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MATHEMATICAL PAPERS.
aonbon: C. J. OLAY and SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
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THE COLLECTED
MATHEMATICAL PAPERS
OF
AETHUE CAYLEY, Sc.D., F.E.S.,
LATE 8ADLERIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE.
VOL. IX.
CAMBRIDGE :
AT THE UNIVERSITY PRESS.
\_All Rights reserved.^
>'M
CAMBKIDUE :
I'BINTED BY J. AND C. F. CLAY,
AT THE UNIVBBBITY PRESS.
ADVEKTISEMENT.
T
HE present volume contains 74 papers, numbered 556 to 629, published
for the most part in the years 1874 to 1877.
The Table for the i
aine volumes is
Vol
. '1. Numbers
1
to 100,
11.
101
„ 158,
in.
159
„ 222,
IV.
223
„ 299,
V.
300
„ 383,
VI.
384
„ 416,
'
VII. „
417
„ 485,
VIII. „
486
„ 555,
TX.
556
„ 629.
A. E. FORSYTH.
17 December 1895.
Digitized by the Internet Archive
in 2007 with funding from
IVIicrosoft Corporation
http://www.archive.org/details/collectedmathema09cayluoft
VII
CONTENTS.
PAGE
556. On Steiner's siir/ace .......... 1
Proc. Lond. Math. Society, t. v. (1873—1874), pp. U— 25
557. On ceHain constructions for hicircular quartics . . . . 13
Proc. Lond. Math. Society, t. v. (1873—1874), pp. 29—31
558. A geometrical interpi'etation of the equations obtained by equating
to zero the resultant and the discriminants of two binaiy
quantics ........... 16
Proc. Loud. Math. Society, t. v. (1873—1874), pp. 31—33
559. [Note on inversion] ....... . . 18
Proc. Lond. Math. Society, t. v. (1873—1874), p. 112
560. [Addition to Lord Rayleigh's paper " On the numencal calcu-
lation of the roots of fluctuating functions"] . , , . 19
Proc. Lond. Math. Society, t. v. (1873—1874), pp. 123, 124
561. On the geoTnetrical representation of Cauchy's theorems of root-
limitation . .......... 21
Camb. Phil. Trans., t. xii. Part ii. (1877), pp. 395—413
562. On a theorem in maxima and minima : addition [to Mr Walton's
paper] by Professor Cayley ....... 40
Quart. Math. Jour., t. x. (1870), pp. 262, 263
563. Note on the transformation of two simultaneous equations . . 42
Quart. Math. Jour., t. xi. (1871), pp. 266, 267
564. On a theorem in elimirMtion ....... 43
Quart. Math. Jour., t. xii. (1873), pp. 5, 6
VUI CONTENTS.
PAQE
565. Note on the Cartesian ........ 45
Quart Math. Jour, t xii. (1873), pp. 16—19
566. On the transformation of the equation of a surface to a set
of chief axes .......... 48
Quart Math. Jour., t. .\ii. (1873), pp. 34—38
567. On an identical equation connected with the theoi'y of invariants 52
Quart. Math. Jour., t. xii. (1873), pp. 115—118
568. Note on the integrals I cos afdx and I sina;*c?a; . . . 56
Jo jo
Quart Math. Jour., t. xii. (1873), pp. 118—126
569. On the cyclide 64
Quart. Math. Jour., t. xii. (1873), pp. 148—16.5
570. On the superlines of a quadric surface in five-dimensional space 79
Quart. Math. Jour., t xii. (1873), pp. 176—180
571. A demonstration of Dupin's theorem . . . . . . 84
Quart Math. Jour., t xii. (1873), pp. 185—191
572. Theorem in regard to the Hessian of a quaternary function . 90
Quart. Math. Jour., t. xii. (1873), pp. 193—197
573. Note on the (2, 2) correspondence of two variables ... 94
Quart Math. Jour., t xii. (1873), pp. 197, 198
574. On WronskHs theorem ........ 96
Quart Math. Jour., t xii. (1873), pp. 221—228
575. On a special quartic transformation of an elliptic function . 103
Quart. Math. Jour., t xii. (1873), pp. 266—269
576. Addition to Mr Walton's paper " On the ray-planes in hiaxal
crystals "........... 107
Quart Math. Jour., t. xii. (1873), pp. 273—275
577. Note in illustration of certain general theorems obtained by
Dr Lipschitz HO
Quart Math. Jour., t xii. (1873), pp. 346—349
CONTENTS. XX
PAGE
578. A memoir on the transforntation of elliptic functions . 11&
PhU. Ti-ans., t. CLXiv. (for 1874), pp. 397—456
579. Address delivered by the President, Professor Cayley, on pre-
senting the Gold Medal of the [Royal Astronomical] Society
to Professor Simon Newcomh . . . . . 17&
Monthly Notices R. A.st. Society, t. xxxiv. (1873—1874), pp. 224—233
580. On the number of distinct tei'ms in a symmetrical or partially
symmetriccd determinant; with an addition . . . . 185
Monthly Notices R. Ast. Society, t. xxxiv. (1873— 1874), pp. 303—307;
p. 335
581. On a theorem in elliptic motion. . . . .■ . . 191
Monthly Notices R. Ast. Society, t. xxxv. (1874—1875), pp. 337—339
582. Note on the Tlieory of Precession and Nutation . . . 194
Monthly Notices 'B. Ast. Society, t. xxxv. (1874—1875), pp. 340—343
583. On spheroidal trigonometry ....... 197
Monthly Notices R. Ast. Society, t. xxxvii. (1876—1877), p. 92
584. Addition to Prof R. S. Ball's paper " Note on a transfor-
mation of Lagrange's equations of motion in generalised
coordinates, tvhich is convenient in Physical Astronomy" . 198
Monthly Notices R. Ast. Society, t. xxxvii. (1876—1877), pp. 269—271
585. A new theorem on the equilibrium of four forces acting on a
solid body .......... 201
Phil. Mag., t. XXXI. (1866), pp. 78, 79 ; Camb. Phil. Soc. Proc, t. I.
(1866), p. 235
586. On the mathematical theory of isom,ers . . . . . 202
Phil. Mag., t. XLVii. (1874), pp. 444—467
587. A Smith's PHze dissertation [1873] 205
Messenger of Mathematics, t. ill. (1874), pp. 1 — 4
588. Problem, [on tetrahedra] ....... . 209
Messenger of Mathematics, t. iii. (1874), pp. 50 — 52
C. DC h
CONTENTS.
PAOB
389. On residuation in regard to a cubic curve . . . . 211
Memnger of Mathematics, t. iii. (1874), pp. 62 — 65
590. Addition to Prof. Hall's paper " On the motion of a particle
toward an attracting centre at which the force is infinite. " . 215
Messenger of Mathematics, t. ill. (1874), pp. 149 — 152
591. A Smith's Prize paper and dissertation [1874]; solutions and
remarks ........... 218
Messenger of Mathematics, t. ill. (1874), pp. 165 — 183; t. iv.
(1875), pp. 6—8
592. On the Mercator's projection of a skew hyperholoid of revo-
lution . . . . . . . . . . , 237
Messenger of Mathematics, t. iv. (1875), pp. 17 — 20
593. A Sheepshanks' problem (1866) 241
Messenger of Mathematics, t. iv. (1875), pp. 34 — ^36
594. On a differential equation in the theory of elliptic functions . 244
Messenger of Mathematics, t. iv. (1875), pp. 69, 70
595. On a Senate-House problem 246
Messenger of Mathematics, t, iv. (1875), pp. 75 — 78
596. Note on a theorem of Jacobi's for the transformation of a double
integral 250
Messenger of Mathematics, t. iv. (1875), pp. 92 — 94
597. On a differential equation in the theory of elliptic functions . 253
Messenger of Mathematics, t. iv. (1875), pp. 110 — 113
598. Note on a process of integration 257
Messenger of Mathematics, t. iv. (1875), pp. 149, 150
599. A Smith's Prize dissertation ....... 259
Messenger of Mathematics, t iv. (1875), pp. 157 — 160
600. Theorem on the n-th Roots of Unity 263
Messenger of Mathematics, t. iv. (1875), p. 171
601. Note on the Cassinian ........ 264
Messenger of Mathematics, t. iv. (1875), pp. 187, 188
CONTENTS. xi
PAGE
602. On the potentials of polygons and polyhedra .... 266
Proc. Lond. Math. Society, t. vi. (1874—1875), pp. 20—34
603. On the potential of the ellipse and the circle .... 281
Proc. Lond. Math. Society, t. vi. (1874—1875), pp. 38—58
604. Determination of the attraction of an ellipsoidal shell on an
exterior point .... ...... 302
Proc. Lond. Math. Society, t. vi. (1874—1875), pp. 58—67
605. Note on a point in the theoiy of attraction . . . . 312
Proc. Lond. Math. Society, t. vi. (1874—1875), pp. 79—81
606. On the expression of the coordinates of a point of a quartic
curve as functions of a parameter . . . . . 315
Pi-oc. Lond. Math. Society, t. vi. (1874—1875), pp. 81—83
607. A memoir on prepotentials. . . . . . . . 318
Phil. Ti-ans., t. oixv. (for 1875), pp. 675—774
608. [Extract from a] Report on Mathematical Tables . . . 424
Brit. Assoc. Report, 1873, pp. 3, 4
609. On the analytical forms called factions ..... 426
Brit. Assoc. Report, 1875, Notices of Communications to the Sections,
p. 10
610. On the analytical forms called Trees, with application to the
theory of chemical combinations ...... 427
Brit. Assoc. Report, 1875, pp. 257—305
611. Report on mathematical tables . . . . . . . 461
Brit. Assoc. Report, 1875, pp. 305—336
612. Note sur une formule d'integration indefinie .... 500
Comptes Rendiis, t. Lxxvni. (1874), pp. 1624—1629
613. On the group of j)oints Gt on a sextic curve with five double
points ........... 504
Math. Ann., t. viii. (1875), pp. 359—362
614. On a problem of projection ....... 508
Quart. Math. Jour., t. xiii. (1875), pp. 19—29
62
XU CONTENTS.
PAOX
615. On the conic tones ......... 519
Quart. Math. Jour., t. xiii. (1875), pp. 127—129
616. A geometrical illustration of the cubic transfoiyncUion in elliptic
functions .......... 522
Quart. Math. Jour., t. xiii. (1875), pp. 211—216
617. On the scalene transformation of a plane curve . . . 527
Quart Math. Jour., t. xiii. (1875), pp. 321—328
618. On the mechanical descmption of a Cartesian .... 535
Quart MatL Joui-., t xm. (1875), pp. 328—330
619. On an algebraical operation ....... 537
Quart Math. Jour., t xiii. (1875), pp. 369—375
620. Correction of two numerical errors in Sohnke's paper respect-
ing modular equations . . . . . . . . 543
Crelle, t. Lxxxi. (1876), p. 229
621. On the number of the univalent radicals C„H5„+i . . 544
Phil. Mag., Ser. 5, t iii. (1877), pp. 34, .35
622. On a system of equations connected vnth MalfattHs problem, . 546
Proc. Lond. Math. Society, t vii. (1876), pp. 38—42
623. On three-bar mx>tion ......... 551
Proc. Lond. Math. Society, t vii. (1876), pp. 136—166
624. On the bicursal sextic 581
Proc. Lond. Math. Society, t vii. (1876), pp. 166—172
625. Ori the condition for the existence of a surface cutting at
right angles a given set of lines ...... 587
Proc. Lond. Math. Society, t viii. (1877), pp. 53 — 57
626. On the general differential equation -,-|r + -7^=0, where X, Y
are the sanfie quartic functions of x, y respectively . 592
Proc. Lond. Math. Society, t viii. (1877), pp. 184—199
CONTENTS. Xlll
PAOE
627. Geometrical illustration of a theorem relating to an irrational
function of an imaginary variable ...... 609
Proc. Lond. Math. Society, t. viii. (1877), pp. 212—214
628. On the circular relation of Mohitis . . . . . . 612
Proc. Lond. Math. Society, t. viii. (1877), pp. 220—226
629. On the linear transformation of the integral \.jt • • • 618
Proc. Lond. Math. Society, t. viii. (1877), pp. 226—229
I
Plate -to face p. 460
XV
CLASSIFICATION.
Geometry :
Miibius' circular relation, 628
Malfatti's problem, 622
Inversion, 559
Projections, 592, 614
Mechanical construction of curves, 557, 618
Quurtic curve and functions of a single parameter, 606
Cartesian, 565
Cassinian, 601
Sextic curves, 613, 624
Correspondence, residuation and transformation, 566, 573, 589, 617
Conformal representation, 627
Illustrations of algebraical theorems, 558, 561
Three-bar motion, 623
Spheroidal Trigonometry, 583
Dupin's theorem, 571
Cyclide, 569
Steiner's surface, 556
Surface orthogonal to set of lines, 625
Conic torus, 615
Quadrics in hyfierdimensional space, 570
Astronomy and Dynamics :
Presidential address, 579
Elliptic motion, 581
Precession and nutation, 582
General equations of dynamics, 577, 584, 590
Potentials, attractions and prepotentials, 602, 603, 604, 605, 607
XVI CLASSIFICATION.
AKALV8I8 :
Determinant, number of terms in, 580
Elimination, 564
Invariants and covariants, 567, 572, 619
Mathematical tables, 608, 611
Trees, and their applications to chemistry, 586, 610, 621
Roots of Unity, 600
Transformation of equations, 563
Maxima and minima, 562
Factions, 609
Fluctuating functions, 560
Wronski's theorem of expansion, 574
Integration and definite integrals, 568, 596, 598, 612
Elliptic functions, 626, 629
Transformation of elliptic functions, 575, 578, 594, 597, 616, 620
Smith's Prize Dissei-tations and Solutions, 587, 591, 599
Miscellaneous, 576, 585, 588, 593, 595
556]
556.
ON STEINER'S SURFACE.
[From the Proceedings of the London Mathematical Society, vol. v. (1873 — 1874),
pp. 14—25. Read December 11, 1873.]
I HAVE constructed a model and drawings of the symmetrical form of Steiner's
Surface, viz. that wherein the four singular tangent planes form a regular tetrahedron,
and consequently the three nodal lines (being the lines joining the mid-points of
opposite edges) a system of rectangular axes at the centre of the tetrahedron. Before
going into the analytical theory, I describe as follows the general form of the surface :
take the tetrahedron, and inscribe in each face a circle (there will be, of course, two
circles touching at the mid-point of each edge of the tetrahedron; each circle will
contain, on its circumference at angular distances of 120°, three mid-points, and the
lines joining these with the centre of the tetrahedron, produced beyond the centre,
meet the opposite edges, and are in fact the before-mentioned lines joining the mid-
points of opposite edges). Now truncate the tetrahedron by planes parallel to the
faces so as to reduce the altitudes each to three-fourths of the original value, and
from the centre of each new face round off symmetrically up to the adjacent three
circles; and within each circle scoop down to the centre of the tetrahedron, the
bounding surface of the excavation passing through the three right lines, and the
sections (by planes parallel to the face) being in the neighbourhood of the face nearly
circular, but as they approach the centre, assuming a trigonoidal form, and being close
to the centre an indefinitely small equilateral triangle. We have thus the surface,
consisting of four lobes united only by the lines through the mid-points of opposite
edges, these lines being consequently nodal lines; the mid-points being pinch-points
of the surface, and the faces singular planes, each touching the surface along the
inscribed circle. The joining lines, produced indefinitely both ways, belong as nodal
lines to the surface; but they are, outside the tetrahedron, mere acnodal lines not
traversed by any real sheet of the surface.
C. IX, 1
••r
2 ON steiner's surface. [556
We may imagine the tetrahedron placed in two different positions, (1) resting with
one of its faces on the horizontal plane, (2) with two opposite edges horizontal, or say
Mrith the horizontal plane passing through the centre of the tetrahedron and being
parallel to two opposite edges ; or, what is the same thing, the nodal lines form a
system of rectangular axes, one of them, say that of z, being vertical. And I proceed
to consider, in the two cases respectively, the horizontal sections of the surfece.
In the first case, the coordinates x, y, z, w may be taken to be the perpendicular
distances of a point from the faces of the tetrahedron, w being the distance from the
base. We have*, if the altitude be h,
x + y+ z + w = h ;
an equation which may be used to homogenize any equation not originally homogeneous;
thus, for the plane w = X, of altitude X, we have
or, what is the same thing,
The equation of the surface is
«» = ^ _ ^ (« + y + z)-
and if we herein consider w as having the last-mentioned value, the equation will
belong to the section by the plane w = X. I remark that the section of the tetra-
hedron, by this plane, is an equilateral triangle, the side of which is to an edge of
the tetrahedron as h—\ : h. For a point in the plane of the triangle, if X, Y, Z
are the perpendiculars on the sides, then
X^Y^Z = P,
(if for a moment P is the perpendicular from a vertex on the opposite side of the
triangle, viz. we have P = — r — p. if 1p be the perpendicular for a face of the tetra-
hedron). And it is clear that x, y, z are proportional to X, Y, Z\ we consequently
have, for the equation of the section,
VZ -)- VF-F V^+ y^^ ^ ^ (Z -H F+ Z) = 0,
* I take the opportunity of remarking that in a regular tetrahedron, if < be the length of an edge,
p the perpendicular from a summit on an edge (or altitude of a face), 7i the perpendicular from a summit
on a face (or altitude of tlic tetrahedron), and q the distance between the mid-points of opposite edges, then
The tetrahedron can, by means of planes through the mid-points of the edges at right angles thereto, be
divided into four hexahedral figures (8 summits, 6 faces, 12 edges, each face a quadrilaterEil) ; viz. in each
such figure there are, meeting in a summit of the tetrahedron, three edges, each =J«; meeting in the centre
three edges, each =ih\ and six other edges, each =\iy.
556] ON steiner's surface. 3
where the coordinates X, Y, Z are the perpendicular distances from the sides of the
triangle which is the section of the tetrahedron. To simplify, I write
that is,
_ 2\ - h
* " 2A - 2\ '
the equation then is
^JX + s/Y + ^/Z + 'J{2q+\){X -{-Y + Z) = 0;
or, proceeding to rationalize, we have first
q(X+Y + Z) = 'JYZ+^ZX + ^/XY,
and thence
q'{X+Y-^Zf-{YZ+ZX + XY) = 1^/XYZ{^X + ^Y+^JZ)■
and finally
[q^{X + Y + Zf - YZ - ZX - XYY = 'i{2q + \)XYZ{X -{■ Y + Z).
This is a quartic curve, having for double tangents the four lines X = 0, F=0, Z = 0,
X + Y+ Z=0, the last of these being the line infinity touching the curve in two
imaginary points, since obviously the whole real curve lies within the triangle. This
is as it should be : the double tangents are the intersections of the plane w = \ by
the singular planes of the surface.
To find the points of contact, writing for instance Z = 0, the equation becomes
q'(X+Yy-XY = 0,
that is,
whence
^=(-l+2i^±\/5^=)^'
giving the two points of contact equi-distant from the centre ; these are imaginary if
q>^, but otherwise real, which agrees with what follows. (See the Table afterwards
referred to.)
The nodal lines of the surface are (x-y = 0, z — w = 0), (y — z = 0, w — w = 0),
(2 — x = 0, y — w = 0). Considering the first of these, we have for its intersection with
the plane w=\,
X = Y. Z^j^(X+Y+Z).={2q + l){X + Y + Z),
and the last equation gives
Z = (2q + l)i2X + Z),
that is,
0 = (2q + l)X+qZ,
1—2
4 ON steiner's surface. [556
80 that for the point in question we have X : Y : Z = — q : —q : 2^ + 1; and taking
the perpendicular from the vertex on a side as unity, the values —q, —q, 25' + 1 will
be absolute magnitudes. We thus see that the curve must have the three nodes
(23+1, -q, -q), (-q, 2q + l, -q), (— q, —q, iq + 1); and it is easy to verify that this
is so.
The curve will pass through the centre X=Y=Z, if
(V-3)»= 12(23+1),
that is, if
3(3g»-l)'-4(2g + l) = 0,
or if
(3q + iy(q-l) = 0.
It q=l, that is, X = 3 (A — X), or X = f A, the equation is
(X* +Y^ + Z^ + YZ + ZX + XVy -UXYZiX + Y + Z) = 0,
where the curve is, in fact, a pair of imaginary conies meeting in the four real points
(3, - 1, - 1), (- 1, 3, - 1), (- 1, - 1, 3), (J, J, i). To verify this, observe that, writing
A = (Y-Z){2X+ Y+ Z),
B = {Z-X)( X + 2Y+ Z),
C=(X-Y)i X+ Y + 2Z),
and therefore
A + B + G=0,
the function in (X, Y, Z) is = ^ (A^ + 3^ + G% and thus the equation may be written
in the equivalent forms
each of which shows that the curve breaks up into two imaginary conies. The fore-
going value 9=1, or X = fA, belongs to the summit or highest real point of the surface.
, 2\-h ^ ,.
In the case 3^ + 1 = 0, that is,
the equation is
{{X -{-Y + Zy-^ {YZ-\- ZX + XY)Y=10SXYZ{X +Y + Z),
which is, in fact, the equation of a curve having the centre, or point X= Y=Z, for a
triple point. ,
To verify this, write
X = /3 - 7 + M,
F = 7 — a + M,
Z =a -/3 + m;
556] ON steiner's surface.
also
Then we have
and the equation is
that is,
or finally
2A = (/S - yY + (y- a)= + (a _ 0y,
n = (/9-7)(7-a)(a-^).
X+Y+Z=3u,
7Z+ZX + XY=3u'- A,
XYZ= u'-uA + n,
{du" - 9 (.Sw^ - A)}'' - 324m (u' - mA + O) = 0,
(- 2m2 + A)' - 4m (m= - mA + fl) = 0,
A^ - 4!tn = 0,
where the lowest terms in 0-y, 7 -a, a-/3 are of the order 3, and the theorem
is thus proved. The case in question, q = -^ or X = iA, is where the plane passes
through the centre of the tetrahedron.
When q = i = n, _„ , or X = §A, the equation is
(X^ +Y^ + Z'- 2YZ - 2ZX - 2XYy = 128XYZ(X + Y+ Z).
Here each of the lines X = 0, F=0, Z=Q is an osculating tangent having with the
curve a 4-pointic intersection.
When 5=0 = ai _a-^ . or X = \h, the equation is
{YZ-\-ZX-^XYy-^XYZ{X+Y+Z) = Q,
that is,
Y-Z^ + Z^X^ + X-Y-'- 2XYZ (Z + F+ ^) = 0 ;
viz. each angle of the triangle is here a cusp.
When q = — \, or X = 0, the curve is
[X^+Y^ + Z^ -2{YZ+ ZX + XY)Y=^Q,
viz. the plane is here the base of the tetrahedron, and the section is the inscribed
circle taken twice.
For tracing the curves, it is convenient to find the intersections with the lines
Y—Z = 0, Z — X = 0, X—Y=0 drawn from the centre of the triangle to the vertices;
each of these lines passes through a node, and therefore besides meets the curve in
two points. Writing, for instance, Y = X, the equation becomes
{q'(2X + Zy-2XZ-X^Y-i(2q+l)X'-Z(2X+Z) = 0;
viz. this is
{qZ +{2q + l)XY {q'Z" + {^^ -2q-t)XZ + (iq' -4:q + l)X"-} = 0,
6 ON steiner's surface. [556
where the first factor gives the node. Equating to zero the second factor, we have
|52r+(29-l-?)z|' = Z'|(29-l-?y-4?' + 45-l}
= Z«i(l-9)(l + 23);
or, finally,
9^= |- 2g + 1 + 1 ± ? sf{i^)'(1^2q)^ X,
giving two real values for all values of q from q = l to q = — ^. (See the Table
afterwards refeiTed to.)
We may recapitulate as follows :
5 > 1, or X > |A ; the curve is imaginary, but with three real acnodes, answering
to the acnodal parts of the nodal lines:
q = l, or \r=|A; the summit appears as a fourth acnode:
q < 1 > ^, ov \ < f A > §A ; the curve consists of three acnodes and a trigonoid lying
within the triangle and having the sides of the triangle for bitangents of imaginary
contact :
5 = J, or \ = |/t ; the curve consists of three acnodes and a trigonoid having the
sides of the triangle for osculating tangents :
</ < ^ > 0, or \<^h> i^h; the curve consists of three conjugate points and an in-
dented trigonoid having the sides of the triangle for bitangents of real contact:
q = 0, or \ = iA; curve has the summits of the triangle for cusps:
q < 0 > — ^, 01 \< ^h> \h; curve has three crunodes, or say it is a cis-centric trifolium:
3 = — J, or X = ^h; curve has a triple point, or say it is a centric trifolium :
g < — J > — ^, or X < JA > 0 ; curve has three crunodes, or say it is a trans-centric
trifolium :
q = — ^, or X = 0 ; curve is a two-fold circle :
g < — ^, or X < 0 ; the curve becomes again imaginary, consisting of three acnodes
answering to the acnodal parts of the nodal lines.
For the better delineation of the series of curves, I calculated the following Table,
wherein the first column gives a series of values of X : h; the second the corre-
2X — h
spending values of q, =^t — k^', the third the positions of the point of contact, say
with the side ^=0, the value of X : F being calculated firom the foregoing formula,
^-y=-^+l±^/^-h
556]
ON STEINERS SURFACE.
and the fourth the apsidal distances, say for the radius vector X = Y, the value of
Z : X being calculated from the foregoing formula
The Table is:
I
q q- q\ \q J\ ql
\:h
Q
Contact,
Z=0, X : Y=
-v=
Apses,
Y; X:Z=
•75
1-00
im
poss.
1
•70
•666
•6666
•5
im
)X)SS.
-032
0-
o
r 7-968
16-
1
•
•65
•4285
•320 o
r 3-124
-006
22-44
•6
•25
•0721
13-9279
•059
67-941
•55
•1111
•Oil
78^988
-15
337-85
•5
0^
0
or 00
-25
00
•45
- 0909
•005
118-99
-39
457-61
•4
-•1666 '•
•029
33971
•496
127-504
•35
- •2308
•060
1672
•648
61-79
•3
- ^2857
•099
10^151
-812
37-187
•25
- 3333
•141
6-854
1-
25
•2
-•375
•207
4-904
1-218
17-89
•15
-•4118
•276
3-622
1-48
13-25
•10
- ^4444
•372
2-690
1-813
9-937
•05
- -4737
•515 0
r 1-941
3-15
0
r 6-46
■0
-•5
1
•
4
•
where the asterisks show the critical values of \ : h.
It is worth while to transform the equation to new coordinates X', Y', Z' such
that X' =0, Y' — 0, Z' =0 represent the sides of the triangle formed by the three nodes.
Writing for shortness X-^Y+Z=P. YZ+ZX + XY = Q, XYZ= R, the equation is
(fP-Qy='t(2q+l)PR
The expressions of X, Y, Z in terms of the new coordinates are of the form X' + 0P',
Y' + OF, Z + eP', where F = X'^-Y' + Z' ; writing also Q' = Y'Z'+Z'X'+XT, R = X'Y'Z',
then the values of P, Q, R are
(i + .3^)P', Q' + (2^ + 3^•')F^ R'+eFq'+{e^->r0>)F,
8 ON steiner's surface. [556
and the transformed equation is
[{{•(I + Sey -20- 3^1 F^ - Q'p = 4 (2? + 1) (1 + 35) P {(^ + ^) F' + ^p-Q- + ij'},
which is satisfied by Q' = 0, R = 0, if only
(j" (1 + Wy -20- 3(9»j» = 4 (2g + 1) (1 + 35) (0" + 0^),
or, if for a moment q(l + S0)=il, the equation is
(fl» - 25 - 35')= = 4 (5» + 5=) (2n + 1 + 35),
that is,
n* + fi»(-65= - 45) + n(-855 - 85*) - 35^ - 45^ = 0,
that is,
(fi + 0f (n= - 25n - 35= - 45) = 0.
If the new axes pass through the nodes, then Xi + 5 = 0; that is, g'(l + 35) + 5 = 0,
which equation gives the value of 5 for which the new axes have the position in
a
question ; substituting in the first instance for q the value ^^ — - , the equation becomes
on + 1
{25 (1 + 0)1^^ + Q'Y = 4 (1 + 5) P' {5* (1 + 5) P'» + 0^^ + R'},
that is,
g^ = 4(1 + 5) P'i2';
or, finally, substituting for 5 its value in terms of q, the required equation is,
that is,
(Y'Z' + Z'X' + X'YJ = 4 |?^J XTZ' (X' +Y' + Z').
Sq + 1
In particular, for q = 0 the equation is
( Y'Z' + Z'X' + X'Y'y - ^X'Y'Z' (X' +Y' + Z') = 0,
which is right, since, in the case in question (the tricuspidal cui-ve), we have
X, Y, Z^X', F, Z'.
I remai'k, in passing, that, taking the equation to be
{Y'Z' + Z'X' + X'Y'y = mX'Y'Z' (X' + F' + Z'),
we may write herein
Z' = i- X.
I
556] ON steiner's surface.
where
2 v^7^(m - 3) , 2(m-3) „.
a; = ); cos G ^^-;r COS 20,
2Vm(m-3) . - 2 (m - 3) . „ .
y = ^ ^ sin e + — 5-g ^ sin 20,
which are the formulae for the description of the trinodal quartic as a unicursal curve.
I consider now the second position ; viz. the horizontal plane now passes through
the centre of the tetrahedron, and is parallel to two opposite edges. The equations of
the nodal lines are here (y = 0, z = 0), (^ = 0, a; = 0), {x = 0, y = 0); and if for convenience
we assume the distance of the mid-points of opposite edges to be = 2, or the half
of this = 1, then the equations of the faces are
X= a: + y + z-l=0,
Y=-x-y + z-l = 0,
Z = x-y-z-l=0,
W = -x + y-z-l=0,
and the equation of the surface is
Proceeding to rationalise, this is
X + Y+ 2 s/XY=Z+ W + 2 'JZW,
VIZ.
we thence have
or, since
this is
whence
or reducing.
22 + VZr = VZF;
\z^^^z^XY^XY=ZW;
ZW-XY=4,z + ixy,
z + xy— z" = z \XY;
{z + xy- zj = z"- \{z -\f-(x^ yf\ ;
2xyz + fz'' + z'^a? + x^ = 0,
a form which puts in evidence the nodal lines. Considering z as constant, we have
the equation of the section ; this is a quartic having the node {x = 0, y = 0), and two
other nodes at infinity on the two axes respectively; moreover, the curve has for
bitangents the intersections of its plane with the faces of the tetrahedron ; or what
is the same thing, attributing to z its constant value, the equations of the bitangents are
a; + y + 2— 1=0,
~x — y-\-z — \ = 0,
X — y — z —1=0,
— x + y — z — l=0.
C. IX. 2
10 ON steiner's surface. [556
These lines form a rectangle which is the section of the tetrahedron ; observe that
this is inscribed in the square the corners of which are x= ±1, y = ±l; viz. z = + l
(highest section), this is the dexter diagonal (considered as an indefinitely thin rect-
angle), and as z diminishes, the longer side decreases and the shorter increases until
for a; = 0 (central section) the rectangle becomes a square ; after which, for z negative
it again becomes a rectangle in the conjugate direction, and finally, for z = —l (lowest
section) it becomes the sinister diagonal (considered as an indefinitely thin rectangle).
But on account of the symmetry it is sufficient to consider the upper sections for which
z Ls positive. The sides ±(x + y) + z—i=0 parallel to the dexter diagonal of the
square may for convenience be termed the dexter sides, and the others the sinister
sides. In what follows I write c to denote the constant value of z.
We require to know whether the bitangents have real or imaginary contacts ; and
for this purpose to find the coordinates of the points of contact.
Take first a dexter bitangent x + y+c — 1 = 0; the coordinates of any point
hereof are
x=^(i-c + e), y = ^(i-c-^),
where 6 is arbitrary ; and substituting in the equation of the curve, we should have
for 0 a twofold quadric equation, giving the values of 6 for the two points of
contact respectively. We have
^+f=h{a-cy+n xy=i[(i-cy-0'}.
And thence
8c= {(1 - cf + e']+8c {(1 - c)= - ^^1 + {(1 - cY- - &f = 0,
viz. this equation is
{^-(l-c)(l + 3c))^ = 0,
a twofold quadric equation, as it should be ; and the values of 0 being = + V(l — c)(l + 3c),
we see that these, and therefore the contacts, are real from c = 1 to c = — J.
In exactly the same way for a sinister bitangent ±{x — y) — c — l=0, we have
a; = J(l+c + <^), -y = ^(l + c-<l>), and <^ = + %/(! -3c)(l + c),
viz. the values of <f>, and therefore the contacts, are real from c = ^ to c = — 1.
That is.
Contacts of Contacts of
Dexter Bitangents. Sinister Bitangents.
c= 1 to ^ real, imaginary,
c= ^ to — J^ real, real,
c=— ^ to — 1 imaginary, real ;
or say c = 1 to |, the contacts are real, imaginary ; but c = ^ to 0, they are real, real.
In the transition case, c = |, the sinister bitangents become osculating (4-pointic) tangents
touching at points on the dexter diagonal. This can be at once verified.
556] ON steiner's surface. 11
Observe that when c=l, we have
so that the only real point is a; = 0, y = 0; viz. this is a tacnode, having the real
tangent x + y={). For c = 0 (central section) the equation becomes xhf = 0 ; viz. the
curve is here the two nodal lines each twice.
It is now easy to trace the changes of form.
c = 1 ; curve is a tacnode, as just mentioned, tangent the dexter diagonal.
c < 1 > ^ ; curve is a figure of 8 inside the rectangle, having real contacts with
the dexter sides, but imaginary contacts with the sinister sides.
c=|; curve is a figure of 8 having real contacts with the dexter sides, and
osculating (4-pointic) contacts with the sinister sides.
c < ^ > 0 ; curve is an indented figure of 8 having real contacts as well with the
sinister as the dexter aides.
c = 0 ; curve is squeezed up into a finite cross, being the cruuodal parts of the
nodal lines ; and joined on to these we have the acnodal parts, so that the whole
curve consists of the lines a' = 0, 2/ = 0 each as a twofold line.
