the successive numbers
are given by the equation
^i - iV*_, -(k-iy N,_, + ^(k-l)(k-2) {iVi_3 + (k-Z) N,-,} = 0,
k being positive and N^ being taken equal to unity.
Solution by Professor Cayley.
It is a curious coincidence that the question of determining the number of distinct
terms in a symmetrical determinant has been recently solved by Captain Allan
73—2
580 PROBLEMS AND SOLUTIONS. [705
Cunningham in a paper in the last number of the Quarterly Journal of Science*;
and the question having been proposed to me by Mr Glaisher, I have also solved
it in a paper [580] printed in the April Number of the Monthly Notices of the Royal
Astronomical Society. I there obtain
iVi= 1 .2 ... Arcoeff. a^ in 7^ .,,
(i. — X)'
viz. writing
l+^'l— 2
« = i^o+-Z^iT + -^»¥-o +
I show that u satisfies the differential equation
giving when the constant is determined
Writing the differential equation in the form
2(l-x)g = (2-^)«,
we at once obtain for Ni the equation of differences
Nt - kNt-, + Uk-l){k-2) Nt-^ = 0,
which is in fact a particular first integral of Mr Roberts's equation ; viz. from the
above equation we have
Nt-^-(k-l)Nt., + ^(k-2)(k-3)Ni^=^0,
and multiplying this last by fc — 1 and adding, we have
Nt - N,., -(k-iyN,_, + i{k-l) {k - 2) {i^i_, + (k-3) Nt^} = 0,
which is the equation obtained by Mr Roberts. It thence appears that the general
first integral of his equation is
Nt-kNt-, + ^ (k - !)(&- 2) Nt^^i-)" Cl . 2 ... (k-1).
The equation
N„ = kN,_, -^(k-l){k-2) N,_,
gives very readily the numerical values, viz.
17 = 4.5 - 3.1
73 = 5.17- 6.2
388 = 6.73-10.6
1
= 1
1-
-0
2
= 2
1-
-0
5
= 3
2-
-1.
1
2461 = 7.388 -15.17
18155 = 8.2461-21.73.
• I have not the volume at hand to refer to, but he obtains an equation of differences, and gives the
numbers 1, 2, 6, 73, 398 (should be 388),...
705] PROBLEMS AND SOLUTIONS. 581
[Vol. XXII., July to December, 1874, pp. 20, 21.]
4354. (Proposed by R. Tucker, M.A.)— Solve the equations
— ce' + a!y + xz = a = 4i (1),
-f + xy + yz = b=-20 (2),
- 2- + xz + yz = c = - 8 (3).
Note on Question 4354. By Professor Cayley.
A question of simple algebra such as this, becomes more interesting when inter-
preted geometrically : thus, writing the equations in the form
— a?+ixy + xz = aw", yx — y^ + yz = bw', zx + zy — z^ = eu/',
and then putting for shortness
a = — a + b + c, ^ = a — b + c, y = a + b — c,
the solutions obtained are
X : y : z : w = aa : b^ : cy : {a^yi',
X : y : z : w = aa : b^ : cy : — {a.^y) ;
say these are
{aa, 6/3. cy, (a/Sy)*} and [aa, b^, cy, -(<x^y)*].
But the equations are also satisfied by
(x = 0, y = z, w=0), {y = 0, z = x, w = 0), (z = 0, x = y, w= 0),
or what is the same thing, (0, 1, 1, 0), (1, 0, 1, 0), (1, 1, 0, 0). The three equations
represent quadric surfaces, each two of them intersecting in a proper quadric curve,
and the three having in common 8 points; viz. these are made up of the first
mentioned two points each once, and the last mentioned three points each twice:
2 + 3.2, =8.
To verify this, observe that, at each of the three points, the tangent planes of
the surfaces have a common line of intersection ; this line is the tangent of the
curve of intersection of any two of the surfaces, and the curve of intersection therefore
touches the third surface ; wherefore the point counts for two intersections. In fact,
taking {X, Y, Z, W) as current coordinates, the equations of the tangent planes at
the point (x, y, z, w) are
X{2x-y-z)-Yx -Zx +2aFw = 0,
-Xy +Y(-x+2y-z)-Zy +2bWw = 0,
-Xz -Yz +Z(-x-y + 2z) + 2cWw =0:
hence at the point (0, 1, 1, 0) these equations are
-2Z=o, x+Y-z = o, -x-r+z=o,
which three planes meet in the line X = 0, Y—Z = 0; and similarly for the other
two of the three points.
582 PROBLEMS AND SOLUTIONS. [705
[Vol. XXII., pp. 60—64.]
4458. (Proposed by Professor Cayley.) — Find (1) the intersections of the two
quartic curves
\ (ab — xyf = ahx (a — y){h — y), fi {ah — xyf = aby (a — x){b — x);
and (2) trace the curves in some particular cases ; for instance, when a = 1, 6=2,
X=l, fji = -2.
Solution by the Proposer.
1. The 16 intersections are made up as follows : 5 points at infinity on the
line x=0, 5 at infinity on the line y = 0, the two points (x = a, y = b), (x = b, y — a),
and 4 other points, 16 = 5 + 5 + 2 + 4. As to the points at infinity, observe that, as
regards the first curve, the point at infinity on the line a; = 0 is a flecnode having
this line for a tangent to the flecnodal branch ; and, as regards the second curve,
the same point is a cusp, having this line for its tangent ; hence the point in
question counts as 2 + 3, =5 intersections; and the like as to the point at infinity
on the line ^ = 0. It remains to find the coordinates of the 4 points of intersection.
Assume xy = aba), then the equations become
\ (1 - «)' = a; + cay — (a + 6) w, /jl(1 — coy = ax + y — (a -i- b) to ;
hence, eliminating successively y and x, the factor 1 — w divides out, — this factor
belongs to the points (x = a, y = b), (x = b, y=a) for which obviously co = 1 — , and the
equations become
(\ — fjLti)){l—o)) + {a+b)o) = {l+a))x, {jx — \a>) (1 —w) + (a + b) a> = (1 + to) y.
Multiplying these two equations together, and substituting for xy its value ahto,
we find
{(X - (la) (fi - \o>) + (a + b)(X+fj.)(o}(l- w)' + (a + bf w' - (1 + w)" aab = 0.
Write, for shortness, p=(\ + fi){a + b)-\' — fi^, then, dividing by w', and writing
« + - = n, the equation is
(\fin+p)(n-2) + {a + by--ab{n + 2) = 0;
viz. this is a quadric equation for fl. But, instead of fi, it is convenient to introduce
the quantity ^, = o TTg > ~ ( T ) ' '^^^ equation thus becomes
Wlil+P
40 4
or
{2Xfi{l + 6) + p{l - 0)} id + (a + by {I ~ ey - iab{l - 6) = 0,
or
&> [{a + by-4,(p- 2XfjL)} +d{- 2a' - 2¥ + 4 (;) + 2 V)) +{a-by = 0;
705] PROBLEMS AND SOLUTIONS. 58S
viz. substituting for p its values, this is
^(a + 6-2\-2/i)2 + 2^{-a=-6=+2(\ + /i)(a + 6)-2(\-;ii)y+(a--6)2=0;
or if we write
A = a? -2a{\ + ^l) + {\ - iJ.f, B = b^ -2b(\ + /i) + {X-fiy,
this is
0'(a+b-2\-2fj,y-2(A+B)d + {a-by=O,
whence
{{a- by -{A +B) 0}" = ^ [{A+By -{a-by(a + b-2\- 2^)=}
= 0'{(A+ By -(A- By} = ^ABO' ;
viz. taking for convenience the sign — on the right-hand side, this is
{a-by-{A+B)e = -2e'JAB;
and we have thus
(a - by
e =
(^/A-^By
that is,
^_o) — 1_ a — b _ s/A — \JB + a — b
We may write
x = ^L{eo-l) + ^(a + b) + i(a + b-2\-2fj,)'"~^\,
CO -\- 1
y = \(m-l) + ^(a + b) + ^ia + b-2\-2ij,)~^;
CO + 1
whence also x — y = (jj, — X)(a> — l), as is also seen at once from the original equations;
then we have
Ha..-2X-2,)"^;=H^-^)(.^._^-^2X-2.)
^(a-b){fi-\) + b>^A-a>/B
^JA-^B-a + b
2\(a — b) ^ , , A i-n 7s
_ {a-b){\- n) + b-JA-a»JB
s/A->jB-a + b '
which may be expressed in the more simple form
x = -^(a + \-fi + ^/A){b + \-fi+s/B),
y = j-{a-\ + f^ + 'JA)(b-\ + fj. + s/B),
and the values are
584 PROBLEMS AND SOLUTIONS. [705
the transformations depending on the identity
8V|oz*) =a6-(\ + //)(a + 6) + (X-M)' + V^(6-X-M)+V5(a-X-M) + V45,
tJA — V-o — a + 0
which is easily verified. Of course, since the signs of >JA, >jB are arbitrary, we have
4 systems of values of {x, y), which is right.
In the original equations, for a, h, X, fi, x, y, write 1, fc-", V, -/i.^ a?, -/;
then the equations become
X,« (1 + l<^a?fy = a^ (1 + f) (1 + l^y% iJi? (1 + ^a^')' = fO--a?){\- Jt^x'),
and we thence have
. X VpT+y') (1 + A:y) + ^y V(l - ar^) (1 - k'ai')
'^+'*''~ 1 + kVy'
viz. assuming a; = sn a (sinam a), iy = sQi^, this is \+/ii = sa(a+/3t); viz. the problem
is (for a given modulus k, assumed as usual to be real, positive, and less than 1)
to reduce a given imaginary quantity \ + /xi to the form sn (a + /3t). The proper
solution is that in which the signs of the radicals are each — , viz. it may in this
case be shown that the value of x^ is positive and less than 1, that of y' positive.
The values thus are
where
A = 1 -2\= + 2/*= + (V + /iO, -8 = |5-|x= + |/i= + (\= + /i=y.
The solution is really equivalent to that given by Richelot (Crelle, t. XLV., 185.S, p. 225).
To verify this partially, observe that, writing a, r for Richelot's tan^<^, tan ^i^, we
have
y'=^^(^-^'-i''-'^^^{h-^''-f''-'^^)'
a
givmg
•-y\ = -V^;
hl)n
i\\ 1
+ x' + M
giving
whence
{^-\)h-^B;
\ 1
2<rX=l + \= + ;u^-V^, 2t^ = ^ + V + ^=-V5,
or the above value of a? is =^--'o■T, agreeing with his; the value of y"^ is, however,
presented under a somewhat different form.
+ 1-65
+ 0-94,
+ 1-22
+ 213,
- 012
- 3-4.9.
705] PROBLEMS AKD SOLUTIONS. 585
2. The curves are
{2-xyy = 2x{l-y){2-y), -{2-xyy = y{l-x){2-x) (1, 2),
each passing through the points (1, 2) and (2, 1) ; the four points of intersection
found by the foregoing general theoiy are all real, viz. these are
a; = i (2 + V3) (5 + -v/17), y = - ^ (- 1 + ^3) (- 1 + ^17), say +17-00 and - 0-57,
-V3, +V17
+ V3, -V17
-V3, -V17
The equation of the curve (1) may also be written iu the forms
f (ic^ - 2x) + 2yx - 4a; + 4 = 0, a;y + « ( - 2^/2 + 2i/ - 4) + 4 = 0.
The original form shows that, if y is between 1 and 2, x is negative — (but by a
further examination it appears that there is not in fact any branch of the curve
between these limits of y) — but y being outside these limits, then x is positive; in
fact, the whole curve lies on the positive side of the axis of y. And then the inspec-
tion of the first quadric equation shows that the lines a; = 0 and a; = 2 are each an
asymptote.
The point at infinity on the axis of y is in fact a flecnode, the tangent to the
flecnodal branch being a; = 0, and that of the ordinary branch a; = 2.
Similarly, from the second quadric equation, it appears that the line 3/ = 0 is an
asymptote ; the point at infinity on the axis of a; is in fact a cusp, the axis in
question y = 0 being the cuspidal tangent.
The equation of the curve (2) may also be written in the forms
a?^ + («2 _ 7^ 4. 2) y + 4 = 0, (2/= + 2/) «' - 7ya; + 2y + 4 = 0.
The original form shows that, if a; is between 1 and 2, y is positive ; but that x
being beyond these limits, y is negative ; and as regards the first case, x between
1 and 2, we at once establish the existence of an oval, meeting the line y = 1 in
the points a; = 2 and f, and the line y=2 in the points a; = l and |; it is further
easy to see that the horizontal tangents of the oval are 2/ = ^(25 + Vll3), =say 22
and 0-9.
The remainder of the curve lies wholly below the line y = 0. The first quadric
equation shows the asymptote a; = 0 ; the point at infinity on the axis of y is in
fact a cusp, having the axis itself for the cuspidal tangent. The second quadric
equation shows the asymptotes y = 0, y = — l; the point at infinity on the axis of
a; is in fact a flecnode, having the line y = 0 for the tangent to the flecnodal branch,
and y= — l for that of the other branch. It is further seen that there are two
vertical tangents a; = J (11 ± VI 13)= 10-8 or 0-2; the former of these touches a branch
C. X. 74
586
PROBLEMS AXD SOLUTIONS.
[705
lying wholly between the two asymptotes y = 0, y = — l\ the latter one of the branches
belonging to the cuspidal asymptote a; = 0 ; this last branch cuts the asymptote a; = 0
at y = — 2, and then, cutting the asymptote y = — l and x = — ^{= — 0'3), goes on to
touch at infinity the asymptote y = 0. It is now easy to trace the curve.
The figure shows the two curves. The curve (1) is shown by a continuous line,
the curve (2) by a thick dotted line; the points 1, 2, 3, 4 show the above mentioned
four intersections of the curves ; the point 1 and the dotted branch through it are
of necessity drawn considerably out of their true positions; viz. as above appearing,
the ^-coordinate of 1 is =17"00, and the equation of the vertical tangent to the
branch is a; = 108.
[Vol. XXII., pp. 78, 79.]
4620. (Proposed by A. B. Evans, M.A.) — Find the least integral values of x and
y that will satisfy the equation x''-Qb^y^ = -\.
Solution by Professor Cayley.
The values are given in Degen's Tables, viz.
X = 2746864744, y = 88979677.
The work referred to is entitled "Canon Pellianus, sive Tabula simplicissimara sequa-
tionis celebratissimse y' = aa? + l solutionem pro singulis numeri dati valoribus ab 1
usque ad 1000 in numeris rationalibus iisdemque integris exhibens. Auctore C. F. Degen,
Hafnise (Copenhagen), 1817."
705] PROBLEMS AND SOLUTIONS. 587
Table I., pp. 3 — 106 gives, for all numbers 1 to 1000, the denominators, (?) the
quotients of the convergent fraction of tja, and also the least values of x, y which will
satisfy the equation x-— ay- = + \. Thus
953 30, 1 , 6, 1 , 2 , 1 , 3 , 8, 1 , 1 , (4 , 4 ),
1 , 53, 8, 41, 17, 37, 16, 7, 32, 29, (13, 13),
488830275367615376, 15090531843660371073.
Table II., pp. 109 — 112, is described as giving for all those values of a between
1 and 1000, for which there exists a solution of the equation a? — ay^ = — 1, the least
values of x and y which satisfy this equation : thus 953, x and y as above. It is,
however, to be noticed that the values of a = /9^ + l, for which there is the obvious
solution a; = /S, y=\, are omitted from the table. The reason for this appears, but
the heading should have been different.
[Vol. XXIII., January to July, 1875, pp. 18, 19.]
4528. (Proposed by Professor Cayley.) — A lottery is arranged as follows : — There
are n tickets representing a, h, c pounds respectively. A person draws once ; looks
at his ticket ; and if he pleases, draws again (out of the remaining n—1 tickets) ;
looks at his ticket, and if he pleases draws again (out of the remaining n- 2
tickets); and so on, drawing in all not more than k times; and he receives the
value of the last drawn ticket. Supposing that he regulates his drawings in the
manner most advantageous to him according to the theory of probabilities, what is
the value of his expectation?
Solution by the Proposer.
Let the expression "a or a " signify " a or a, whichever of the two is greatest,"
and let if, (a, 6, c, ...) denote the mean of the quantities (a, b, c, ...), viz. their sum,
divided by the number of them.
To fix the ideas, consider five quantities a, b, c, d, e, and write
M, (a, b, c, d, e) — M^ (a, b, c, d, e),
Mt{a, b, c, d, e) = Mi {a or Mi(b, c, d, e), h or il/, (a, c, d, e), ..., e or Jfj (a, b, c, d)],
Maia, b, c, d, e) = Mi [a or M^ib, c, d, e), b or M^ia, c, d, e), ..., e or Mi{a, b, c, d)},
and so on. And the like in the case of any number of quantities a, b, c, ....
Then the value of the expectation is =Mi;{a, b, c, ...).
For, when k = l, the value is obviously =Mi{a, b, c, ...).
74—2
588 PROBLEMS AND SOLUTIONS. [705
When i=2, if a is drawn, the adventurer will be satisfied or he will draw again,
according as a or ilf, (6, c, ...) is greatest, viz. in this case the value of the expectation
is "a or M,(b, c, ...)."
So if b is drawn, the adventurer will be satisfied or he will draw again, according
as 6 or Mi(a, c, ...) is greatest; viz. in this case the value of the expectation is
"b or M,(a, c, ...)"; and so on: and the several cases being equally probable, the
value of the total expectation is
= Mi {a or Mi(b, c, ...), b or il/i(a, c, ...), ...) =Mi{a, b, c, ...):
and the like for k = S, ^• = 4, &c.
For instance, a, b, c, d = 1, 2, 3, 4, M, (1, 2, 3, 4) = ^,
M,(l. 2. 3, 4) = ilf,(l or §, 2 or f , 3 or |, 4 or |) = i\/.(§, |, |, ^) = ^,
i¥,(2, 3, 4) = i¥,(2 or |, 3 or f , 4 or ^y=M,q, f. f) = ^.
M,(l, 3, 4) = M,(1 or I 3 or f , 4 or ^) = M,(i, f, f) = ^.
M,{1, 2, 4) = ilf,(l or f, 2 or f, 4 or f) = Jf/(f, f, f) = J^.
M,(l, 2, 3) = i¥,(l or f, 2 or f 3 or f) = i¥,(f, f, |) = J^,
3/3(1, 2, 3, 4) = il/,(l or V, 2 or ^, 3 or J^, 4 or i^) = J)/, (^, i^, J^, J^) = ff,
M,il, 2, 3) = 3&c.,
Jlf4(l, 2, 3, 4) = ifi(l or 4, 2 or 4, 3 or 4, 4 or 3) = iV,(4, 4, 4, 4) = 4.
Or finally
Mu M„ Jf„ M, = ^, ft, H. 4 = M. M. M. If-
Cor. If the a, 6, c, ... denote penalties instead of prizes, then the solution is
the same, except that "a or a " must now denote " a or a, whichever of them is
least."
[Vol. xxin., pp. 47, 48.]
4581. (Proposed by the Rev. M. M. U. Wilkinson.) — A witness, whose statement is
what he opines once in m times, and whose opinion is correct once in n times, asserted
that the number of a note, issued by a bank universally known to have issued notes
numbered from B to B-\-A — \ inclusive, was B + P, where P is either 0, 1, 2, ...,
or A —1. Prove (1) that the probability that the note in question was that note is
1 [ (m-l){n-l))
mn\ ^ A-1 y
The above witness also said that the note was signed by X, it being universally
known that X has signed one note, and Y the remaining A — 1 notes ; find (2) the
probability that this last statement was correct.