For tracing the curve, it is convenient to turn the axes through an angle of
45° ; viz. writing " > /o ^^ place of x, y respectively, the equation becomes
c (y— a?) + cH2/= + ^) + i (2/— *-7 = 0 ;
a; = 0 gives y" = 0 or y^ = — 4c (1 + c), *
y = 0 gives (i? = 0 ov or = 4c (1 — c).
Moreover, we have
i{c-<f){f-a?) + %dhj^ + {y^-a?f=0,
viz.
(ar^ - 2/' + 2c= - 2c)- = 4c^ {(c - 1)- - 2y%
and similarly
(2/= - a:= + 2c= + 2c)= = 40^ {(c + 1 )= - 2a;=i ,
putting in evidence the bitangents, now represented by the equations c — 1 = + y \/2 and
c + 1 = + a: \/2 respectively. And for the first of these, or c-l = ±y s/% we have for the
points of contact a? = ^(l — c)(l + 3c); and for the second of them, or c+l = + a;V2,
the points of contact are y' = ^ (1 + c) (1 — 3c).
I consider the circumscribed cone having its vertex at a point (0, 0, 7) on the
nodal line (a;=0, y = 0). Writing in the equation of the surface x = \{z — y),y = fji,(z — y),
the equation, throwing out the factor {z — yy, becomes
2\fiz + (X^ + fi-) z- + \V (■^ - 7)' = 0,
that is,
(\>- + X- + /i=) z-
+ 2(-y\/ji, + l)zXfi
-l- y- . X'/ur = 0 ;
* y always imaginary when c is positive.
2—2
12 . ON steinee's surface. [556
and equating to zero the discriminant in regard to z, we have
7> (\V + X' + /*') - (- 7X/* + 1)' = 0,
that is,
7>(X» + At') + 27X/i-l = 0;
and substituting herein the values X = — - and /*=—£—. we have the equation of
the cone, viz. this is
or, what is the same thing,
y (a? + 2^ - 1 ) + 27 (icy + « ) - ^- = 0 ;
viz. this is a quadric cone having for its principal planes z — 'f = (i, x + y = 0, x — y = 0,
these last being the planes through the nodal line and the two edges of the tetrahedron.
In the particular case y=x>, the cone becomes the circular cylinder ar' + y» — 1 =0.
The cone intersects the plane 2 = 0 in the conic
'f(a^ + f - 1) + ^xy = 0,
which is a conic passing through the comers of the square (a; = 0, y = + 1), (a; = + 1, y = 0).
For 7 > 1, that is, for an exterior point, the conic is an ellipse having for the squares
of the reciprocals of the semi-axes 1 + - , 1 — (this at once appears by writing in
the equation .^ , —.a in place of x, y respectively). In particular, for 7 = x , the
curve becomes the circle 3?-\-y'^ —\—(i. We have thus the apparent contour of the
surface as seen from the point 2 = 7 on the nodal line, projected on the plane 2^ = 0
of the other two nodal lines.
To find the curve of contact of the cone and surface, or say the surface-contour
from the same point, write for a moment
F = 7 (ar= -I- 2/^ - 1) + 27 («y 4- 2) - 2=,
{/■ = (a;y -h 2)- + 2:» (ar" -I- y' - 1) ;
then, substituting for «^-|-y'-I its value in terms of V from the first equation, we find
-2\2
V={xy^z~'-)^'^y,
z^
and the equations 17^=0, F=0 give therefore xy-\-z- =0, or say 7(a;y + 2)-z» = 0.
7
The cone and surface therefore touch along the quadriquadric curve
7= (ar* + y» - 1 ) -I- 27 (a;y -h 2) - 2= = 0,
7(a;y-|-2)-«» = 0,
equations which may be replaced by
7 (a:» + y' — l)-|-art/ + 2=:0,
7»(a^+y^-l) + 2» =0.
In the case 7 = » , the equations are ar" + y^ — 1 = 0, xy-\-z = ^, viz. the curve is
the intersection of the hyperbolic paraboloid xy + 2 = 0 by the cylinder a;" + y= - I = 0.
557] . 13
557.
ON CERTAIN CONSTRUCTIONS FOR BICIRCULAR QUARTICS.
[From the Proceedings of the London Mathematical Society, vol. v. (1873 — 1874),
pp. 29—31. Read March 12, 1874.]
I CALL to mind that if F, G are any two points and F', G' their antipoints ;
then the circle on the diameter FG and that on the diameter F'G' are concentric
orthotomics, viz. they have the same centre, and the sum of the squared radii is
= 0. Moreover, if the circles B, B' are concentric orthotomics, and the circle A is
orthotomic to B, then it is a bisector of B", viz. it cuts B' at the extremities of a
diameter of B" ; and R is then said to be a bifid circle in regard to A.
Given two real circles, these have an axial orthotomic, the circle, centre on the
line of centres at its intersection with the radical axis, which cuts at right angles the
given circles ; viz. this axial orthotomic is real if the circles have no real intersection ;
but if the intersections are real, then the axial orthotomic is a pure imaginary, and
insteafJ thereof we may consider its concentric orthotomic, viz. this is the axial bifid
of the two circles, or circle having its centre on the line of centres at the inter-
section thereof with the radical axis or common chord of the two circles, and having
this common chord for its diameter.
If one of the circles is a pure imaginary, then we have still an axial orthotomic;
viz. the pure imaginary circle is replaced by the concentric orthotomic ; and the axial
orthotomic is a bisector of the substituted circle ; and so if each of the circles is a
pure imaginary, then we have still an axial orthotomic, viz. each circle is replaced by
the concentric orthotomic, and the axial orthotomic is a bisector of the substituted
circles. And in either case the axial orthotomic of the original circles (one or each of
them pure imaginary) is real; viz. this is given either as the axial bisector of one
real circle and orthotomic of another real circle ; or as the axial bisector of two circles,
from which the reality thereof easily appears. Or we may verify it thus: Suppose
14 ON CERTAIN CONSTRUCTIONS FOR BICIRCULAR QUARTIC8. [557
that the two circles are (a; — ay + y- = 0*, (x— a')" + y = I3'°, and their axial orthotomic
(x - my + y* = L", then we have (»i— a)^=/3'+A:', (m — a'Y = $'- + Ic' ; subtracting, it appears
that m is i-eal ; and then if either ff' or y9'' is negative, the equation containing this
quantity shows that k' Is positive ; viz. the circle (x — vif + y" = k- is real.
The above remarks have an obvious application to the theory of bicii'cular quartics ;
viz. a bicircular quartic is the envelope of a variable circle, having its centre on a
conic, and orthotomic to a circle : it may be that this circle is a pure imaginary.
We then replace it by the concentric orthotomic, and say that the curve is the
envelope of a variable circle having its centre on a conic and bisecting a circle. We
have thus a real form for cases which originally present themselves under an imaginary
form.
The Bicircular Quartic with given vertices.
First, if the vertices are real ; let the vertices taken in order be F, G, H, K.
First construction: On FG as diameter describe a circle, and on HK as diameter
a circle ; on the line terminated by the two centres (as tran.sverse or conjugate axis)
describe a conic 0,, and describe the axial orthotomic circle 2, of the two circles
(viz. the centre of 2, is on the axis of symmetry at its intersection with the radical
axis of the two circles) ; then the curve is the envelope of a variable circle having
its centre on 0, and orthotomic to 2i.
Second construction : On FH as diameter describe a circle, and on GK as diameter
a circle. On the line terminated by the . two centres (as transverse or conjugate axis)
describe a conic 0a, and describe the axial bifid circle 2/ of the two circles (viz. the
centre of 2./ is on the axis of symmetry at its intersection with the radical axis or
common chord of the two circles, and its diameter is this common chord) ; then the
curve is the envelope of a variable circle having its centre on 0. and bisecting 2/.
Third construction: On FK as diameter describe a circle, and on OH as diameter
a circle; and then, as in the first consti-uction, a conic 0, and a circle 2,; the curve
is the envelope of a variable circle having its centre on 0, and orthotomic to 23.
Observe that in the three constructions the conies have always the same centre ;
and if the three conies are taken with the .stime foci, then the three constructions give
one and the same bicircular quartic. The first and third constructions form a pair,
and there is no reason for selecting one of them in preference to the other; but the
second construction is unique; it is on this account natural to make use of it in
discussing the series of curves with the given vertices.
In the particular case where the points F, G and H, K are situate symmetrically
on opposite sides of a centre 0 {0F= OK, OG = OH), then in the thii-d construction
the centres each coincide with 0, or the axis of the conic vanishes ; hence the con-
struction fails: the first and second constmctions hold good, and in each of them the
557] ON CERTAIN CONSTRUCTIONS FOR BICIRCULAR QUARTIC8. 15
conic and circle are concentric. The ciu-ve is in this case quadrantal: having, besides
the original axis of symmetry, another axis of symmetry through 0, at right angles
thereto.
Secondly, if the vertices are two real, two imaginary, say f, g = a±^i: h, k, we
modify the first or third construction ; viz. if F', G' are the antipoints of F, G ; then
on F'G' as diameter describe a circle, and on HK as diameter a circle. On the line
terminated by the two centres (as transverse or conjugate axis) describe a conic 0i,
and describe the axial bisector-orthotomic circle 2i of the two circles ; viz. this is the
circle (centre on the axis of symmetry) which bisects the circle F'G' , and cuts at right
angles the circle HK ; then the curve is the envelope of the variable circle having
its centre on 0, and orthotomic to 2i.
Thirdly, if the vertices are all imaginary, say _/",</ = a + /3t ; h, f<: = y ± St, we modify
the first or third construction. Take F', G' the antipoints of F, G, and H', K' the
antipoints of H, K ; then on F'G' as diameter describe a circle, and on H'K' as
diameter a circle ; on the line terminated by the two centres (as transverse or conjugate
axis) describe a conic ©, and describe the axial bisector-circle 2 of the two circles
(viz. this is a circle, centre on the axis of symmetry, bisecting each of the circles) :
the curve is the envelope of a variable circle, centre on the conic 0 and cutting at
right angles the circle 2.
16 [558
558.
A GEOMETRICAL INTERPRETATION OF THE EQUATIONS OB-
TAINED BY EQUATING TO ZERO THE RESULTANT AND
THE DISCRIMINANTS OF TWO BINARY QUANTICS.
[From the Proceedings of the London Mathematical Society, vol. v. (1873 — 1874),
pp. 31—33. Read March 12, 1874]
CoNSlDEK the equations
V =={a, h,...\t, 1)*=0,
U' = {a', h',...Jt, 1)^' = 0;
and equating to zero the discriminants of the two functions respectively, and also the
resultant of the two functions, let the equations thus obtained be
A =(a, 6,...)-^-2=0,
A' = (a', 6',...)-^-^ = 0.
R = {a, h,...f{a, b,...Y = 0.
I take (o, b,...), (a', b',...) to be linear functions of the coordinates (x, y, z); and
t to be an indeterminate parameter. Hence U=0 represents a line the envelope
whereof is the curve A = 0, or, what is the same thing, the equation U = 0 represents
any tangent of the curve A = 0 ; this is a unicursal curve of the order 2\ — 2 and
class \, with 3 (X — 2) cusps and ^ (\ — 2) (\ - 3) nodes. Similarly U" = 0 represents a
line the envelope of which is the curve A' = 0 : this is a unicursal curve of the order
2X'-2 and class V, with 3(\'-2) cusps and |(\'-2)(\'-3) nodes; the equation
J/' = 0 represents any tangent of this curve.
The equations U = 0, U' = 0 considered as , existing simultaneously with the same
value of t, establish a (1, 1) correspondence between the tangents (or if we please,
between the points) of the two curves. The locus of the intersection of the corre-
558] A GEOMETRICAL INTERPRETATION OF SOME EQUATIONS. 17
spending tangents is the curve R = 0, a unicursal curve of the order \ + V, with
^(X + X'— l)(\ + \'-2) nodes and no cusps; consequently of the class 2(\+\'-l).
It is to be shown that the curve R=0 touches the curve A = 0 in \' + 2X - 2
points, and similarly the curve A' = 0 in 2\' + \ — 2 points.
In fact, consider any tangent T' of the curve A' ; let this meet the curve A in
a point A, and let Q be the tangent at A to the curve A ; suppose, moreover, that
T is the tangent of A corresponding to the tangent T' of A'. Then if Q and T
coincide, the corresponding tangent of T' will be Q, and the curve R will pass
through A. It is easy to see that in this case the curves R, A will touch at A.
Again, if P be a tangent from A to the curve A, then, if P and T coincide, the
corresponding tangent of T' will be P, and the curve R will pass through A ; but
in this case the point A will be a mere intersection, not a point of contact, of the
two curves.
The tangents T, Q each correspond to T', and they consequently correspond to
each other. For a given position of T we have a single position of 2", and therefore
2X — 2 positions of A, or, what is the same thing, of Q ; that is, for a given position
of T we have 2X — 2 positions of Q. Again, to a given position of Q corresponds a
single position of A, therefore X' positions of T', therefore also X' positions of T; that
is, for a given position of Q we have X' positions of T. The correspondence between
T, Q is thus a (X', 2X — 2) correspondence, and the number of united tangents is
therefore X' + 2X — 2, or the curves R, A touch in X' + 2X — 2 points.
The tangents T, P each correspond to T', and they therefore correspond to each
other. For a given position of T we have a single position of 2", and therefore 2X — 2
positions of A, and thence (2X — 2)(X— 2) positions of P; that is, for a given position
of T we have (2X — 2) (X — 2) positions of P. Again, to a given position of P corre-
spond 2X — 4 positions of A, therefore (2X— 4)X' positions of 7" or of T; that is,
for a given position of P we have (2X — 4) X' positions of T. The correspondence
between T, P is thus a [2X'(X — 2), 2(X — 1)(X— 2)] correspondence, and the number of
united tangents is 2(X-|-X' — 1)(X — 2); or the curves R, A meet in 2(X + X' -1)(X- 2)
points.
Reckoning the contacts twice, the total number of intersections of R, A is
2X' + 4X-4-|-2(X+X'-l)(X-2), =(X-I-X')(2X- 2),
as it should be.
In the particular case X = X' = 2, the curves A, A' are conies, and the curve R
is a quartic curve touching each of the conies 4 times ; this is at once verified, since
the equations here are
ac-b' = 0, aV-6'= = 0, 4 (ac - &■) (aV - 6'=) - (ac' + a'c - 266')' = 0-
C. IX.
18 [559
559.
[NOTE ON INVERSION.]
[From the Proceedings of the London Mathematical Society, vol. \'. (1873 — 1874), p. 112.]
The inverse of the anchor ring (in the foregoing paper* called the cyclide) is in
fact the general binodal cyclide or binodal bicircular quartic; viz. assuming it to be
a cyclide (bicircular quartic), to see that it is binodal, it need only be observed that
the anchor ring is binodal (has two real or imaginary conic points, viz. these are the
intersections of the circles in the several axial planes) ; and to see that it is the
general binodal cyclide, we have only to count the constants ; viz. the general cyclide
or surface
{af + y^ + z^y+{a^ + f + z^)(cuc + ^y + yz)+{a, b, c, d, f, g, h, I, m, n)(x, y, z, iy = 0
contains 13 constants, and therefore the binodal cyclide 13 — 2, =11 constants. But the
anchor ring, irrespective of position, contains 2 constants; centre of inversion, taken in
given axial plane, has 2 constants ; radius of inversion, 1 constant ; in all 2 + 2 + 1, =5
constants; or taking the inverse surface in an arbitrary position, the number of constants
is 5 + 6, =11.
* By Mr H. M. Taylor: Inversion, with special reference to the Inversion of an Anchor Bing or ToruB,
{Lond. Math. Soc. Proc., same volume, pp. 105 — 112).
560] 19
560.
[ADDITION TO LORD RAYLEIGH'S PAPER "ON THE NUMERICAL
CALCULATION OF THE ROOTS OF FLUCTUATING FUNCTIONS."]
[From the Proceedings of the London Mathematical Society, vol. v. (1873 — 1874),
pp. 123, 124. November 22.]
Prof. Cayley, to whom Lord Bayleigh's paper was referred, pointed out that a similar result may be
attained by a method given in a paper by Encke, "Allgemeine Anfloaung der numerischen Gleiohungen,"
CrelU, t. xxji. (1841), pp. 193—248, as follows:
Taking the equation
0=l—ax + ba? — ai?+ da::* — eaf +faf — gx' + /t** — . . . ;
if the equation whose roots are the squares of these is
0=1— «,« + hiO? — CiO^ + ...,
then
a, = a" — 26,
b, =¥- 2ac + 2d,
c,2 = c= - 2bd + 2ae- 2/,
d,» = d- - 2ce + 26/- 2ag + 2h, &c. ;
and we may in the same way derive Uj, 63, Ca, &c. from ai, 61, Ci, &c., and so on.
As regards the function
<^"(^) = 2n.p(^ + l) \^ ~ 2T2ft + 2'*"2.4.2ii + 2.2M + 4~""J '
3—2
20 ON ROOTS OF FLUCTUATING FUNCTIONS. [560
we have as follows:
a-' =2'.ji + l,
b-' = 2» . n + 1 . n 4- 2,
c-' =2'. 3.n + l ...n + 3,
d-' =2".3.n + l ...71 + 4,
«-' =2".3.5.n + l ...n + 5,
/-' =2".3».5.n+l ... tt + 6,
f = 2" . 3= . 5 . 7 . 71 + 1 ... n + 7,
A-> =2«'.3».5.7.n + l ...n + 8,
ai-' = 2* .(n + iy.n+2,
br' =2' . (n + 1 . w + 2)^ W + 3 . n + 4,
cr' = 2" . 3 . (n + 1 ... n + 3)^ n + 4 ... n + 6,
d,-' = 2"' . 3 . (n + 1 ... ?i + 4)\ w + 5 ... »j + 8,
5n + ll
a,=
^^ = W
2» . (n + ly (n + 2)» Ji + 3 . n + 4 '
25ji''+231n + 542
03 =
If n = 0,
whence
2".(re+l .7i+2/(»!, + 3.?H-4)-n+ 5 ... n + 8'
429«' + 764071* + 537o2w» + 185430w° + 311387w + 202738
2«(n + 1)8 (w + 2)'' (71 + 3 . 71 + 4)^71 + 5 . 7i + 6 . h + 7 . ?; + 8 '
_. _,, 101369
2p »« = a,=— — — -^=^, ■», suppose;
2'^.3».5.7
Pi = 2-404825.
[The quantities p^, p^,... are the roots of the function Jni^:) in increasing order
of magnitude, so that, as these roots are all real, it follows that for Jo(^),
a = 2^1-2, a^^lpr*, a, = lpr% a, = Sjsr", • • •]
561] 21
561.
ON THE GEOMETRICAL REPRESENTATION OF CAUCHY'S
THEOREMS OF ROOT-LIMITATION.
[From the Transactions of the Cambridge Philosophical Society, vol. xii. Part ii. (1877),
pp. 39.5—413. Read February 16, 1874.]
There is contained in Cauchy's Memoir "Calcul des Indices des Fonctions,"
Joum. de I'Ecole Polytech. t. xv. (1837) a general theorem, which, though including
a well-known theorem in regard to the imaginary roots of a numerical equation,
seems itself to have been almost lost sight of. In the general theorem (say Cauchy's
two-curve theorem) we have in a plane two curves P = 0, Q = 0, and the real inter-
sections of these two curves, or say the "roots," are divided into two sets according as
the Jacobian
d^P.dyQ-d^Q.dyP
is positive or negative, say these are the Jacobian-positive and the Jacobian-negative
roots: and the question is to determine for the roots within a given contour or
circuit, the difference of the numbers of the roots belonging to the two sets respectively.
In the particular theorem (say Cauchy's rhizic theorem) P and Q are the real part
and the coefficient of i in the imaginary part of a function of x + iy with, in general,
imaginary coefficients (or, what is the same thing, we have P + iQ =f(x + iy) + i<f> {x + iy),
where /, </> are real functions of a; -I- iy) : the roots of necessity are of the same set :
and the question is to determine the number of roots within a given circuit.
In each case the required number is theoretically given by the same rule, viz.
P
considering the fraction py, it is the excess of the number of times that the fraction
changes from -|- to — over the number of times that it changes from — to -f-, as
the point {x, y) travels round the circuit, attending only to the changes which take
place on a passage through a point for which P is = 0.
22 ON THE GEOMETRICAL REPRESENTATION OF [561
In the case where the circuit is a polygon, and most easily when it is a rect-
angle the sides of which are parallel to the two axes respectively, the excess in
question can be actually determined by means of an application of Sturm's theorem
successively to each side of the polygon, or rectangle.
In the present memoir I reproduce the whole theory, presenting it under a com-
pletely geometrical form, viz. I establish between the two sets of roots the distinction
of right- and left-handed: and (availing myself of a notion due to Prof. Sylvester*)
I give a geometrical form to the theoretic rule, making it depend on the " inter-
calation" of the intersections of the two curves with the circuit: I also complete the
Sturmian process in regard to the sides of the rectangle : the memoir contains further
researches in regard to the curves in the case of the particular theorem, or say as
to the rhizic curves P = 0, Q = 0.
The General Theory. Articles Nos. 1 to 19.
1. Consider in a plane two curves P = 0, Q = 0 (P and Q each a rational and
integral function of x, y), which to fix the ideas I call the red curve and the blue
curve respectively f : the curve P = 0 divides the plane into two sets of regions, say
a positive set for each of which P is positive, and a negative set for each of which
P is negative : it is of course immaterial which set is positive and which negative,
since writing —P for P the two sets would be interchanged : but taking P to be
given, the two sets are distinguished as above. And we may imagine the negative
regions to be coloured red, the positive ones being left uncoloured, or say they ai-e
white. Similarly the curve Q = 0 divides the plane into two sets of regions, the
negative regions being coloured blue, and the positive ones being left uncoloured, or
say they are white. Taking account of the twofold division, and considering the
coincidence of red and blue as producing black, there will be four sets of regions,
which for convenience may be spoken of as sahle, gules, argent, azure: viz. in the figures
we have
— — sable, shown by cross lines,
— + gules, „ „ vertical lines,
+ -I- argent, left white,
+ — azure, shown by horizontal lines,
sable and argent (— — and -|- +) being thus positive colours, and gules and azure
(- + and -I ) negative colours. See figures [pp. 32, 38] towards end of Memoir.
• See his memoir, "A theory of the Syzygetic relations, &c." Phil. Tram., 1853. The Sturmian process
is by Storm and Canchy applied to two independent functions <px, fx of a variable x ; but the notion of
an intercalation as applied to the order of succession of the roots of the equations 0 (a;) = 0, /(x) = 0 is due
to Sylvester, and it was he who showed that what the Sturmian process determined was in fact the inter-
calation of these roots: but, not being concerned with circuits, he was not led to consider the intercalation
of s circuit.
+ It is assnmed throughout that the two curves have no points (or at least no real points) of multiple
intersection; i.e. they nowhere touch each other, and neither curve passes through a multiple point of the
other curve.
561] cauchy's theorems of root-limitation. 23
2. Consider any point of intersection of the two curves. There will be about
this point four regions, sable and argent being opposite to each other, as also gules
and azure; whence selecting an order
sable, gules, argent, azure;
if to have the colours in this order we have to go about the point, or root, right-
handedly, the root is right-handed: but if left-handedly, then the root is left-handed:
or, what is more convenient, going always right-handedly, then, if the order of the
colours is
sable, gules, argent, azure,
the root is right-handed: but if the order is
sable, azure argent, gules,
the root is left-handed.
3. The distinction of right- and left-handed corresponds to the sign of the Jacobian
^^^(=d,P.dyQ-d,Q.d,P);
we may (reversing if necessary the original sign of one of the functions) assume that
for a right-handed root the Jacobian is positive, for a left-handed one, negative.
4. I consider a trajectory which may be either an unclosed curve not cutting
itself, or else a circuit, viz. this is a closed curve not cutting itself. A circuit is
considered as described right-handedly : an unclosed trajectory is considered as described
according to a currency always determinate pro hdc vice : viz. one extremity is selected
as the beginning and the other as the end of the trajectory: but the currency may
if necessary or convenient be reversed : thus if an unclosed trajectory forms part of a
circuit the currency is thereby determined: but the same unclosed trajectory may form
part of two opjwsite circuits, and as such may have to be taken with opposite
currencies. It is assumed that a trajectory does not pass through any intersection of
the P and Q curvea
5. A trajectory has its P- and Q-sequence, viz. considering in order its inter-
sections with the two curves, we write down a P for each intersection with the red
curve and a Q for each intersection with the blue curve, thus obtaining an inter-
mingled series of P'a and Q'b, which is the sequence in question. In the case of a
circuit, the sequence is considered as a circuit, viz. the first and last terms are con-
sidered as contiguous, and it is immaterial at what point the sequence commences.
The sequence will of course vanish if the trajectory does not meet either of the curves.
6. A P- and Q-sequence gives rise to an "intercalation," viz. if in the sequence
there occur together any even number of the same letter, these are omitted (whence
also any odd number of the same letter is reduced to the letter taken once): and if
by reason of an omission there again occur an even number of the same letter, these
24 ON THE GEOMETRICAL REPRESENTATION OF [561
are omitted : and so on. The intercalation contains therefore only the letters P and Q
alternately: viz. in the case of an unclosed trajectory the intercalation may contain an
even number of letters, beginning with the one and ending with the other letter, and
80 containing the same number of each letter — or it may contain an odd number of
letters, beginning and ending with the same letter, and so containing one more of
this than of the other letter; say the intercalation is PQ or QP, or else PQP or
QPQ. The intercalation may vanish altogether; thus if the sequence were QPPQ, this
would be the case.
7. In the case of a circuit the intercalation cannot begin and end with the same
letter, for these, as contiguous letters, would be omitted; and since any letter thereof
may be regarded as the commencement it is PQ or QP indifferently. A little con-
sideration will show that the whole number of letters must be evenly even, or, what
is the same thing, the number of each letter must be even. Thus imagine the circuit
beginning in sable, and let the intercalation begin with PQ ; viz. P we pass from
sable to azure, and Q we pass from azure to argent : in order to get back into sable
we must either return the same way (Q argent to azure, P azure to sable), but then
the sequence is PQQP, and the intercalation vanishes: here the number of letters
is 0, an evenly even number: or else we must complete the cycle of colours P argent
to gules, Q gules to sable: and the sequence and therefore also the intercalation then
is PQPQ, where the number of letters is 4, an evenlj' even number.
8. In the case of any trajectory whatever, the half number of letters in the inter-
calation is termed the " index," viz. this is either an integer or an integer -I- i. But
in the case of a circuit the index is an even integer, and the half-index is therefore
an integer. The index may of course be = 0.
9. But we require a further distinction: instead of a P- and Q-sequence we
have to consider a + P- and Q-sequence. To explain this, observe that a passage
over the red curve may be from a negative to a positive colour (azure to sable or
gules to argent), this is +P, or from a positive to a negative colour (sable to azure
or argent to gules), this is —P. And so the passage over the blue curve may be
from a negative to a positive colour (gules to sable or azure to argent), this is + Q,
or else from a positive to a negative colour (sable to gules or argent to azure), this
is —Q. The sequence will contain the P and Q intermingled in any manner, but
the signs will always be H alternately ; for + (P or Q), denoting the passage into
a positive colour, must always be immediately succeeded by — (P or Q), denoting the
passage into a negative colour. Whence, knowing the sequence independently of the
signs, we have only to prefix to the first letter the sign -I- or — as the case may
be, and the sequence is then completely determined.
10. Passing to a + intercalation, observe that in omitting any even number of
P's or Qs, the omitted signs are always -I- — -I- — &c. or else — H -I- &c., viz. the
omitted signs begin with one sign and end with the opposite sign. Hence the signs
being in the first instance alternate, they will after any omission remain alternate :
and the letters being also alternate, the intercalation can contain only -|- P and — Q
561] cauchy's theorems of root-limitation, 25
or else — P and + Q. Hence in the case of a circuit the intercalation is either
(+ P - Q), say this is a positive circuit, or else (- P + Q), say this is a negative circuit.
There is of course the neutral circuit {PQ\ for which the intercalation vanishes.
11. Consider a circuit not containing within it any root; as a simple example let
the circuit lie wholly in one colour, or wholly in two adjacent colours, say sable and
gules: in the former case the sequence, and therefore also the intercalation, vanishes:
in the latter case the sequence is +Q — Q, and therefore the intercalation vanishes :
viz. in either case the intercalation is (PQ)o.
12. Consider next a circuit containing within it one right-handed root ; for instance
let the circuit lie wholly in the four regions adjacent to this root, cutting the two
curves each twice ; the sequence and therefore also the intercalation is + P — Q-\- P — Q;
viz. this is a positive circuit {+ P - Q)i, where the subscript number is the half-index,
or half of the number of P's or of Q's. Similarly if a circuit contains within it one
left-handed root, for instance if the circuit lies wholly in the four regions adjacent
to this root, cutting the two curves each twice, the sequence and therefore also the
intercalation is — P + Q — P + Q, viz. this is a negative circuit {— P + Q)i : and the
consideration of a few more particular cases leads easily to the general and fundamental
theorem :
13. A circuit is positive {+P—Q)s or negative (—P+Q)s according as it contains
vxithin it more right-handed or more left-handed roots ; and in either case the half-index
8 is equal to the excess of the number of one over that of the other set of roots. If
the circuit is neutral (PQ)o, then there are within it as many left-handed as right-
handed roots.
14. The proof depends on a composition of circuits, but for this some preliminary
considerations are necessary.
Imagine two unclosed trajectories forming a circuit, and write down in order the
intercalation of each. The whole number of letters must be even: viz. the numbers
for the two intercalations respectively must be both even or both odd. I say that if
the terminal letter of the first intercalation and the initial letter of the second inter-
calation are different, then also the initial letter of the first intercalation and the
terminal letter of the second intercalation will be different : if the same, then the
same. In fact, the intercalations may be each PQ or each QP, or one PQ and the
other QP: or each PQP, or each QPQ, or one PQP and the other QPQ. Supposing
the letters in question are different, then the intercalations may be termed similar;
but if the same, then the intercalations may be termed contrary.
15. In the first case, that is when the intercalations are similar, the two together
form the intercalation of the circuit ; the sum of their numbers of letters (that is,
twice the sum of their indices) will be evenly even, and the half of this, or the sum of
the indices, will be the index of the circuit ; each intercalation will be (+ P — Q) or
else each will be (-P + Q); and the circuit will be (-t- P - Q) or {- P -\- Q) accordingly.
C. IX. 4
26 ON THE GEOMETRICAL REPRESENTATION OF [561
In the second case, that is, when the intercalations are contrary, they countei-act
each other in forming the intercalation of the circuit : it is the difference of the
numbers of letters, or twice the difference of the indices, which is evenly even, and
the half of this, or the difference of the indices, which is the index of the circuit:
one intercalation is (+ P — Q), and the other is (— P + Q) : and the circuit will agree
with that which has the larger index.
In particular if the circuit consist of a single unclosed trajectory, taken forwards
and backwards ; then the trajectory taken one way is (+ P - Q), taken the other way
it is (— P + Q) ; the number of terms is of course equal, and the circuit is {PQ\.
16. Consider now two circuits ABC A and ACDA, having a common portion CA,
or, more accurately, the common portions AG and CA : write down in order the inter-
calations of
ABC, CA, AC, CDA:
the two mean terms destroy each other, and we can hence deduce the intercalation
of the entire circuit ABC DA,
Suppose first, that ABC and CDA are similar ; then if CA is similar to ABC
it is also similar to CDA, that is, AC is contrary to CDA : and so if CA is contrary
to ABC, then AC is similar to CDA.
To fix the ideas suppose CA similar to ABC, but AC contrary to CDA, then
ABCA is similar to CA ; but ACDA will be similar or contrary to AC, that is, contrary
or similar to CA, that is, to ABCA, according as index of AC> or < index of CDA.
Suppose Ind. AC < Ind. CDA, then ACDA is similar to ABCA.
Now Ind. ABCDA = Ind. ABC + Ind. CDA,
Ind. ABCA = Ind. ABC + Ind. AC,
Ind. ACDA = Ind. CDA - Ind. AG,
and thence
Ind. ABCDA = Ind. ABCA + Ind. ACDA,
the whole circuit being in this case similar to each of the component ones.
But if Ind. .4(7 > Ind. CDA, then ACDA is contrary to ABCA.
And Ind. ABCDA = Ind. ABC + Ind. CDA,
Ind. ABCA = Ind. ABC +Ind. GA,
Ind. ACDA = - Ind. CDA + Ind. AC,
and thence
Ind. ABCDA = Ind. ABCA - Ind. ACDA ;
and the investigation is like hereto if GA is contrary to ABC but AC similar to CDA.
17, Secondly, if ABC and CDA are contrary, then if CA is similar to ABC it is
contrary to CDA, that is, ^C is similar to CDA ; and so if CA is contrary to ABC
it is similar to CDA, that is, .40 is contrary to CDA.
561] cauchy's theorems of root-limitation. 27
Suppose CA similar to ABC, aud AO similar to CD A ; then ABC A is also
similar to ABC, and AGDA similar to CDA ; viz. ABC, GA and ABCA are similar
to each other, and contrary to AC, CDA, AC DA which are also similar to each other.
Also Ind. ABCDA =Ind. ABC ~ Ind. CDA,
Ind. ABCA = Ind. ABC + Ind. CA,
Ind ACDA = Ind. CDA + Ind. AG,
and thence
Ind. ABCDA = Ind. ABCA ~ Ind. ACDA,
and the investigation is like hereto if CA is contrary to ABC and AC contrary to CDA.
18. It thus appears that in every case
Ind. ABCDA = Ind. ABCA + Ind. ACDA,
or =Ind. ABCA -Ind. AGDA,
according as the component circuits are similar or contrary, and in the latter case
the entire circuit is similar to that which has the largest index.
Moreover, any circuit whatever can be broken up into two smaller circuits, aud
these again continually into snialler circuits until we arrive at the before-mentioned
elementary circuits, and the theorem as to the number of roots within a circuit is
true as regards these elementarj' circuits ; wherefore the theorem is true as regards
any circuit whatever.