705] PROBLEMS AND SOLUTIONS. 589
Remark by Professor Cayley.
There is a serious difficulty in the question, or the answer ; I think, in the
question. Try the answer in numbers m = 10, n = 10. The witness says what he
opines once out of 10 times — he is in fact an atrocious liar ; and he opines rightly
once out of 10 times, that is, wrongly 9 times out of 10; he is therefore a blunderer —
but a remarkably ingenious one, in that the chances are so greatly against his blunder-
ing upon a right result.
He says that the note was signed by X, and the chance of this being so is found
to be 1^+^ = -^, or more than ^; the larger part -^ of this is obtained as
follows : — the witness having said that the note was signed by X, the chances are 9
out of 10 that he thought the reverse ; and, thinking the reverse, the chance is 9
out of 10 that he thought wrongly, viz. that the note was signed by X. But can
the statement of such a witness create any probability in favour of the event ?
The fallacy seems to consist in the assumption that n can have a determinate
value irrespective of the nature of the opinion. Suppose there are 500 notes, and
that the opinion is that the note was a definite number 99 ; it is quite conceivable
that, in forming a series of such opinions, the witness may be wrong 9 times out
of 10. But let the opinion be that the note was not 99 ; no amount of ingenuity
of blundering can make him wrong 9 times out of 10 in a series of such opinions.
If it could, a friend who knew the true opinion of the witness, would be able 9
times out of 10 to know the number of the note, from the mere fact that the
witness opines that the note is not a named number.
[Vol. XXIII., p. 58.]
4638. (Proposed by Professor Cayley.) — Find the equation of the surface which is
the envelope of the quadric surface aod' + by"- + cz" + dvfi = 0, where a, b, c, d are variable
parameters connected by the equation .46c + Bca + Cab + Fad + Gbd + Hcd = 0 ; and
consider in particular the case in which the constants A, B, C, F, 0, H satisfy the
condition
{AFf + {BGf + (CiT)* = 0.
[Vol. XXIV., July to December, 1875, p. 41.]
4694. (Proposed by Professor Cayley.) — Taking F, F' a pair of reciprocal points in
respect to a circle, centre 0; then if F, F' are centres of force, each force varying as
(distance)"", prove that (1) the resultant force upon any point P on the circle is in
the direction of a fixed point S on the axis OFF' ; and if, moreover, the forces at
the unit of distance are as (Oi^)^'"~" to (Oi^')*'"~". then (2) the resultant force is
proportional to
(SP)-*'"-".(PF)-*"'*",
where PV is the chord through <S'.
590 PROBLEMS AND SOLUTIONS. [705
[Vol. XXIV.. pp. 72—74.]
4793. (Proposed by Professor Wolstenholme, M.A.)— If y = a^(loga;)', where n and r
are integers, prove that
rfn+ry r{r-l) _ d^^'y ^ r(r- l)(r - 2) (3r- 5) d^-^y
the coeflScients being
Ar-ilr-i ^r-j^r-i ^r-Slr-l Al*^' , ,
^ . — - , -— — TV- , and 1 ;
|r-l ' |r-2 ' |r-3 ' |1
so that tlie result may be symbolically written
\ dec"*') X •
d d^v
where D denotes -r- and operates on -r-f only, and A operates on l*-' only, the
terms after the rth all vanishing since A'" x" = 0, when in is an integer > n. The
calculations involved prove that, when x=l,
I'^ + l » , -, 3w-2
24 •
^.-3,„^|,^l.(»-l)(«-2.).
Solution by Professor Cayley.
Since y = x^ (log xf, therefore {xdx — n)y = rx^ (log a;)*""' ; by repeating the same
operation, we have
(xdx -nyy = [rYx^ ; whence dx^ (xdx - nfy = [r]*" [»]" .
Now, for any value whatever of the function y, we have
dj" (xdx - nYy = Ax'-dj+^y + 5a;'-'rf/+"-'?/ + Car^d/+''-^-y + &c.,
the coeflScients .4, B, C, ... being functions, presumably of r, n, but independent of
the form of the function y. It will, however, appear that A, B, C, ... are, in fact,
functions of r only.
To see how this is, observe that (xdx — n)" consists of a set of terms
(xdxf, (^ = 0 to r),
where (xd^* denotes 6 repetitions of the operation xdx', by a well-known theorem, this
is =[a;di + 5 — 1]*, where, after expansion of the factorial, (xd:^ is to be replaced by
afdx', thus
(xdxf = {xdx + 1]' = ci?dx^ + xdx, (a^*)' = M* + 2]» = (B»d,» + ^a?dx^ + 2.xdx, &c. ;
705] PKOBLEMS AND SOLUTIONS. 591
thus {xdx — ny consists of a series of terms scfidx^, {0 = 0 to r), and, operating with
dx", this last, =(dx + dx'}^, consists of a series of terms such as dx^d^^^'", where the
unaccented symbol operates on the «*, and the accented symbol on the y ; the term
is thus a^da;"+*~", or observing that 0 — a is at most =r, and putting it =r — k,
the term is a;'^*(/j;"+*, viz. dj^ (xdx — ny consists of a series of terms of the form
x'~''dx'^*'' ; or, what is the same thing, d^^ (xdx — nYy is a series of the form in
question.
To understand how it can be that the coefiScients A, B, C, ... are independent
of n, take the particular case r = 2 ; then we have here
4» (xdx - nfy = Aa^dx''+^y + Bxdx^+'y + Cdx^'y.
The right-hand side is
dx" {afdx' - (2w - 1) xdx + «') y,
which is
[a^d''+^ + 2nxdx''+' + {n^ - n) dx"] y
- (2?i - 1) { xdx"+' + ndx""} y
+ «'{ dx"}y;
hence
A = l, 5 = 2n - (2n -i 1), = 1, C=(n' - n)-n(2n- I) + n\ = 0 ;
and we thus see also how in this particular case the last coefficient is = 0, viz. that we
have
4" M« - nfy = a^dj'^'y + xdx^^'y,
without any term in d^y.
To find the coefficients A, B, C, ... generally, write 3/ = «"■+"+*, then xdx — n = r + 0,
and consequently
dx" (xdx -nYy, =(r + dYdx''af+''+^, =(r + 0y[r + n + efx''+^;
whence
(r + ey[r + n + 0Y = A [r + e + n]»+'- + B [r + ^ + n]»+'-' + ;
or, what is the same thing,
(r + 0y = A[r + 0y + B[r + ef\'-'+
Since the left-hand side, and every term [r + 0]' on the right-hand side, contains
the factor r + 0, there is not on the right-hand side any term [r + 0]" ; dividing the
equation by r+ 6, it then becomes
(r+0y-' = A[r+0-lY-' + B[r + d-l]'-'+ ,
and we thus have
viz. writing r + 0=1 +x, Ux=={l +x)^\ and taking the terms in the reverse order,
the series is the well-known one
x,x—l
1 — ' '"172"
Ux = Mo + T ^"0 + 1 0 '^^"0 + &C.
592 PROBLEMS AND SOLUTIONS. [705
Hence, in general,
where observe that the last term is =xdx^^y.
For the function 3/ = a;»(loga;)», the value of each side is =[r]''[»]».
[Vol. XXIV., pp. 89—91.]
4752. (Proposed by Professor Cayley.)— Mr Wolstenholme's Question 3067 may
evidently be stated as follows : —
If (a, h, c) are the cooi-dinates of a point on the cubic curve
a» + 6» + c* = (6 + c) (c + a) (a + h),
and if
then {x, y, z) are the coordinates of a point on the same cubic curve.
This being so, it is required to find the geometrical relation of the two points
to each other.
Solution by Professor Cayley.
1. On referring to Professor Wolstenholme's Solution of the original Question
3067 (Reprint, Vol. Xlli., p. 70), it appears that the coordinates (x, y, z) of the point
in question may be expressed in the more simple form
X : y : 2=a(— a + 6 + c) : b{a — h + c) : c(a + 5 — c);
viz. the given relation between {a, b, c) being equivalent to
iabc + (-a + b + c){a-b + c)(a + b-c) = 0,
we have
, ., .„ -4a6c
a - (o - c)' = T ,
^ ^ -a+b+c
and thence
\ -a + b + cJ \-a + b + cJ'
and consequently
' a {-a + b + c)
whence the transformation in question.
2. Writing for greater symmetry {x, y, z) in place of {a, b, c), and (a', y', z')
in place of {x, y, z), the coordinates {x, y, z) and {x', 'i/, z') of the two points are
connected by the relation
x' : y' : z' = x{—x + y + z) : y{x — y + z) : z{x+y - z).
705] PROBLEMS AND SOLUTIONS. 593
and we thence at once deduce the converse relation
X : y : z = x' {—x' + y' + z') : y {x' —y'+ zf) : z' {x + y' — z').
Hence, writing
{-x +y -^z, X -y +z, x + y - z) =(^, r), ^),
and similarly
(- x+y'+ z, x'-y' + z', x' -\- y - z') = (f , ,,', ?'),
we have
x : y' : z' = x^ : yr] : z^, x : y : z = x'^' : y'r)' : z'^',
and thence also f^' = i??;' = ff ; so that, regarding (^, t], f ), (|', rf, f) as the
coordinates of the two points, we see that these are inverse points one of the other
in regard to the triangle ^ = 0, rj=0, f = 0.
To complete the solution, we must introduce these new coordinates into the
equation of the cubic curve. Writing this under the form
Hxyz +2{— x + y + z){x — y + z){x + y — z) = 0,
and observing that
the equation is H
('? + r) (?+?)(? + 7,) + 2?,7?=0;
viz. this is a cubic curve inverting into itself. And the two points in question are
thus any two inverse points on this cubic curve.
3. In regard to the original form, that the point {x, y, z) defined by the equations
X (- a" +b'' + c') = y (a' -b'' + c'') = z (a" + 6= - c^,
lies on the cubic curve
a^+ l^ + c^-(b+c)(c + a){a + b) = 0,
Professor Sylvester proceeds as follows: — Writing
(x, y, z) = {a* - (¥ - d'Y, b* - (c- - a?)"-, (f-{a^- h')% = {A, B, C),
suppose; and
F{a, b, c) = w> + b'+<^-(b + c){c + a)(a + b),
he observes that the truth of the theorem depends on the identity
F(A,B,C) + F (a, b, c)F{a, - b, c) F{a, b, - c) F(a, -b,-c) = 0,
and that, in oi-der to prove the identity generally, it is sufficient to prove it for the
three cases a'=0, a^ — b' + c', a'' = b', which may be effected without difficulty.
4. But, for a general proof of the identity, write
\ = b- + c", fj, = b" — C-,
so that
A=a*-fi\ B = (a'+fi)(-a^ + \), C = (- a= + \) (a^ - /i),
c. X. 75
594 PROBLEMS AND SOLUTIONS. [705
whcncG
-F{A,B,C)=-{a*- fi'Y + 2 (a» - Xf (a« + 3a>0 - 8a-6^c= (a* - fi') (a» - \),
= a" - 6\a" + (6V + 9fi- - Sb'if) a' + X (- 2\» - 18^" + 86V) a«
+ /t' (18X= - 3/i= + Sb^c') a* + \fj? (- 6X» - 86'c'') a» + /*«.
Moreover
i!'(a, &, c) = o {a» - (6 + c)»] - (6 + c) (o' - (6 - cf},
therefore
J!'(a, - 6. - c) = a (a' - (6 + c)'} + (6 + c) {a^ - (6 - c)-) ;
whence
F {a, b, c) F (a, - 6, - c) = a» {a" - (6 + c)t - (^ + c)' {a^-{b- cy}'
= a« - 37'a^ + 7= (7= + 28^) a'^ - 728^,
if 7 = 6 + c, 5 = 6 — c. By changing the sign of c, we interchange 7 and S, and we
thus have
F(a, b, - c) F (a, - b, c) = a' -SB'- a* + B\2y' + S") a' -YS",
and the identity to be verified is thus
[a" - Sya* + r" (7= + 25^) a'' - rfB*} [a' - SS'a* ■hS'(2'f + S') a" - y S*}
= a"-6Xd"'+ +fJL\ ut suprd;
the values of X, /it in terms of 7, B are X = ^ (7^ + 8-), /x = 7S ; substituting these values
on the right-hand side, the verification can be completed without difficulty.
[Vol. XXV., January to July, 1876, p. 82.]
4946. (Proposed by Professor Cayley.) — Show that the attraction of an indefinitely
thin double convex lens on a point at the centre of one of its faces is equal to that
of the infinite plate included between the tangent plane at the point and the parallel
tangent plane of the other face of the lens.
[Vol. XXVI., July to December, 1876, pp. 41, 42.]
5020. (Proposed by W. S. B. Woolhouse, F.R.A.S.)— Let 1, S„ S^, S^, ...,S,» be
the first differences of the coefficients of the expansion of the binomial (1 + a;)*" taken
as fai' as the central or maximum coefficient ; also let
v = i(n+ 1) re, v' = ^n (n - 1), v" = H« - 1) ('» - 2), &c. ;
then show that the algebraic function
.■c" - 5,af' + B.jX'" - B^x'"' + &c.
is divisible by (a; — 1)" without a remainder; and that the sum of the numerical
coefficients of the quotient is equal to 1 . 3 . 5 . . . 2h — 1.
[See Solution to Question 1894, Reprint, vol. v., p. 113.]
705] PROBLEMS AND SOLUTIONS. 595
Solution by Professor Cayley.
Mr Woolhouse's elegant theorem depends ultimately on the property of triangular
numbei-s <f> (n), = ^ (n° — n) ; then <f) (n + I) = <l} {— n), so that, writing down the series
of triangular numbers backwards and forwards,
.... 10, 6, 3, 1, 0, 0, 1, 3, 6, 10,...
, a, b, c, d, e, f, g, h,
we have, in fact, a continuous single series obtained by giving to n the different
negative and positive integer values, zero included.
Thus a particular case is
(1 + a;)« - 5 (1 + «)» + 9 (1 + «) - 5 = 0 (mod. a^) = 1 . 3 . Sar* + &c. «* + ...,
where, on the left-hand side, the exponents are the triangular numbers ^(«+l),
n = 0 to 3 ; and the coefficients, after the first, are the differences of the binomial
coefficients of the power 2n (in the particular case, n = 3) ; viz. the binomial coefficients
being
1, 6, 15, 20, 15, 6, 1,
the differences taken as far as thej- are positive are
5, 9, 5.
Expanding the several terms and writing down only the coefficients, we have a diagram
1, 6, 15, 20, 15, 6, 1,
-5 1, 3, 3, 1,
+ 9 1, 1,
-5 1,
The theorem in the particular case depends on the identities
1- 5 + 9-5 = 0,
6-15 + 9=0,
5-15 =0,
20-5 =1,3.5;
or writing, as above, h, g, f, e, to denote the triangular numbers 6, 3, 1, 0, these may
be replaced by
h' -5^ +9/o_5eo ^0,
h -5g + 9y _ 5e = q,
ih{h~l) -5.y(g-l) +... =0,
ih{h-l)(h-2)-5.ig(g-l)(ci-2) + ... =1.3.5;
75—2
596 PROBLEMS AND SOLUTIONS. [705
or, reducing each equation by those which precede it, these become
hP-5g' + 9/» - oef = 0,
h^ - og' + 9/' - 5e' = 0,
h*-5f + 9/^-0^ = 0,
A»-o5r'+9/'-5e» = 1.2.3,l .3.5.
Consider any one of these, for instance the third ; the function on the left-
hand is
lh'-(6-l)g' + ilo-6)f'-(20-15)e%
or, introducing the values 6, c, d as above,
W - 6(7' + 15/^ - 20e» + Ud' - 6c= + lb',
which is, in fact, = 0, if b, c, d, e, f, g, h are any successive triangulai- numbers ;
viz. this is an immediate consequence of the well-known theorem
1 (^ + e)'" - 6 (^ + S)"* + 1 5 (^ + 4)™ - 20 (^ + S)*" -H5 (^ + 2)™ - 6 (^ -I- 1)'" + ^"'
_ ^fffm^ — 0 foj. j^jjy value of m up to m = 5, and
= 1.2.3.4.0.6 for ?ft = 6.
We have thus all the equations except the last; and as regards the last equation,
observe that the equation to be verified is
1 [H^ + 6) {e + 5)Y -6[^(e + 5)(e + 4)]' + ... = 1 . 2 . 3 . 1 . 3 . .5,
viz. this may be replaced by
l(^ + 6)«-6(e + 5)« + ... = 2M.2.3.1.3.5 = 2.4.6.1.3.5 = 1.2.3.4.5.6,
which is right.
It is clear that the proof, although worked out on a particulai- case, is perfectly
general; and Mr Woolhouse's theorem is thus proved.
[Vol. XXVI., pp. 77, 78.]
5079. (Proposed by Professor Caylev.) — Show that the curve
{(/9 - 71? - s"!* {(^ - /8t)' + f]i + 9 K/9 + yty - s=}i {(x + my + f}^
= {(1 - ?=) f f V- - (7 - S^?)i {{a^-y- Sif + y%
where t=(V-l) as usual, is a real bicircular quartic having the axial foci
$i, — /St, 7 + Si, y — Bi,
705] PBOBLEMS AND SOLUTIONS. 597
Solution by the Proposer.
Consider the equation
(I + mi)i [(x - ^iy + y-]i +q{l- «w')i [(x + /3i)- + y-]* = (\ + fii}^ [(« - 7 - Bi)- + y']K
This is
{I + mi) {af + y-- (8- - 20xi} + </- {I - mi) {x' + t/^ - /S'- + 2^xi}
- (\ + fii) [x- +if- jS- + /3- + 7- - S- - 27a; - 2 (« - 7) Bi]
+ 2q (l' + ?«,-)* [(of + y'- ^■'f + 4/3-'a.-]i = 0,
where, putting the imaginary part equal to zero, we have
/» (1 - f/) (a;^ + 2/= - ^-) - 2« (1 - 9O /8^ - M {«" + y' - /S' + (/3- + 7' - 8"-) - 27«) + 2\ (*• - 7) S = 0,
which will be true identically if
m(l-f/)-/x = 0,
-i(l-f/)/8 + M7+^S =0,
- A (/S" + f - 8') - 2X78 = 0.
The last gives
and then
so that, putting
we have
Therefore
\=e{^- + r- ?r), M = - 2^78, e arbitrary ;
i (1 - 5^) yS = dh i^' + 'f-B'- 27-^) = 08 (^' - 1 - B'),
m{l-q')=-2eBj;
eB = il-,f)^, or 0 = (l-f/)|,
l = ff>-y"-- B\ m = - 2/^7,
^ = (l-'Z")f(/3-^ + 7=-n M = (l-?^)f(-278).
i ±mi={^T 'py — S-,
\±^i = (i-50f[/8-^ + (7 + 8^?];
and the equation is
{(y8 - r-y - ^]^ {(* - W + 2/¥ + ? {(/8 + 71? - «=)» {(^ + AT + 2/»}i
= {(1 - 1) If {^' - (7 - Si?}* K^' - 7 - «0^ + 2/^)^
which is a real curve having the axial foci +/3t, —^i; 7 + 8*; 'y—Bi; viz. 7 + 1
being a focus, and the curve being real, it is clear that 'f — Bi is also a focus.
598 PROBLEMS AND SOLUTIONS. [705
[Vol. XXVII., January to June, 1877, p. 20.]