19. In the case where a trajectory is a finite right line, y is a given linear
function of x, or the coordinates x, y can if we please be expressed as linear functions
of a parameter u, so that as the describing point passes along the line, u varies
between given limits, say from m=0 to u=\. The functions P, Q thus become given
rational and integral functions of a single variable u (or it may be x or y), and the
question of the P- and Q-sequence and intercalation relates merely to the order of
succession of the roots of the equations P = 0, Q = 0, where P and Q denote functions
of a single variable as above. To fix the ideas, let the trajectory be a line parallel
to the axis of x; and in this case taking x as the parameter, and supposing that
yo is the given value of y, P and Q are the functions of x obtained by writing y^
for y in the original expressions of these functions. Of course the theory will be precisely
the same for a line parallel to the axis of y : and by combining two lines parallel
to each axis we have the case of a rectangular circuit. We require, for each side of
the rectangle considered according to its proper currency, the intercalation PQ, QP, PQP
or QPQ as the case may be, and also the sign + or — of the initial letter of the
first intercalation ; for then writing down the intercalations in order, with the signs for
the several letters, -|- and — alternately (the first sign being -1- or — as the case may
be), we have or deduce the intercalation of the circuit, and thus obtain the value of
the difference of the numbers of the included right- and left-handed roots. We thus
see how the whole theory depends on the case where the trajectory is a right line.
4—2
28 ON THE GEOMETRICAL REPRESENTATION OF [561
Intercalation-theory for a right line. Articles Nos. 20 to 31.
20. Considering then the case where the trajectory is a line parallel to the axis
of X, P and Q will denote given rational functions of x ; the curves P = 0, Q = 0 being
of course each of them a set of right lines parallel to the axis of y : the regions
will be bands each of them included between two such lines; and colouring them as
explained in the general case, the colours will be as before, sable, gules, argent, azure,
each region having in the neighbourhood of the trajectory (what we are alone con-
cerned with) the same colour that it had in the original case where P and Q were
functions of {x, y). We may regard the trajectory as described according to the
currency x = — <x> toa; = +oo: we have in regard to the trajectory a P- and Q-sequenee
and intercalation, a + P- and Q-sequence, &c., as in the original case. The inter-
calation may be as before PQ, QP, PQP or QPQ, and in each of these cases it may
be positive, that is, (+ P — Q), or else negative, that is, (— P + Q).
21. The question of sign may in the present case be disposed of without difficulty.
For the initial point of the trajectory, we know the signs of P, Q, that is, the colour
of the region: suppose for example that we have P = — , Q = +, or that the region is
gules : then if the intercalation begin with P, this means that we either firet pass a
red line, or before doing so we pass an even number of blue lines : but in the last
case the colours are sable, gules, sable, gules, . . . always ending in gules ; and the passage
over the red line is gules to argent, viz. this is + P ; and so in general the initial
P or Q of the intercalation has the sign opposite to that of the P or Q belonging to
the commencement of the trajectory.
22. For the solution of the problem we connect with P, Q a set of functions
R, S, T, &c. : the intercalation is in fact given by means of the gain or loss of
changes of sign in these functions on substituting therein the initial and final values
of the variable x. It is convenient to consider the functions as arranged in a column
P
Q
R
8
«ay this is the column PQRS. . . , and to connect therewith a signaletic bicolumn : viz.
the left-hand column is here the series of signs of these functions for the initial value
of X, and the right-hand column is the series of signs for the terminal value of x:
the bicolumn thus consisting of as many rows each of two signs, as there are functions.
But such a bicolumn may be considered apart from any series of functions, as a set
of rows each of two signs taken at pleasure.
We say that the " gain " of a bicolumn is
= — (No. of changes of sign in left-hand column) + (No. in right-hand ditto),
the gain being of course positive or negative ; and a negative gain being regarded as
a loss. Also if a positive gain be converted into an equal negative gain or vice versd,
we may speak of the gain as reversed.
-/
»
561] cauchy's theorems of root-limitation. 29
23. A bicolumn may be divided in any manner into parts, taking always the last
row of any part as being also the first row of the next succeeding part. This being
so, the gain of the whole bicolumn is equal to the sum of the gains of its parts.
In a bicolumn of two rows, if we reverse either row (that is, write therein — for
+ and + for — ), we reverse the gain : and hence dividing a bicolumn into bicolumns
each of two rows, viz. first and second rows, second and third rows, and so on, it at
once appears that if we reverse alternate rows (viz. either the first, third, fifth, &c.,
rows, or the second, fourth, sixth, &c., rows) we reverse the gain. It of course follows
that reversing all the rows, we leave the gain unaltered.
24. If to any bicolumn we prefix at the top thereof the second row reversed, we
either leave the gain unaltered or we alter it by + 1. In fact, as regards either
column, if this originally begin with a change, the process introduces no change therein ;
but if it begins with a continuation, then the process introduces a change. Hence if
the columns begin each with a change or each with a continuation, the gain is
unaltered : but if one begins with a change, and the other with a continuation, then
the gain is altered by + 1 ; viz. the left-hand column beginning with a continuation,
the gain is altered by — 1 : and the right-hand column beginning with a continuation,
the gain is altered by -f 1.
The column PQRST. . . is tfiken to satisfy the following conditions : two consecutive
terms never vanish together (that is, for the same value of the variable): if for a
given value of the variable, any term vanishes, the preceding and succeeding terms
have then opposite signs ; the last term, say V, is of constant sign.
25. Considering P, Q as given functions without a common measure, such a column
of functions is obtained by the well-known process of seeking for the greatest common
measure, reversing at each step the sign of the remainder: viz. we thus derive a set
of functions R, S, T, ... where
P = \Q-R,
Q=^iR-S,
R = vS-T,
8=pT-U,
the degrees of the successive functions R, S, T, ... , being successively less and less,
80 that the last of them, say V, is an absolute constant : or we may stop the process
as soon as we arrive at a function V, the sign of which remains unaltered for all
values between the initial and final values of the variable. It may be observed that
the process may be regarded as applicable in the case where the degree of Q exceeds
that of P : viz. we then have X = 0, R = -P, and the column begins (P, Q, - P, 8,...),
the subsequent terms being, except as to sign, the same as if P, Q had been inter-
changed.
Reversing the sign of P or Q, we reverse in the bicolumn a set of alternate
rows, and thus reverse the gain : and reversing both signs we reverse all the rows,
30
ON THE GEOMETRICAL REPRESENTATION OF
[561
and leave the gain unaltered — of course the intercalation (considered irrespectively of
sign) is in each case unaltered. It is convenient to take the signs in such manner
that for the initial value of x, the signs of P, Q shall be each positive : or, what
is the same thing, taking P, Q with their proper signs, we may in the bicolumn, by
reversing if necessary each or either set of alternate rows, make the left-hand column
to begin with the signs + +.
26. The complete rule now is — for a given trajectory form the bicolumn for
PQRS..., and if necessary, by reversing each or either set of alternate rows, make the
left-hand column to begin with + + : then if there is a gain the intercalation begins
with P, if a loss with Q, the gain or loss showing the number of P's. To find the
number of 0*8 prefix at the top of the bicolumn the second row reversed — then the
gain or loss (equal to or differing by unity from the original value) shows the number
of Q's. It may happen that for P the gain is =0; then for Q the gain is 0 or + 1,
and the intercalation vanishes or is Q.
27. I give some simple examples.
0 2 4
P=ar-1
-
+
+
Q=a:-S
-
-
+
i?= -1
_
—
—
0
2
4
P=x-S
—
—
+
Q=x-1
-
+
+
i?= +1
-f-
-1-
+
0
Q
0
Q
In the left-hand example taking the intervals to be successively 0 — 2, 0 — 4, 2 — 4,
the bicolumns modified as above are
0-
-2
0-
-4
2
-4
-
-
-
+
-
+
+
-
+
—
-1-
+
+
-t-
4-
-
+
-
+
-1-
+
+
-
—
viz.
Interval 0-2; for P gain = 1, P first; for Q gain =0; Intercalation is P;
0-4
= 1,
2-4 „ „ =0 „ loss
And similarly in the right-hand example we have
0-2 0-4 2-4
- -I-
= 1;
= 1;
PQ;
Q.
+ +
+ -
561]
CAUCHYS THEOREMS OF ROOT-LIMITATION,
31
0-4
., = - 1, Q first, „ „ =
-1; „
.. 2-4
„ = -1- 1, P first, „ „ =
0; „
28.
Or to take a slightly more complicated example,
1 3 5 + e 7
9
P =x'- 8a; + 12
+ - - +
+
Q=a^-12a: + 32
+ + - -
+
R = - x+ 5
+ + + -
-
S = + I
+ + + +
+
P Q P
Q
QP;
P.
0 12 3 4
And hence for the several intervals,
1_3 i_5 1-7 1-9 3_5
3-7 3-9
7 5-9 7-9
- - ' - + [ - +
- -
- +
- +
_ _ _ _
- + 1 - +
+ - + -
+ +
1
+ +
+ +
+ -
+ -
+ -
+ - : + +
+ + + - ' + -
+ +
+ -
+ -
+ +
+ +
+■ - + -
+ + + T
+ + + +
+ -
+ +
+ -
+ +
- ±
+ +
- +
+ +
- +
+ +
+ +
± + ; - -
gP PQ
PQP
PQPQ
<2
QP
Qm
p
PQ i <2
For instance: —
Interval 1 — 9 for P gain = 2, P first, for Q gain = 2 : Intercalation is PQPQ.
It may be added that P being + for x = 1, the + intercalation is + PQPQ.
29. As an example of circuits take the following : curves are P = 0, Q = 0, where
P = x' + f-'i,
Q=y -X -1;
viz. P = 0 (see figure) is a circle, centre the origin, radius = 2 : the inside hereof
(P = — ) being coloured red : and Q = 0 is a right line cutting the axes of x, y at
the points (—1, 0) and (0, 1) respectively, or say running N.E. and S.W., the lower
region (Q = — ) being coloured blue: the square is an arbitrary circuit («=±3, y=t3)
surrounding the circle, and the regions within the square are coloured by what precedes
sable, gules, azure, argent, as shown in the figure : the line and circle intersect in
two points M, N. Going right-handedly round these respectively, for M the order is
sable, gules, argent, azure, viz. JIf is a right-handed root ; while for N the order is
32
ON THE GEOMETRICAL REPRESENTATION OF
[561
sable, azure, argent, gules, viz. iV is a left-handed root: the two points are accordingly
in the figure denoted hy + M and — N respectively.
30. Now considering successively the four smaller squares of the figure, say these
are the squares N.E., S.E., S.W., N.W. : and going right-handedly round each of these :
In the square N.E., the sequence and therefore also the intercalation is+F — Q + P— Q,
viz. this is an intercalation (+ P — Q), showing an excess 1 of right-handed roots, and
of course consisting with the single right-handed root M.
In the square S.E., the sequence is —P + P, viz. this is an intercalation (PQ)o,
showing an equality of right- and left-handed roots, and consisting with no root.
In the square S.W., the sequence and therefore also the intercalation is —P + Q — P + Q:
viz. this is an intercalation {-P + Q)i, showing an excess 1 of left-handed roots, and
consisting with the single left-handed root N.
And in the square N.W., the sequence is —Q + P — P+Q, viz. this is an inter-
calation (PQ)o, showing an equality of right- and left-handed roots, and consisting with
no root.
Again take the whole large square : the sequence is —Q+Q: viz. the intercalation
is {PQ)o, showing an equality of right- and left-handed roots, and consisting with there
being one of each.
So taking the squares N.E. and N.W. conjointly, the sequence and therefore also
the intercalation is -Q + P-Q + P, viz. this is an intercalation (+ P - Q\, as for the
single square N.E.
561]
CAUCH Y S THEOREMS OF ROOT-LIMITATION.
S3
31. As regards the analytical determination it will be sufficient to consider a
single square, say N.E. : going round right-handedly, the trajectories will be
(1) a^ = 0, y = 0 to 3 ;
(2) y = 3, « = 0 to 3 ;
(3) x = S, 2/ = 3 tu 0 ; or if y' = - ?/, then y' = - 3 to 0 ;
(4) y = 0, a; = 3 to 0 ; or if x' = -x, then *•' = - 3 to 0.
And we thus have
(1) P= 3/^-4
«= y-1 1 -
R^ -1
0 3
(2) P = ar" + 5
Q=-x +2
R= -1
(3) P =
i2 =
y'^ + o
-y +4
-1
(4) P =
Q =
x"' - 4
a:' -1
+ 1
0
3
, that is.
0
3
—
+
—
+
—
+
+
+
+
—
for P gain = - 1, Q begins, IP ;
., Q „ =-1, „ IQ.
Intercalation is QP, or since at
origin P = — , P = — , or region
is sable, it is —Q-\-P.
0 3
+
+
, tkat is,
—
+
+
—
+
+
+
—
-3 0
-3 0
+
+
, that is.
—
-
+
+
- 1 -
+
+
-3 0
-3 0
+
—
, that is,
-
-
+
_
4- 1 +
+
+
+
+
, for P gain = 0,
„ Q ,. =-1.
Intercalation is
-Q-
, for P gain = 0,
,. Q „ =0.
Intercalation vanishes.
, for P gain = + 1, P first,
,. Q „ = 0.
Intercalation is
+ P.
Hence for the four sides, combining the intercalations, we have —Q + P — Q+P,
and since there are no terms to be omitted, this is the intercalation of the N.E.
square : which is right.
C. IX.
34 ON THE GEOMETRICAL REPRESENTATION OF [561
The Rhizic Theory. Articles Nos. 32 to 38.
32. Consider now F{z) = (*)(z, 1)" a rational and integral function of z, of the
order n with in general imaginary (complex) coefficients, or, what is the same thing,
let F(z)=/(z) + i<l>{z), where the functions /, <f> are real*. Writing herein z = x + iy,
let P, Q he the real part and the coefficient of the imaginary part in the function
F(a; + iy): or, what is the same thing, assume
P + iQ =f{oo + iy) + i^ {x + iy),
then it is clear that to any root a + i/8 (real or imaginary) of the equation F{^z) = 0,
there corresponds a real intersection, or root, a; = a, y = /3, of the curves P = 0, Q = 0.
The functions, P, Q, as thus serving for the determination of the roots of the equation
F{z) = 0, are termed "rhizic functions," and similarly the curves P = 0, Q = 0 are "rhizic
curvea" The assumed equation shows at once that we have
or, what is the same thing,
And we hence see that
dy{,P + iQ) = id^{P + iQ,),
dyP^-d^Q. d^P^dyQ.
^^ , = (d^py + (dyPy, or (4Q)= + (dyQY,
is positive : viz. that the roots P = 0, Q = 0 are all of them right-handed (the essential
thing is that they are same-handed; for by reversing the signs of P and Q they
might be made left-handed : but it is convenient to take them as right-handed) :
hence the theorem — which in the general case, where P and Q are arbiti-ary functions,
serves to determine the difference of the numbers of the right- and left-handed roots —
in the particular case, where P and Q ai-e rhizic functions, serves to determine the
number of intersections of the curves P = 0, Q = 0 : or, what is the same thing, the
number of the (real or imaginary) roots of the equation F(z) = 0: viz. we thus deter-
mine the number of roots within a given circuit.
33. The rhizic curves P = 0, Q = 0 have various properties. 1". Each curve has
n real points at infinity, or, what is the same thing, n real asymptotes: and the P
And Q points at infinity succeed each other, a P-point and then a Q-point, and so
on, alternately.
In fact, from the equation
P + iQ = (a' + ia") (x + iy)" +... + {k' + k"i),
writing herein a' -(- la" = a (cos a -H i sin a), and x-\-iy = p(cosd + isind), we have
P + tQ = o/a" [cos (n^ -f- a) -I- i sin (w^ -I- o)] + . . . + ^' + Fi.
* It is assumed that the equation F(z) = (i has uo equal roots: this being so, the curves P=0, Q = 0,
will have no point of multiple intersection ; which accords with the assumption made in the general case of
two arbitrary curves.
561] cauchy's theorems of root-limitation. 85
It thus appears that for the curve P = 0, the points at infinity are given by the
equation cos(n0 + o) = O; while for the curve Q = 0, they are given by the equation
sin (nO + a) = 0 : which proves the theorem.
Representing infinity as a closed curve or circuit, each point at infinity must be
represented by two opposite points on the circuit; so that writing down P for each
P-point and Q for each Q-point, we have 2n P's and 2w Q's succeeding each other,
a P-point and then a Q-point, and so on, alternately.
It may be assumed that taking the circuit right-handedly, the P's are -f- and
the Q's -, (this depends only on the colouring, but it corresponds with the foregoing
assumption that the roots P = 0, Q=0 are right-handed): the theorem just obtained
then really is that for the circuit infinity, the intercalation is i+P-Q),,: and we have
herein a proof of the theorem that a numerical equation of the order n with real
or imaginary coefficients has precisely n real or imaginary roots. But the force of this
will more distinctly appear presently.
34. 2». Neither of the curves P = 0, Q = 0 can include as part of itself a closed
curve or circuit.
The foregoing relations between the differential coefficients give
d^''P.+ dy'P = 0, d^'Q + dy'Q = 0,
which equations for the two curves respectively lead to the theorem in question. For
as regards the curve P = 0, take z a coordinate perpendicular to the plane of xy,
and consider the surface z=P: if the curve P = 0 included as part of itself a closed
curve, then corresponding to some point (x, y) within the curve we should have z a
proper maximum or minimum, viz. there would be a summit or an imit; at the point
in question we should have rfj,P = 0, dxQ = Q; and also (as the condition of a summit
or imit) dx'P .dy'P -(dxdyPy = +, implying that d^^P and rf/P have at this point
the same sign : but this is inconsistent with the foregoing relation dJ'P + dy'P = 0.
35. 3°. The curves P = 0, Q = 0 have not in general any double (or higher mult-
iple) points. A point which is a double (or higher multiple) point on one of these
curves is not of necessity a point on the other curve : but being a point on the other
curve it is on that curve a point of the same multiplicity. For changing if necessary
the coordinates, the point in question may be taken to be at the origin: forming the
equation
P + iQ = (a' + a"i) {x + iyf -I- . . . -i- (A/ -H Ft) (a; -|- iy^ -f- {I' -f- l"i) {x + iy) -f- m' -f- m'i = 0,
the point a; = 0, y = 0 will not be a double point on the curve P = 0, unless we have
m =0, r = 0, I" — 0 ; these conditions being satisfied, it will not be a point on the
curve Q = 0 unless also m" = 0 ; but this being so, it will be a double point on the
curve Q = 0 : and the like for points of higher multiplicity. But a point which is a
multiple point on each curve, represents four or more coincident intersections of the
curves P = 0, Q = 0, that is, four or more equal roots of the equation F{z) = Q; so
that assuming that the equation has no equal roots, the case does not arise : and we
in fact exclude it from consideration.
5—2
36 ON THE GEOMETRICAL REPRESENTATION OF [561
To fix the ideas assume that the curves P = 0, Q=0 are each of them without
double points. As already seen, neither of them includes as part of itself a closed
curve. Hence in the figure the curve P = 0 must consist of n branches each drawn
from a point P in the circuit (viz. the circuit infinity) to another point P in the
circuit; and in such manner that no two branches intersect each other: this implies
that the two points P of the same branch must include between them an even
number (which may of course be =0) of points P. And the like as regards the curve
<2=o.
36. 4°. No branch of the P-curve can meet a branch of the Q-curve more than
once. In fact, drawing the two branches to meet twice, the colouring would at once
show that of the two intersections or roots, one must be right, the other left-handed :
whence, the roots being all right-handed, the branches do not meet twice. And in exactly
the same way it appears that no P-branch can meet two Q-branches, or any Q-branch
meet two P-branches. And under these restrictions it requires only a consideration of
a few successive cases to show that the n P-branches, and the n Q-branches can only
be drawn on the condition that each P-branch shall intersect once and only once a
single Q-branch ; which of course implies that eaeh Q-branch intersects once and once
only a single P-branch : and further, that there shall be precisely n intersections : viz.
the n P-branches and the n Q-branches must satisfy the conditions just stated. And
the theorem of the »i roots is thus obtained as a consequence of the impossibility
{except under the same conditions) of drawing the n P-branches and the n Q-branches,
so as to give rise to right-handed roots only. But the case of double or higher
multiple points would need to be specially considered
37. It is interesting for a given value of n to consider <j>{n), the number of
different ways in which the P-branches and the Q-branches can be drawn. We have
2n points P and 2n points Q, in all 47t points : starting from any point P, these may
be numbered in order 1, 2, 3, ...,4«, the points P bearing odd numbers and the points
Q even numbers. We may consider the P-branch which joins I with some P-point
y3, and (intersecting this) the Q-branch which joins some two Q-points a and y : the
numbers 10)87 ^® t^^'i ^^ order of increasing magnitude : and excluding these four
points there remain the points corresponding to numbers between 1 and a, between
a and y9, between /3 and 7, and between 7 and 1. Now since the P-branch 1^ meets
the Q-branch 07, no branch from a point between 1 and a can meet either of these
curves; hence these points form a system by themselves, capable of being connected
together by P-branches and Q-branches: the number of them must therefore be a
multiple of 4: and the like as to the points between a and /S, between /3 and 7, and
between 7 and 1. Taking the number of the points in the four systems to be
ix, 4y, 4z, and 4w respectively, we have x+i/ + z + tv = 7i — l, and the first-mentioned
four points bear the numbers
1.
a = 4a; -f 2,
7 = 4a; + 4y + 4z -h 4.
561] cauchy's theorems of root-limitation. 37
For the four systems the number of ways of drawing the P- and Q-branches are
<f>w, <f>i/, <f>z, <f>w respectively : that is, x, y, z, w being any partition whatever of n—\
(order attended to), and ^(0) being = 1, we have
<l>{n) = t<i>{x)4,{y)4>{z)<l,{w\
which is the condition for the determination of <^?i.
Taking then 6 for the value of the generating function
l+<(^(l) + «''<^(2)...+<»^0i)+ ...,
it hei-eby appears that we have
or >vriting this for a moment d=u + tO*, and expanding by Lagrange's theorem, but
putting finally «= 1, we have the value of d, that is of the generating function,
that is,
and generally
= 1 -^ [4P I 4- m 1^2 + [12? 1^3 + M- j-£
= 1 + i + 4f-= + 22t» + UOf + ... ,
^(1) = 1, <^,(2) = 4, <^(3)=22, <^(4) = 140,...
, , , r4nl"~' 4w . 4» — 1 . . . 3« + 2
. n
+ ...
The results are easily verified for the successive particular cases ; thus n = l, the
points are 1, 2, 3, 4, and the P- and Q-branches respectively are 13, 24: <f>(l)=l.
Again « = 2, the points are 1, 2, 3, 4, o, 6, 7, 8 : we may join 13, 24 or 13, 28 or
17, 28 or 17, 68, leaving in each case four contiguous numbers which may be joined
in a single manner : that is, <p (2) = 4. Or, what is the same thing, the partitions of
1 are 0001, 0010, 0100, 1000, whence ^ (2) = 4 {(^(0)j»<^(l) = 4. Again n = 3, the
partitions of 2 are 0002, &c. (4 of this form) and 1100 (6 of this form): that is,
(f> (3) = 4 {^(0)j»^(2) + 6 {4> (0)]- {<t> (1)1=, = 4 . 4 + 6 . 1 = 22, and so on.
38. Starting from the 4n points P and Q, and joining them in any manner
subject to the foregoing conditions, we have a diagram representing two rhizic curves ;
and colouring the regions we verify that the n roots are all of them right-handed.
We have for instance the annexed figure (n = 3).
Having drawn such a figure we may, by a continuous variation of the several
lines, in a vaiiety of ways introduce a double point in the P-curve, or in the Q-curve :
and by a continued repetition of the process introduce double points in each or either
curve: thus for instance we may from the last figure derive a new figure in which
the P-curve has a node at N. It will be observed that here it is no longer the case
that each P-branch intersects one and only one Q-branch : the P-branch 1 — 9 does
not meet any Q-branch, but the P-branch 7—11 meets two Q-branches. But looking
at the figure in a different manner, and considering the P-branches through If as
38
ON THE GEOMETRICAL REPRESENTATION OF
[561
being either 11-JV-l and 7-iV-9, or l-iV-7 and 9-iV-ll, then in either
case each P-branch intersects one and only one Q-branch : and in this way, in a
diagram in which the two curves have each or either of them double points, but
neither curve passes through a double point of the other curve, the theorem may be
regarded as remaining true — we in fact consider the diagram as the limit of a diagram
wherein the curves have no double points. It will be lecollected that, the equation
F(z) being without equal roots, we cannot have either curve passing through a multiple
point of the other curve. And we thus see that the various figures drawn as above
without double points are, so to speak, the types of all the different forms of a system
of rhizic curves P = 0, Q = 0.
561] cauchy's theorems of root-limitation. 39
In connexion with the present paper I give the following list of Memoirs : —
Cauchy. Calcul des Indices des fonctions. Jour, de VEcole Polyt. t. xv. (1837),
pp. 176 — 229. First part seems to have been written in 183.3 : second part is
dated 20th June, 1837. Kefers to a memoir presented to the Academy of Turin
the 17th Nov. 1831, wherein the principles of the "Calcul des Indices des fonctions"
are deduced from the theory of definite integi-als : I have not seen this.
Sturm and Iilouvllle. Demonstration d'un theoreme de M. Cauchy relative aux racines
imaginaires des equations. Liowv. t. i. (1836), pp. 278 — 289.
Sturm. Autres demonstrations du meme thdoreme. Liouv. t. i. (1836), pp. 290 — 308.
These two papers contain proofs of the particular theorem relating to the roots
of an equation F(z) = 0, but do not refer to the general theorem relating to the
intersection of the two curves P = 0, Q = 0: the special theorem of the existence
of the n roots of the equation F{z) = 0 is considered.
Sylvester. A theory of the syzygetic relations of two rational integral functions, com-
prising an application to the theory of Sturm's functions and that of the greatest
algebraical common measure. Phil. Trans, t. CXLIII. (18.53), pp. 407 — 548.
De Morgan. A proof of the existence of a root in every algebraic equation, with an
examination and extension of Cauchy's theorem on imaginary roots, and remarks
on the proofs of the existence of roots given by Argand and Mourey. Camb.
Phil. Trans, t. X. (1858), pp. 261—270.
Contains the important remark that the two curves P = 0, Q=0 are such
that two branches, one of each curve, cannot inclose a space ; also that the two
curves always [i.e. at a simple intersection] intersect orthogonally.
Airy, G. B. Suggestion of a proof of the theorem that every algebraic equation has
a root. Camb. Phil. Trans, t. x. (1859), pp. 283—289.
Cayley, A. Sketch of a proof of the theorem that every algebraic equation has a
root. Phil. Mag. t. xviil. (1859), [248], pp. 436—439.
"Walton, W. On a theorem in maxima and minima. Quart. Math. Jour. t. x. (1870),
pp. 253—262. Cayley, A. Addition thereto, [562], pp. 262, 263. (Relates to the
curves P = 0, Q = 0.)
Walton, W. Note on rhizic curves. Qiuirt. Math. Jour. t. XI. (1871), pp. 91—98.
First use of the term " rhizic curves : " relates chiefly to the configuration of each
curve at a multiple point, and of the two at a common multiple point.
Walton, W. On the spoke-asymptotes of rhizic curves. Quart. Math. Jour, t, xi.
(1871). pp. 200—202.
Walton, W. On a property of the curvature of rhizic curves at multiple points.
QuaH. Math. Jour. t. XI. (1871), pp. 274—281.
BJorling. Sur la separation des racines d'^quations alg^briques. Upsala, Nova Acta
Sac. Sci. (1870), pp. 1 — 35. (Contains delineations of some rhizic curves.)
40 [562
562.
[ADDITION TO MR WALTON'S PAPER "ON A THEOREM IN
MAXIMA AND MINIMA."]
[From the Quarterly Journal of Pure and Applied Mathematics, vol. x. (1870),
pp. 262, 263.]
In what follows I write x, y, z in place of Mr Walton's u, v, w : (so that if
i = V(— 1). as usual, we have
f{x + iy) = P-^iQ):
and I attend exclusively to the case where the second differential coefficients of P, Q
do not vanish.
There are not on the surface z = P any proper maxima or minima ; but only level
points, such as at the top of a pass : say there are not any summits or imits, but
only cruxes ; and moreover at any crux, the two crucial (or level) dii'ections intersect
at right angles. Every node of the curve Q = 0 is subjacent to a crux of the
surface z = P: and moreover the two directions of the curve Q = 0 at the node are
at right angles to each other; hence, considering the intersection of the surface z = P
by the cylinder Q = 0, the path Q = 0 on the surface has a node at the crux ; or say
there are at the crux two directions of the path ; these cross at right angles, and are
consequently separated the one from the other by the crucial directions ; that is to
say, there is one path ascending, and another path descending, each way from the
crux. And the complete statement is ; that the elevation of the path is then only a
maximum or minimum when the path passes through a crux ; and that at any crux
there are two paths, one ascending, the other descending, each way from the crux.
The analytical demonstration is exceeding simple ; we have
(dP^^dQ\./dP_^.dQ\
\dy dyj \dx dxj'
562] ON A THEOREM IN MAXIMA AND MINIMA. 41
that is,
dP^_dQ dQ^dP
dy dx' dy dx'
and passing thence to the second diflferential coefficients, we may write
dx dy ' dy~ dx
d'P ^_dHi ^d^^
dxdy da?~ dy^~ "'
dxdy da? ~ dy^
so that we have
hP = Lhx + Mhy, SQ = - MZx + Uy,
S'P = (b, a, - b^Sx, %)^ S'Q = (- a, b, a^Sx, Syf.
Hence, for the maximum or minimum elevation of the path, we have 0 = 8P, where
BQ = 0; that is, 0= -^ — Sx, and therefore L'' + M'- = 0; that is, Z=0, M = 0; and
at any such point Bz = 0, that is, there is a crux of the surface z = P; and BQ = 0,
that is, there is a node of the curve Q = 0. Moreover the crucial directions for the
surface z = P are given by the equation {b, a, - b'^^Bx, Byf = 0, or these are at right
angles to each other ; and the nodal directions for the curve Q = 0 are given by
(— a, b, a^Sx, Byy = 0 ; or these are likewise at right angles to each other.
C. IX.
42
[563
563.
NOTE ON THE TRANSFORMATION OF TWO SIMULTANEOUS
EQUATIONS.
{From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871), pp. 266, 267.]
Writing in Mr Walton's equations (1) and (2)
a 6 c a /3 7
d' d' d' S ' 8 ' S
instead of a, b, c, a, yS, 7 respectively ; and putting for shortness
4 = 67 - c/3, F = aS- da,
B = ca-ay, 0=bS- d^,
(7 = a/3 - 6a, H = cS-dy, •
the equations become
a (b - c) b {c - a) c(a - b) _
F ^ G ^ H ~ '
«(<g-7), ^(7-a) , y(cL-fi)_
F ^ 0 ^ H ~^'
Multiplying by FGH and effecting some obvious transformations, the equations become
aAF+ bBG+cCH = 0)
aAF+^BG + yCH = Ol ^^^^'
whence also
AF''+ BG'-{- Cfr» = 0 (19).
Now regarding (a, j8, 7, 0) as the coordinates of a point in space, the equations
{18) and (19) represent each of them a cone having for vertex the point a : /S : 7 : S
= a : 6 : c : d, viz. (18) is a quadric cone, (19) a cubic cone ; they intersect therefore
in six lines; and it may be shown that these are
the line a : yS : 7 = ti : 6 : c (twice) 2
0 : y : S = b : c : d 1
„ y : OL : B = c : a : d 1
„ a:/3:8=a:6:d 1
» 0 — y: y-a: a — ^i 8 = b — c:c — a: a — b: d 1
agreeing with Mr Walton's result
564] • 43
564.
ON A THEOREM IN ELIMINATION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873), pp. 5, 6]
I FIND among my papers the following example of a theorem in elimination com-
municated to me by Prof Sylvester. Writing
^ =aaf + Sbafy + Scxi/' + dy',
(^ = bid' + 2cxy + dy^,
4>3= ex +dy ,
<j>3= d ;
f =ba^+ Scafy + Sdxy^ + ey^,
/i = ca^ + 2dxy + ey\
/j = dx +ey,
/a = e ,
then we have
A„ . R (/, </,) = A/. R {cj>„ f,y R {<!>„ f,y,
viz. R(f, <j>) is the resultant of the functions (/, (f>), and similarly R{^^, f), R(<j)^, f.).
Moreover, A/ is the discriminant of /; and AaJ2 (/, (f)) is the discriminant of R (f, (p)
in regard to a. The equation thus is
Aa [(ae -4,bd + 3c»)» - 27 (ace -ad^-b^e-d' + 2bcdY\
= (ft'e' + 46d' + 4c'e - Sc^* - Qbcdef (d» - 2cde + 6e')» ;
or, what is the same thing, reversing the order of the letters (a, h, c, d, e), it is
A« [{ae - 4M + Sc^ - 27 (ace - ad'' - b^e - c= + 2bcd)]
= {a'd' + 4,ad> + 4:b'd - Sb^c- - Qcd)cdf (b" - 2abc + a^d)-,
6—2
■44 ON A THEOREM IN ELIMINATION. [564
viz. arranging in powers of e, the function is
+ 3e= . - a" (46d - 3c') - 9 (ac - 6«)»
+ 3e . a (46d - Sc')' + 18 (ac - 6«) (ad» - 2bcd + c»)
+ 1 .- (4-M -Sc'Y- 27 {ad'' -2bcd + c'y,
which last coefficient is
= - d» (27a-^d'' + 5400^ + 646'd - 36&'c» - 108a6cd),
and the discriminant of this cubic function of e is
= (a'd^ + 4ac5 + 46'd - Sb'd' - Qaicdy (6" - 2a6c + a'd)=.
The occurrence of the factor
a»d» + 4ac' + 4:b'd - Sb^c? - 6abcd
is accounted for as the resultant in regard to e of the invariants /, J; we, in fact,
have
(ac - ¥)I-aJ=-{ac - bf) (- 46d + 3c») -a (- ad^-d'^ 2bcd)
= a^d'' + 4ac' + 4<¥d - 36V - 6a6cd,
and the identity itself may be proved without any particular difficulty.
565]
45
565.
NOTE ON THE CARTESIAN.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. Xli. (1873),
pp. 16—19.]