5130. (Proposed by Professor Cayley.)— Show that the envelope of a variable
circle, having its centre on a given conic and cutting at right angles a given circle,
is a bicircular quartic; which, when the given conic and the circle have double contact,
becomes a pair of circles; and, by means of the last-mentioned particular case of the
theorem, connect together the porisms arising out of the two problems —
(i) Given two conies, to find a polygon of n sides inscribed in the one and
circumscribed about the other.
(ii) Given two circles, to find a closed series of n circles each touching the
two circles and the two adjacent circles of the series.
[Vol. XXVII., pp. 81—83.]
5208. (Proposed by Professor Sylve.ster.) — Let the magnitude of any ramification
signify the number of its branches, and let its partial magnitudes in respect to any
node signify the magnitudes of the ramifications which come together at that node.
If at any node the largest magnitude exceeds by k the sum of the other magnitudes,
let the node be called superior by k, or be said to be of superiority k ; but if no
magnitude exceeds the sum of the other magnitudes, let the node be called subequal.
Then the theorem is, in any ramification, eithei- there is one and only one subequal
node; w else there are two and only two nodes each superior by unity, these two
nodes being contiguous.
Solution by Professor Cayley.
The proof consists in showing that (1) there cannot be more than one subequal
node ; (2) thei"e cannot be more than two nodes each superior by unity : and if
there is one such node, then there is, contiguous to it, another such node ; (3) starting
from a node which is superior by more than unity, there is always a contiguous
node which is either of smaller superiority, or else subequal; for, these theorems
holding good, we can, by (3), always amve at a node which is either subequal or
else superior by unity ; in the former case, by (1), the subequal node thus arrived
at is unique ; in the latter case, by (2), we have, contiguous to the node arrived
at, a second node superior by unity ; and we have thus a unique pair of nodes each
superior by unity.
I will prove only (3), as it is easy to see that the like process applies to the
proof of (1) and (2).
Let the whole magnitude be n ; and suppose at a node P which is superior by
k, the largest magnitude is o, and that the other magnitudes are, say, /3, 7, B. We
705] PROBLEMS AND SOLUTIONS. 599
have a = /3 + 7 + 8 + ^; and since n = a + /3 + y + S, we have thence n = 2a — k, or
a= ^(n + k), 0 + y + 8 = ^(n- ^•) : clearly k is even or odd, according as n is even
or odd.
Suppose now that we pass from P, along the branch of magnitude a, to a
contiguous node Q; and let the magnitudes for Q be a', 0', <y', B', e, where a'
denotes the magnitude for the branch QF. We have a' = yS + 7 + S + 1 : for the
ramification consists of the branch QP and of the ramifications of magnitudes 0, 7, 8
which meet in P. We have thus
and thence
a = i (n - k) + I = ^n - ^ (k - 2) ;
/8'+ 7' + S' +e'= in + H^' - 2).
Supposing here that k is greater than 1, viz. that it is = or > 2, ^•— 2 is 0 or
positive ; and if a' be the greatest magnitude belonging to the node Q, this is a
subequal node. But it may be that o' is not the greatest magnitude ; supposing then
that the greatest magnitude is /9', we have
0' = ^n + ^{k-2)-y'-B'-,',
a +y' + B' + e' = ^n- ^{k-2) + y' -^ B' + 6,
and thence
/9'-(a' + 7' + S' + €') = i-2-2(7'+8' + 6');
viz. either the node is subequal, or else, being superior, the superiority is at most
= ^• — 2 ; that is, if from the node P, of superiority = or > 2, we pass along the
branch of greatest magnitude to the contiguous node Q, this is either subequal, or
else of superiority less than that of P; which is the foregoing proposition (.3).
The subequal node, and the two nodes of superiority 1, in the cases where they
respectively exist, may be termed the centre and the bicentre respectively; and the
theorem thus is, every ramification has either a centre or else a bicentre. But the
centre and the bicentre here considered, due (as remarked by Professor Sylvester) to
M. Camille Jordan, and which may for distinction be termed the centre and the bicentre
of magnitude, are quite distinct from the centre and the bicentre discovered by Professor
Sylvester, and considered in my researches upon trees, British Association Report,
187.5, [610]. These last may for distinction be termed the centre and the bicentre of
distance: viz. we here consider, not the magnitude, but the length of a ramification, such
length being measured by the number of branches to be travelled over in order to
reach the most distant terminal node. The ramification has either a centre or else
a bicentre of distance : viz. the centre is a node such that, for two or more of the
ramifications which proceed from it, the lengths are equal and superior to those of
the other ramifications, if any; the bicentre a pair of contiguous nodes such that,
disregarding the branch which unites the two nodes, there are from the two nodes
respectively (one at least from each of them) two or more ramifications the lengths
of which are equal to each other and superior to those of the other ramifications,
if any.
600 PROBLEMS AND SOLUTIONS. [705
It is very noticeable how close the agreement is between the proofs for the
existence of the two kinds of centre or bicentre respectively. Say, first as regards
distance, if at any node the length of the longest branch exceeds by k the length
of the next longest branch or branches, then the node is superior by k, or is of
the superiority k; but, if there are two or more longest branches, then the node is
subequal. And say next, in regard to magnitude, if at any node the largest magnitude
exceeds by k the sum of all the other magnitudes, the node is superior by k, or
has a superiority k; but if the laigest magnitude does not exceed the sum of the
other magnitudes, then the node is subequal. Then, whether we attend to distance
or to magnitude, the three propositions hold good: (1) there cannot be more than
one subequal node; (2) there cannot be more than two nodes each superior by
unity : and if there is one such node, there is contiguous to it another such node ;
(3) starting from a node which is superior by more than unity, there is always a
contiguous node which is of smaller superiority or else subequal ; whence, as in the
solution just referred to, there is always, as regards distance, a centre or a bicentre ;
and there is always, as regards magnitude, a centre or a bicentre.
[Vol. XXVII., pp. 89, 90.]
On Mr Artemas Martin's First Question in Probabilities. By Professor Cayley.
The question was, " A says that B says that a certain event took place : required
the probability that the event did take place, ;>i and jx^ being A's and B's respective
l)robabilities of speaking the truth."
The solutions, referred to or given on pp. 77 — 79 of volume xxvii. of the Reprint,
give the following values for the probability in question : —
Todhunter's Algebra pip^-\- (\—p^){\—p^.
Artemas Martin JOi I>iP2 + (l -|Ji)(l -pa)].
American Mathematicians and Woolhouse... pip^.
It seems to me that the true answer cannot be expressed iu terms of only
Pi and Pi, but that it involves two other constants /9 and k; and my value is —
Cayley p,p,-^ ^{\-p,){l -p,) + k{\- ^)(\-p,).
In obtaining this I introduce, but I think of necessity, elements which Mr Woolhouse
calls extraneous and imperfect.
£ told A that the event happened, or he did not tell A this; the only evidence
is A'b statement that B told him that the event happened; and the chances are
Pi and 1-^1. But, in the latter case, either B told A that the event did not
happen, or he did not tell him at all ; the chances (on the supposition of the
incorrectness of .4*8 sUtement) are /9 and 1-/3; and the chances of the three cases
705] PROBLEMS AND SOLUTIONS. 601
are thus p^, /3(1— pi), and (1 — /3) (1 — ^i). On the supposition^ of the first and second
cases respectively, the chances for the event having happened are p^ and l—p.^; on
the supposition of the third case (viz. here there is no information as to the event)
the chance is k, the antecedent probability ; and the whole chance in favour of the
event is
p,p, + ^ (1 - pO (1 -p,)+k(l- /S) (1 - p,).
If /3 = 1, we have Todhunter's solution; if /3 = 0, and also k = 0, we have the solution
preferred by Woolhouse; but we do not (otherwise than by establishing between k
and /3 a relation which is quite arbitrary) obtain Martin's solution. The error in
this seems to be as follows : — A says that B told him as to the event, and says
further that B told him that the event did happen ; the probability of the truth of the
compound statement is taken to be =j5/; whereas, in calling the probability of A'&
speaking the truth pi, we mean that if A makes the statement, "B says that the
event took place," this is to be regarded as a simple statement, and the probability
of the truth of the statement is = pi ; viz. I think that Martin introduces into his
solution a hypothesis contradictory to the assumptions of the question.
I remark further that in my solution I assume that the event is of such a
nature that, when there is any testimony in regard to it, the probability is determined
by that testimony, irrespectively of the antecedent probability. This is quite consistent
with the antecedent probability being, not zero, but as small as we please ; so that,
if k is (as it may very well be) indefinitely small, the whole probability is the same
as if A were =0. But there is absolutely no reason for assigning any determinate
value to /3; so that the solutions p^p, + (1 — pO (1 — p^) and p^p^, which assume
respectively /9 = 1 and yS = 0, seem to me on this ground erroneous.
[Vol. XXVIII., June to December, 1877, p. 17.]
6306. (Proposed by Professor Cayley.) — If a, /3, 7, S; Oj, A, y^, S], are such
that
(«:-S,)(A-7i)=(«-S)(^-7),
(/3,-8i)(7.-ai) = (/9-S)(7-«). (7.-SJ(ax - A) = (7- «)(« -^);
show that the three equations
'i^: - <A3w^) K' - ">' '"' - '>' - <' - "^ <' - '«■■
'i^l - (;;t:ssW^) '('^ - '>' <* - ''' - <' - -*' <^ - *"•
C. X.
76
I
602 PROBLEMS AND SOLUTIONS. [705
are equivalent to each other; and show also that, consistently with the foregoing
relations between the constants, the differences fli — Sj, ySj — Si, 71 — Si may be so
determined that the equations in (x, a;,) constitute a particular integral of the
differential equation
dx dxi
{(x-a)(x-ff)(x-y)(x-i^)]i ~ [(x, - a,) (rr, - A) (x, - 7,) (x, - S,)}* *
[Vol XXIX., January to June, 1878, p. 20.]