The following are doubtless known theorems, but the form of statement, and the
demonstration of one of them, may be interesting.
A point P on a Cartesian has three "opposite" points on the curve, viz. if the
axial foci are A, B, C, then the opposite points are Pa, Pb, Pc where
Pa is intersection of line PA with circle PBG,
Pb „ „ PB „ PGA,
Pc „ „ PC „ PAB.
And, moreover, supposing in the three circles respectively, the diameters at right angles
to PA, PB, PC are aa', y9/3', 77' respectively, then the points a, «', /8, /S', 7, 7' lie by
threes in two lines passing through P, viz. one of these, say Pa^y, is the tangent,
and the other Pa'^'y the normal, at P; and then the tangents and normals at the
opposite points are Pad and Pa"^', Pb^ and Pt/S', Pc7, and Pc7' respectively.
There exists a second Cartesian with the same axial foci A, B, G, and passing
through the points P, Pa, Pb, Pe (which are obviously opposite points in regard
thereto) ; the tangent at P is Pa'^'y and the normal is Pa^y ; and the tangent and
the normal at the other points are P^a' and PaO, Pj/S' and Pj/S, P<;7' and P^y respec-
tively: viz. the two curves cut at right angles at each of the four points.
Starting with the foci A, B, G and the point P, the points Pa, Pb, Pc are con-
structed as above, without the employment of the Cartesian ; there are through P
with the foci A, B, G two and only two Cartesians ; and if it is shown that these
pass through one of the opposite points, say Pb, they must, it is clear, pass through
46
NOTE ON THE CARTESIAN.
[565
the other two points P^, Pe- I propose to find the two Cartesians in question. To
fix the ideas, let the points C, B, A be situate in order as shown in the figure, their
distances from a fixed point 0 being a, b, c, so that writing a, p, y = b — c, c — a, a — b
respectively, we have a + /3 + 7 = 0, and a, y will represent the positive distances CB
and BA respectively, and — y8 the positive distance AC. Suppose, moreover, that the
distances PA, PB, PC regarded as positive are R, 8, T respectively ; and that the
distances P^A, P^B, PbC regarded as positive are R', S', T' respectively.
Suppose that for a current point Q the distances QA, QB, QG regarded as
indifferently positive, or negative, are r, s, t respectively; then the equation of a
bicirculai' quartic having the points A, B, G for axial foci is
Ir + ms + nt = 0,
where I, m, n are constants ; and this will be a Cartesian if only
a P 7
We have the same curve whatever be the signs of I, m, n, and hence making the
curve pass through P, we may, without loss of generality, write
lR + mS + nT==0,
R, S, T denoting the positive distances PA, PB, PC as above. We have thus for
the ratios I : m : n, two equations, one simple, the other quadric ; and there are thus
two systems of values, that is, two Cartesians \vith the foci A, B, C, and passing
through P.
I proceed to show that for one of these we have -IR' + mS' + nT' = 0, and for
the other IR' + mfif — nT' = 0, or, what is the same thing, that the values of Z : to : 71 are
I : VI : n = - (ST' + S'T) : TR' + T'R : RS' - R'S,
and
I : m m^ (ST' - S'T) : -(TR' + T'R) : RS' + R'S;
viz. that the equations of the two Cartesians are
r ,
s ,
t
R.
s,
T
-R\
S'.
r
= 0, and
r , 8 , t
R, S, T
R', S', -r
= 0,
565] NOTE ON THE CARTESIAN. 47
respectively ; this being so each of the Cartesians will, it is clear, pass through the
point Pb, and therefore also through P^ and Pg.
The geometrical relations of the figure give
oB' + ^^' + yT' = - a/Sy,
aR' 4 ^S'^ + yT'" = - a/37,
RT' + R'T = - ^ (S + S'),
yoL = SS',
yTT = aRR,
to which might be joined
R'S + rf{S + 8') + RS' = SS' {S + S'),
T''S+ci>{S+S') + T'S' = SS'iS + S'},
SRT' = S'RT,
SP'R = S'PR,
but these are not required for the present purpose.
Ab regards the first Cartesian, we have to veiify that
(Sr + S'ly (TR -f TRY (RS' - Rsy
a /3 7
The left-hand side is
= 0.
ST' + gf'T' + 2yaTT' 0^ (^ + S'' + 2ya) S'R' + S''R' - 2yedtR
a + ;S + 7
viz. this is
7 / \ a 7
which ia
= ^f' (^ + iS + — ") + S'^ (~+^ + -) + 2a;87 + 2 (7^2" - a-R^'),
= S' (~^^] + S'' (^^1 + 2a/97 + 2 (yTT' - aRR),
and since the first and second terms are together = — 2 — S'S'-, that is, = — 2a/87,
yQ[
the whole is as it should be = 0.
In precisely the same manner we have
(Sr - S'T)^ (TR + T'Ry {RS' + Rsy
which is the condition for the second Cartesian: and the theorem in question is thus
proved
48
[566
5Qe.
ON THE TRANSFORMATION OF THE EQUATION OF A SURFACE
TO A SET OF CHIEF AXES.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiL (1873),
pp. 34—38.]
We have at any point P of a surface a set of chief axes (PX, PY, PZ), viz.
these are, say the axis of Z in the direction of the normal, and those of X, Y in
the directions of the tangents to the two curves of curvature respectively. It may
be required to transform the equation of the surface to the axes in question; to
show how to effect this, take {x, y, z) for the original (rectangular) coordinates of the
point P, x + hx, y + Sy, z + Sz for the like coordinates of any other point on the
surface, so that {Bx, By, Sz) are the coordinates of the point referred to the origin P;
the equation of the surface, writing down only the terms of the first and second
orders in the coordinates Sx, By, Bz, is
ABx + BBy + CBz + ^ (a, b. c, f, g, h) (Bx, By, hzf + &c. = 0,
where {A, B, C) are the first derived functions and (a, b, c, f, g, h) the second derived
functions of U for the values {x, y, z) which belong to the given point P, if U =0
is the equation of the surface in terms of the original coordinates (a;, y, z); we have
X, Y, Z linear functions of (Bx, By, Bz) ; say
X
Y
Z
that is, X = fliSa; + jSiSy + 7i8^, &c. and Bx = tt.iX -\-ai^ ■'tolZ, &c. where the coefficients
satisfy the ordinary relations in the case of transformation between two sets of rect-
angular axes; and the transformed equation is therefore
4 (a,Z + 0,7 + a^ + £ (AZ + /8, F + /SZ) + C (7,X + 7, F+ 7.^
+ (a, b, c,/, g, A)(a.Z + a»F+a^, A^-f- ^F-l-yS-?, 7,Z-|-7,F+7Z)» = 0,
8a;
8y
Oj
/3,
1%
a
)3
y
566] ON THE TEAN8F0RMATI0N OF AN EQUATION.
or, as this may be written,
X (Aa^ + 5/3, + Cy,) + YiAa, + B^, + Gy,) + Z{Aa + B^ + Gy)
+ iX= (a,. ..)(«!. A. 7i)'
+ iF= (a,...) (a,, /S,. y^f
+ ZF(a,...)(ai. A, 7i)(a2. -S^. 7.)
+ ZZ (a, ...)(«!, A, 7.)(a. /3, 7)
+ rZ(a,...)(«2, A. 72)(«. /8. 7)
+ i^^ (a,...)(a, /3, 7)' +&c.=0.
where the &c. refers to terms of the form (X, Y, Zf and higher powers.
But in order that the new axes may be chief axes, we must have
Aa, + 5/3, + C71 = 0,
Aou, + BA + C72 = 0,
(a, ...)(<*!, A, 7i)(aj, /3j, 72) = 0,
80 that putting for shortness
. A<x+B^ + Gy=V,
the equation becomes
V^: + iZ= (a,...)(a„ A. 7.)' + iI^n«.---)(a„ A, 7=)'
+ Z^ (a, ...)(«!, A, 7i)(a. A 7)
+ F^(a,...)(a„ /3„ 7.)(a. )8. 7)
+ iZ« (a,...)(a,y3, 7)» +&c. = 0.
4&
We have
that is,
and thence
A : B : C = /3,72-/S27i : 7i«s-7»«i = "A-tt^du
- a : ^ : 7 ,
a, /9, 7=4, V' V- ^=^(^^ + £= + (7=).
I write
and also for a moment
- = (a, ...)(«!. A, 7i)'.
Pi
V pi
<2=( A
i2 = ( g
b-
h
f
9 j («i. A. 7i).
/ ) («.. A, 70,
c )(ai, Ai 7i)-
C, IX.
50
We find
ON THE TRANSFORMATION OF THE EQUATION OF A
Pa, + QA + Ay, = (a, ...) (a„ /3., <y,)» - - , = 0,
Pi
[566
■P«. + QA + i27, = (o, . . .) («„ )8„ 7,) (a,, /8„ 7,) - - (0.0, + /9,/8, + 7,7,). = 0,
and thence
P : Q : -R = A7a-/337i : 7i««-7a«i = "i/Sj-fl^,
a : /3 : 7 ,
or say
P, Q, R = e,A, 6,B, e,G;
we have thus the equations
(a - - , h , S' ) («i. /9». 7i) = 6,A,
{ h . ^-^. / )(«., A, 7.)=^i5,
and joining hereto
(4,5. C)(a„ A, 70 = 0.
we eliminate Oi, A, 71 and obtain the equation
1
a —
Pi
h , 9 , A
and in like manner writing
h , b--, f , B
Pi
9 , f , c--, C
Pi
A , B , 0,0
1
= 0.
P'i
= (0, ...)(a„ /9„ y,y,
we have the same equation for p^; wherefore p,, p, are the roots of the quadric equation
«--. h , g , A
h . b-l, f , B
r
9 , f . c--, C
A , B , (7,0
= 0.
566]
SURFACE TO A SET OF CHIEF AXES.
51
Moreover, pi, p^ being thus determined, we have, a^, ^i, 71, ^1 proportional to the
determinants formed with the matrix
1
h ,
9 •
A
h ,
Pi
f .
B
9 •
/ .
1
c ,
Pi
C
say, a„ A, 7i. di = k%, A;S3,, A;Si, k£l, where SI,, S,, g,, fij are the determinants in
question ; and then 1 = fc= (Sir + 33i' + Si'), or we have
^1 =
fii
But we find at once
that is.
(o, ...)(o,, ^,, 7,) (a. A 7)^
vn,
and in the same manner
(o, ...)(«>, A, 7,) (a, A 7) =
Hence the transformed equation is
V(2li»+S3i' + (5i»)'
V(3l2' + 33,' + e2')'
Pi PS
+ x^
vn,
+ YZ-
va
-H^z»<°--><^;^-^.4-&c. = o,
where it will be recollected that V = /v/(4' + B' + C). The &c. refers as before to the
terms {X, Y, Zf and higher powers, which are obtained from the corresponding terms
in %x, hy, hz, by substituting for these their values hx = aL^X -k-OL^ ■\-olZ, &c., where the
coefficients have the values above obtained for them. It will be observed, that the
radii of curvature are Vp,, Vpj, and that the process includes an investigation of the
values of these radii of curvature similar to the ordinary one ; the novelty is in the
terms in XZ, YZ, and Z^. But regarding X, Y as small quantities of the first order,
Z is of the second order, and the terms in XZ, YZ are of the third order, and that
in Z^ of the fourth order.
7—2
52 [567
567.
ON AN IDENTICAL EQUATION CONNECTED WITH THE THEORY
OF INVARIANTS.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873),
pp. 115—118.]
Write
a = g — h,
h^h-f,
equations implying a fourth equation forming with them the system
. —h-\-g — a = 0,
h . -/-6=0,
-g+f ' -c =0,
a + 6 + c . =0,
and also
a/+ bg + ch = 0.
Then, putting for shortness
P = (bg- ch) (ch - af) {af- hg),
Q = a'g'h' + b'h'f + <^fy + a'b'cf',
R = a«/' (a» +/'') + h'g^ (6' + g"") + c^h" (c" + h%
we have
2P + Q-R = 0,
viz. substituting for a, b, c their values g — h, h—f, f—g, this is an identical equation.
567] AN IDENTICAL EQUATION CONNECTED WITH THE THEORY OF INVARIANTS. 53
The direct verification is however somewhat tedious, and the equation may be
proved more easily as follows:
In the terms a?+p, h^+g^ c' + h^ of R, substituting for a, h, c their values, we
find
B = {f' + g^ + ¥) (a?p + hy + c*)
-^fghia-f+h-'g + c'h),
which may be written
R = -2{f-\-g'-\- h') (bcgh + cahf+ ab/g)
-2fgh(a'/+b'g + <fh).
We have then
2P = - 2bcgh (bg - ch) - 2cahf{ch - af) - 2abfg {af- bg),
and thence
2P - ii = 2bcgh {f + g^ + h' -bg + ch)
+ 2cahf (p + g^ + h' - ch + af)
+ 2abfg (p +g"- + h'-af-¥bg)
+ 2/gh(a''f+b'g + d'h),
which is at once converted into '
2P-R= 2bcgh [a' +f (/+ g + h)}
+ 2cahf{b'+g(f+g + h)}
+ 2abfg[d'+h(f+g + h)}
+ 2fgh{a?f+h'g + c%);
or, what is the same thing,
2P-R = 2fgh {(6c + ca+ ab){f-\-g + h) + a?f+ ¥g + c'h] + 2abc (agh + bh/+ cfg),
where, since
agh + bh/+cfg = — ahc,
the last term ia
= - 2a'bV.
But from the equation last written down we deduce at once
Q = 2a'6»c^ - 2fgh (bcf+ cag + abh),
and we thence have
2P + Q - iJ = 2fgh {(6c + ca + ab) if+g + h) + {a^f+ b'g + d'h) - 6c/- cag - abh],
which is
= 2fgh(a + b + c){af+bg+ch),
and consequently =0, the theorem in question.
54 AN IDENTICAL EQUATION CONNECTED WITH THE THEOKY OF INVARIANTS. [567
Instead of a, b, c, f, g, h, I write aTT-r YZ, bW-i-ZX, cW-i-XY, f-i-X, g^Y,h-rZ:
we have therefore
. -hY + gZ-aW=0,
hX . -fZ-bW = 0,
-gX+fY . -cW=0,
aX + bY+cZ . =0,
and as before
af+ bg + ch = 0.
Moreover, omitting a common factor, the new values of P, Q, R are
P = XYZW{bg- ch) (ch - of) (af- bg),
Q = ayh^X* + h'PpY* + c^fyZ* + a?b-c?W\
R = ay (a«X» Tr» +/^ Y^Z^) + by (¥ Y' W- + g^Z^X") + c%^ {c^Z^ TT^ + h^X* Y%
and the identical equation is, as before,
2P + Q - iJ = 0.
Consider the operative symbols
*!, > rtjr,, Ota:,, ttxji
%!' %.' "j/i' %»'
and write a = d«,dy, - dy^dx, = 12, &c., that is
a = 23, /=14,
6 = 31, ^ = 24,
c = 12, A = 34,
and also X = xdx, + ydy^, &c. say
Z=V„ F=V„ Z=V3, F=V«.
These values of a, b, c, f, g, h, X, Y, Z, W satisfy the above written equations of
connexion, and therefore the identical equation 2P + Q — R = 0. Hence taking U to
denote the quartic function U — {a, b, c, d, e){x, y)*, and therefore Ui = {a, ...){xx, yi)*, &,c.,
we have
{2P-^q-R)U,UJJ,U, = 0,
where, after the differentiations, (a-i, y^ (xt, yt) are to be each of them replaced by
(«. y)-
Observe that P is the sum of three positive and three negative terms, but that
after the omission of the suffixes each term taken with its proper sign becomes equal
to the same quantity, and the value of P is =6 times any one term thereof. Thus
omitting for the moment the factor ViVjVjV^, two of the terms are —{afybg-\-af{bgf,
that is,
- (14 . 23)' (24 . 31) + (14 . 23) (24 . 31)»,
567] AN IDENTICAL EQUATION CONNECTED WITH THE THEORY OP INVARIANTS. 55
and, if in the first term we interchange 3 and 4, it becomes —(13. 24)" (23. 41), that
is, +(14. 23) (24. 31)^ viz. it becomes equal to the second term. As regards Q the
teiins are all positive and become equal to each other; and the like as regards R:
hence we have
(12 V,V2V,V4(14 . 23) (24 . 31)» + 4Vi'(23)^ (34)= (42)=' - 6 V,^V,»(43)« (14)=} U^U^U.U, = 0,
which, omitting a numerical factor 6.2. 12". 2 . 24*. 4, =3". 2", is in fact the well-known
equation
n + JU-IH = 0,
where
U = (a, b, c, d, e)(x, y)*,
ft = disct. (ax + hy, bx + cy, ex + dy, dx + ey) (f , rjf
= (aa; + byf (dx + eyf + &c.,
/=ae-46d + 3c=,
J=axx-ad^-ll'c-c? + 2hcd,
viz. attending only to the coefiBcient of a^, this equation is
a»d» + 4ac' + 46»d - 36'c= - Qabcd + a(ace-ad'- b^e - c' + 2bcd) + (ac - 6=) (ae - 46d + Sc') = 0.
i.
56 [568
568.
NOTE ON THE INTEGRALS co&a?dx AND sinx'dx.
I co&a^dx AND I £
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xri. (1873),
pp. 118—126.]
Mr Walton has raised, in relation to these integi-als, a question which it is very
interesting to discuss. Taking for greater convenience the limits to be — oo , + x , and
writing
/•oo 1*00
2m=I co&a?dx, 2d=I sina;"da;,
J -00 J — QO
then we have
I COS (a^ + y') dx dy,
— oo J —00
Too /"oo
Sitt; =1 I sin (av' + y") da; dy,
and writing herein x = r cos ^, y = r sin 0, and therefore da;dy = rdr d6, it would thence
appear that we have
4 («' — v*) = I I cos r' . rdrdd — iir\ cos r* , rdr,
Jo Jo Jo
/•oo rin Too
8uv =1 I sin 7^ . rdrdO = 27r | sin r" . rdr,
J 0 J 0 Jo
or, finally
4- (m' — t;*) = TT sin oo ,
8m?; = 7r (1 - cos X );
that is, either the integrals have their received values -jeach =9 //o\r. and then
sin 00 = 0, cos 00 = 0 ; or else the integrals, instead of having their received values, are
indeterminate.
568] NOTE ON THE INTEGRALS | COSX'dx AND I S.m.as' dx. 57
The error is in the assumption as to the limits of r, 6; viz. in the original
expressions for 4,(u^-i/'), 8uv, we integrate over the area of an indefinitely large square
(or rectangle); and the assumption is that we are at liberty, instead of this, to
integrate over the area of an indefinitely large circle.
Consider in general in the plane of xy, a closed curve, surrounding the origin,
depending on a parameter k, and such that each radius vector continually increases
and becomes indefinitely large as k increases and becomes indefinitely large : the curve
in question may be referred to as the bounding curve; and the area inside or outside
this curve as the inside or outside area. And consider further an integral jlzdxdy,
where z is a given function of x, y, and the integration extends over the inside area.
The function z may be such that, for a given form of the bounding curve, the integi-al,
as k becomes indefinitely large, continually approaches to a determinate limiting value
(this of course implies that z is indefinitely small for points at an indefinitely large
distance from the origin); and we may then say that the integral taken over the
infinite inside area has this determinate value ; but it is by no means true that the
value is independent of the form of the bounding curve ; or even that, being determ-
inate for one form of this curve, it is determinate for another form of the curve.
I remark, however, that if' z is always of the same sign (say always positive)
then the value, assumed to be determinate for a certain form of the bounding curve,
is independent of the form of this curve and remains therefore unaltered when we
pass to a different form of bounding curve. To fix the ideas, let the first form of
bounding curve be a square {x = ±k, y=±k), and the second form a circle (oa' + y'' = k^).
Imagine a square inside a circle which is itself inside another square ; then z being
always positive, the integral taken over the area of the circle is less than the integi-al
over the area of the larger square, greater than the integral over the area of the
smaller square. Let the sides of the two squares continually increase, then for each
square the integral has ultimately its limiting value; that is, for the area included
between the two squares the value is ultimately = 0, and consequently for the circle
the integral has ultimately the same value that it has for the square. When z is
not always of the same sign the proof is inapplicable ; and although, for certain
forms of z, it may happen that the value of the integral is independent of the form
of the bounding curve, this is not in general the case.
We have thus a justification of the well known process for obtaining the value
of the integral ( tr^dx, viz. calling this u, or writing
Jo
= r e-^'dx,
4m>= e-^'^-^^ dxdy = \ e-'^'rdrdd
= 2v.^, or M = ^ ^(v),
2u
then
C. IX.
58
NOTE ON THE INTEGRALS I COS af dx AND ( siusc'dx. [568
J« Jo
but in consequence of the alternately positive and negative values of cos a* and sinx*,
we cannot infer that the like process is applicable to the integrals of these functions.
To show that it is in fact inapplicable, it will be sufficient to prove that the
integrals in question have determinate values ; for this being so, the double integrals
1 1 cos (a:^ + i^)dx dy and 1 1 sin (a? + 'if)dx dy, taken over an infinite square (or, if we
please, over a rectangle the sides of which are both infinite, the ratio having any value
whatever), will have determinate values ; whereas, by what precedes, the values taken
over an infinite cii'cle are indeterminate. The thing may be seen in a very general
sort of way thus : consider the surface z = sin {a? + y^), and let the plane of xy be
divided into zones by the concentric circles, radii s/iir), •\J{2it), V(37r), &c then in
the several zones z is alternately positive and negative, the maximum (positive or
negative) value being ± 1 ; and though the breadths of the successive zones decrease,
the areas and values of the integral remain constant for the successive zones; the
integral over the circle radius \/(»wr) is thus given as a neutral series having no determ-
inate sum. But if the plane xy is divided in like manner into squares by the lines
x = ± 'J{n-ir), y=± i\J{nir), then in each of the bands included between successive squares,
z has a succession of positive and negative values; the breadths continually diminish,
and although the areas remain constant, yet, on account of the succession of the
positive and negative values of z, there is a continual diminution in the values of
the integral for the successive bands respectively, and the value of the integral for
the whole square is given as a series which may very well be, and which I assume
is in fact, convergent. Observe that I have not above employed this mode of integration
(but by considering the single integral have in effect divided the square into indefinitely
thin slices, and considered each slice separately); it would be interesting to carry out
the analytical division of the square into bands, and show that we actually obtain a
convergent series; but I do not pursue this inquiry.
Consider the integral
v= \ sm a^ dx,
Jo
and taking for a moment the superior limit to be (n+l)Tr, then the quantity under
the integral sign is positive from a^ = 0 to a? = -n; negative from a^ = ir to a^ = 2ir, and
so on ; we may therefore write
where
Jo
(n+l|ir
sin a^ da; = ^0 - -4, + .^2 . . . + (-)» A^,
Ar, =(-)*■ I sin a;" da;,
is positive. Writing herein «• = m- + m, we have
f '^ sin udu
0 ^{rtr + m) '
568] NOTE ON THE INTEGRALS I co&a? dx AND I sin or" c^cc. 59
which, for r large, may be taken to be
-i/'
sin vdu 1
viz. r being large, we have Ar differing from the above value -j — - by a quantity
of the order -j,
r*
It is obviously immaterial whether we integrate from a^ = 0 to (w + l)7r or to
(n+l)7r+e, where e has any value less than tt; for by so doing, we alter the value
of the integral by a quantity less than ^„+i, and which consequently vanishes when n
is indefinitely large. And similarly, it is immaterial whether we stop at an odd or
an even value of n.
We have therefore
Jo
or, taking n to be odd, this is
== Ao — Ai + Aj . . . — An,
or, say it is
= {A,-A,) + {A,-A,)...+ (A^, - A„),
viz. n here denotes an indefinitely large odd integer.
If instead of Ao — Ai + A^ — Ai + Sic., the signs had been all positive, then the
term A being ultimately as -77—,, the series would have been divergent, and would
have had no definite sum : but with the actual alternate signs, the series is convergent,
and the sum has a determinate value. To show this more distinctly, observe that we
have
A 4 -/ \r-i L r sia(r7r + u)du _ _ i f" sin udy
or, taking the integral from — tt to 0 and from 0 to tt, and in the first integral
writing — m in place of u, then
where, r being large, expanding the term in { } in ascending powers of u, then
Ar-i — Ar is of the order -=: and the series {Aa — Ai) + {A2 — A3)... + (An-i — A„) is
r*
■therefore convergent, and the sum as w is increased approaches a definite limit. Hence
the integral v has a definite value : and similarly, the integral u has a definite value.
8—2
60
NOTE ON THE INTEGRALS I COS Ot? dx AND I Bin ofdx. [568
The values of u, v being shown to be determinate, I see no ground for doubting
that these are the values of the more general integrals
I e-"** COS a!» etc, / er«^ sin of' dx,
Jo Jo
(a real and positive) when a is supposed to continually diminish and ultimately become
= 0. We have, in fact, (o as above)
/,
^ e 2/ «y (a» + 6»)i»'
Jo
where 6 = tan~' - , an angle included between the limits — ^tt, + ^tt. Writing herein
» = i, 6 = 1, y = oi?, then
^ "^ 2(a»+l)i'
where ^ = tan~' - , an angle included between the limits — \it, + \it ; or, putting herein
a = 0, we have 6 = \ir, and therefore
/■
Jo
that is, equating the real and imaginary parts,
which are the received values of the integrals
M = I cos a?" dx, v—\ Bin a^dx.
Jo Jo
An important instance of the general theory presents itself in the theory of elliptic
functions, viz. the integral
dxdy
II
{ilx + Tyf
the ratio il : T being imaginaiy, will, if the bounding curve be symmetrical in regard
to the two axes respectively, have a determinate value dependent on the form of the
hounding curve; if for instance this is a rectangle x=±ak, y = ±hk, then the value
of the integral will depend on the ratio a : 6 of the infinite sides; and so if the
bounding curve be an infinite ellipse, the value of the integral will depend on the
ratio and position of the axes. See as to this my papers " On the inverse elliptic
* For brevity I take the integral under this form, but the real and imaginary parts might have been
considered separately ; and there would have been some advantage in following that course. The like remark
applies to a subsequent investigation.
568] NOTE ON THE INTEGRALS I C0S3?dx AND ( Sm 31? dx. 61
Jo Jo
functions," Camb. Math. Jour., t. iv. (1845), pp. 257—277, [24]; and "Mdmoire sur les
fonctions doublement pdriodiques," Liouv. t. X. (1845), pp. 385 — 420, [25].
A like theory applies to series, viz. as remarked by Cauchy, although the series
A0 + A1+A2 + ... and Bo + Bi + Bi + &c. are respectively convergent, then arranging the
product in the form
AJBo + AoB, + AA+ ■■•
+ A,Bo + A,B, + A,B^+...
+ A^o + AiB, + AA + —
+ ...,
say the general term is Cm,n> then if we sum this double series according to an
assumed relation between the suffixes m, n (if, for instance, we include all those terms
for which m° + n^ < k', making k to increase continually) it by no means follows that
we approach a limit which is equal to the product of the sums of the original two
series, or even that we approach a determinate limit.
Mr Walton, agreeing with the rest of the foregoing Note, wrote that he was
/•cO
unable to satisfy himself that the value of I e"'' dx is correctly deduced from that of
Jo
I g(-a+bi)y yn^i ^y Writing ti = ^, the question in fact is whether the formula
•'0
g(-<n-W)» y-idy = /^ /^.. 6 = tan"' - , angle between ^ir, - Jtt ) ,
Jo (ffl + tr)* \ O' '
which is true when a is an indefinitely small positive quantity, is true when a = 0 ;
that is, taking b positive, whether we have
Write in general
JO
then, differentiating with respect to b, we have
db Jo
or, integrating by parts,
db -a + U-^ 2(-a + 6t)Jo^ ^
62 NOTE ON THE INTEGRALS I coa oc^ dx AND I sinar'rfa;. [568
where the first term is to be taken between the limits oo , 0 ; viz. this is
db L-a + 6i^ Jo 2(-a + W)"-
When a is not =0, the first term vanishes at each limit, and we have
du —i
db~2{-a + U)'"'
The doubt was in effect whether this last equation holds good for the limiting value
du
db
a = 0. When a is = 0, then in the original equation for -^ the first term is indeterm-
inate, and if the equation were true, it would follow that -jj- was indeterminate; the
original equation for jt is not true, but we trtiiy have
db
du _ 1
db~~2b^'
the same result as would be obtained from the general equation, rejecting the first
term and writing a = 0.
To explain this observe that for a = 0, we have
u= \ y-ie"^ dy,
Jo
which for a moment I write
u= I y-ie"^dy,
Jo
where, as before, b is taken to be positive. Writing herein hy = x, we have
1 r**
V(o) J 0
fit ^
and assuming only that the integral I arie'^dx has a determinate limit as M becomes
Jo
indefinitely large* then supposing that k is indefinitely large, the integral in the last-
mentioned expression for u has the value in question
(= j" x-i e^ dxY
* This is in fact tlie theorem I e'*'dx= a, determinate value { = iij{r)e^''}, proved in the former part
/ 0
of the present Note.
568] NOTE ON THE INTEGRALS I COS x' dx AND ( SUlSC'dx. 63
which is independent of b, say this is
G
and thence differentiating in regard to h, we find
du _ 1
But the value of -jr cannot he obtaitied hy differentiating under the integral sign,
the theorem in question.
(
dh
for this would give
and this integral is certainly indeterminate.
64 [569
S
569.
ON THE CYCLIDE*.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. Xli. (1873),
pp. 148—165.]
The Cyclide, according to the original definition, is the envelope of a variable
sphere which touches three given spheres, or, more accurately, the envelope of a variable
sphere belonging to one of the four series of spheres which touch three given spheres.
In fact, the spheres which touch three given spheres form four series, the spheres of
each series having their centres on a conic; viz. if we consider the plane through the
centres of the given spheres, and in this plane the eight circles which touch the
sections of the given spheres, the centres of these circles form four pairs of points,
or joining the points of the same pair, we have four chords which are the transverse
axes of the four conies in question.
It thus appears, that one condition imposed on the variable sphere is, that its
centre shall be in a plane ; and a second condition, that the centre shall be on a
conic in this plane ; so that the original definition may be replaced first by the
following one, viz.:
The cyclide is the envelope of a variable sphere having its centre on a given
plane, and touching two given spheres.
Starting herefrom, it follows that the locus of the centre will be a conic in the
given plane: the transverse axis of the conic being the projection on the given plane
of the line joining the centres of the given spheres; and it, moreover, follows, that if
in the perpendicular plane through the transverse axis we construct a conic having
for vertices the foci, and for foci the vertices, of the locus-conic, then the conic so
constructed will pass through the centres of the given spheres.
* I use the term in its original sense, and not in the extended sense given to it by Darboux, and
employed by Casey in his recent memoir "On Cyclides and Spheroquartics," PMl. Tram. 1871, pp. 582 — 721.
With these authors the Cyclide here spoken of is a Dnpin's or tetranodal Cyclide.
569] ON THE CYCLIDE. 65
Two conies related in the manner just mentioned are the flat-surfaces of a system
of confocal quadric surfaces; they may for convenience be termed anti-conies (fig. 1); one
of them is always an ellipse and the other a hyperbola; and the property of them is
that, taking any two fixed points on the two branches, or on the same branch of the
Fig. 1.
hyperbola, and considering their distances from a variable point of the ellipse: in the
first ease the sum, in the second ease the difference, of these two distances is constant.
And similarly taking any two fixed points on the ellipse, and considering their distances
from a variable point of the hyperbola, then the difference, first distance less second
distance is a constant, -|- a for one branch, — a for the other branch of the hyperbola.
And we thus arrive at a third, and simplified definition of the cyclide, viz. con-
sidering any two anti-conics, the cyclide is the envelope of a variable sphere having
its centre on the first anti-conic, and touching a given sphere whose centre is on the
second anti-conie.
And it is to be added, that the same cyclide will be the envelope of a variable
sphere having its centre on the second anti-conic and touching a given sphere whose
centre is on the first anti-conic, such given sphere being in fact any particular sphere
of the first series of variable spheres. And, moreover, the section of the surface by the
plane of either of the anti-conics is a pair of circles, the surface being thus (as will
further appear) of the fourth order.
In the series of variable spheres the intersection of any two consecutive spheres
is a circle, the centre of which is in the plane of the locus-anti-conic, and its plane
perpendicular to that of the locus-anti-conie, this variable circle having for its diameter
in the plane of the locus-anti-conie a line terminated by the two fixed circles in that
plane. The cyclide is thus in two different ways the locus of a variable circle; and
investigating this mode of generation, we an-ive at a fourth definition as follows : —
Consider in a plane any two circles, and through either of the centres of symmetry
draw a secant cutting the two circles ; in the perpendicular plane through the secant,
draw circles having for their diameters the chords formed by the two paii-s of anti-
parallel points on the secant (viz. each pair consists of two points, one on each circle,
such that the tangents at the two points are not parallel to each other): the locus
of the two variable circles is the cyclide.
Before going further it will be convenient to establish the definition of "skew anti-
points": viz. if we have the points Ki, K^ (fig. 2), mid-point R, and i,, Z,, mid-point
iS', such that K^K^, JiS and L^L^ are respectively at right angles to each other, and
c. IX. 9
66 ON THE crycLiDE. [569
KxI^+Ii^+SLj^=0, &c. ; or, what is the same thing, the distances LiKi= LiKi = L^Ki^ L-tK^
are each = 0, so that the points iT, , K.^ and Z, , L^ ai-e skew anti-points. Observe that
the lines of the figure and the points R, 8 are taken to be real ; but the distances
RKi = RK^ and SL-^ = SL^ cannot be both real : it is assumed that one is real and
Fig. 2.
the other a pure imaginary, or else that they are both of them pure imaginaries. To
fix the ideas we may iu the figure consider the plane through K^K^, RS as horizontal,
and that through RS, LiL^ as vertical.
Reverting now to the cyclide, suppose that we have (in the same plane) the two
circles C, C intersecting in K^, K^, and having S for a centre of symmetry, and let
R be the mid-point of K^, K^.
The construction is: — through 8 draw a secant meeting the two circles in A, B
and A', B' respectively, where A, A' and B, B' are parallel points, (therefore A, B'
and A', B anti-parallel points), then the cyclide is the locus of the circles in the
perpendicular plane on the diameters AR and A'B respectively.
The two circles have their radical axis passing through 8, and not only so, but
the points of intersection Lj, Xj of the two circles are situate at a distance SLi = 8L2,
which is independent of the position of the secant: the points Z,, L^ and Ki, K^
being in fact a system of skew anti-points. And, moreover, the two circles have a
centre of symmetry at the point where the plane of the two circles meets the line K^K^.
Consider in particular the two ch-cles D, JD' which are situate in the perpendicular
plane through SR ; these have the radical axis L^L^, and a centre of symmetry R ;
and if with these circles B, D' as given circles, and with R as the centre of symmetry,
we obtain in a plane through KJ{^2 two circles having K^K^ for their radical axis,
and having for a centre of symmetry the intersection of their plane with LiL^, the
locus of these circles is the same cyclide as before ; and, in particular, if their plane
passes through RS, then the two circles are the before-mentioned circles G, C, having
8 for a centre of symmetry.
It will be noticed that, starting with the same two circles C, C or D, Z)', we
obtain two diJBFerent cyclides according as we use in the construction one or other of
the two centres of symmetry.
The cyclide is a quartic surface having the circle at infinity for a nodal line:
viz. it is an anallagmatic or bicircular quartic surface ; and it has besides the points
569] ON THE CYCLIDE. 67
jfiTi, Ki, Li, L.,, that is, a system of skew anti-points, for nodal points; these determine
the cyclide save as to a single parameter. In fact, starting with the four points
Z,, Zj, K^, K^, which give 8, and therefore the plane of the circles C, C"; the circle
C is then any one of the circles through K^, K^; and then drawing from S the two
tangents to C, there is one other circle C passing through Ki, K^ and touching these
tangents ; C is thus uniquely determined, and the construction is effected as above.
Hence, with a given system of skew anti-points we have a single series of cyclides,
say a series of conodal cyclides.
If in general we consider a quartic surface having a nodal conic and four nodes
A, B, G, D, then it is to be observed that, taking the nodes in a proper order, we
have a skew quadrilateral A BCD, the sides whereof AB, BC, CD, DA, lie wholly on
the surface. In fact, considering the section by the plane ABC, this will be a quartic
curve having the nodes A, B, G and two other nodes, the intersections of the plane
with the nodal conic ; the section is thus made up of a pair of lines and a conic ;
it follows that two of the sides of the triangle ABC, say the sides AB, BC, each
meet the nodal conic, and that the section in question is made up of the lines
AB, BC, and of a conic through the points A, C and the intersections of AB, BC
with the nodal conic. Considering next the section by the plane through ACD, here
(since AC is not a line on the surface) the lines CD, DA each meet the nodal conic,
and the section is made up of the lines CD, DA and of a conic passing througli
the points A, C and the intersections of the lines CD, DA with the nodal conic.
Thus the lines AB, BC, CD, DA each meet the nodal conic, and lie wholly on the
surface; the lines AC, BD do not meet the conic or lie wholly on the surface.
A quartic surface depends upon 34 constants; it is easy to see that, if the surface
has a given nodal conic, this implies 21 conditions, or say the postulation of a given
nodal conic is = 21, whence also the postulation of a nodal conic (not a given conic)
is =13. Suppose that the surface has the given nodes A, B, C, D; the postulation
hereof is =16; the nodal conic is then a conic meeting each of the lines AB, BC,
CD, DA, viz. if the plane of the conic is assumed at pleasure, then the conic passes
through 4 given points, and thus it still contains 1 arbitrary parameter; that is, in
order that the nodal conic may be a given conic (satisfying the prescribed conditions)
the postulation is = 4. The whole postulation is thus 16 + 13 + 4, = 33, or the quartic
surface which satisfies the condition in question (viz. which has for nodes the given
points A, B, C, D, and for nodal conic a given conic meeting each of the lines
AB, BC, CD, DA) contains still 1 arbitrary parameter: which agrees with the foregoing
result in regard to the existence of a series of conodal cyclides.
It is to be added that, if a quartic surface has for a nodal line the circle at
infinity and has four nodes, then the nodes form a system of skew anti-points and
the surface is a cyclide. In fact, taking the nodes to be A, B, C, D, then each of
the lines AB, BC, CD, DA meets the circle at infinity; but if the line AB meets
the circle at infinity, then the distance AB is = 0, and similarly the distances BC,
CD, DA are each = 0 ; that is, the nodes {A, C) and {B, D) are a system of skew
anti-points.
9—2
68 ON THE CYCLIDE. [569
Reverting to the cyclide, and taking (as before) the nodes to be K,, K, and
i,, L,, the line RS which joins the mid-points of K^Kt and L^L^ may be termed the
axis of the cyclide, and the points where it meets the cyclide, or, what is the same
thing, the circles C, C or D, D', the vertices of the cyclide, say these are the points
F, 0, H, K. Supposing that the distances of these from a point on the axis are
/, g, h, k, the origin may be taken so that f-\-g+h-\-k = 0; the origin is in this case
the "centre" of the cyclide. It is to be remarked, that given the vertices there are
three series of cyclides: viz. we may in an arbitrary plane through the axis take for
C, C the circles standing on the diameters FG and HK respectively ; and then, according
as we take one or the other centre of symmetry, we have in the plane at right angles
hereto for D, D' the circles on the diameters FH and GK, or else the circles on the
diameters FK and GH respectively; there are thus three cases according as the two
pairs of circles are the circles on the diameters
FH, KG and FK, GH,
FK, GH „ FG, HK,
FG, HK „ FH, KG.
The equation of the cyclide expressed in terms of the parameters /, g, h, k assumes
a peculiarly simple form ; in fact, taking the origin at the centre, so that /+ g + h + k = (i,
the axis of x coinciding with the axis of the cyclide, and those of y, z parallel to
the lines K^K^ and L^L^, or ijZj and KJ^^ respectively: writing also
fg^hk^G,
fh + kg = H,
fk + gh = K,
then the equation of one of the cyclides is
{y^ + zy + 2af (y^ + z') + Gy^ + Hz^ + {a;-/)(x- g) {x - h) {x - k) = 0,
which we may at once partially verify by observing that for ^ = 0 this equation becomes
b' + (^ -/) i<^ - 9)1 [y' + (x-h)(x- k)] = 0,
and for y = 0 it becomes
[z' +{x-f)ix- h)] \f + (^ - k) {x - g)-\ = 0,
viz. the equations of the circles C, C are
t^{x-f){x-g)^^, f + (x-f,)(x-k)=0,
and those of D, U
z" + {x -f) {x -h) = 0, z"-+{x-k)(x-g) = 0.
Starting from these equations of the four circles, the points K^, K^ are given by
Y^=^-(P-f)(P-g) = -{P-h)(P-k),
569] ON THE CYCLIDE. 69
and the points X,, L.^ by
Z'=-{Q-fm-h) = -{Q-h){Q-g).
Now writing for a moment
^=f^g = -h-k,
y=f+h=-k-g,
S=f+k = -g-h,
we have P = — ^-4-, Q = — ^ — , and thence PQ = ^B\ Moreover
p 7
2F'' + 2^^ + 2(P-Q)''
= -(P-/)(P-5r)-(P-A)(P-Z;)
-{Q-f){Q -h)-(Q-k)(Q-g) + 2(P-Qy
= -(fg + hk+fh+gk)-iPQ
= a»-4PQ
= <^'
that is,
Y^ + Z^ + (P- Qf = 0,
which equation expresses that the four points are a system of skew anti-points.
The point x = Q should be a centre of symmetry of the circles G, C ; to verify
that this is so, transforming to the point in question as origin, the equations are
f + [x + Q-Hh + k)Y-iik-hy=^Q,
that is;
y»+|a;-i^(S + 7)}' -Hf-9y = 0,
But S + 7 =/- g, B — y=k — h, so that these equations are
f + \cc-^^{k-h)^''=H^-hy.
which are of the form
f + (x- af = c^
y^ + (x- may - m?c\
and consequently x = Q is a centre of symmetry of the circles G, G' ; and in like
manner it would appear that x = P is a centre of symmetry of the circles D, D'.
70 ON THE CYCLIDE. [569
If in the last-mentioned equations of the circles G, C we write x = n cos 0,
y = fi sin 6, and put for shortness
p =a cos 0 — V , a =m(a cos 0 — V),
p'=acos0 + V , a = in (a cos ^ + V ),
where V = v'(c' — a' sin' 5), then the values of fi for the first circle are p, <r, and those
for the second circle are p', a. Hence the equations of the generating circles are
^» + (r - p ) (r - o-') = 0,
^^ + (r + p') (»■ - <^ ) = 0.
where r is the abscissa in the plane of the circles, measured from the point a; = Q.
Attending say to the first of these equations, to find the equation of the cyclide, we
must eliminate 0 from the equations
«'' + (»■ — p)(r — cr') = 0, a; = rcos^, y = 7*sin^;
the first equation is
2% + ,.2 + „i (0(2 _ c") -r{p + a) = 0,
and we have
p + a-' ={m+\)aco%0-{m-\) ^/{c' - a^ sin= 0),
and thence
(p + <r')r= (m + l)ax- {m - 1) Vic- (a^ + f-) - ay},
BO that we have
z- + cd' + y^+ni (a^ - c^ - (m + 1) ow + {m - 1) \/{c» (af + f) - ay] = 0,
viz. this is the equation of the cyclide in terms of the parameters a, c, m, the origin
being at the point x=Q, the centre of symmetry of the circles G, G'.
Reverting to the former origin at the centre of the cyclide, we must write x — Q
for X ; the equation thus is
{f + z'- + {x- QY - (m + l)a{x-Q) + m (a' - c')Y - {ni ~ 1)« [{c' (a; - QY + (c" - a') f]] = 0,
where
whence also
After all reductions, the equation assumes the before-mentioned form
(y' + zy + 2a? (y» -I- 2») + Gy^ + Hz^ ■^{oo-f){x- g) {x -h){x-k) = 0.
The equation may be written
{af'-\-y^ + z''y + {G + H + K)a? + Gy^ + Hz^- ^yBx +fghk = 0,
569] ON THE CYCLIDE. 71
and if we express everything in terms of /3, 7, S by the formulae
2/= /3 + 7 + S, 2G = ^--y^-B'-,
2g= 0-y-S, 2H =-^ + y^-B^
2^=-/3 + 7-S, 2K =-0"--.y^ + 8\
2k = -^-y + B, 2{G + H + K) = -^'-y^-h'-
then we have
- ^yhx + ^ (/3^ + r/ + 54 _ 2^Y _ 2/3=S2 - 27=8^ = 0 ;
or, what is the same thing,
{a? + f + z'' + l^ + li'-\?^y-{0' + y^)x'-yY-^z"--^yix-l^-i' = (i.
An equivalent form of equation may be obtained very simply as follows : the
surface
(sd' + f-irz^'f + 2Aa? + 2By- + 2Cz'+2Kx + L = 0
will be a cyclide if only the section by each of the planes y = 0, z=Q breaks up into
a pair of circles. Now for y = Oi.the equation is
(a? + ^f+ 2Ax' + 26V + 2Kx + L = 0,
that is,
z* + 2z''(x'+C) + x* + 2Ax^+2Ka; + L = 0,
or
(z' + x' + C)' = 2(0 - A)!,^- 2Kx + C- L,
which will be a pair of circles if only
2(G-A)(C'-L) = K'';
and similarly writing z=0, we obtain
2{B-A)(B=-L) = K\
These equations give
L =(B + Cy-{BC+CA + AB),
K^ = - 2 (B - A) (G ~ A)(B + C),
so that L, K having these values the surface is a cyclide; there are two cyclides
corresponding to the two different values of K, which agrees with a former result.
Reverting to the equation in terms of /8, 7, 8 this may be written
^-'f+ V{(27^ + /9S)» - 4 (^ - 7=) y'} + V{(2/3a; + ySy + 4 (^^ - 7^) z'} = 0.
[Compare herewith Rummer's form
b' = ^f{(ax - eky + b'y''] + ^/{(ex - ahf - H'z''], where If = ci" - el]
72 ON THE CYCLIDE. [569
In fact, representing this for a moment by
/3^-y' + V(0) + V(<I>) = O.
we have
(/S' - 7^)= + 0 - <& = - 2 (/3» - 7») V(e),
or, substituting and dividing by ^ — 'f, we have
^' - rf + B- - 4 (of + f + z') + 2 ^{(2yx + ^Sy- i{^ -rf)y'-} =0,
or, similarly
;32 _ y _ g= + 4 (a^ + y2 + ^2) + 2 V ((2/3a: + 78)= + 4 (y3= -'f)'^}= 0,
either of which leads at once to the rational form.
The irrational equation
^-'f+ Vl(27aJ + 0By - 4 (/3' - y) f] + sj[{2^x + 78)= + 4 (^— 7=) z-] = 0
is of the form
which belongs to a quartic surface having the nodal conic p = 0, qr —st = 0 (in the
present case the circle at infinity), and also the four nodes {q = 0, r = 0, p^ — st = 0)
and (s = 0, < = 0, ^^^ — ^r = 0), viz. these are
«' = -if. 2/ = o, ^=±iJv((^-7^)(r-8%
and
and we hence again verify that the nodes form a system of skew anti-points, viz. the
condition for this is
«'(f-lJ-(^=-^)('-D-(^-^)(i-|)-o,
that is,
8^ {^- - 7O + ^' (r - S'O - 7'^ (^ - 8') = 0,
which is satisfied identically.
The cyclide has on the nodal conic or circle at infinity four pinch-points, viz
these are the intersections of the circle at infinity with the planes ^y" + 'fz' = 0.
If /3=0, the equation becomes
^ + ^{a^ + f) + V(i8» - 2=) = 0,
viz. the cyclide has in this case become a torus; there are here two nodes on the
axis (« = 0, y= 0), and two other nodes on the circle at infinity, viz. these are the
circular points at infinity of the sections perpendicular to the axes, and the pinch-
points coincide in pairs with the last-mentioned two nodes; viz. each of the circular
points at infinity = node + two pinch-points.
569] ON THE CYCLIDE. 73
The Parabolic Cyclide.
One of the circles C, C and one of the circles Z>, D' may become each of them
a line; the cyclide is in this case a cubic surface. The easier way would be to treat
the case independently, but it is interesting to deduce it from the general case. For
this purpose, starting from the equation
(y2 ^ 2^y 4. 2a^ (y= + 2'.) + Gf + Hz- + (x -f) {x -g){x- h) {x-k) = 0,
where /+£r+A + A' = 0, G =/g + hk, H=fh+gk, I write x — a for x, and assume a+f,
a + ff, a + h, a+k, equal to /', g", h', k' respectively; whence 4ia=f'+g' + h'-\-k'; and
the equation is
{f + z'J+2{x- af {y^ + z^) + {f'g' + h'k' - 2a^) y^ + {f'h' + g'k' - 2oe) 2^
+ {x-f'){x-^){x-h')[x-k') = 0,
or, what is the same thing,
iy^ -irZ^'fJr{2a?- ^ax) (y' + z^) + {fg + h'k') f + (fh' + g'k') ^
+ (x-f')(x-g')ix-h'){x-k') = 0.
Now assuming k' =: 00 , we have 4a = i'=oo, or writing 4a instead of k', and attending
only to the terms which contain ia, we have
X (f + Z1')- h'y' - g'z' + {x-f)(x-g')(x-h') = 0,
or, what is the same thing,
{x -f) (x-g') (x - h') + (x- h') f + {x- g') z' = 0,
where by altering the origin we may make f = 0.
It is somewhat more convenient to take the axis of z (instead of that of x) as
the axis of the cyclide ; making this change, and writing also 0, y8, 7 in place of the
original constants, I take the equation to be
z{z-^)(z-y)+iz-y)f + (z-0)a^^O,
viz. this is a cubic surface having upon it the right lines {z = y, x = 0), (z = ^, y = 0);
the section by a plane through either of these lines is the line itself and a circle ;
and in particular the circle in the plane x=0 is z(z — ^) + y- = 0, and that in the
plane y = 0 is z{z — y) + x' = 0. And it is easy to see how the surface is generated :
if, to fix the ideas, we take /3 positive, 7 negative, the lines and circles are as shown
in fig. 3; and if we draw through Gy a plane cutting the circle CO and the line
Bx in P, Q respectively, then the section is a circle on the diameter PQ; and
similarly for the sections by the planes through Bx. It is easy to see that the whole
surface is included between the planes z = ^, z = y; considering the sections parallel to
these planes (that is, to the plane of xy) z = fi, the section is the two-fold line y = 0 ;
z = any smaller positive value, it is a hyperbola having the axis of y for its transverse
axis; z=0, it is the pair of real lines yy' + ^ic'^O; z negative and less in absolute
C. IX. 10
74
ON THE CYCLIDE.
[569
magnitude than —7, it is a hyperbola having the axis of x for its transverse axis;
and finally z = 7, it is the two-fold line a; = 0. It is easy to see the forms of the
cubic curves which are the sections by any planes x = const, or y = const.
Fig. 8.
The before-mentioned circles are curves of curvature of the surface; to verify this
<i posteriori, write
for the equation of the surface ; and put for shortness P = Sz'-2z (/3 +7)+ ^y, P + x' + y^ = L,
so that diU = P+x' + y\ =L. The differential equation for the curves of curvature is
2x(z-^) , 2y(2-7) , P+a^ + f
xdz + (z — ^) dx, ydz + iz — y) dy, ^P'dz + xdx + ydy
dx , dy , dz
= 0,
or, say this is
a. = da?.2xy{z-y)- dy^ . 2xy{z - /3) -t- dzK 2xy{y-^)
+ dzdy .x[-2(z-^) {2z -fi) + L]
+ dxdz.y[ 2(z-y){2z-y)-L]
+ dxdy. [(y-0)P + {2z-^-y){f-x^)] = O.
But in virtue of the equation U = 0, we have identically
{2{z-0)xdx + 2{z-y)ydy + Ldz}xi^-^^^ydx + ^y^xdy + ^^^^^^^^dz^
= (7 - j8) |2 - .^--^w— - J X {xydz' - y{z - y) dzdx -x{z-S) dzdy+ (z -0){z- 7) dxdy\.
569] ON THE CYCLIDE. 75
Hence in virtue of the equations U = 0, dU=0 the equation fl = 0 becomes
xydz^—y{z — 7) dzdx - x {z - ^) dzdy + {z — 0) (z — 7) dx dy = 0,
that is,
{xdz — (z — y) dx} [ydz — (2 — yS) dy] = 0,
whence either a; — C (z - 7) = 0 or y — C (z - /3) = 0; viz. the section of the surface by
a plane of either series (which section is a circle) is a curve of curvature of the surface.
The equation of the cyclide can be elegantly expressed in terms of the ellipsoidal
coordinates (\, fi, v) of a point (x, y, z) ; viz. writing for shortness a = 6^ — c=, 0 = c' — a',
y = a? — h-, the coordinates (\, ^, v) are such that
- y37.«- = {a^ + \) (a- + fi) (a' + v),
- yay- = {b' + \) (6^ + ^) (6- + v),
- a^z' = (c» + \) (c^* + /jl) (c^ + v),
(see Roberts, Comptes Rendus, t. Liii. (Dec, 1861), p. 1119), whence
x^ + y' + z'=a' + b- + c' + \+ /i + v,
(6' + c»)ar'+ (c^ +a=)2/» + {a^ + lf)z^ = 6V + c'a" + a?h-- fiv - v\-\/j,.
The equation of the cyclide then is
V(a= + X) + v'(a'' + H') + \/(a' + ") = V(S).
In fact, starting from this equation and rationalising, we have
(3a» + X + A* + .- - «)» = 4 [V{(a» + /i) (a' + ")} + V{(a' + ") («" + ^)} + \f{{a- + X) (a= + /*)}]»
= 4 [3a* + 2a^ {\ + fjL + v) + fj,v + v\ + \fi + 2 V{(a' + X) (a" + /t) (a= + v)} \/(S)],
which, substituting for
X + fx + v, fiv + v\ + \fi and >^{{a' + X) (a^ + fi) (a^ + j/)}
their values, is
(ai' + y* + z' + y-0-By = 4,{(y-^)a^-0y^ + yz''-/3y- 2x V(- /3yB)},
or, writing — Jy, {^, \B- in place of /3, 7, 8 respectively, this is
(a;» + y" + «•-■ + Jr" + i^- i^T = (y' + ^)«^ + ^z' + 'ff + 1^ + /SySa;.
which agrees mth a foregoing form of the equation.
The generating spheres of the cyclide cut at right angles each of a series of
spheres; viz. each of these spheres passes through one and the same circle in the
plane of, and having double contact with, the conic which contains the centres of the
generating spheres; the centres of the orthotomic spheres being consequently in a line
meeting an axis, and at right angles to the plane of the conic in question. Or, what
is the same thing, starting with a conic, and a sphere having double contact therewith,
the cyclide is the envelope of a variable sphere having its centre on the conic and
cutting at right angles the fixed sphere.*
* I am indebted for thia mode of generation of a Cyclide to the researches of Mr Casey.
10—2
76 ON THE CYCLIDE. [569
It may be remarked, that if we endeavour to generalize a former generation of
the cyclide, and consider the envelope of a variable sphere having its centre on a conic,
and touching a fixed sphere, this is in general a surface of an order exceeding 4 ;
it becomes a surface of the fourth order, viz. a cyclide, only in the case where the
fixed sphere has its centre on the anti-conic. But if we consider the envelope of a
variable sphere having its centre on a conic and cutting at right angles a fixed
sphere, this is always a quartic surface having the circle at infinity for a double line ;
the surface has moreover two nodes, viz. these are the anti-points of the circle which
is the intersection of the sphere by the plane of the conic. If the sphere touches
the conic, then there is at the point of contact a third node ; and similarly, if it has
double contact with the conic, then there is at each point of contact a node ; viz. in
this case the surface has four nodes, and it is in fact a cyclide.
There is no difficulty in the analytical proof: consider the envelope of a variable
sphere having its centre on the conic Z = 0, — -pH =1, and which cuts at right
angles the sphere (x—iy + {y — my + {z — ny= Ii^.
Take the equation of the variable sphere to be
(x-Xr + (y-Yy + z^^c^
then the orthotomic condition is
{X-iy + (Y-my + n^ = c* + k',
or, substituting this value of c^ the equation of the variable sphere is
(x-Xy + iy- Yy + z' = -k' + (X-iy + (Y-my + n\
all which spheres pass through the points
x = l, y = m, z = ± \Hji? — Ar*) ;
that is,
!ia' + y^ + z^ + lc'-P-m?-n-'-2{x-l)X-2{y-m)Y=0,
X" Y-
and considering F, F as variable parameters connected by the equation — -| =1,
the equation of the envelope is
(af + y^ + z^ + Itr'-l^-m'-ny + 4!fi{x-iy-4!a(y-my = 0,
viz. this is a bicircular quartic, having the two nodes x=l, y = m, z=± \/{ii- — ]<^); these
are the anti-points of the circle {x — iy+{y — my = k^ — n', which is the intersection of
the sphere {x — ly + (y- my + {z — ny = k' by the plane of the conic.
The constants might be particularised so that the equation should represent a
cyclide ; but I treat the question in a somewhat different manner, by showing that
the generating spheres of a cyclide cut at right angles each of a series of fixed
spheres. Write a, 0, y = b^-c', c--a^, a'-b''; then if
the points {X, Y, 0) and (X,, 0, Z,) will be situate on a pair of anti-conics.
569] ON THE CYCLIDE. 77
Consider the fixed sphere
then if this is touched by the variable sphere
the last-mentioned sphere will be a generating sphere of the cyclide. The condition
of contact is
that is,
= -^Z'-2ZZ,-^Xj2
P 7
= n»,
if for a moment
that is, c = — Ci+ n, and the equation of the variable sphere is
{x-Xy + {y-Y)'+z^ = (c, - n)»,
where X, Y are variable parameters connected by
Suppose that the variable sphere is orthotomic to
{x-Xif->rf^{z-Z^ = c^,
the condition for this is
{X-Xif+Y''^-Z} = c'^-c^-,
or combining with the identical equation
{X-X,y^Y^^ZC- = {c^c,)\
we have
- IX {X, - Z) + Z./ - Zr + Z,^ - Z:- = c^ - c,= - 2ci (- c, + n)
= c/ + c,^ - 2c,n,
or, substituting for ^i, fi their values, this is
- 2Z (Z, - ZO + Z,^ - Z' + Z,^ - a (-'' - l) = c/ + c,» - 2c, {z y/(- ^) + ^i -^ (" ^)} >
78 ON THE CYCLIDE. [569
viz. this will be identically true if
X,' + Z^ - c.' = - 1 Z.» + 2ciZ, ^(- ^) - a + c.»,
or, as this last equation may be written
^.-0.= = ^.- c,= | + 2C.Z. {^(-f) +y/(-|)}.
The equation of the orthotomic sphere is thus found to be
or, what is the same thing,
o? + f^z^-lzZ^-tc\x^ + c^J(-'^^-xA-tc^X^J(-^^ + c,= - a = 0,
or, as this may be written
viz. this is
where Z^ is arbitrary. We have thus a series of orthotomic spheres; viz. taking any
one of these, the envelope of a variable sphere having its centre on the conic
— 0+ - — 1=0. and cutting at right angles the orthotomic sphere, is a cyclide. The
centre of the orthotomic sphere is a point at pleasure on the line
<r = Z. + c,^(-|), y=0;
and the sphere passes through the circle z = 0,
viz. this is a circle having double contact with the conic — 3 + — = 1 ; or, what is
p a
the same thing, the orthotomic sphere is a sphere having its centre on the line in
question, and havmg double contact with the conic - -g + — = 1.
P «
570]
79
570.
ON THE SUPERLINES OF A QUADRIC SURFACE IN FIVE-
DIMENSIONAL SPACE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873),
pp. 176—180.]
In ordinary or three-dimensional space a quadric surface has upon it two singly
infinite systems of lines, such that each line of the one system intersects each line
of the other system, but that two lines of the same system do not intersect.
In five-dimensional space* a quadric surface has upon it two triply infinite systems
of superlines, such that each superline of either system intersects each superline of
the same system ; a superline of the one system does not in general intersect a
superline of the opposite system, but it may do so, and then it intersects it not in
a mere point, but in a line.
The theory will be established by an independent analysis, but it is, in fact, a
consequence of the correspondence which exists between the lines of ordinary space
and the points of a quadric surface in five-dimensional space. Thus the correspondence is
In five-dimensional space.
Point on quadric surface.
Points which lie in tangent plane at
In ordinary space.
Line.
Lines meeting a given line.
Pair of intersecting lines.
Lines meeting each of two given lines.
given point.
Two points such that each lies in the
tangent plane at the other, or say, pair of
harmonic points.
Points lying in the sub-plane common
to the tangent planes at two given points.
* In explanation of the nomenclature, observe that in .5 dimensional geometry we have: space, surface,
snbsnrface, supercnrve, curve, and point-system, according as we have between the six coordinates 0, 1, 2, 3, 4,
or 5 equations : and so when the equations are linear, we have : space, plane, subplane, superline, line, and
point. Thus in the text a quadric surface is the locus determined by a single quadric equation between the
coordinates ; and the superline and line are the loci determined by three linear equations and four linear
equations respectively.
80 ON THE SUPEBLINES OF A QUADRIC SURFACE [570
But in ordinary space if the two given lines intersect, then the system of lines
meeting these, breaks up into two systems, viz. that of the lines which pass through
the point of intersection, and that of the lines which lie in the common plane of
the two given lines. It follows that in the five-dimensional space the intersection of
the quadric surface by the subplane common to the tangent planes at two harmonic
points must break up into a pair of superlines, viz. that we have on the quadric
two systems of superlines; a superline of the one kind answering in ordinary space
to the lines which pass through a given point, and a superline of the other kind
answering to the lines which lie in a given plane. (Observe that, as regards the
five-dimensional geometry, this is no distinction of nature between the two kinds of
superlines, they are simply correlative to e£ich other, like the two systems of generating
lines of a quadric in ordinary space.)
Moreover, considering two superlines of the first kind, then answering thereto in
ordinary space we have the lines through one given point, and the lines through
another given point; and these systems have a common line, that joining the two
given points; whence the two superlines have a common point. And, similarly, two
superlines of the second kind have a common point. But taking two superlines of
opposite kinds, then in ordinary space we have the lines through a given point, and
the lines in a given plane : and the two systems have not in general any common
line; that is, the two superlines have no common point. If, however, the given point
lies in the given plane, then there is not one common line, but a singly infinite
series of common lines, viz. all the lines in the given plane and through the given
point; and coiTCsponding hereto we have as the intersection of the two superlines, not
a mere point, but a line.
Passing now to the independent theory, I consider, for comparison, first the case
of the lines on a quadric surface in ordinary space; the equation of the surface maj'
be taken to be
(u, V, X, y ordinary quadriplanar coordinates) and the equations of a line on the
surface are
u = ax +^y,
v=tt.'x + ^y,
where a, /9, a', /3' are coefficients of a rectangular transformation, viz. we have oP + ^=\,
a!- + /3'» = 1, aa' -I- iS/3' = 0 ; and therefore {a^ - a'0f = 1, consequently a^' - a'/3 = + 1 ; and
the lines will be of one or the other kind, according as the sign is + or — . It is
rather more convenient to assume always ay3' — a'/9 = -|- 1, and write the equations
u= ax + ^y,
v = k (a'x + /9'y),
k denoting ± 1, and the lines being of the one kind or of the other kind, according
as the sign is -f- or -.
570] IN FIVE-DIMENSIONAL SPACE. 81
Thus considering any two lines, the equations may be written
M = aw; + /Sy , u = — {ax +by),
V = a'x + ^'y, V = — k {a'x + h'y),
where the lines will be of the same kind or of different kinds, according as k is
= + 1 or = — 1. Observe that k is introduced into one equation only ; if it had been
introduced into both, there would be no change of kind. If the lines intersect we have
(a + a)a; + (/3+ 6)y = 0,
(a' + ka!) a; + (/3' + ^6') y = 0,
viz. the condition of intersection is
a + ra, j8 + 6
o' + te', ^' + kh'
= 0,
that is,
a^ -a'^ + k (ab' - a'b) + a/3' -a'0 + k (ab' - a'b) = 0,
or, what is the same thing,
l + a^-a'^ + k{l+ab'-a'b) = 0.
But we have, say
o = cos 9, /8 = sin 0, a = cos 0, 6 = sin 0,
a' = — sin 6, /3' = cos 6, a' = — sin <f>, V = cos <^,
and thence
a^ - a'/3 = cos (0-<f>) = ab' - a'b,
and the equation is
(l + i'){l-|-cos(^-<^)} = 0,
viz. this is satisfied if ^= — 1, i.e. if the lines are of opposite kinds, but not if A; = -|-l.
And it is important to remark that there is no exception corresponding to the other
factor, viz. if k = + l, and 1+ cos (^ — (^) = 0, for we then have d—<^ = iT, cos<f> = — cosd,
sin if> = — sin 0, and consequently the two sets of equations for u, v become identical ;
that is, for lines of the same kind a line meets itself only.
Passing to the five-dimensional space, the equation of the quadric surface may be
taken to be
u" + v^ + iif - uy' - y'' - z^ = 0,
and for a superline on the surface we have
u =ax + ^y +yz ,
V =a.'x + /S'y + y'z ,
w = a"x + 0"y + y"z,
where (a, /8, 7), &c., are the coefficients of a rectangular transformation; the determinant
formed with these coefficients is = ± 1, and the superline is of the one kind or the
C. IX. 11
82 ON THE 8UPERL1NES OF A QUADRIC SURFACE [570
other, according as the sign is + or — . It is more convenient to take the determ-
inant to be always +, and to write the equations in the form
u —k{ax + /3y +72 ),
V =k (a'x + ^y + y'z ),
w = k(oi"x + ^'y + y"z),
where k= ±1, and the superline is of the one or the other kind, according as the
sign is + or — . 1
Now considering two superlines, we may write
M = OCT + ySy + 72 , u = — k {ax + by + cz ),
D = o'a; + /S'y + V'^ > '» = — k {a'x + b'y + c'z ),
w = a"x + fi"y + i'z, w = -k {a"x + h"y + c"z).
If the superlines intersect, then
(a +ka)x + {& + it ) 2/ + (7 + Arc ) 2 = 0,
(a' +ka')x-{-{0 ■\-kh')y-\-{',' +kc')z^O,
(a" + ka") x + (^' + kb") y + (7" + kc") z = 0,
viz. the determinant formed with these coefficients must be =0. The condition is at
once reduced to
l + ]^ + {k + i^)(aa + b0 + cy + a'a' + 5'/3' + cV + «"«" + 6"/9" + c'V") = 0,
viz. it is satisfied when k = —\, that is, when the superlines are of the same kind;
but not in general when A; = + 1.
If A; = + 1 the condition will be satisfied if
1+ aa + 6y9 + C7 + a'a' + 6'/3' + c'7' + a"a" + 6"/3" + c'y = 0,
and it is to be shown that then the three equations reduce themselves not to two
equations, but to a single equation.
It is allowable to take the second set of equations to be simply u=—kx, v = — ky,
w= — kz; for this comes to replacing the analytically rectangular system cuc + by + cz,
a''x + b'y + c'z, a"x + b"y + c"z by x, y, z. Writing also k = + l, the theorem to be proved
is that the equations
(o-l-l)a; + /3y +yz =0,
c^x +(0'+l)y + y'z =0,
a"x+ I3"y +7"^'=0,
reduce themsselves to a single equation, provided only 1 + a + /S' + 7" = 0 ; or, what is
the same thing, we have to prove that the expressions /S" — 7', 7 — a", a' — /8 each
vanish, provided only 1 + a + /9' + 7" = 0. This is a known theorem depending on the
570]
IN FIVE-DIMENSIONAL SPACE.
88
theory of the resultant axis, viz. the rotation round the resultant axis is then 180°,
and we have OX = OX', OY=OY', OZ=OZ\ and thence we have evidently YZ'=Y'Z,
ZX' = Z'X, XY'^X'Y.