4870. (Proposed by Professor Cayley.) — Given three conies passing through the
same four points ; and on the first a point A, on the second a point B, and on
the third a point C. It is required to find on the first a point A', on the second
a point R, and on the third a point C, such that the intersections of the lines
A'R and AC, A'C and AB, lie on the first conic ;
B'O' and BA, RA' and BC, lie on the second conic;
C'A' and CB, C'B' and CA, lie on the third conic.
[Vol. XXIX., pp. 96, 97.]
5625. (Proposed by Professor Cayley.) — The equation
{q'^ (x + y + zf — yz — zx —xy]^ = 4 (2q + 1) xyz {x + y+z)
represents a trinodal quartic curve having the lines a; = 0, y = 0, z = 0, x+y + z = {i
for its four bitangents; it is required to transform to the coordinates X, Y, Z, where
X = 0, F=0, Z=0 represent the sides of the triangle formed by the three nodes.
[Vol. XXXI., January to June, 1879, p. 38.]
6387. (Proposed by Professor Cayley.) — Show that a cubic surface has at most
4 conical points, and a quartic surface at most 16 conical points.
[Vol. XXXII., July to December, 1879, p. 35.]
5927. (Proposed by Professor Cayley.) — If {a + /3 + 7 + ...}* denote the expansion of
(a + /3 + 7+ ...)'', retaining those terms Na^^''^^''' ... only in which
6 + c+d+... :|>p-l, c+d+ ...is'p-2, &c. &c.;
prove that
«» = (a; + a)» - («), {«)' {x + a + /3)»-' + " ^J^^^ {a + ^Y(x + a + ^ + 7)"-=^
-"^""•^2^3~^-^!a + ^+7lH«'+« + ^ + 7 + g)"-' + &c....(l).
705] PROBLEMS AND SOLUTIONS. 603
[Vol. xxxiii., January to July, 1880, p. 17.]
6155. (Proposed by Professor Cayley.) — Given, by means of their metrical co-
ordinates, any two lines ; it is required to find their inclination, shortest distance, and
the coordinates of the line of shortest distance.
N.B. — If \, n, V are the inclinations of a line to three rectangular axes, and
a, /3, 7 the coordinates to the same axes of a point on the line, then the metrical
coordinates of the line are
a, h, c, f, g, h,
= cos X, cos fi, cos V, /3 cos v — y cos fi, y cos \ — a cos v, a cos yu. — jS cos X,
satisfying identically the relations
a'' + 6" + c= = 1, af+bg + ch = 0.
[Vol. XXXVI., 1881, p. 21.]
6470. (Proposed by Professor Cayley.) — It is required, by a real or imaginary
linear transformation, to express the equation of a given cubic curve in the form
[Vol. XXXVI., p. 64.]
6766. (Proposed by Professor Cayley.) — Find the stationary and the double tangents
of the curve ar* + 3/* + 2* = 0.
Solution by the Proposer.
Take I a fourth root of — 1 ; m and n fourth roots of + 1 ; then the 28 double
tangents are the lines x = ly, x = Iz, y = Iz, (4 + 4 + 4 =) 12 lines; and the lines
x + my + nz=0, 16 lines; and the first 12 of these, each counted twice, are the 24
stationarj- tangents. In fact, any one of the 12 lines is an osculating tangent, or
line meeting the curve in 4 coincident points; it counts therefore once as a double
tangent, and twice as a stationary tangent. There should consequently be 16 other
double tangents ; and it only needs to be shown that these are the 16 lines
x + my+nz = 0. Consider any one line x + viy + nz = 0; for its intersections with the
curve a!^ + y* + z* = 0, we have
(my + nz)* + y* + z* = 0,
or, as this may be written,
viz. this is
or, what is the same thing.
(my + nz)* + m*if + n*z* = 0 ;
2(1, 2, 3, 2, \\my, nzf^O,
2[(1, 1, \^my, nzn = 0:
76-
604 PROBLEMS AND SOLUTIONS. [705
SO that the line is a double tangent, the two points of contact being given by means
of the equation (1, 1, l'$_my, nz)' = 0; viz. w being an imaginary cube root of unity,
we have nz = amy or m'my : and thence, for the points of contact,
X : y : z =1 : — : — , or = I : — : —;
" m n m n
values which satisfy, as they should do, the two equations
X + my + nz = 0 and x* + y*+z*=0.
[Vol. XXXVI., pp. 106, 107.]
6800. (Proposed by W. J. C. Miller, B.A.)— Prove that, if
ayz _ bzx _ cxy _
y'' + z'~ ?Ta^ ~ ^ry ~ '
then
a- + If + c^ = ahc + 4.
Note on Question 6800. By Professor Cayley.
The identity given by the solution is a very interesting one. Instead of a, b, c,
writing (a, b, c)-i-d, we have
4cZ« -d{a'+b' + c") + abc = 0,
satisfied by
a : b : c : d = x{y^ + z^) : y (z' + x') : z{a? + y^) : xyz ;
or, considering (a, b, c, d) as the coordinates of a point in space, and (x, y, z) as
the coordinates of a point in a plane, we have thus a correspondence between the
points of the cubic surface 4d* — d (a" + &" + c*) + abc = 0, and the points of the plane.
To a given system of values of {x, y, z) there corresponds, it is clear, a single system
of values of (a, b, c, d); and it may be shown without difficulty that, to a given
system of values of (a, 6, c, d) satisfying the equation of the surface, there cor-
respond two systems of values of {x, y, z); the plane and cubic surface have thus a
(1, 2) correspondence with each other.
[Vol. xxxviL, 1882, p. 74.]
5244. (Proposed by Professor Cayley.) — Writing for shortness
F=a? + ^-^--y\ G = /S=4-S=-7=-a% 7/ = 7=+ S^^-a' - ^3*,
A=a=+/3^ + 7= + 8^;
705]
PROBLEMS AND SOLUTIONS.
605
show that the equation
LMN {a^ + y*-\-2' + vf)^- MN{F^ + 2L) (y^z^ + afw^) + NL ((?A + IM) {z'^a? + y'^w^)
+ LM(HA + 2N) {afy^ + z=w^) - 2a^yB FGHA xyzw = 0
belongs to a 16-nodal quartic surface, having the nodes
X == a
a
a
a
ys
;8
^
ys
7
7
7
7
8
8
8
8
y=/9
-^
-/3
^
a
— a
— a
a
8
-8
-8
8
7
-7
-7
7
2 =7
-7
7
-7
8
-8
8
-8
a
— a
a
— a
;S
-/3
/3
-/3
w = 8
S
-S
-S
7
7
-7
-7
ys
/3
-^
-^
a
a
— a
- a
and the sixteen singular tangent planes represented by the equations
(a, 0, y, 8) (x, y, z, w) = 0, &c.
[VoL XXXVIII., 1883, pp. 87—89.]
7190. (Proposed by Professor Wolstenholme, M.A.) — If x, y, z be three quantities
satisfying the two symmetrical equations
yz + zx + xy = 0, x^ -\- y' + z^ + ^xyz = 0 ;
prove that (1) they will al.so satisfy one of the two pairs of semi-symmetrical
expressions
yz + z^x + x'y =(j/-z){z- x) {x -y), = + xyz,
yz^ + za? + xy^ =(y — z)(z — x) (x — y), = — xyz ;
and (2) one set of the following equations will also be satisfied : —
{a? + yz - y^ = 0, y^ + zx - z' = 0, z"- + xy - x" = 0) ;
{af + yz- z' = 0, z^ +zx-ay' = 0, z-+xy-y'^ = 0).
Solution by Professor Catley.
The two symmetrical equations represent a conic and a cubic respectively; they
intersect therefore in 6 points, and if we denote by a a root of the equation
w' + u''-2u-l = 0,
then the other two roots of this equation are
;3, =-1--, y = ^-^;
viz. if a^ + a'' — 2a — 1 = 0, then we have
{u-a)(u+l+-](u+ — - j =u' + u'-2u-l,
606 PROBLEMS AND SOLUTIONS. [705
an identity which is easily verified. It may be remarked that, if
then
-1
the left-hand side of the last mentioned equation thus is (u — a) {u — (f>a) (u — ijy'a.),
which remains unaltered when a is changed into ^a or (fy'a. Then the coordinates
of the six points of intersection can be expressed indifferently in terms of any one
of the roots (a, 0, y), viz. the coordinates are
(a»-l, -a, -1), (-1, a»-l, -a), (- a, -1, o^-l), ... (1, 2, 3),
(a» - 1, - 1, - a), (- a, a= - 1, - 1), (- 1, -a, a=- 1), ... (4, 5, 6);
or they are equal to the like expressions in /3 and in y respectively ; say these are
the coordinates of the points 1, 2, 3, 4, 5, 6 respectively, as shown by the attached
numbers. Thus, writing
x,y, 2 = 0."-!, -a, -1,
we find
yz + zx + xy= a- a' + 1- of + a = -(a^ + aP -2a- 1) = 0,
a?+y> + !* + 4!xyz = (a' -ly - c^ - l + 42(a'- 1)
= a« - 3a« + 3a» + 3a= - 4a - 2 = (a^ + a^' - 2a - 1) (a' - a= + 2) = 0,
which verifies the formulas for the six points of intersection. Take, again,
x,7/,z = a''-l, -a, - 1 ;
then we find
yz" + zai' + xy^ = - a - (a= - 1)^ + a= (a^ - 1) =«=-«- 1,
y-'z + z^x + a^y = - ci' + {a!' - 1) -a (a= - l)^ = -a»+2a'-a- 1.
Or, since o^ = — a^ +.2a + l, and thence
o* = 3a''-a-l, a' = -4a= + 5a + 3,
the last equation becomes
y^z + z"-x + sd'y = 2a? - 2a - 2.