But to prove it analytically, writing P, Q, R for 0"-y', 7 -a", a'-/8 respectively,
and fl for 1 + o + ^' + 7", observe that we have identically
(^' + y)[l = QR,
(7 + a' ) fi = iiP,
(a' +y9")n = PQ,
(y9" + 7')P = (7 + «") Q = («' + ^) «.
(a -l)n = - 7Q + ^R
o-n =- y'Q +(l+y3')i2
a"n =-(l+y")Q+ 0"R
m =-(l + a}R+ yP
(/S'-l)n = - a'R + y'P
ff'il =- a'R +(l+y")P:
70 = - /3P + (1 + a ) Q
7'n =-(l+/9')i^+ «'Q
(7"-l)n = - /3"P + a"Q
whence fl being =0, we have also P = 0, Q = 0, R = 0. The final conclusion is that
the two Buperlines of opposite kinds, when they intersect, intersect in a line.
11—2
84 [571
571.
A DEMONSTRATION OF DUPIN'S THEOREM.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873),
pp. 185—191.]
Thk theorem is that three families of surfaces intersecting everywhere at right
angles intersect along their curves of curvature. The following demonstration puts in
evidence the geometrical ground of the theorem.
I remark that it was suggested to me by the perusal of a most interesting paper
by M. L^vy, " Memoire sur les coordonn^es curvilignes orthogonales et en particulier sur
celles qui comprennent une famille quelconque de surfaces de second degrd," (Jour, de
I'Ecole Polyt., Cah, 43 (1870), pp. 157—200). It was known that a family of surfaces
p=/(x, y, z) where the function is arbitrary, does not in general form part of an
orthogonal system, but that p considered as a function of {x, y, z) must satisfy a
partial diflferential equation of the third order. M. Ldvy obtains a theorem which, in
fact, enables the determination of this partial differential equation; he does not himself
obtain it, although he finds what the equation becomes on writing therein t^ = 0,
-J- = 0; but I have, in a recent communication to the French Academy, found this
equation.
Proceeding to the consideration of Dupin's theorem, on a surface of the first family
take a point A and through it two elements of length on the surface, AB, AC, at
right angles to each other; draw at A, B, C the normals meeting the consecutive
surface in A', E, C and join A'F, A'C. It is to be shown that the condition in
order that RA'C may be a right angle is the same as the condition for the inter-
section of the normals A A' and BB' (or of the normals A A' and CC); for this being
80, since by hypothesis EA'C is a right angle, it follows that AA', BB" intersect;
571]
A DEMONSTRATION OF DUPIN S THEOREM.
85
that is, that AB is an element of one of the curves of curvature through the point
A of the surface. And, similarly, that AA', CC intersect ; that is, that AC is an
element of the other of the curves of curvature through the point A on the surface.
Take x, y, z for the coordinates of the point A; a, /9, 7 for the cosine inclinations
of AA' ; a,, /3i, 71 for those of AB; and a^, /Sa, 72 for those of AG. Write also
S = a dx + yS dj, + 7 d^,
S] = ajdx + ^idy + 7,^2,
^3 = Oorfj; + /Sady + 72^2 ;
then it will be shown that the diondition for the intersection of the normals AA', BF is
a2Sia + /3A/3 + 728,7 = 0,
the condition for the intersection of the normals AA', CC is
0,820 + ^82^ + 7.827 = 0,
and that these are equivalent to each other, and to the condition for the angle B'A'C
being a right angle.
Taking I, i,, l^ for the lengths AA', AB, AC, the coordinates of A', B, C measured
from the point A are
{la, l^, ly), (i,a„ l^i, i,7,), (Ia, IA> hi) respectively.
The equations of the normal at A may be written
X = x + da,
where X, Y, Z are current coordinates, and ^ is a variable parameter. Hence for the
normal at B, passing from the coordinates x, y, z to x-\-lfl^, y + lSx, ^ + ?i7i, the
equations are
Z=a;+^a, + ^A (6'a),
F = 2/ + iA + ^.8i(^/8),
Z = 2 + ?,7, + lA {O-i),
86 A DEMOXSTBATION OF DUPIN's THEOREM,
and if the two normals intersect in the point (X, V, Z), then
Oi + ah,d + ^S,o = 0,
7i+ 781*9 + 6'S,7 =0,
viz. eliminating 0 and h^O the condition is
[571
a, , a , 8,a
7i . 7 > 8,7
= 0;
or, smce
this is
We have
«s, A, 72 = /37i - ^i7. 7'*i - yi«> "A - a,/3,
OjSia + ^ihS + 728,7 = 0.
Similarly the condition for the intersection of the normals AA', CC is
OiS^a + /SiSj/S + 71827 = 0.
ajS,a + /9s8,/3 + 7,817 = OiSja + ASj/S + 71827 ;
in fact, this equation is
(0,81 - ai8,) a + (;S,8i - ^A) /3 + (7281 - 718.) 7 = 0,
which I proceed to verify.
In the first term the symbol OoSi — a^h^ is
Ok (aidic + /9 A + 7i4) - Oi (Mx + My + lA),
viz. this is
(aojSi - Bi^Sj) dy + (7103 - 7,0,) dt ;
or, what is the same thing, it is
^dz — fdy,
and the equation to be verified is
(/84 - idy) a + (74 - adz) /8 + (arfy - ^d^) 7 = 0,
Z 1' -?
viz. ^vriting
a. /3. 7 =
ii' E' iJ'
where if p=f{x, y, 2) is the equation of the surface X, Y, Z are the derived functions
-4-, -T- , -f , and R = JiX^ +Y^ + Br), the function on the left-hand consists of two
ax dy dz
parts ; the first is
i {{^dz - ydy) X + (74 - adz) y + (ad, - ^4) ^,
571] A DEMONSTRATION OF DUPIN S THEOREM. 87
that is,
-^ {a {dyZ- d,Y) + ^ {d,X - d,Z) + 7 {d,Y- dyX)],
which vanishes; and the second is
- -^ {a (ySrf, - r^dy) + ^ (7d« - ad,) + 7 {ady - ^d^)] R,
which also vanishes; that is, we have identically
OjSia + /SaSijS + 7A7 = "i^aa + ^i^S + 71S27,
and the vanishing of the one function implies the vanishing of the other.
Proceeding now to the condition that the angle B'A'C shall be a right angle,
the coordinates of jB' are what those of A' become on substituting therein x + lid^,
y + ^ojSi , z + Iffi in place of x, y, z; that is, these coordinates are
a; + Za + iiOj + l^h-^ (la), &c.,
or, what is the same thing, measuring them from A' as origin, the coordinates of B' are
li (a, + ISia + a 8^1),
I, (A + IB^8 + ^B,l),
h (71 + '^7 + T^iO.
and similarly those of C measured from the same origin A' are
4 (a, + IZ^a + a hj,),
k{8^ + lS^ + ^B,l),
l,(y,+lS,y+yU).
Hence the condition for the right angle is
(«! + lS,a + aBil ) (a, + lS,a + aBJ, )
+ (A + IB,0 + /38i0 (8, + IB^ + /SSjO
+ (71 + l^iy + y^il ) (72 + ^S.7 + y^^l ) = 0.
Here the terms independent of I, BJ, B.J, vanish; and writing down only the terms
which are of the first order in these quantities, the condition is
tti (IB^a + aBJ,) + a^ (IS.a + aBJ)
+ 8,ilS^ + ^BJ) + 8,(i^i0 + ^^ii)
+ 7i (18^ + 78.0 + 72 (^^17 + 7S1O = 0,
where the terms in BJ, BJ vanish ; the remaining terms divide by I, and throwing
out this factor, the condition is
(aM + ^A^ + 7.S27) + ("Aa + ^2^,^ + 7A7) = 0,
88 A DEMONSTRATION OF DUPIN'S THEOREM. [571
viz. by what precedes, this may be written under either of the forms
fliSja + fiiBi0 + 71847 = 0,
a,S,a + ^M + 7=8.7 = 0,
and the theorem is thus proved.
It may be remarked that if we had simply the first surface, and two other surfaces,
or say a second and a third surface, cutting the first surface and each other at right
angles, that is, cutting each other in AA' the element of the normal at A, and cutting
the first surface in the elements AB, AC at right angles to each other, then the
tangent plane of the second surface will be the plane A'AB, not in general passing
through B"; and the tangent plane of the third surface will be the plane A' AC, not
in general passing through C. The condition, that the elements A'B" and A'C on the
surface consecutive to the first surface are at right angles, makes CC and BB' each
intersect A A' ; and we then have, the tangent plane of the second surface is the plane
through the elements AA', BR, the tangent plane of the second surface is the plane
through the elements A A', CC.
As already remarked, a family of surfaces p =/(«, y, 2) where the function is
arbitrary cannot form part of an orthogonal system. In fact, if the surfaces do belong
to an orthogonal system, we have AA', BB' in the same plane, and consequently AB
and A'R intersect; and, similarly, AC and A'C intersect; that is, if firom a point A
on a given surface of the family we pass along the normal to the point A' on the
consecutive surface; and if the lines AB, AC are the tangents to the curves of
curvature at A, and A'B', A'C the tangents to the curves of curvature at A', then
AB intersects A'R, or, what is the same thing, AC intersects A'C; and, conversely,
when this condition is satisfied in general (that is, for every surface of the family and
the surface consecutive thereto), then the family forms part of an orthogonal system;
this is, in fact, the fundamental theorem of M. Levy's memoir. The analytical form
of the condition, viewed in this manner, is
o,Sai + /3M + 72^71 = 0. or a.8a, + /9.8A + 7:87= = 0 ;
or, as it is convenient to write it,
a,8a. + /3,S/3, + 7,87, - (o,Sa., + 0,B0, + 7,87,) = 0 ;
and it was by means of it that I obtained the partial dififerential equation of the
third order above referred to. The condition written in the form
X,SXi + FjS F, + Z,BZ, = 0, or Z,SZ, + F,S F, + Z,BZ^ = 0,
presents itself in the proof of Dupin's Theorem by R. L. Ellis, (given in Gregory's
Examples, Cambridge, 1841), but the geometrical signification of it is not explained.
Closely connected with Dupin's, we have the following theorem: if two surfaces
intersect at right angles along a curve which is a curve of curvature of one of them,
it is a curve of curvature of the other of them. I remark hereon as follows :
571]
A DEMONSTRATION OF DUPIN S THEOREM.
89
Let the intersection be a curve of curvature on the first surface ; the successive
normals intersect, giving rise to a developable, and the intersection of the two surfaces,
say /, is an involute of the edge of regression of this developable, say of the curve G.
The successive normals of the second surface are the lines at the different points of
/ at right angles to the planes of the developable, that is, to the osculating planes
of C; or, what is the same thing, they are lines parallel to the binomials of G (the
line at any point of a curve, at right angles to the osculating plane, is termed the
" binormal "). But if the intersection / is a curve of curvature on the second surface,
then the successive lines intersect; that is, starting from the curve G, the theorem in
effect is that at each point of the involute drawing a line parallel to the binormal
of the corresponding point of the curve, the successive lines intersect, giving rise to a
developable. To prove this, let the arc s be measured from any fixed point of the
curve, and the coordinates x, y, z be considered as functions of s ; and let «', x" , x"'
dsc d'X o?3C
denote -j- , -t^< j^j and the like as regards y and z. Measuring off on the tangent
at the point {x, y, z) a length l — s, the locus of the extremity is the involute ;
that is, for the point (x, y, z) on the curve, the coordinates of the corresponding point on
the involute are x + {l — s) x', y + {l — s)y', z + (l — s) z'. Moreover, the cosine inclinations
of the binormal are as y'z"-y"z!, zfx" - z"x', x'y"-x"y'. Hence taking X, Y, Z as
current coordinates, the equation^ of the line parallel to the binormal may be written
X = x-^{l-s)x' + e {y'z" - y"z'\
Y = y + {l-s)y' + 6 (z'x" - z"x'),
Z =z + il-8)z' + 0 ix'y" - x"y'),
and the condition of intersection is therefore
of', T/z"-y"z'. {^z"-y"z')' 1=0.
f, /a!"-/V, {z^x" -z"x')' I
z\ x'f-^y, {^'y"-«^'y')' I
Foi-m a minor out of the first and second columns, e.g.
y" {x'y" - x"y') - z" {z'x" - z"x'),
this is,
^.' (x"' + y". + y'«) _ x" (x'x" + y'y" + !!z"),
or the last term being =0, and the factor a;"» + y"^ + /'' being common, the minors
are as a^ : jr' : /. Moreover {y'z" -y"z')' =y'z"' -y"'i^, &c., hence the determinant is
a/ {y'z'" - y" V) + y {z'a/" - z"'x') + z' {x'y'" - x"'y'),
viz. this is =0, or the theorem is proved.
C, IX,
12
90
[572
572.
THEOREM IN REGARD TO THE HESSIAN OF A QUATERNARY
FUNCTION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873),
pp. 193—197.]
I WISH to put on record the following expression for the Hessian of P* + XP**',
where P, P' are quaternary functions of {x, y, z, w) of the degrees I, I' respectively,
and X is a constant; the demonstration is tedious enough, but presents no particular
diflSculty.
I write {A, B, C, D) for the first derived functions of P; and (a, h, c, d,f, g, h, I, m, n)
for the second derived functions ; and similarly for P'. The Hessian of P is thus
a, h, g, I
h, h , f, m
9, f , c, n
I, m, n, d
which is denoted by (abed); moreover, if in this determinant we substitute the accented
letters for the letters of each line successively, the result is denoted by (abed'); and
so if we substitute the accented letters for the letters of each pair of lines successively,
the result is denoted by (abc'd'). Observe that
abed' = (a'Ba + b'Bb + ...) abed and abc'd' = ^ (a'Sa + b'Bi,+ ...y abed.
The notation (oicZ)'') is used to denote the determinant
A', B', C, D'
A', a , h , g , I
B' , h , b , f , m
C , g , f , c , n
U, I , m, n , d
572] THEOREM IN REGARD TO THE HESSIAN OF A QUATERNARY FUNCTION. 91
and from it we derive the expression {ahc'D'-), viz.
ahc'iy"- = {a'K + h% + . . . ) abcD\
The final result is expressed in terms of the several functions ahcd, abed', abc'd', ab'c'd',
a'b'c'd', abcD'-, a'b'c']}', abc'D'-, a'b'cBf, viz. we have
f A- (A: - 1) f -JL + U ifc'P»*-3 J ^'""' • «^^'^' 1 1
^J \k-\ 1-1} \+{k'-\)F'''-Kabcm
"T* ^1 f
^-"(^-l)^^'
[r {l'-\) + P{k-\)'\ P=*-^ P'*' . a6cd
{i-iy
( k'k'^ P*-2 P'^-= abc'd'
+ Ifik'' (k'-l) P^-^ F'*'-' oic'i)'''
+ )t-"^-'» (A; - 1) P*-' P*'-" a'6'ci>
^
+ V
+ ;fc=A;'' (k-l) {k' - 1) P^-' F^'-' -
J"
(l-l)(l'-l)
I
„ ab'c'd'. P'
abed' . PF
-abcd'.P''
+ \'
( /I V \,T.,i.. \ -P*"' • a<}'c'd 1
A'» (F - 1) rAr + yr- 1 kP'^ ' \ \
'\k-l l-\) \+ (A- - 1) P*-2 . a'b'c'D"]
P^-K a'b'c'd 1 1
l)P*-2
_ ^'' (f^_A)i [i (i - 1) + i'^ ('fc' - 1)] P* P''*'-* . a'b'c'd'
+ V . ¥* {k' - 1) (jpi^j + ^^j) F^'- . a'6'c'ci'.
In verification, I remark that, X = 0, the formula becomes
I
^ (P*) = k^ik-l) (^^ J + ^^ P^-* . a6cd,
that is
= ^!^^Lz})p4k-*_abcd.
12—2
92 THEOREM IN REGARD TO THE HESSIAN OF A [572
Hence, writing P' = P, which implies k' = k and I' = I, we ought to have
JQ {(1 + X) P*} = (1 + xy . ^^^^y^ P"'-* . abed.
But writing in the formula P' = P, it is to be observed that abed' = 4o6cd, abed' = Gabcd,
ab'c'd' = iiahcd, a'b'c'd' = ahcd : moreover that abciy and a'b'c'iy are each = abcIP, but
that abc'jy^ and a'b'cD' are each =3aicZ)', and (as is easily shown to be the case)
abcl>' = j—j P. abed.
Thus the whole coefficient of \ becomes
where the numerical factor is
■ P^-* . abed,
or, finally, it is
The coefficient of V is
~ l-l •
+ t je {k - 1) + ^-^^£^} i^-' . abc]> ;
or, substituting for abcLy its value = -. — - P . abed, the expression is equal to P*'^~*ahed
into a numerical coefficient, which is
viz. this is
, 2l-{k-\r f6(k-l)l 2(k-iyi^\)
*^*'~ (I -if ^[ i-i '^ (i-iy Jl'
= 6
k*{kl-l)
l-l '
and the coefficients of X', and X* are equal to those of \ and X" respectively. Hence
the formula gives, as it should do,
^ {(1 + \) P*) = (I + \)* ^!iM^) p**-. . abed.
572]
QUATERNARY FUNCTION.
93
Attending only to the form of the result, and representing the numerical factors by
A, B, &c., we may write
J&(P* + \P'*)= A P^-*abcd
+ \ . B fP'^-'P''^-'abcd' "I
\+ (k'-l) P»*-' P'^-- abcD'^j
+ G P^-'P'^abcd
+ XK D P^-- P'*-= abc'd'
+ E P^-' P'-^-' abc'D"
+ E' P^-^ F"^-^ a'b'cD^
+ P P^-»P''*'-»(AP + A'P')
+ \^ C" P^P'^-^a'b'c'd'
+ B' (P^-^F^-^a'b'c'd ^
1+ (^ - 1) P*-^ P'*-» a'b'c'iy^
+ X*. A' P'*^-*a'b'c'd',
where, for shortness, certain terms in V have been represented by AP + A'P'.
Suppose t = ^•' = 2 ; then attending only to the terms of the lowest order in P, P'
conjointly, we have
^ (F + \F') = XB .PK abcD'^
+ V . PP' (AP + A'P')
+ \'B' . P'' . a'b'c'D\
If the function operated upon with ^ had been UF+ VF', the lowest terms in
P, P' would have been of the like form; and it thus appears that for a surface of
the form UF + UF"^ = 0, the nodal curve P = 0, P' = 0 is a triple curve on the Hessian
sur&ce.
If i = 2, k' = 3, then attending only to the terms of the lowest order in P, P'
conjointly, we have
^ (P" + XF') = A.P*. abed
+ X.2B.FP' .abcD'^;
and the like result would be obtained if the function operated upon with Jq had been
UF + UF'\ It thus appears that for a surface of the form UF + UF'^ = 0, the
cuspidal curve P = 0, P' = 0 is a 4-tuple curve on the Hessian surface, the form in
the vicinity of this line, or direction of the tangent plane, being given by
P'iA.P. abed + 2BX.F. abcD'') = 0,
viz. there is a triple sheet P' = 0, coinciding with the direction of the surface in the
vicinity of the cuspidal line ; and a single sheet
A.P. abed + 2BX . P' . abcB'' = 0.
At the points for which the osculating plane of the curve P = 0, P' = 0 coincides with
the tangent plane of P = 0 (or, what is the same thing, with that of the surface), we
have abeiy^ = 0, and the triple and single sheets then coincide in direction.
94 [573
573.
NOTE ON THE (2, 2) CORRESPONDENCE OF TWO VARIABLES.
[From the Qimrterly Journal of Pure and Applied Matliematics, vol. xil. (1873),
pp. 197, 198.]
In connection with my paper " On the porism of the in-and-circumsciibed polygon
and the (2, 2) correspondence of points on a conic," Quar. Math. Jour., t. XI. (1871),
pp. 83 — 91, [489], I remark that if {&, <f>) have a symmetrical (2, 2) correspondence, and
also (<^, x) *^® same symmetrical (2, 2) correspondence, then (d, x) will have a (not in
general the same) symmetrical (2, 2) con-espondence. In fact, to a given value 6 there
correspond, say the values <^, <^a of <f>; then to <^i correspond the values 6, Xi of X
(viz. one of the two values is = 6), and to </)o the values 0, x^ of X (^*^- one of the
values is here again = 6) ; that is, to the given value 0 there correspond the two
values Xi, x^i of X' ^"^^ similarly to any value of x there correspond two values of 6;
viz. to Xi t^® value 0 and say ^, ; to X' ^^^ value 0 and say 0.<; that is, the
correspondence of 0, x ^^ ^ (2, 2) correspondence and is symmetrical.
Analytically, if we have
(a, b, c,f,g, hl0<t>, 0 + <p.l)r = O,
and
(a, b, c, f, g, h\^X' <^ + X. 1)' = 0,
then writing
(a, ...^(^M, (^ + u, \f = 0,
the roots hereof are u = 0, u = x', ie. we have
(o,...$^M, <}> + u, l)= = (a,...$<^, 1, Oy{u-0)iii-x)i
573] NOTE ON THE (2, 2) CORRESPONDENCE OF TWO VARIABLES. 95
or, what is the same thing, we have
1 :-(e + x) : ex = {a,...\<i>, 1, 0)' : 2(a,...][<^, 1, Op, <i>, 1) : (a, ...]10, </., 1)=
= a<^^ + 2A</) + 6 : 2{h(f>-'Jrh + 'g^+f) : b<j>' + 2f<}> + c,
giving (f)" : (f) : 1 proportional to linear functions of 1, 0 + x, ^X' ^"^^ therefore a quadric
relation {*^dX' ^ + X' 1)'' = 0, with coefficients which are not in general (a, b, c, f, g, h).
Suppose, however, that the coefficients have these values, or that the correspondence is
(a, b, cf.g, h^0X' ^ + X' 1? = 0.
we must have
(a, b, c. f, g, h^a^ + 2A<^ + b, - 2 {h(f>' + b + g(j> +/), b(}>' + 2/0 + c)= = 0,
that is,
or, we have
(ac + b'+2bg-ifh)(a, b, c, /, g, h\<\>\ - 2<^, iy = 0,
ac + b^ + 2bg - ifh = 0,
as the condition in order that the symmetrical (2, 2) correspondence between 0 and x
may be the same correspondence . as that between 0 and <f>, or between <f> and x-
96 [574
574.
ON WRONSKI'S THEOREM.
[From the Qiixirtei'ly Journal of Pure and Applied Mathematics, vol. xil. (1873),
pp. 221—228.]
The theorem, considered by the author as an answer to the question " En quoi
consistent les Math^matiques ? N'y aurait-il pas moyen d'embrasser par un seal problfeme,
tous les problfemes de ces sciences et de resoudre g^ne'ralement ce problfeme universel?"
is given without demonstration in his Refutation de la Theorie de Fonctions Analytiques
de Lagrange, Paris, 1812, p. 30, and reproduced (with, I think, a demonstration) in the
Philosophie de la Technie, Paris, 1815 ; and it is also stated and demonstrated in the
Supplement d la Reforme de la Philosophie, Paris, 1847, p. cix et seq. ; the theorem,
but without a demonstration, is given in Montferrier's EncyclopMie Mathematique (Paris,
no date), t. iii. p. 398,
The theorem gives the development of a function Fx of the root of an equation
0 =fx + XifiX + Xif^X + &c.,
but it is not really more general than that for the particular case 0 =fx + x^f^x ; or
say when the equation is 0 = <^ + \fx* Considering then this equation
<^ + \fx — 0,
let a be a root of the equation <^ie = 0; the theorem is
Fx = F
_\ \
If
+ ^-
^1.2^"
* For in the result, as given in the text, instead of \fx write xJ-^x\x,Jx-\-&<i., then expanding the several
powers of this quantity, each determinant is replaced by a sum of determinants of the same order, and we
have the expansion of ¥x in powers of %, x^
(//-f')' 1
I
f, (//^F'X
1
"'»
<^". (//^^')"
1
574]
ON WRONSKI S THEOREM.
97
1 . 2 . 3 </)'«
+ &C.,
where ^, /, F', &c. denote Fa, fa, Fa, &c. and the accents denote differentiation in
regard to a; the integral sign / is written instead of /„; this is introduced for
symmetry only, and obviously disappears; in fact, we may equally well write
Fx = F
■•" 1 . 2 </>'*
X" J.
1.2.3 f
+ &c.
</>', PF |1
4>". ifF')' ! ^
1
1.1.2
I stop for a moment to remark that Laplace's theorem is really equivalent to
Lagrange's ; viz. in the first mentioned theorem we have x = <^{a + Xfx), that is
^~' a; = a + \f4> . <^~' «, and then Fx = F<f) . c^"' x, viz. by Lagrange's theorem
Fx = F<f> + ^ Fcf>' ./^ + ^ {Fcf>' . {f<f>y\' + &c.,
where on the right hand F^ and /4> are each regarded as one symbol, the argument
being always a and the accents denoting differentiation in regard to a, thus F<ji' is
da . F<f>a = F'(f)a . ^'a, &c.,
viz. this is Laplace's theorem.
Suppose in Wronski's theorem if)x = x — a; that is, let the equation be
x — a + \<})x — 0,
then each determinant reduces itself to a single term : thus the determinant of the
third order is
(x-a)', {{x-ayy , pF
(x-a)". {{x-ary, ipF')'
{x-a)'", [{x-afX", (f'F)"
where in the first and second columns the accents denote differentiation in regard to
X, which variable is afterwards put = a ; the determinant is thus
1, * , •
0, 1 . 2, *
0, 0 , i/'F)"
C. IX.
13
98
viz. it is
ON WR0NSKI8 THEOREM.
[574
= 1.1.2 {pF'f,
and so in other cases ; the formula is thus
Fx^F- \fF' + j^ {pF'y - j-|^ if^F')" + &c.,
agreeing with Lagrange's theorem.
Suppose in general <^ = (a; — a) -^x, or let the equation be
{x — a) y^x + \fx = 0,
that is,
! - a + X •(- = 0 :
we have then by Lagrange's theorem
Consider for example the term l-^'f,) [ > tbis is
the accents denoting differentiation in regard to x, and x being ultimately put = o ;
or, what is the same thing, it is
the accents now denoting differentiation in regard to 6, and this being ultimately put
= 0. This is
^.y
\de.
F' {a + e)
{f{a+m
This may be written \^'f^-rA > where
it being understood that as regards F' f^, which is expressed as a function of a only
(^ having been therein put = 0), the exterior accents denote differentiations in respect
to a, whereas in regard to A, =</>' + ^^(^" + &c., they denote differentiation in regard
to d, which is afterwards put =0. And the theorem thus is
^'■"^-i(^^-3)+,-2(^/'i)'-il3(^:^-i-.)"**=-
574]
ON WRONSKIS THEOREM.
99
This must be equivalent to Wronski's theorem; it is in a very different, and, I think,
a preferable form ; but the results obtained from the comparison are very interesting,
and I proceed to make this comparison.
Taking the foregoing coefficient (F'/^ -jg J this should be equal to Wronski's term
f , (<fry . pF'
<t>", m", (pF'Y
4>"'< (<f>T> {pF')"
1 . 1 . 2 ^'«
or, what is the same thing, the determinant should be
= 1 . 1 . 2,/,'« \rF' (1)" + 2 (/'i-y ( I3)' + ipn" ^
that is, the values of
1.1.2,^"]3, 1.1.2f«2(i-3)', 1.1.2f«(l3)"
should be
=f (<^)"-<^"(<^% f"W-fW". <i>"(<f>r'-<i>"'(<f>r
respectively. Or, what is the same thing, if
1
' . ^ J." • ^ A'" J. V
{l>'4l>' •2.S
= A, + \A,d + ^^A,e=+...,
then the last mentioned functions should be
1.1.2<^'M„, 1.1. 2^2^, 1.1.2^ A.
We have
Ao=-^,, 'd;--2^-4, ^2- ^'4-t-^'S'
or the identities are
+ 6</."=<^' - 2</."' </.'= = <^" (0=)'" - <^"' (<^=)", = <t>" i^H'" + 6f <^") - r i^H' + ^4>'%
which is right. And in like manner to verify the coefficient of V, we should have
to compare the first four terms of the expansion of
1
/ , ^ . //
[^'^y
■^2:3'^'"+
13—2
100 ON WRONSKI'S THEOREM. [574
with the determinants formed out of the matrix
1/ Aft ittt ttni
(,^')'. (<^')". m"> m""
(,/,')', (.^»)", m'\ m""
The series of equalities may be presented as follows, writing as above A to denote the
function
1
A
-r^
1
1
-1
^, 1
1
A^
"f'
<^'. f
1'
1
A'
+ 1
1
1
•1.1.5
1
A*
-1
id .
1
(•^^x, (^n",
(<^o"',
(<^r'
(•^o; m",
(.^O'".
(^0""
1.1.2.1.2.3'
&e.,
where in each case the function on the left hand is to be expanded only as far as
the power of 0 which is contained in the determinant: the numerical coefficients in
the top-lines of the several determinants are the reciprocals of
7i(n-l)...2.1, n(n-l)...2, n(n-l), n, 1,
where n is the index of the highest power of 0. The demonstration of Wronski's
theorem therefore ultimately depends on the establishment of the foregoing equalities
As a verification, in the fourth formula, write </> = e" (a = 0), we have
{?^r
or
(l + i0 + i^ + ^^ +...)*
= -iV
^^, i^, ^0, 1
1, 1, 1, 1
2, 4, 8, 16
3, 9, 27, 81
where the right hand is
= - 3ij(- 1 • 12 + i^- 72-^^.132 + ^^.72)
i-2^ + -y-^-^".
574]
ON WRONSKIS THEOREM.
101
and expanding the left hand as far as 6', this is
= 1 =1
- 4(^5 + ^^ + ^^3)
+ 10(
-20(
-26- |^-^<
i^+ i<
+ |(9^ + |^3
i^')
— ^0
1-26 + i^e^ - 6\
which agrees.
Reverting to the above equations, and expanding the several terms {^'^)' = 20^',
(<^2y' _ 2<^(^" + 2^'^, &c., then, since in each case the left-hand side contains ^', (/>", <^"',
&c. but not </), it is clear that on the right-hand side the terms involving ^ must
disappear of themselves; and assuming that this is so, the equality takes the more
simple form obtained by writing in the foregoing expressions </> = 0, viz. we thus have
(<^-)' = 0, (<^'-)" = 2<f>'^, &c. In order to simplify the formulae, I replace the series <^', \<^",
i4>"'' A"^'"' "^^^ ^y ^' '^' ^' ^' ^-y ^°^ ^ *^"^ ^^^ *^^*' *^®y assume the following
simple form, viz. writing
@ = b + c6 + de' + ed' + &c.,
then we have
1 1
e~ b-
1,
12 6,
c
)
03- + 6a ^^
c
, d
>
6^
, 2bc
1 4
&* b">
b,
c
P, ^6,
, d,
e
i
^ 26c,
6^
26d -1- c'
Sb'o
viz. for 0~" the right-hand gives the development as far as 6"^\ It will be observed,
that in the determinants the several lines are the coefficients in the expansions of
0, 0", ©», &c. respectively.
The demonstration is very easy; it will be sufficient to take the equation for ^.
Assume
^=...r6' + q6' +p6* + ^6' + ^yd" + ^86 + ^e,
4
where clearly e = ^, and write also
0 =B, + C,6 + B,6' + E,6' + ....
0^= B, + C,6 +0,6"+...,
03= B, +0,6 +...,
102
ON WR0N8KIS THEOREM.
[574
where fi, = 6, Bj = &', 5, = 6'; we wish to show that
/95, + 7C. + SA + e^i = 0,
75j + BC, + eDj = 0,
8B, + eC, = 0,
for this being the case, neglecting the terms in 0*, 9°, &c., and writing
then eliminating /9, 7, S, e, we have
^-. *^. i-d
60*
A
= 0,
■Si. C^ii A.
Ai C?2 ,
A,
in which equation the term which contains
14
and the equation thus is ^u = ~ /To multiplied by the determinant without the term in
question (that is, with J for its corner term).
To prove the subsidiary theorems, multiply the expression of ^ by ^^ , and
differentiate in regard to 0, we have
(6®)' ~^^''~^'*'^"^^'^^ + 0»-
Multiplying by
e® = B,e + Ci^ + A^' + E,e*,
we see that A/3 + O17 + AS + A« is the coefficient of ^ in At^ttt ; ^.nd similarly
1 4 (^0)' 1 4 (^0)'
A7+C'jS + ^s6 is the coefficient of ^ in A^.^ , and AS + Cje that of g in 7^^^-
Now, m being any positive integer, .^r^^ expanded in ascending powers of 6 contains
negative and positive powers of 6, but of course no logarithmic term ; hence differ-
entiating in regard to 6, 7^a\m+i contains no term in ^;* and the expressions in
question are thus each = 0 ; which completes the demonstration.
The foregoing formulse giving the expansion of >,„ up to ^~' in terms of the
coefficients in the expansions of 0, 0^ ... 0"~' are I think interesting.
* Thi8 is a well-known method made nse of by Jacobi and Murphy.
575] 103
575.
ON A SPECIAL QUARTIC TRANSFORMATION OF AN ELLIPTIC
FUNCTION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873),
pp. 266—269.]
It is remarked by Jacobi that a transformation of the order n'n" may lead to a
modular equation
A' _ n' Z'
£!L~n" K'
and in particular when n' = n", or the order is square, then the equation may be
A' K'
— = -^ ; viz. that instead of a transformation we may have a multiplication. A quartic
A K.
transformation of the kind in question may be obtained as follows : writing
X={a, b, c, d, e'^x, iy = a{x-a){x- ^)(x-y)(x-B),
H the Hessian, <I> the cubi-covariant, / and J the two invariants, then there is a
well known quartic transformation
leading to
dz _ 2 V(- 2) dx
V(^- V(^) '
where Z =2^ — 7^ + 2/. In fact we have
Z = J-, (4JEf » -IH'X + JX% = ~.f <^\
that is,
V(^ = ^^^VW,
104 A SPECIAL QUABTIC TRANSFORMATION. [575
80 that, by Jacobi's general principle, it at once appears that we have a transformation
of the form in questioa
Now we may establish a linear transformation
such that to the roots z^, z^, z, of the equation :!^ — Iz ■\-2J=0 correspond the values
a, /9, 7 of 2/ ; and this being so, we have between y, z the relation
dz ^V(-2)dy
V(^) V(F) ■
where Y — a{y — a){y — ^){y — 'y){y — h), =(a, b, c, d, e\y, 1)*; that is, we have
py + q_yS
y-B -X'
such that
dy _ 2dx
which is a quartic transformation giving a duplication of the integral. The foundation
of the theorem is that we can determine p, q in such wise that the functions
pa + q
p0 + q
py + q
a-B '
yS-S '
y-B
shall be the roots Zj, z^, Z3 of the equation z^ — Iz + 2J=0. For writing
A=(^-y)ia-B),
B=(y-a)(^-B),
C=(a-/3)(7-S),
and observing the equations
I^^iA-' + B' + O), =-~(BC+CA+AB),
^'r = -^iB-C){C-A){A-B),
(since A+B + G = 0) and
the equation in 2 is
{z-ia(B-C)}{z-isa(C-A)}{z-^a(A-B)].
and the equations for the determination of p, q thus are
pa +q = ^a(a -B)(B-C). =ia(a - S){2iaB + ^y)-{a + B){^ + y)},
pfi + q = ia{^-B)(C-A), =ia{^-B){2i^B + y<i)-(fi + B)iy+a)},
py+9 = ia(7-S)(4-fi), = |a (7 - S) f2 (7S +a;9) -(7 + S) (a + /3)],
575] OF AN ELLIPTIC FUNCTION. 105
giving
jB = ^a {- 38= + 28 (a + /3 + 7) - /37 - 7a - a;8},
^ = ^a {S- (a + /3 + 7) - 28 (/37 + 7a + ay9) + 3a/37),
or, as these may also be written
^ = ^a{(/3-S)(7-8) + (7-8)(a-8)+(a-8)(^-S)},
r? = ia{a(/3-8)(7-8)+/9(7-8)(a-6) + 7(a-S)(^-8)); .
and observe also
;>8 + ? = ia(a-8)(/8-8)(7-8).
Taking X in the standard form ={\- a?)(\— Ic^a?), and writing
7 = -l, 8 = 1, «=+;[.. ^=~l'
we have
_pj + q _ -i\2k^l + if) (1 + k^af) + (1 - lOA;^ + 1) a^}
i A = -
-I-.i
B =
T 2 1
C =
4
z,= J (1 + 6k + k'),
z„= i (1 - &k + h?),
^3 = - J (1 + A:^) ;
Z = s= - tV (1 + 14i» + k')z + T7S (1 + ^) (1 - 34A;^ + k")
= (Z- Z^ (Z - Z^ (Z - Z3),
p = Hl-5n 'Z = i(5-n p + q^l-k";
giving as they should do
i-1 ---\ ~
Write for shortness
- ^ {2/fc^ (1 + A;») (1 + iP*) + (1 - 10X-= + i*) a;^} = Q,
so that
i'y + g _ Q
y-1 ~^'
C. IX. 14
106 QUARTIC TRANSFORMATION OF AN ELLIPTIC FUNCTION. [575
then
X '~k-l- y-l'
Q_,_P+Jl A-y + 1
X '~;t+l" y + l '
^ , -P+1 y+l
X ^•- 2 -y-l-
The last of these is
that is,
l-fc" y + l ^ ^ (1 - ^y ar-
2 ^ i/-l~(l-a;')(l-ifc'a:')'
y + l _ J^^)J^
y - 1 "(1 + a!»)(l - Jfc»a?) '
from which the foregoing equation
dy _ 2dx
V(F)~VW
may be at once verified.
576] 107
576.
ADDITION TO MR WALTON'S PAPER "ON THE RAY-PLANES
IN BIAXAL CRYSTALS."
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873),
*• pp. 273—275.]
Instead of Mr Walton's a', b\ (^ write a, b, c, and assume a, /3, 7 = 6 — 0, c — a,
a—b; S, e, 5"= 6 + c, c + a, a + b. Also instead of his scf, y', z"^ write x, y, z. Then
instead of the octic cone we have the quartic cone, or say the quartic curve
oi'^'f sec= d.xyz(x + y + z)
= a^z [(be -a^)x + a (% - yz)Y
+ ^zx {{ca -¥)y+b (yz - ax)}-
+ rfxy {(ah-(i')z + c{ax- ^y)}",
viz. we may herein consider x, y, z as trilinear coordinates, the ratios x : y : z being
positive for a point within the fundamental triangle.
The curve passes through the angles of the triangle, and it touches the sides in
the points (x = 0, ^y — yz= 0), (y=0, yz-ax = 0), (^ = 0, ax — ^y = 0) respectively.
Moreover, the tangents at the angles of the triangle lie each of them outside the
triangle. Hence, supposing a, b, c each positive, and a>b> c, we have a and 7 each
positive, yS negative, and the form of the curve is as shown in the figure, or else the
like form with the oval lying outside the triangle. And it is hence clear that, if the
side AC (y = 0) instead of touching the curve meets it in a node, this is a conjugate
point arising from the evanescence of the oval ; and in this case no part of the curve
lies within the triangle. Now considering any point x : y : z = l : m : n, we obtain a
tetrad of points x : y : z=± >J(l) : ± \/{m) : ± \/(n) on the octic cone or curve ; and in
order that the point on the octic curve may be real, we must have I, m, n all of the
14—2
108
ADDITION TO MR WALTON S PAPER
[576
same sign ; that is, the point on the qnartic curve must lie within the triangle.
Hence, when in the quartic curve the oval becomes a conjugate point, the octic curve
has no real branch, but it consists wholly of conjugate points ; viz. it consists of the
points A, B, C as conjugate points; two imaginary conjugate points answering to the
point a of the figure, two other imaginary conjugate points answering to the point 7;
and two conjugate points answering to the point /8, these last being not ordinary
conjugate points, but conjugate tacnodal points, or points of contact of two imaginary
branches of the curve.
The case in question, /9 a conjugate point on the quartic curve, answers to
Mr Walton's critical value of sec= d, viz. in the present notation sec" 0= ^ . To
show this I consider the intersection of the curve by the line 72 - ax = 0 ; and I
write for convenience yz = aa; = yau, that is, x — yu, z = aw. Substituting these values,
the equation divides by yii, or omitting this factor it is
e<f^- sec^ ^ . 2< (r/ + (a + 7) m}
= a' I7W (6c — a" — aa.) + a^y\-
4- ^ay . uy (ca — b")-
+ y'{au{ab-Cf' + cy)- c^y]",
or observing that we have a + 7 = - /3, bc-a^-ai=^^, ab-d' + cy = - B0, this becomes
oY setf &u {y - fiu)
= 0-' (y^u + ayY
+ ay {ca — ¥)• uy
+ 7" (aSit + cyy,
viz. this is
M" j(r'7=?- + aV«- + aY sec' 0 ■ /Sj
+ uy {2a»a7? + 27»caS + 07 (ca - b^f - aV sec' 0\
i
576] "ON THK RAY-PLANES IN BIAXAL CRYSTALS." 109
The required condition is that the coefficient of u' shall vanish ; viz. we then have
- a/37 sec= ^ = a^ + 78=
= {b-c){a+by+(a-b){b + cy
= (a - c) {36^ + b(a + c)- ac}
that is,
ory sec- 6 = 4fr- + ay,
agreeing with Mr Walton's value. Giving sec'' 0 this value, and throwing out the
factor u, the equation becomes
u {2a'ay^ + 2yca8 + 07 (ca — b-f — ory' (46- + OLy)\
+ y (ctV + cV) = 0 ;
or, what is the same thing,
ayu {2a (a + b) {b - cY + 2c (c -I- 6) (b - af + {ca - bj- -{b-c){a- b) 46= -{b-cy(b- af]
+ y{a^oi'+cY) = 0.
say
ay Ku + (a-a^ + c'y^) y = 0,
viz. this equation determines the remaining intersection of the curve by the line
7^ — aa? = 0 ; the point in question lies outside the triangle, that is, u : y should be
negative ; or a, 7, aV + (^7^ being each positive, we should have K positive ; we in fact
find
if = 46^ + i' (a" + c' - 6ac) -f 4a V
= 4 (6= - acf + ¥ (a + cf,
which is as it should be.
110
[577
&77.
NOTE IN ILLUSTRATION OF CERTAIN GENERAL THEOREMS
OBTAINED BY DR LIPSCHITZ.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873),
pp. 346—349.]
The paper by Dr Lipschitz, which follows the present Note [in the Quarterly
Journal, l.c.\ is supplemental to Memoirs by him in Crelle, vols. LXX., LXXil., and LXXiv. ;
and he makes use of certain theorems obtained by him in these memoirs ; these theorems
may be illustrated by the consideration of a particular example.
Imagine a particle not acted on by any forces, moving in a given surface ; and
let its position on the surface at the time t be determined by means of the general
coordinates x, y. We have then the vis-viva function T, a given function of x, y, x, y' ;
and the equations of motion are
ddT_dT^^ ddT_dT^^
dt dai dx ' dt dy dy ~ '
which equations serve to determine x, y in terms of t, and of four arbitrary constants ;
these are taken to be the initial values (or values corresponding to the time t = <„)
of X, y, x', y' ; say the values are a, /3, a', yS'.
We have the theorem that a;, y are functions of o, y9, a' (t — 1„), y8' (t — <„).
Suppose for example that x, y, z denote ordinary rectangular coordinates, and that
the particle moves on the sphere a? -V y"^ ■>(■ z^ = (? \ to fix the ideas, suppose that the
coordinates z are measured vertically upwards, and that the particle is on the upper
hemisphere ; that is, take z = -\- \/{c- — x^ — y"^), we have
T^\{x'' + y'' + z^%
where z' denotes its value in terms of x, y, x, y ; viz. we have xx' + yy' + zz' = 0, or
, _ xx' + yy' _ xx' + yy'
' z ' ~ ~ s/id'-a^-y') '
577] NOTE IN ILLUSTRATION OF CERTAIN GENERAL THEOREMS. Ill
the proper value of T is thus
but it is convenient to retain z, z , taking these to signify throughout their foregoing
values in terms of x, y, x', y.
The constants of integration are, as before, a, y3, a', P' ; but we use also 7, 7'
considered as signifying given functions of these constants, viz. we have
7 = \/(c- — a- — p') and 7 = —
(in fact, a= + /8= + 7* = c' and aa' + /3/3' + 77' = 0 ; 7, 7' being thus the initial values
of z, /).
Now, writing
c '
the required values of x, y and the corresponding value of z are
y = ^cos.. + ^^^,,_^'^|,^^,,^sin<r.
To verify that these are functions of a, /3, a' (< — Q, ^ (t — to), write a (t - 1^) = u,
ff (t-U) = v; and take also rf'(t-Q-w\ we have aw + /Su + 7W = 0, viz. w, = - - (aw + ^Sv),
is a function of a, ^, u, v; and then
V(m^ + V- + w=)
^ = c •
and
u .
x= a cos ff + - sin a,
<r
V = o COS o- + - 8in a,
o-
2 = 7 COS <r + - sin o- ;
80 that X, y (and also z) are each of them a function of a, /9, m, v, that is a, 0, a' (< — t^),
^(t — to), which is the theorem in question.
The original variables are x, y; the quantities a'{t-Q, ^'{t-Q, or u, v are
Dr Lipschitz' " Normal- Variables," and the theorem is that the original variables are
functions of their initial values, and of the normal-variables.
112 NOTE IN ILLUSTRATION OF CERTAIN GENERAL THEOREMS. [577
The vis-viva function T may be expressed in terms of the normal-variables and
their derived functions; viz, it is easy to verify that we have
_, , / 1 sin'' <r\ , , ,
where w denotes — (aw + ^v) and consequently w denotes — (au' + $v') ; introducing
7 7
herein differentials instead of derived functions, or writing
4> (du) = i ( ^n — ^-j- ) {udu + vdv + wdwf
. sin* <T , , „ , „ , .,,
+ i a {du- + ctf -1- dw),
where w, dw denote — (au + ffv), — (adu + ^dv) respectively ; then <f> (du) is the
function thus denoted by Dr Lipschitz : and writing herein t — t„ = 0, and thence m = 0,
v = 0, w = 0, o- = 0, the resulting value of <f) (du) is
/„ {du), = i (dw" -I- dv^ + d-uf),
where ftidu) is the function thus denoted by him; the corresponding value of /„(«) is
= ^ (m' + 1;* + vf). We have thus an illustration of his theorem that the function ^ {du)
is such that we have identically
4> {du) - {d v{/o {u)\y = 2^^- [/, {du) - {d ^/{^u)n
where m is a function of u, v independent of the differentials du, dv; the value in
the present example is in fact m' = c' sin- <r ; or the identity is
<!> {du) - {d ^{f,u)Y = f^'^^^ [/, {du) - {d v(/o«))=],
in verification whereof observe that we have
dfo{u) udu + vdv + ivdw
d V(/ow) =
2V(/o«) ^/{u^ + v^ + iv')
= — {udu + vdv + wduif,
CO-
The value of the left-hand side is thus
= - -^ {udu + vdv + wdwf + J ^^ *^ {du^ + dif + dw%
viz. this is
c* sin" o" f 1 )
= - , < ^ {du" ->rdi^ + dw^) - -5—, {udu + vdv + wdiuf \ ;
or, finally, it is
which is right.
578] 113
578.
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
[Frona the Philosophical Transactions of the Royal Society of Lmidon, vol. CLXiv. (for the
year 1874), pp. 397—456. Received November 14, 1873,— Read January 8, 1874.]
The theory of Transformation in Elliptic Functions was established by Jacobi in
the Fundamenta Nova (1829); and he has there developed, transcendentally, with an
approach to completeness, the general case, n an odd number, but algebraically only the
cases n = 3 and n = 5 ; viz. in the general case the formulae are expressed in terms of
the elliptic functions of the nth part of the complete integrals, but in the cases w = 3
and n=b they are expressed rationally in terms of u and v (the fourth roots of the
original and the transformed moduli respectively), these quantities being connected by
an equation of the order 4 or 6, the modular equation. The extension of this alge-
braical theory to any value whatever of n is a problem of great interest and difficulty:
such theory should admit of being treated in a purely algebraical manner ; but the
difficulties are so great that it was found necessary to discuss it by means of the
formulae of the transcendental theory, in particular by means of the expressions
the exponential of =^ 1 , or say by means of the ^'-transcendents.
Several important contributions to the theory have since been made : — Sohnke, " Equa-
tiones modulares pro transformatione functionum ellipticarum," Grelle, t. xvi. (1836),
pp. 97 — 130, (where the modular equations are found for the cases n = 3, 5, 7, 11, 13,
17, and 19); Joubert, " Sur divers Equations analogues aux equations modulaires dans
la throne des fonctions elliptiques," Gomptes Rendus, t. XLVii. (18.58), pp. 337 — 345,
(relating among other things to the multiplier equation for the determination of
Jacobi's M) ; and Konigsberger, " Algebraische Untersuchungen aus der Theorie der
elliptischen Functionen," Grelle, t. LXXii. (1870), pp. 176 — 275 ; together with other
papers by Joubert and by Hermite in later volumes of the Gomptes Rendus, which need
not be more particularly referred to. In the present Memoir I carry on the theory,
algebraically, as far as I am able ; and I have, it appears to me, put the purely
c. IX. 15
114 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
algebraical question in a clearer light than has hitherto been done; but I still find
it necessary to resort to the transcendental theory. I remark that the case n = 7
(next succeeding those of the Fundamenta Nova), on account of the peculiarly simple
form of the modular equation (l—u'){l—if) = (l—tivf, presents but little difficulty;
and I give the complete formulae for this case, obtaining them as well algebraically as
transcenden tally ; I also to a considerable extent discuss algebraically the case of the
next succeeding prime value n = 11. For the sake of completeness I reproduce Sohnke's
modular equations, exhibiting them for gieater clearness in a square form, and adding
to them those for the non-prime cases w = 9 and w = 15 ; also a valuable table given
by him for the powers of /(q) ; and I give other tabular results which are of assistance
in the theory.
The General Problem. Article Nos. 1 to 6.
1. Taking n a given odd number, I write
l-y_l-x/P-Qx\''
1+y l+x\P+Qx)'
where P, Q are rational and integral functions of ar', P ± Qx being each of them of
the order ^(n— 1), or, what is the same thing, (1 ±x)(P ±Qxy being each of them
of the order n ; that is,
w = 4p+l, n = 4p-|-3,
Order of P in ar* is ^ , p,
Q ,, p-1; p;
whence in the first case the number of coefficients in P and Q is (p + l)+p, =^(w-|-l),
and in the second case the number is (p + l) + (p+l), =^(tt-|-l), as before. Taking
P=a +yoi^ + 6X* + ...,
Q=^ + Sx' + ^x'+...,
the formula is
1 — y _1 — X /a — fix + yx^ —
1 — y _\ — X fa. - px + fx^ — ...y
T+y ~ 1+x \a + fix+yx'+ ..J '
the number of coefficients being as just explained. Starting herefrom I reproduce in
a somewhat altered form the investigation in the Fundanwnta Nova, as follows.
2. If the coefficients are such that the equation remains true when we therein
change simultaneously x into r- and y into — , then the variables x, y will satisfy
the differential equation
Mdy dx^
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 115
(Ma constant, the value of which, as will appear, is given by -li- = 1 + - ) ; and the
problem of transformation is thus to find the coefficients so that the equation may
remain true on the above simultaneous change of the values of x, y.
In fact, observing that the original equation and therefore the new equation are
each satisfied on changing therein simultaneously x, y into —x, —y, it follows that the
equation may be written in the four fonns
1- y = {\- x)A^{-^), 1+ y = {\+ x)B^(^),
l-Xy=(l-kx)C(-r), l+\y=(l + kx)D^-i-),
the common denominator being, say E, where A, B, C, B, E are all of them rational
and integral functions of x; and this being so, the differential equation will be
satisfied.
3. To develop the condition, observe that the assumed equation gives
a; (P= + 2PQ + QV) x%
where SI, S3 are functions each of them of the degree ^(?i— 1) in x'^. (Hence, if
1 1 / 20\ 2i3
with Jacobi -v^ denotes the value (y-r «),;=„, we have 7i? = (l + "^) . =1 + — . as
mentioned.)
Suppose in general that U being any integral function (1, a?y>, we have
viz. let U* be what U becomes when x is changed into t- and the whole multiplied
by (li-'a?)p.
Let y* be the value of y obtained by writing ^r for «; then, observing that in
the expression for y the degree of the numerator exceeds by unity that of the
denominator, we have
1 ^
kx^*'
whence
yy ~ k 3333* '
and the functions 21, 93 may be such that this shall be a constant value, =-; viz.
this will be the case if
k " 8121* '
which being so, the required condition is satisfied.
15—2
y* = -
116 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
4. I shall ultimately, instead of k, X introduce Jacobi's u, v {u = V^, v = v^X) ; but
it is for the present convenient to retain k, and instead of X to introduce the
quantity CI connected with it by the equation X = kCl'' ; or say the value of il is
= D* -r it'. The modular equation in its standard form is an equation between m, v,
which, as will appear, gives rise to an equation of the same order between u", t^; and
writing herein if = ilu\ the resulting equation contains only integer powers of «*, that
is, of k, and we have an fli-form of the modular equation, or say an fli-modular
equation, of the same order in fl as the standard form is in v; these fli-forms for
n = 3, 5, 7, 11 will be given presently.
6. Suppose then, fl being a constant, that we have identically
1
iljfcl(»-»
93*
this implies
(In fact, if
then
95= ^ 21*
?l = a + aB= + . . . + qx'^' + saf^-^,
93 = 6 + (ir=+... + »•«"-'+ te"-',
?1* = s + qk^a? + ... + ck^-' a;"-' + ai"-'a;"-',
33* = t + rk'^af + . . . + dk^'-^af^' + hk''-^x^-\
and the assumed equation gives
_ 1 k? k^-' J _ A;"^ .
that is,
and therefore 93 = ,,,„ „ 21*.)
^ (t»— 1)
9393* . X .
From these equations g,5,7» = CI", that is, =-j , as it should be ; so that fl signifying
as above, the required condition will be satisfied if only 21 = Q;i|n_i) 93* ; or substituting
for 21, 93 their values, if only
(P" + 2PQx' + QV)* = flifci'"-" (P" + 2PQ + QV),
where each side is a function of ar* of the order ^ (n — 1), or the number of terms is
^ (w + 1), the several coeflScients being obviously homogeneous quadric functions of the
J(n+1) coefficients of P, Q, We have thus ^(n+1) equations, each of the form
U — SIV, where U, V are given quadric functions of the coefficients of P, Q, say of
the ^(re+1) coefficients a, y9, 7, S, &c., and where fl is indeterminate.
578]
A MEMOIE ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
ii:
6. We may from the ^(«+l) equations eliminate the J(n— 1) ratios a : ^ -.y: ... ,
thus obtaining an equation in fi (involving of course the parameter k) which is the
fii-modular equation above refeiTed to; and then H denoting any root of this equation,
the ^{n + 1) equations give a single value for the set of ratios a : /9 : 7 : S : ... , so
that the ratio of the functions P, Q is determined, and consequently the value of y
as given by the equation
1-y ^ {l-x)(P-Qxf ^ x(P^ + 2PQ + Q'x')
l+y (l+a;){P+Qxy' °^ ^ R- + 2PQay' + Q'x^ '
The entire problem thus depends on the solution of the system of ^(n+1) equations,
(i>- + 2PQaf + Q'af)* = ilki'"-'^ (P' + 2PQ + Q'a?).
The Clk-Modular Equations, ft = 3, 5, 7, 11, Article No. 7.
7. For convenience of reference, and to fix the ideas, I give these results, calculated,
as above explained, from the standard or uv-forms.
A'
k
1
0*
+ 1
'.
o*
-4
o»
+ 6
n
-4
ly
1 ^'
= 0
w = 3 ;
n = 1, we have -i{k- 1)= = 0.
k" k"
Of
+ 1
o»
-16 1
+ 10
o«
!
+ 15
o*
-20
a»
i
+ 15
0
+ 10
-16
o*
+ 1
-16
+ 32
-16
= 0
n = 5 :
n = l, we have - 16 (/fc^- 1)^ = 0.
118 A MEMOIR ON THE TBANSFORMATION OF ELLIPTIC FUNCTIONS. [578
*•
*»
k"
*•
*«
k
1
o»
I* >,
1
CF
-64
+ 66
1
0»
-112
+ 140
0»
-112!
+ 56
o«
+ 70!
O*
+ 56
1-112
0"
+ 140
-112
Q
+ 56
-64
o»
+ 1
= 0
n-7
n= 1, we have
U{k-lY(W + 3k+l){K'+3k + i) = 0.
-64 -112 0 +352 0
71 = 11:
112 -64
*«♦
*•
A^
¥
k'
/fc»
/f
/fi
At
k'
/tf
o«
+ ll
1
1
O"
-1024
+ 1408
- 396
1
1
0"»
-5632
+ 4400
+ 1298
o»
- 16192
+ 16368
- 396
o»
- 18656
+ 19151
O'
- 16016
- 1144
+ 16368
0»
+ 4400
- 7876
+ 4400
O"
+ 16368
- 1144
- 16016
o«
+ 19151
- 18656
n»
- 396
+ 16368
-16192
o»
+ 1298
+ 4400
-5632
o>
- 396
+ 1408
- 1024
o»
+ 1
i
+ 1
- 12
+ 66
- 220
+ 495
-792
+ 924
-792
+ 495
- 220
+ 66
- 12
+ 1
- 32208 - 18656 - 1936 - 7876 - 1936 - 18656 - 32208
+ 1408 + 8800 + 32736 + 40900 + 32736 + 8800 + 1408
- 1024 - 5632 - 30800 - 9856 + 30800 + 33024 + 30800 - 9856 - 30800 - 5632 - 1024 + 94624
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 119
Equation-systems for the cases n = S, b, 7, 9, 11. Art. Nos. 8 to 10.
8. u = 3, cubic transformation. k = u*, fi = — (here and in the other cases).
P = a, Q = /8. The condition here is
k'afa' + (2a;3 + ,S-) = ilk {(or + 2a/3) + ^'af},
and the system of equations thus is
2a/3 + /3==flA;(a= + 2a/S),
and similarly in the other cases ; for these it will be enough to write down the
equation-systems.
n = 5, quintic transformation.
2o7 + 2a/3 + /9» = n (207 + 2&y + fi'),
7= + 2/37 = ^^ («" + 2a/3).
n = 7, septic transformation.
P = o + 7a?, Q = /8-|-Sa:». ^
^• (207 + 2a/3 + ^) = n(y'+ 2jB + 2/3S),
y + 2/87 + 2o8 + 2/3S = ni (2a7 + 2^7 + 2a8 + ^),
g»+27S = ^^•^(a^-|-2a/3).
n = 9, enneadic transformation.
P = a + 7a;» + eoj*, Q = y3 -H aar".
A:*a= = He',
ifc= (207 + 2a/3 + /8^) = n (276 + 2€S + S^'),
2ae + y2 + 2ag + 27,8 + 2/38 = H (2ae + 7- + 278 + 26/8 + 2^88),
27€ + 278 + 2e/3 + 8= = nt (237 + 2a8 + 2y^ + ^),
6=i-l-28e = fii-*(a-+2a;8).
n = 11, endecadic transformation.
P=:0+7ar' + ex*, Q = /3+8a!2+5ar*.
^•= (207 + 2a;8 + ^0 = " (e' + 2e?+ 28S),
jfc (2ae + 7» + 2a8 + 27^ + 2/38) = fi (276 + 27?+ 2e8 + 2/38 + 8^,
276 + 2of+ 278 + 2e/3 + 2/3?+ 8= = n^• (2ae + 7- + 2a? + 278 + 2e^ + 2^8),
r + 27?+ 2e8 + 28?= ill<? (iay + 2a8 + 27/8 + ^0.
2£?+?= = n^»(a= + 2a^).
And so on.
120 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
9. It will be noticed that if the coefficients oi P+Qx taken in order are
a, ff, . . . , p, a,
then in every case the first and last equations are
W»-'» of = n<r2,
2p<T + or- = ni-i"»-» (a= + 2a/3).
Putting in the first of these k = u\ II = — , the equation becomes
where each side is a perfect square ; and in extracting the square root we may without
loss of generality take the roots positive, and write i("a = va.
This speciality, although it renders it proper to employ ultimately u, v in place of
k, CI, produces really no depression of order (viz. the IlA^-form of the modular equation
is found to be of the same order in il that the standai-d or ww-form is in v), and
is in another point of view a disadvantage, as destroying the uniformity of the several
equations: in the discussion of order I consequently retain il, k. Ultimately these are
to be replaced by u, v; the change in the equation-systems is so easily made that
it is not necessary here to write them down in the new form in u, v.
10. The case a = 0 has to be considered in the discussion of order, but we have
thus only solutions which are to be rejected ; in the proper solutions a is not = 0,
and it may therefore for convenience be taken to be = 1. We have then <r = m" -;- v.
The last equation becomes therefore
7"(^^-V>5--C-^»^
or recollecting that /3 is connected with the multiplier M by the relation t? = 1 + 2/9,
that is,
and substituting for 1 + 2/9 its value, the equation becomes
that is, the first and the last coefficients are 1, — , and the second and the penultimate
coefficients are each expressed in terms of v, M. The cases n = 3, « = 5 are so far
peculiar, that the only coefficients are a, /9, or a, /9, 7 ; in the next case n = 7, the
only coefficients are a, /9, y, h, and we have in this case all the coefficients expressed
as above.
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 121
The Clk-form — Order of the Systems. Art. Nos. 11 to 22.
11. In the general case, n an odd number, we have D, and ^(?i + l) coefficients
connected by a system of i(n + l) equations of the form
^ U'~ V'~"-'
where U, V,.., U', F', ... are given quadric functions of the coefficients. Omitting the
U V
(n =), there remains a system of ^(n—1) equations of the form yp ===, = .., or say
( U, V, W,.. ) = 0,
\U', V, W',.. I
which determine the ratios a : ^ : <y : ... of the coefficients ; and to each set of ratios
there corresponds a single value of il. The order of the system, or number of sets of
ratios, is =^(n + l) .2*'"~", =(«+ 1) . 2*'"""; and this is consequently the number of
values of il, or the oi-der of the equation for the determination of fl ; viz. but for reduction,
■ the order in f2 of the flZ;-modular equation would be =(w + 1). 2*'""". In the case
n = 3, this is = 4, which is right, but for any larger value of n the order is far too
high ; in fact, assuming (as the case is) that the order is equal to the order in v of
the M«-form, the order should for a prime value of n be = m + 1, and for a composite
value not containing any square factor be = the sum of the divisors of n. I do not
attempt a general investigation, but confine myself to showing in what manner the
reductions arise.
12. I will first consider the cubic transformation; here, writing for convenience
IS gi
-q = d, the equations give
2(9 + 1 kid' + 26)
and
k&' = n.
, that is, ]c'e'ie + 2)-(2d + l) = 0,
The equation in 0 gives {Ic^O* - ly - 'iO^ (i^ff' -iy=0, and we have thence
k (ft" - ly - 4n (kn -ly = o,
that is,
kn* - 4i-^n' + 6kn^ - 4n + ^■ = o,
the modular equation ; and then k'O* -1 + 26 (k^ff' - 1) = 0, that is, fi= - 1 -f 26{kn - 1 )=0,
or 26 = — -jy: — J, which is = -^ , say we have a=fl=— 1, ^ = 2(1— kil); consequently
1-y _ 1-x p-l+2(m-l)a;]'
'l+y~l+!c\iP-l-2(kn-l)a;}'
which completes the theory.
C. IX, 16
122 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
13. Reproducing for this case the general theory, it appears d priori that fi is
detemnined by a quartic equation ; in fact, from the original equations eliminating il,
we have an equation
U, V
U', v
= 0,
where U, U', V, V are quartic functions of a, /3 ; that is, the ratio a : /9 has four
values, and to each of these there corresponds a single value of il ; viz. il is deter-
mined by a quartic equation.
14. Considering next the case n = 5, the quiutic transformation; the elimination of
il gives the equations
V'~V'~W"
where U, U', &c. are all quadric functions of a, /3, 7. We have thence 4.4 — 2.2, =12
sets of values of a : yS : 7 ; viz. considering o, /S, 7 as coordinates in piano, the curves
UV — U'V = Q, UW — U'W = 0 are quartic curves intersecting in 16 points; but among
these are included the four points U = 0, U' = 0 (in fact, the point a = 0, 7=0 four
times), which are not points of the curve VW —V'W = 0 ; there remain therefore
16 — 4, = 12 intersections, agreeing with the general value {n + 1) . 2*'""". Hence fi
is in the first instance determined by an equation of the order 12 ; but the proper
order being = G, there must be a factor of the order 6 to be rejected. To explain
this and to determine the factor, observe that the equations in question are
A!»a^ (207 + 2y37 + ^')-'f (207 + 2a/3 + fi') = 0,
k*a.»(a + 2^) _y(,y + 2/3) =0;
at the point a = 0, 7 = 0, the first of these has a double point, the second a triple
point ; or there are at the point in question 6 interaections ; but 4 of these are the
points which give the foregoing reduction 16 — 4 = 12; we have thus the point a = 0,
7 = 0, counting twice among the twelve points. Writing in the two equations /9 = 0,
the equations become k?a?^ — a'f=0, k*a* — '/* = 0, viz. these will be satisfied if k'oP—'Y'=0,
that is, the curves pass through each of the two points (/9 = 0, 7 = + ka), and these
values satisfy (as in fact they should) the third equation
k' (2a7 + 2a/3 + /J") a (a + 2y9) - 7 (7 + 2/3) (2a7 + 2/3 + /3=) = 0.
It is moreover easily shown that the three curves have at each of the points in question
a common tangent; viz. taking A, B, G as current coordinates, the tangent at the
point (a, /9, 7) of the second curve has for its equation
^ (2a» + 3a2/3) )fc* + jS (^a> - 7») - C (27» + 37^/3) = 0 ;
and for ;8 = 0, 7=±Aa, this becomes 2i-.4 +iJ (i+ 1) + 2C = 0, viz. this is the line from
the point (/8 = 0, f=±ka) to the point (1, —2, 1). And similarly for the other two
curves we find the same equation for the tangent.
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 123
Hence among the 12 points are included the point (7 = 0, a=0) twice, and the
points (yS = 0, 7 = + ka.) each twice : we have thus a reduction = 6.
15. Writing in the equations 7 = 0, a = 0, the first and third are satisfied
identically, and the second becomes ^ = £1^-, that is, the equations give fl = 1 ; writing
/3 = 0, they become
h-a!^ = Qp/^, ay = flay, y- = flyt^a",
viz. putting herein y"^ = k'a!', the equations again give fl = 1 ; hence the factor of the
order 6 is (fl - 1)", and the equation of the twelfth order for the determination of D, is
(n-i)«{(n, i)«j=o,
where (fl, 1)'=0 is the H^-modular equation above written down.
16. Reverting to the equation
\+y {\+x){P + Qxf'
it is to be observed that for a = 0, 7 = 0, that is, P = 0, this becomes simply y =x,
which is the transformation of the order 1 ; the corresponding value of the modulus
X is \ = k, and the equation \=,D.^k then gives n* = l, which is replaced by fl — 1 = 0.
If in the same equation we write /3 = 0, that is, Q = 0, then (without any use of
the equation 7* = k^d-) we have y = x, the transformation of the order 1 ; but although
this is so, the fundamental equation
(P= + 2PQx^ + Q^a?)* = n.¥{P' + 2PQ + Q'x'),
which, putting therein Q = 0, becomes (P^)* = nA~'P^ that is, {kVa + yf = nk"- (a. + yxj,
is not satisfied by the single relation 0 — 1=0, but necessitates the further relation
The thing to be observed is that the extraneous factor (il — If, equated to zero,
gives for il the value fl = 1 corresponding to the transformation y = xo{ the order 1.
17. Considering next n = 7, the septic transformation ; we have here between a, 13, 7, S
a fourfold relation of the form
( U, V, W, Z ) = 0,
I U', V, W, Z' I
where, as before, U, V, &c. are quadric functions, and the number of solutions is here
8.2', = .32 ; to each of these corresponds a single value of D., or fl is in the first
instance determined by an equation of the order 32. But the order of the modular
equation is =8; or representing this by [(Q, 1)'}=0, the equation must be
(a i)«i(n, iy}=o,
viz. there must be a special factor of the order 24.