We have also
xyz = a' — a, = — a- + a + 1 ;
hence the point in question is situate on each of the cubics
yz'' + Z3? + xy^ + xyz = 0, y^z + z'^x + x^y + 2ocyz = 0,
y''z + z'^x + a^y - 2{yz^ + za^ + xy^) = 0 ;
and this, of course, shows the points 1, 2, 3 are all three of them situate upon
each of the three cubics; and in precisely the same manner it appears that the
points 4, 5, 6 are all three of them situate on each of the three cubics
yz^ + za? + xy^ •{■ 2xyz = 0, y'z + z'x +a?y + xyz = 0,
yz^-\-za? + xy^-2{y''z + z''x + a?y) = 0.
705] PEOBLEMS AND SOLUTIONS. 607
Again, from the values x,y, z = a^ — \, —a, —1, we have
a? + yz—y- = 0, y"^ +zx — z'^ = Q, z'^ + xy — ci? = 0;
viz. the point 1 lies on each of these conies; similarly the point 2 lies on each of the
same conies ; and the point 3 lies on each of the same conies ; that is, the conies in
question have in common the points 1, 2, 3.
In like manner, the conies
sfi-\-yz — z- = 0, y- + zx-x^ = 0, z^-^xy -y'- = 0,
have in common the points 4, 5, 6.
The general result is that the given conic and the cubic meet in six points forming
two groups of points (1, 2, 3) and (4, 5, 6) ; through the points (1, 2, 3) we have
three cubics and three conies; and through the points (4, 5, 6) we have three cubics
and three conies.
If in the equation a? + x- — 2x — \ = 0, whose roots are a, 0 (a), <^^ (a), we put
a;=2cos^, the equation becomes
2 (3 cos ^ + cos 3^) + 2 (1 + cos 2^) - 4 cos ^ - 2 = 0,
or
2co8 3^ + 2cos2^+2cos^ = 0, or ^!-f^ = 0;
sin J P
or the three roots are 2 cos ^tt, 2 cos ^tt, 2 cos f tt. The two equations
yz+ zx + xy = 0, (t? + y^ + z' + 3xyz = 0,
are satisfied if x:y:z= these three roots in any order, giving the six solutions. The
semi-symmetrical systems are satisfied, the one by
X : y : z, or y : z : x, or z : x : y, =cos^7r : cos^tt : cos^tt;
and the other by
z : y : X, or y : x : z, or x : z : y, =cosf7r : cos f tt : cosfTr.
[Vol. XXXIX., 1883, p. 31.]
5689. (Proposed by Professor Cayley.) — Show (1) that the apparent contour of a
Steiner's surface {2xyz -\- y^z^ + z'^a? + x^y^ = Q), as seen from an exterior point on a nodal
line (say the axis of z), projected on the plane of the other two nodal lines, is an ellipse
passing through the four points (+1, 0) and (0, ±1); and (2) find the surface-contour,
or curve of contact, of the cone and surface.
608
PROBLEMS AND SOLUTIONS.
[705
[Vol XXXIX., p. 49.]
4722. (Proposed by Professor Cayley.)— 1. Show that the conditions in regard
to the reality of the roots of the equation
(x'-ay+16A{x-m,) = 0,
(4wi» - 3a)« - (8m» - 9ma - 27^)" = - ,
then the roots are two real, two imaginary ; but if
(4j»-^ - 3a)' - (8m» -9ma-27Ay = + ,
a = + , A(ma-9A) = + ,
the roots are all real, but otherwise they are all imaginary.
2. If the roots of the foregoing equation are all imaginary, then for any real
value whatever of y, the roots of the equation
{a^ + f- a)^+ 16^ {x-m) = 0
are all imaginary.
are, if
then, if simultaneously
that is,
[Vol. XXXIX., pp. 69, 70.]
4387. (Proposed by Professor Cayley.) — Using the term "Cassinian" to denote a
bi-circular quartic having four foci in a right line ; show that the equation of a
Cassinian having for its four foci the points x = a, x=b, x = c, x = d on the axis of
X, may be written in the four equivalent forms
( . , T(d-c), a(b-d), p{c-b))iAi,Bi,Ci,Di) = 0,
r(c — d), . , p{d — a), a-{a — c)
a-(d-b), p(a — d), . , t(6 — a)
p(b-c), a-{c-a), T{a — b),
T(d-c)Bi + a{b-d)Ci + p{c-b)Di = 0,
T(c-d)Ai . +p(d-a)Ci+<7(a-c)Di = 0,
&c., &c.,
where A^, B^, CK -D* arc the distances from the four foci respectively, and the
parameters p, a, t are connected by the equation
p^{a-d){b-c) + a^(b-d){c-a) + -T^ic-d)ia-b) = 0.
Show also that the curve has, at right angles to the axis of x, two double tangents,
the equation whereof is any one of the three equivalent forms
(a -\- d - 2x){b + c - 2x) : (b + d-2x)(c + a-2x) : (c4 d-'2x)(a + b-2x) = p^ : a' : t».
705] PROBLEMS AND SOLUTIONS. 609
[Vol. XL., 1884, p. 32.]
7376. (Proposed by Professor Catley.) — Show how the construction of a regular
heptagon may be made to depend on the trisection of the angle cos~' ( ^r—.^ ] .
[Vol. XL., p. 110.]
7352. (Proposed by Professor Cayley.) — Denoting by x, y, z, ^, t), ^ homogeneous
linear functions of four coordinates, such that identically
a; + y + z + |^ + 77 + ?=0, ax + hy + cz +f^ + gv + H= 0,
where af = bg= ch = l; show that
v'(^f) + V(2/^) + VK) = o
is the equation of a quartic surface having the sixteen singular tangent planes (each
touching it along a conic)
x=0, y=0, z=Q, f = 0, ,; = 0, r = 0,
X +y + z = 0, x + 7) + z = 0, ax + by + cz = 0, ax + grj + cz = 0,
^ + y + z = 0, x + y+^=0, J^ +by + cz^O, ax + by + h^=0,
1-bc^ 1-ca ' 1-ab ' l-gh^l -hf 1 -fg
[Vol. XLL, 1884, p. 37.]
5421. (Proposed by Professor Cayley.) — Suppose
8x = mi(x — a,), wij (x - aj), rria (x — a^), rtit (x — aj) ;
where, for any given value of x, we write +, — , or 0, according as the linear function
is positive, negative, or zero, and where the order of the terms is not attended to.
If X is any one of the values Oj, a,, a^, «!, the corresponding S is 0 + + +, 0 ,
0 + + — , or Oh : and if / denote indifferently the first or the second form, and R
denote indifferently the third or the fourth form : then it is to be shown that the four
S's are R, R, R, R, or else R, R, I, I.
[Vol. XLiv., 1886, p. 109.]
8340. (Proposed by F. Morley, B.A.) — Show that (1) on a chess-board the number
of squares visible is 204, and the number of rectangles (including squares) visible is
1,296 ; and (2) on a similar board, with n squares in each side, the number of
squares is the sum of the first n square numbers, and the number of rectangles
(including squares) is the sum of the first n cube numbers.
c. X. 77
610
PROBLEMS AND SOLUTIONS.
Solution by Professor Cayley.
[705
In a board of n' squares, the number of pairs of vertical lines at a distance
from each other of n-r+1 squares is = r ; and the number of pairs of horizontal
1
2
3
4
2
4
6
8
3
6
9
12
4
8
12
16
lines at a distance from each other of n — s squares is =s. Hence the number of
rectangles, breadth n — r + 1 and depth n — s + l, or say the number of
{n-r + l){n-s + l)
rectangles, is =rs.
For instance, n = 4, the number of rectangles 44, 43, 34, &c., is shown in the
diagram; hence the whole number of rectangles is (1 + 2 +3 + 4)' = l' + 2' + 3' + 4', and
80 for any value of n.
The same diagram shows that the whole number of squares is = I'' + 2' + 3° + 4* ;
and 80 for any value of n.
[Vol. XLVi., 1887, pp. 49, 50.]
8636. (Proposed by Professor Mahendra Nath Ray, M.A., LL.B.) — Show that the
following equations are satisfied by the same value of x, and find this value : —
cur (ic= - a')i + bx («» - 6=)* + ca; («' - c")* = 2abc,
2 (ar" - a')i (ar" - 6")* (x* - c=)* = a; (a" + 6= + c" - 2ar').
Solution hy Professor Cayley.
The second equation rationalised gives
4a» - 4ar* (a> + 6» + c») + 4a^ (6V + c^a" + a'i') - 4a»6'c» = 4a;« - 4«r« (a* + 6» + c") + a:» (a' + 6» + (?)> ;
that is.
705] PEOBLEMS AND SOLUTIONS. 611
if, for shortness,
V = - a* - 6* - c* + 2b-c' + 2d'a' + ^a^bK
We thence find
V («2 - a^ = a" (- a- + 6^ + c^f,
V(iv'-lfi) = b^ (a^ -b'- + cj, V {x'' - c') = c" (a" + ¥- cj,
and therefore also
V ^aW {a? - a-) = 4a«6V (- a^ + b^ + c^)-, &c.
Or, assuming the sign of the square roots.
Vox {a?- a^)i = 2abc (- a' + a^b^ + a^c% Vbxix'- 6=)i = 2abc (+ a'^b'' -b*+ ¥c'),
Vex {a?- c=)* = 2abc (cv'd' + b'c' - c%
whence, adding, the whole divides by V and we have
ax (a? - a«)i + ia; (a? - 6')^ + ex (a? - c=)i = 2a6c,
the second equation. Observe that the second equation rationalised gives an equation
of the form {a?, 1)* = 0 ; the foregoing value x^ = ia-b^c^/A is thus one of the four
values of x'.
[Vol. XLVii., 1887, p. 141.]
5271. (Proposed by Professor Cayley.) — If to be an imaginary cube root of
unity, show that, if
(o) — &)') X + <o^a?
^^ 1 - 6)2 (&) - o)-') a? '
then
dy (ft) — w^) dx
(1 - fy (1 + toy')* (1 - x'f ( 1 + wx^)i '
and explain the general theory.
[Vol. L., 1889, p. 189.]
3105. (Proposed by Professor Cayley.) — The following singular problem of literal
partitions arises out of the geometrical theory given in Professor Cremona's Memoir,
" Sulle trasformazioni geometriche delle figure plane," Mem. di Bologna, tom. v. (1865).
It is best explained by an example: — A number is made up in any manner with the
parts 2, .5, 8, 11, &c., viz. the parts are always the positive integers = 2 (mod. 3);
for instance, 27 = 1.11+8.2. Forming, then, the product of 27 factors a>^{bcdefghif,
this may be partitioned on the same type 1.11 + 8.2 as follows,
a^bcdefghi, ab, ac, ad, ae, af, ag, ah, ai.
(Observe that the partitionment is to be symmetrical as regards the letters which
have a common index.) But, to take another example,
37=0.11+3.8 + 1.5 + 4.2=1.11+0.8 + 4.5 + 3.2.
77—2
612
PROBLEMS AND SOLUTIONS.
[705
The first of these gives the product {abcf d' (e/ghy, which cannot be partitioned
(symmetrically as above) on its own type, though it may be on the second type;
and the second gives the product a" (bcdef (fghy, which cannot be partitioned
(symmetrically as above) on its own type, though it may be on the first type; viz.
the partitions of the two products respectively are:
First product on second type,
{abcfdefgh, abcde, abcdf, ahcdg, abcdh, ab, ac, be;
Second product on first type,
a'bcde/g, a?bcdefh, a^bcdegh, abcde, ab, ac, ad, ae;
so that in the first example the type is sibi-reciprocal, but in the second example
there are two conjugate types. Other examples are :
Parts
48
64
55
56
53
55
No.
2
14
3
1
0
3 6
0 2
5
0
2
3
0
6 0
5 0
8
11
U
0
0
0
3
0
1
2
2
0
7
0
0
0 1
0 3
0 0
2 5
0 1
1 0
•3
17
0
0
0
0
1 0
0 0
20
1
0
0
0
0 0
0 0
viz. the first four columns give each of them a sibi-reciprocal type, but the last
two double columns give conjugate types. It is required to investigate the general
solution.
[Vol. L., p. 191.]
3304. (Proposed by Professor Cayley.) — The coordinates x, y, z being proportional
to the perpendicular distances from the sides of an equilateral triangle, it is required
to trace the curve
{y -z)>^x-\-{z- x) ^y-\-{x- y) sjz = 0.
[Prof Cayley remarks that the curve in question is a particular case of that
which presents itself in the following theorem, communicated to him (with a de-
monstration) several years ago by Mr J. Griffiths : —
The locus of a point {x, y, z) such that its pedal circle (that is, the circle
which passes through the feet of the perpendiculars drawn from the point in question
705] PROBLEMS AND SOLUTIONS. 613
upon the sides of the triangle of reference) touches the nine-point circle, is the sextic
curve
■^ a; cos .4 ( V cos 5 — ^r cos C) ( — ^ ^H
{ ^* '^Vcos^ cosOy]
+ lycos B(z cos C —xcoaA) ( — =. z I r
[ Vcos G cos A/J
+ \z cos C (x cos A - V cos B) { . ^-= | \ = 0.
^ Vcos 4 cos BJ]
It would be an interesting problem to trace this more general curve.]
[Vol. L., p. 192.]
3481. (Proposed by Professor Cayley.) — Find, in the Hamiltonian form
dt) _ dH dvr _ dH „
dt dw ' dt dr) '
the equations for the motion of a particle acted on by a central force.
[Vol. LV., 1891, p. 27.]
10716. (Proposed by Professor Cayley.)— In a hexahedron ABGDA'B'G'D' the
plane faces of which are ABCD, A'B'C'D', A'ADD', D'DCC, G'CBB', B'BAA', the
edges A A', BE, CO', DD' intersect in four points, say AA', DD' in a; BB', CO'
in /3; CO', DD' in 7; AA', BB' in S: that is, starting with the duad of lines
0/3, 7S, the four edges AA', BE, GO', DD" are the lines aS, /3S, ^y, ay which join
the extremities of these duads. Similarly, the four edges AB, GD, A'E, C'D' are
the lines joining the extremities of a duad ; and the four edges AD, BG, A'D', EG'
are the lines joining the extremities of a duad. The question arises, "Given two
duads, is it possible to place them in space so that the two tetrads of joining lines
may be eight of the twelve edges of a hexahedron ? " The duad a^, yS is considered
to be given when there is given the tetrahedron a/SyS, which determines the relative
position of the two finite lines a/3 and 7S.
[Vol. LXL, 1894, pp. 122, 123.]
3162. (Proposed by Professor Cayley.) — By a proper determination of the co-
ordinates, the skew cubic through any six given points may be taken to be
X : y :
w\
614 PROBLEMS AND SOLUTIONS. [705
or, what is the same thing, the coordinates of the six given points may be taken
to be
(1, «., «,», «!«) (1, t„ U\ U).
Assuming this, it is required to show that if
and if
V = Qxyzw — ixz' — 4y'M; + 3y V — xW ;
then the equation of the Jacobian surface of the six points is
3 ( ap, + ^pi — 2w ) Sx V -1
+ ( 2^pa - wpi) By V
y = 0.
+ ( aopt-2ypi )B^S7
+ (2xp,- yp, -tt^3)S„V j
[Vol. LXi., p. 123.]
3186. (Proposed by Professor Cayley.) — An unclosed polygon of (m + 1) vertices
is constructed as follows: viz. the abscissae of the several vertices are 0, 1, 2, ..., m,
and corresponding to the abscissa k, the ordinate is equal to the chance of m + k
heads in 2m tosses of a coin ; and m then continually increases up to any very
large value; what information in regard to the successive polygons, and to the
areas of any portions thereof, is afforded by the general results of the Theory of
Probabilities ?
[Vol. LXI., p. 124.]
3229. (Proposed by Professor Cayley.) — It is required to find the value of the
elliptic integral F{c, 6) when c is very nearly =1 and 6 very nearly =i7r; that is,
the value of
fi'-' dd
Jo
where a, b are each of them indefinitely small.
N.B. — Observe that, for a=0, b small, the value is equal log 4/6, and for 6 = 0,
a small, the value is —log cot ^a.
705]
PROBLEMS AND SOLUTIONS.
615
In the following Contents, the Problems are referred to, each by its number and the
proposer's name ; and the subject is briefly indicated. An asterisk shews that no solution
shews that there is no number.
i given.
3002
2\. line
Collins
♦3144
Cayley
3120
)j
3249
)»
3206
}i
3278
»)
3329
}t
♦3356
n
♦3507
It
♦3536
))
♦3591
»»
•2652
jj
♦3677
99
♦3564
»
3875
>»
3430
Miller
4298
Glaisher
4392
Roberts
4354
Tucker
4458
Cayley
4520
Evans
4528
Cayley
4581
Wilkinson
♦4638
Cayley
*4694
If
4793
Wolstenholme
4752
Cayley
♦4946
>»
5020
Woolhouse
5079
Cayley
♦5130
»
5208
Sylvester
Martin
*5306
Cayley
♦4870
»»
♦5625
>)
•5387
»»
♦5927
j»
*6155
9)
»6470
9>
6766
»
6800
Miller
*5244
Cayley
Systems of circles
Locus in piano
Jacobian of quadric surfaces
Sets of four points on a conic .
Arrangements of points and lines
Arrangements of numbers
Substitutions and permutations.
Roots of quartic equation .
Quadric cones through given points
Particle under central forces
Relation between areas of triangles
Parallel surfaces of an ellip.soid
Angle between normal and bisector of chord
Minimum circle enclosing three points
Description of curves
Negative pedals of ellipsoid and ellipse
Quadrilateral inscribable in circle
Symmetrical determinant .
Geometrical interpretation (note)
Two quartic curves .
Use of Degen's tables
Expectation ....
A question in chances
Envelope of family of quadrics
Resultant of forces .
Relation among derivatives of a function
Cubic curve ....
Attraction of lens-shaped liody
Algebraical theorem .
Bicircular quartic
Bicircular quartic
Trees .....
A question in probabilities
A system of equations
Conies and lines
A trinodal quartic .
Conical points ....
Algebraical theorem .
Coordinates of a line
Equation of a cubic curve
Singular tangents of a quartic .
Geometrical interpretation (note)
16 nodal quartic surface .
PAGE
566
568
ib.
569
570
ib.
574
575
ib.
ib.
ib.
ib.
576
ib.
ib.
ib.
578
579
581
682
586
587
588
589
ib.
590
592
594
ib.
596
597
598
600
601
602
ib.
ib.
ib.
603
ib.
ib.
604
ib.
616
PROBLEMS AND SOLUTIONS.
[705
♦5689
Cayley
♦4722
»>
♦4387
91
♦7376
91
*7352
»
•5421
fl
8340
Morley
8636
Nath Ray
♦5271
Cayley
♦3105
»
♦3304
f)
♦3481
»
♦10716
)i
*3162
)»
*3185
»
♦3229
11
A Steiner's surface ....
Reality of roots of a quartic .
Equation of a Cassinian .
Construction of a heptagon
Equation of a quartic surface .
Algebraical theorem
Topology of chess-board
Solution of equations
Elliptic-function transformation
Partitions
Curve of sixth order
Hamiltonian equations of central orbit
Edges of a hexahedron .
Jacobian surface of six points .
Probability .....
Elliptic integi'al ....
PAOI
605
607
608
ib.
609
ib.
ib.
ib.
610
611
ib.
612
613
ib.
ib.
614
ib.
END OF VOL. X.
CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
'h
V
I
QA Cayley, Arthur
3 The collected
C42 mathematical papers of
v.lO Arthur Cayley
Physical &
Applied Sci.
PLEASE DO NOT REMOVE
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