16—2
124 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
18. One way of satisfying the equations is to write therein a=0, 8 = 0; the
equations thus become
7^ + 2/S7 = ni(2/37 + /9»);
or putting 0, •y= a', y9',
ka'^ = n^»,
/3'» + 2a'/3' = nk {la^ + a'%
which (with o', ^ instead of a, 0) are the very equations which belong to the cubic
transformation ; hence a factor is {(fl, 1)*}.
Observe that for the values in question o = 0, 8 = 0, P = 0'a?, Q = a',
(P±Qxy = a^{a'±0'xy, =a?(F±Q'xy, if F = a', Q' = 0',
and therefore
1-y _ l-a; (F - Q'xV
l + ^j~l+x\P'+<^x)'
which is the formula for a cubic transformation.
19. The equations may also be satisfied by writing therein y = ka, S = k0; in fact,
substituting these values, they become
L^a' = ilk'0^,
2kV + k {2a0 + /3=) = nt» (a' + 2a/9) + 2nA;/3=,
^•^a» +2ki0' + 2a/3) = 2nk'' (i» + 2a/3) + m-/9=,
A? (/3^ + 2a/3) = nkr' (a» + 2o/9) ;
the first and last of these are
ka^ = n/S",
/8» + 2a/3 = ^^-(a»+2ay9),
which being satisfied the second and third equations are satisfied identically; and these
are the formulae for a cubic transformation ; that is, we again have the factor {(fl, l)*j.
Observe that for the values in question y = ka, S = A-/3, we have P = a{l+kx'),
Q = 0(l+kx'); so that, writing P' — a, Q' = 0, we have for y the value
1-?/ ^ {\-x){P'-Q'xy
1 + y (\+x){P'+Q'xf'
■which is the formula for a cubic transformation.
20. It is important to notice that we cannot by writing a = 0 or 8 = 0 reduce the
transformation to a quintic one ; in fact, the equation l<?cP = 128^ shows that if either
of these equations is satisfied the other is also satisfied ; and we have then the
foregoing case a = 0, 8 = 0, giving not a quintic but a cubic transformation.
And for the same reason we cannot by writing a = 0, /8 = 0, 7 = 0 or ;8 = 0, 7 = 0,
i = 0 reduce the transformation to the order 1. There is thus no factor D,— \.
578]
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
125
21. As regards the non-existence of the factor XI — 1, I further verify this by
writing in the equations fl = 1 ; they thus become
k {2ari + 2ay8 + ^) = rf + 27S + 2/3S,
7^ + 2/37-1- 2a8 + 2/38 = k (2a7 + W^ + 2aS + /3=),
8-^ + 278 = i»(a^ + 2a/3),
which it is to be shown cannot be satisfied in general, but only for certain values of k.
Reducing the last equation, this is 78 = ^■'a/3, which, combined with the first, gives
07 = y38; and if for convenience we assume a = l, and write also d=±vk (that is, A; = ^),
then the values of a, /8, 7, 8 are a = l, ^ = y6~^, 7 = 7, 8 = ^; which values, substituted
in the second and the third equations, give two equations in 7, 6; and from these,
eliminating 7, we obtain an equation for the determination of 6, that is, of k. In fact,
the second equation gives
^ (27 + 27^' + r"^) = 7' + 27^ + 27 ;
or, dividing by 7 and reducing,
7 (I - ^) = 2^ (d^ - 1) (^^ - (9 + 1),
that is,
7(l + ^) = -26'H^--^ + l),
or, as this may also be written,
{y + d')il + 6^) = - ff'{0 - ly,
that is,
-8^(0 -ly
B' + l ■
7 + 6'» =
that is,
Moreover the third equation gives
rf + Irfd-^ + 26= + 27 = ^'^ (27 + 27=^3 + 2^ + rfe-'),
rf(e* -26' + 26 - l)-2(y + 6') e*{&' -l)='0 ;
or dividing by ^ — 1, it is
whence also
7=((9- 1)^ = 2^ (7 + ^0;
Also
wherefore
or
that is,
or finally
-2^
id'(d'-e+iy = y'(ff' + if.
2(0^-6 + iy = -0{d' + l) or 2(^-'-^ + l)^-t-^(^'' + l) = 0,
e{e' + iy + 2{e'-e + iy=o,
20* - se' +66^-20 + 2 = 0,
{26'-0 + l){0'-6 + 2) = O.
126 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
We have thus (2^+l)» = ^, that is, 4^4- 3^- + 1=0 or 4Ar» + 3^ + 1 = 0, or else
(^»+2)' = ^, that is, ^ + 3^ + 4=0 or ^•''+ 3i + 4 = 0; viz. the equation in k is
(4i-' + 3i- + 1) (t= + Sk + 4) = 0,
these being in fact the values of k given by the modular equation on putting therein
12 = 1.
The equation of the order 32 thus contains the factor {(fl, 1/} at least twice, and
it does not contain either the factor fl — 1, or the factor {(fl, 1)'} belonging to the
quintic transformation ; it may be conjectured that the factor {{il, l)*j presents itself
six times, and that the form is
{(a iy}'(n, i)»=o;
but I am not able to verify this, and I do not pursue the discussion further.
22. The foregoing considerations show the grounds of the difficulty of the purely
algebraical solution of the problem ; the required results, for instance the modular
equation, are obtained not in the simple form, but accompanied with special factors of
high order. The transcendental theory affords the means of obtaining the results in
the proper form without special factors; and I proceed to develop the theory, repro-
ducing the known results as to the modular and multiplier equations, and extending
it to the determination of the transformation-coefficients a, /3,
The Modular Equation. Art. Nos. 23 to 28.
23. Writing, as usual, q = e *' , we have u, a given function of q, viz.
''^^ 1+q.l + q^.l + q'..
= V25* (l-q+ 2q"--3q' + 4q*-6^+9cf- Uq' + ...}
= \2q^f(q) suppose ;
and this being so, the several values of v and of the other quantities in question are
all given in terms of q.
The case chiefly considered is that of n an odd prime ; and unless the contrary
is stated it is assumed that this is so. We have then 71 + 1 transformations coitc-
sponding to the same number n + l of values of v ; these may be distinguished as
Vo, fj, v„...,Vn; viz. writing a to denote an imaginary Jith root of unity, we have
t;, = (-)"«' V29f/(<?"), v, = 'Ji{aq-ff{aq^), v, = 'J2(a'q^ff{a'q-\ &c.,
Vn =
^2q^/(q«).
(Observe (-)»=+ for n = 8/) ± 1, - for n =8p ± 3.)
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 127
The occurrence of the fractional exponent ^ is, as will appear, a circumstance of
great importance; and it will be convenient to introduce the term "octicity," viz. an
expression of the foi-m q^F{q) (/= 0, or a positive integer not exceeding 7, F {q) a
rational function of q) may be said to be of the octicity /.
24. The modular equation is of course
{v-v,){v-v;)....{v- v„) = 0 ;
say this is
»"+' -Av^ + Bv"-^ - . . . = 0,
80 that A=1v„, B = 1voVi, &c. In the development of these expressions, the terms
having a fractional exponent, with denominator n, would disappear of themselves, as in-
volving symmetrically the several nth roots of unity ; and each coefficient would be of the
a
form q^F(q), F a i-ational and integral function of q. It is moreover easy to see that,
for the several coefficients A, B, C g will denote the positive residue (mod. 8) of
n, 2n, Sn, . . . respectively.
Hence assuming, as the fact is, that these coefficients are severally rational and
integral functions of q, it follows that the form is
au^ + bus+^ + CM«'+" +
g having the foregoing values for the several coefficients respectively. And it being
known that the modular equation is as regards u of the order =n+l, there is a
known limit to the number of terms in the several coefficients respectively. We have
thus for each coefficient an identity of the form
A=auP+ bitP+^ + ....,
where A and ic being each of them given in terms of q, the values of the numerical
coefficients a, b, . . can be determined ; and we thus arrive at the modular equation.
25. It is in effect in this manner that the modular equations are calculated in
Sohnke's Memoir. Various relations of symmetry in regard to (u, v) and other known
properties of the modular equation are made use of in order to reduce the number of
the unknown coefficients to a minimum; and (what in practice is obviously an important
simplification) instead of the coefficients 2»„, Xv^Vi, &c., it is the sums of powers Sy„, 2v„=,
&c., which are compared with their expressions in terms of u, in order to the deter-
mination of the unknown numerical coefficients a, b,.. . The process is a laborious one
(although less so than perhaps might beforehand have been imagined), involving very
high numbei-s; it requires the development up to high powers of q, of the high powers
of the before-mentioned function /(q); and Sohnke gives a valuable Table, which I
reproduce here; adding to it the three columns which relate to (j>q.
128
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
[578
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C. IX.
17
130
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
[578
26. I give from Sohnke the series of modular equations, adding those for the
composite cases n = 9 and }i = 15, as to which see the remarks which follow the Table.
V* tfl v^ V 1
u'
u
1
- 1
+ 2
-2
+ 1
1
1
n = 3.
f 2
-2 -1 ={v + iy(v-i).
v' v' V 1
.
-1
+ 4
-5
+ 5
-4
+ 1
w = 5.
+ 4 +5
-5 -4 -1 =(v+lf{v-l).
V* 1^ v' V \
0
+ 1
-8
+ 28
-56
+ 70
-56
+ 28
-8
+ 1
0
n = 7.
+ 28 -56 +70 -56 +28 -8 +1 =(e-l)».
578] A. MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
v^^ v^^ -v^' v" '\^ i^ tf tf -e* v' 1^ V 1
131
0
+ 1
- 16
+ 8
+ 16
+ 10
-16
-24
0
+ 15
+ 48
-84
+ 48
+ 15
0
-24
- 16
+ 10
+ 16
+ 8
- 16
+ 1
-
0
w = 9.
1 _8 +26 -40 +15
^2 ^1 ^0 ^ ^
+ 48 -84 +48 +15 -40 +26 - 8 +1 =(v-iy»(«+lf
l/ 1^ if V* 1^ v' V 1
w=ll.
0
-1
+ 32
- 22
-44
+ 88
+ 22
0
- 165
+ 132
+ 44
-44
- 132
+ 165
0
-22
- 88
+ 44
+ 22
-32
+ 1
0
1 +10 + 44 + 110 + 165 + 132 0
132 - 165 - 110 - 44 - 10 - 1 ={v+\y^(v-l)
17—2
182
A MEMOIR ON THB TRANSFORMATION OF ELLIPTIC FUNCTIONS.
[578
V^* fP* v" V^ l^» t^
e'e'ti**' V* 1^ t^ V I
0
-1
+64
- 52
0
-65
+ 208
0
- 429
+ 520
+ 52
0
-429
+ 208
-208
+ 429
0
-52
-520
+ 429
0
- 208
+ 65
0
+ 52
-64
+ 1
0
n=13.
1+12+65 +208 +429 +572 +429 0 -429 -572 -429 -208 -65-12-1 ={v+l)"(v-l).
2r
t
I
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+
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+
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+
+
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00
+
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i
s
J
o
+
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^»
n s>.
"li
184
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
[578
^4
w4
o
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&
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+
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+
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a
3
»
%
%
s
a
=»
S 3 S ^
136 A MEMOIR ON THE TBAN8F0EMATI0N OF ELLIPTIC FUNCTIONS. [578
Various remarks arise on the Tables. Attending first to the cases n a prime
number ; the only terms of the order n + 1 in v or m are «"+' ± m"+\ viz. n = 3 or
5 (mod. 8) the sign is — , but n = 1 or 7 (mod. 8) the sign is + . And there is in
every case a pair of terms v"u" and vu, having coeflScients equal in absolute magnitude,
but of opposite signs, or of the same sign, in the two cases respectively.
Each Table is symmetrical in regard to its two diagonals respectively, so that
every non-diagonal coefficient occurs (with or without reversal of sign) 4 times; viz.
in the case n=l or 7 (mod. 8) this is a perfect symmetry, without reversal of sign ;
but in the case n=S or 5 (mod. 8) it is, as regards the lines parallel to either diagonal,
and in regard to the other diagonal, alternately a perfect symmetry without reversal
of sign and a skew symmetry with reversal. Thus in the case m=19, the lines parallel
to the dexter diagonal are —1 (symmetrical), +114, —114 (skew), 0, —2584, —6859,
— 2584, 0 (symmetrical), and so on. The same relation of symmetry is seen in the
composite cases n = 9 and n = 15, both belonging to w = 1 or 7, mod. 8.
If, as before, n is prime, then putting in the modular equation m = 1, the equation
in the case n=l or 7 (mod. 8) becomes (t) — 1)"+' = 0, but in the case n=3 or 5
(mod. 8) it becomes (« + l)"(t;- 1) = 0.
27. In the case n a composite number not containing any square factor, then
dividing n in every possible way into two factors n = ah (including the divisions n . 1
and 1 . n), and denoting by /3 an imaginary 6th root of unity, a value of v is
±^/2(/33^)V•(^g^);
so that the whole number of roots (or order of the modular equation) is =1/, if v be
the sum of the divisors of n. Thus n = 15, the values are
V23'ir/-(2«), -^J^\^if{qil -^J2q\-if(^qil »J2q^\-^f{q^)
1 , 3 , 5 , 15 roots;
and the order of the modular equation is = 24. The modular equation might thus be
obtained as for a prime number ; but it is easier to decompose n into its prime
factors, and consider the transformation as compounded of transformations of these
prime orders. Thus n = 15, the transformation is compounded of a cubic and a quintic
one. If the v of the cubic transformation be denoted by d, then we have
e* + 2^u' - 2^M - w< = 0 ;
and to each of the four values of 6 corresponds the six values of v belonging to the
quintic transformation given by
The equation in v is thus
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 137
where 6^, d,, 0^, 6^ are the roots of the equation in 6, viz. we have
e* + ^e'u' - 26u - M* = (0 - d,) {6 - d.^ (6 - 0,) (0 - 0,) ;
and it was in this way that the equation for the case m = 15 was calculated. Observe
that writing u=l, we have {0 +\f{0 -l) = 0, or say 0,^0,_= 0, = -\, 0^ = 4-1. The
equation in v thus becomes {(w— l)°(u+ l)}'(t) + l)'(t) — 1) = 0, that is, (« — 1)" (?; + 1)' = 0.
28. The case where n has a square factor is a little different ; thus n = 9, the
values are
1 , 3 , 9 . , roots ;
but here w being an imaginary cube root of unity, the second term denotes the three
values,
the first of which is =u, and is to be rejected; there remain 1 + 2+9, =12 roots, or
the equation is of the order 12.
Considering the equation as compounded of two cubic transformations, if the value
of t; for the first of these be 0, then we have
0* + 20'u' - 20U - M^ = 0 ;
and to the four values of 0 correspond severally the four values of v given by the
equation
v* + 2v'0'-2v0-0* = O.
One of these values is however v = — u, since the u^-equation is satisfied on writing
therein v = — u; hence, writing
0* + 20>ie - 20U -u* = {0- 0,) {0 - 0.^ {0 - 0,) (0 - 0,),
we have an equation
{v* + 2v>0^^ - 2v0, - 0^) (i;* + . . - 0-}) («^ + . . - 0i) (f^..- 0,*) = 0,
which contains the factor {v + uf and, divested hereof, gives the required modular
equation of the order 12 ; it was in fact obtained in this manner.
Observe that ^v^iting u=l, we have (0+ 1)^(^-1) = 0, or say 0^ = 0., = 0.i — l, 0^=1;
the modular equation then becomes
{(v - ly (v + 1)1' (v + 1)^ {v-l)^{v+ ly = 0,
that is,
(v - ly {v + ly = 0.
C. IX,
18
188 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
The Multiplier Equation. Art. No. 29.
29. The theory is in many respects analogous to that of the modular equation.
To each value of v there corresponds a single value of M ; hence M, or what is the
same thing ^, is determined by an equation of the same order as v, viz. n being
prime, the order is = n + 1. The last term of the equation is constant, and the other
coefficients are rational and integral functions of w', of a degree not exceeding J(« — 1);
and not only so, but they are, wsl (mod. 4), rational and integral functions of m'(1— «'),
and w = 3 (mod. 4), alternately of this form and of the same form multiplied by the
factor (1 - 2m»).
The values are in fact given as transcendental functions of q; viz. denoting by
Jf,, Mu Jfa. •••> -^n the values corresponding to Vo, v,, I'a, ...,Vn respectively, and writing
A. („\ - (i+g)(i+gO(i+g')--a-g') o^jzjt) (i^go^-
•P W - (i _ o) (1 _ fli) (1 _ o.) ... (1 + ^f)-(i + ^) (1 + ^) ...
(1 - 9) (1 - ?») (1 - 5») ... (1 + g') (1 + 3^) (1 + g*)
= 1 + 2^ + 2g^ 4 2?' + 29" + ,
then we have
-Jfo =
(-)V ,^»(g)
M^ = ^-j-, ■ • (a an imaginary nth root of unity)
Mn=
Hence, the form of the equation being known, the values of the numerical coefficients
may be calculated; and it was in this way that Joubert obtained the following results.
I have in some cases changed the sign of Joubert's multiplier, so that in every case
the value corresponding to m = 0 shall be M=\.
The equations are:
n — ^.-TT. M = 0, this is
M*
{m-'Hm-^ ')-'■'
+ -^.-6 M = l, itis
-3 = 0.
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 139
n = 5, ^.; M = 0 or 1, this is
+ i.-io
+.4+35
+i-«o
• ■^jp- + ''
i. -26 + 256(tHl-
-w«)
+.5=0.
, 1
^w'
+ F--2^«
+ ^,. + 224(1-2m0
+ ^,.-140-21. 25(
+ i . {48 + 2048mH1
+ 7 = 0.
w = ll,
1
+ i-«
+ i-66
(s-)"(i-)=»-
M = 0, this is
M = 1, it is
+ i-. + 112(l-2«») (e+0'(^+0 = '-
w = 0, this is
u = 1, it is
+ 1 . + 440(l-2«») (E+ir(i-lO = ^-
18—2
140 A MEadOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
+ ^ . + 3168(1-2m»)
+ } . _ 4620 - 3 . ll^ 256«' (1 - w«)
+ X,.[+ 4752 + 11 . 4096u' (1 - ««)! (1 - 2«»)
+ A. ._3465-3.7.11.512m»(1-m»)
+ ^- . j+ 1760 + 11 . 83 . 2048zt« (I - u')] (I - 2u»)
+ 1. . _ 594 _ 9 . 11 . 37 . 256m»(1 - m')- 3 . 11 . 131072 [w'(l - «»)p
+ 4-{l20 + 15 . 4096m» (1 - ««) - 524288 (({'(1 - ««))»} (1 - 2««)
-11 = 0.
27(6 Multiplier as a rational function of m, v. Art. Nos. 30 to 36.
30. The multiplier M, as having a single value corresponding to each value of v,
is necessarily a rational function of ti, v ; and such an expression of M can, as remarked
by Konigsberger, be deduced from the multiplier equation by means of Jacobi's
theorem,
n k(l- k'j d\ '
viz. substituting for k, \ their values u', if, and observing that if the modular equation
be F(u, v) = 0 so that the value of -r- is = - F' (v) -i- F' {u), this is
~ n (l-u')uF'u'
and then in the multiplier equation separating the terms which contain the odd and
even powers, and writing it in the form 4> (M') + Tlf'^' (il/=) = 0, this equation, substituting
therein for M'' its value, gives the value of M rationally.
The rational expression of M in terms of u, v is of course indeterminate, since
its form may be modified in any manner by means of the equation F(u, v) = 0; and in
the expression obtained as above, the orders of the numerator and the denominator are
fiu: too high. A different form may be obtained as follows : for gi-eater convenience I
seek for the value not of M but of -j>.
M
578] A MEMOIR ON THE TRANSFORMATION OP ELLIPTIC FUNCTIONS. 141
31. Denoting, as above, by M„, M^, ... , M^ the values which correspond to v^, Vi,-..,Vn
respectively, and writing ^ ^= ^ + ]j^+ ••■ + j^ ' &c., we have S^, S^, &c., all of
them expressible as determinate functions of u ; and we have moreover the theorem
that each of these is a rational and integral function of u : we have thus the series
of equations
where A, B,...,H are rational and integral functions of m. These give linearly the
different values of ^; in fact, we have
(Vo — Vt)...(Vo — Vn)irr=H- GSVi + FSViV^ — ... ± AV1V2 ...V^,
where Svi, SviV^, &c. denote the combinations formed with the roots Vi, v^, ... ,Vn (these can
be expressed in terms of the single root Vo) ; and we have also (v,, — Vi}... (v,, — Vn) = F' (v^) :
the resulting equation is consequently F'v^ -r^ = R{u, %), R a determinate rational and
integral function of (u, Vo); but as the same formula exists for each root of the modular
equation, we may herein write M, v in place of M^, «»; and the formula thus is
i
F'v.j^ = Riu, v).
viz. we thus obtain the required value of -p as a rational traction, the denominator
being the detenninate function F'v, and the numerator being, as is easy to see, a
determinate function of the order n as regards v.
32. The method is applicable when M is only known by its expression in terms
of q; but if we know for M an expression in terms of v, u, then the method trans-
forms this into a standard form as above. By way of illustration I will consider the
case n = 3, where the modular equation is
V* + iifu' - 2vu -u* = 0,
1 2m'
and where a known expression of il/ is ■t>=H . Here writing S^i, So (=4), S, &c.
to denote the sum of the powers - 1, 0, 1, &c. of the roots of the equation, we have
S^ = S,+ 2«'S_i, =0 , as appears from the values presently given,
M
si = S, + 2u'S, , =0 ,
M
S^^S, + 2u'S, , =6m;
142
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
[578
and observing that v„ being ultimately replaced by v, we have
Svi = Svo — V, Sv^Vi = SvoVi — vSva + 1^, v,v,b, = Sv^ViVj — vSv^Vi + VjSvo — »*,
that is,
/St;, = — 2m' — V, SviVi = 2u'v + v', ViV^v^ = 2u — 2u'if — v',
we have
F'v.^= (S, + 2ii'S,)
viz. this is
+ {2u' + v)(S^ + 2u'S,)
+ (2u'v + i^){S, + 2u'>So)
+ {-2u + 2?tV + »>) (/So + 2m»/S_,),
But we have
/S_, = --3. S,= i, S, = -2u', S, = ^', S, = 6u-8u'>;
and the equation thus is
(2v' + Sifu' -h)j^ = S (ifu- + 2u'v + 1) « ;
1 2u'
to verify which observe that, substituting herein for ^ its value 1 H , the equation
becomes
(2i;» + SvW - io) (v + 2m') - Svu. (vhi^ + 2u'v + 1) = 0 ;
that is,
2v* + 4u'tt' - ivu. - 2u* = 0,
as it should do.
33. Any expression whatever of M in terms of m, v is in fact one of a system
of four expressions; viz. we may simultaneously change
u
M
n«-l »-l
into V , (— ) * w , (— ) * nM ;
that is, signs are
or
1
u'
i»«-i
i«-i
n = l
■I- + +
+ + +
1 ir-i 1 .^_^
or -. (-) 8 i, (-) « '4'^^; + + +
+
+ + +
+
n = o
+
+ + +
+ - +
n = 7 (mod. 8)
+ + -
+ + +
+ + -
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 143
1 2ii^
Thus n = 3, starting from t# = 1 + > we have
" M V
M V u u*M It' V* v^
and of course if from any two of these we eliminate M, we have either an identity
or the modular equation ; thus we have the modular equation under the six different
forms:
(1, 2) (v + 2u')(u~2v>)+3uv =0,
(1.3) v'(v+2u^}-u{u^ + 2v) =0,
(1.4) (v + 2u^)(v'-2u) + 3u* =0,
(2, 3) (u-2v''){u? + 2v) + 3v* =0,
(2, 4) v{v'-2u)-u»(;u-2v') =0,
(3, 4) (m» + 2v) (v" - 2u) + 3uV = 0.
34. Next n = 5. Here, starting from ^ = .^ _ ^ , the changes give
1 V — V' -M— ^ + ''' '"* _ '''(■"-W") W* „_ M'(M4-t)')
]tf^'"t;(l-Mt;»)' tr(l<+M»!;)' mW ~ m^I - w^^) ' ^ ~ ¥{\ + u^v)'
viz. the third and the fourth forms agree with the first and the second forms respect-
ively; that is, there are only two independent forms, and the elimination of M from
these gives
5mi; (1 - MD^) (1 + u»t)) - (?) - M') (« + ^) = 0,
which is a form of the modular equation,
35. In the case n = 7
post. No. 43), the forms are
. ^ 1 -7m(1-m»)(1-mi; + mV) , , ,,.
35. In the case n = 7, startmg from -^ = ^ ^_^^ (as to this see
1 — 7m (1 — uv) {1—UV + mV) .^.
_^^^ _-1v(l-uv){l-uv + v?v^) ,^.
(3),
V — u'
U*M ~ u'(u — v')
«* y j^ _-1u'{l-uv){l-uv+ mV)
11* v'{v — vl) '
BO that here again the third and the fourth forms are identical with the second and the
third forms respectively ; there are thus only two forms, and the elimination of M gives
(m - v') (v - vl) + liiv (1 - my {1-UV + u-vf = 0,
which is a form of the modular equation.
we make the change u, v, t> into v, ± u, ± nM, it becomes
144 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
36. If in the foregoing equation
M
infr* 41 -1- ii
M
±F'u.nM = R{v, ±u);
combining these equations, we have
Fv R(u, v) '
or, substituting herein the foregoing value
M, 1 (l-^^)vF'v
n(l-ti>}uF'u'
this becomes
_ v(l — if) _R{v, ± w) + for n = 3 or .5 (mod. 8),
u(l-u*)~ R(u, v) - for nn 1 or 7 (mod. 8),
which must agree with the modular equation. Thus in the last-mentioned case w = 3,
we have
^F'v -4 = 3 (v^u" + 2M»t) + 1) It,
R{u, v)= (v-ii' + 2u'v + l)u,
R (v, - u) = (vhi' - 2nv' + l)v;
v(l- •««) _ (iW - 2111^ + I) V
^ W(l-M^) ~ (W"it» + 2m>J) + 1) m '
or, say
and therefore
the equation is
which is right; because Jacobi, p. 82, [Ges. Werke, t. I., p. 137], for the modular equation,
gives
1 -t*' = (l -itV)(t/'it>+2M»y + 1), 1 - «;« = (1 -mV)(«%=-2mi;» + 1).
Observe that the general equation
_ v(l-v') _ R(v, ± u)
M (1 — u') R (m, v)
no longer contains the functions F'v, F'u, which enter into Jacobi's expression of M\
Theorem in connexion with the multiplication of Elliptic Functions. Art. Nos. 37 to 40.
37. The theory of multiplication gives an important theorem in regard to trans-
formation. Starting with the nthic transformation
\-y 1-x fa-^x + yx'- ...y \-x [P- Qx^
_ l-x /a-0x + yx'-...Y _ l-x /P - Qx"^
~l-irx\a-\-^x + yx'+...) ' ~\ + x\P-¥Qxl '
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 145
we may form a like transformation,
1-z ^ \-y /a'-/3'y + 7y--Y = II^^ (P'-Q'vX
\+z 1 + y U' + ;8'y + 7y + .-/ ' \+y\F + Q'y)'
such that the combination of the two gives a multiplication, viz. for the relation
between y, z, deriving w from v as v from u, we have w = u; and instead of M we
have M, = + -vy ; that is, we have
dx Mdy
and thence
Vl-ar'.l-MV Vl-2/M-?;y'
dy _ M'dz
'Jl-y-.l- rfy- Vl -z" .1 - u^z' '
J + -dz
ax ~ n
Vl-ar'.l-wV Vl-^M-
u'z'
n-l
or, writing a; = sn ^, we have z=±snnd; ± is here (— ) ^ , viz. it is — for ?i = 3 or
7 (mod. 8), and + for 7i = 1 or 5 (mod. 8).
Now in part efifecting the substitution, we have
l-z_\-x /P-QxV /P' - Q'yV
l+z~l+x[p+Qx) •[P' + Q'y) '
where y denotes its value in terms of x.
And from the theory of elliptic functions, replacing sn n0, sn 6 by their values
±z, X, we have an equation
1-z l-x(A-Bx + Ga?
\-z _ 1-x /A-Bx + Va?- ...S^
where A — Bx+Cx'— ... , A +Bx + Ca^+ ... are given functions each of the order
^(n'—l); viz. the coefficients are given functions of k, or, what is the same thing, of u*.
Comparing the two results, we see that in the wthic transformation the sought-for
function, a + ^x + yx'+... of the order J (» - 1), is a factor of a given function
A +Bx+ Ca!»+ ... of the order ^ (n^- 1).
38. Considering the modular equation as known, then by what precedes we have
f B w" )
a+^x + ya^ + ... = a\l + -«+... + — a;*'"-" h
B m"
that is, the given function A+Bx + Cx'+... has a factor 1+- x+ ...-{ a;*!"-", of
which one (the last) coefficient — is known, and we are hence able theoretically to
C. IX. 19
146 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
determine all the other coefficients rationally in terms of u, v ; that is, the modular
equation being known, we can theoretically complete the solution of the transformation
problem. I do not, however, see the way to obtaining a convenient solution in this
manner.
39. The formula in question for w = .3 is
1 + sn 3g ^ 1 - an g /I + 2 an g - 2^^ sn' g - &» sn* gy
1 - sn 3^~ 1 + sn (? U - 2 sn ^ + 2A» sn» ^ - jfc» sn^"^/ '
which, putting therein a; = sn ^, z = — sn SO, and replacing k by u*, may be written
l+z(^) = (l+x)(l-2x + 2mV - u'x^y (^),
where the signs (-=-) indicate denominators which are obtained from the numerators by
changing the signs of z, x respectively.
The theorem in regard to n = 3 thus is, 1 H — a; is a factor of 1 — 2a; + 2mW — u'a^ ;
... . . v
VIZ. wntmg in the last-mentioned function a; = ^ , we ought to have
0 = 1 + 24-2^-?^„
u^ u u*
that is,
u* + 2uv - 2mV - tr" = 0,
which is in fact the modular equation.
40. And so for n = 5, if a; = sn^, ^ = sn5^; and for n = 7, if a; = sn^, z= — sn7d;
the formulae are: —
n — 0, n = 7,
l+z = {l+x)[
(-)
1
1+2
= (l+a;){ 1
+ 2
X
(-)
- 4
X
- 4
a?
- 4
X'
- lOu'
a?
+ 4(2 + 7m»)
a^
+ OM*
x"
-14m«
X*
4. 4m« (3 + 2m«) «»
- 28m« (3 + 2m»)
ar"
+ 4m'(1- (
ii^)a»
+ 28u« (4 + M»)
of
- 4m« (2 + 3;
u})aF
+ 4tt» ( 16 + 51w8 +
8m") of
- .5m"
a?
- u" (144 + 305m« +
16m") ««
+ lOet"
a?
- 8w« ( 4+ 25w» +
16m") a;»
+ 4m«
x">
+ Sjt' ( 8 + 57m« +
46m") a?"
- 2w"
a;"
+ 56w" ( 2 + w')
«>•
- «"
a;"}''-^
- 4w" ( 56 + 161u' +
56m") a;"
578] A MEMOIR OK THE TRANSFOBMATION OF ELLIPTIC FUNCTIONS. 147
Term in { j has factor + o6?t" (1 + 2m' ) a^^
8 w°
l + -a;+-a-=; + 8m"(46+ 57^8+ 8M")a^*
« = 1, term in j } is
= (1 + xy (1 - xf.
y
- 8m«( 16 +
25m8 +
4rM")a^
- M«( 16 + 305m« + 144w")«"
+ 4m^(
8 +
51m^ +
16m") a?'
+ 28it^ (
1 +
4m0
«!'
- 28w'2 (
2 +
3m«)
«!»
_14m*>
fl^
+ 4m* (
7 +
2m')
a;^'
- 4m«
a!»
- 4it«
0^
+ M«
s^Y
m in { } has factor
(-)
1 +-a;+^ar' +
a a
'^'^;
D
: It = 1, term in { } is (1 + a;)" (1 - «)".
The transformations n = 3, 5, 7, 11. Art. Nos. 41 to 51.
41. The cubic transformation, n = 3.
I reproduce the results ab-eady obtained; since there are only two coefficients a, ^,
these are also the last but one and last coefficients p, a. Hence, from the values of
a, /3, p, a, we have
a = l, 2a= - -jj.-- ,
1 1mm'' 2m'
the two values of -jg are thus -n-=^~i + ~i' =l-< . giving the modular equation
V* + 2ifu' -2vu-u* = 0;
and we then have
1 — y_l— « fv — u^xV
1 + y~ 1 +x \v + u'xj
19—2
148 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
42. The quintic transformation, n=5.
Here there are the three coefficients a, /3, 7, or /3, 7 are the last but one and
last coefficients p, a ; we have
a=l. 2^ = ^u(}^-g.
2/3 = 1-1. 7 = -.
Comparing the two values of ^, we have -^= q _««mV ^°*^ ^'^^^
80 that only the modular equation remains to be determined.
The unused equation is
2a7 + 2a/9 + /? = -, (207 + 2/87 + ^),
which, putting therein a = l, may be written
(27 + ^) {ii? - If) = 2/3 {r/v' - u') ;
attending to the value of yS, this divides by u^ — ifi; in fact the equation may be written
and then completing the substitution, and integralizing, this becomes
jSvit' (1 - ifuf + {v*- w*y} = ^uv {u? + 1?) (1 - ii?v) (1 - uv"),
viz. this is
4 (1 - ■w'm) uv {2m= {l-iihi)- {n- + v") (1 - vu^)] + (?;*- m*)> = 0 ;
and the term in { } being =—{if^ — tu') (1 + vu"), the whole again divides by ^ — v},
and the equation thus becomes
{v'^ 4- 11?) (if-u*)— 4!uv (1 - ifu) (1 + m') = 0,
which is the modular equation.
43. The septic transformation, n = 7.
I do not propose to complete the solution directly from the fundamental equations
for a, /3, 7, B, but resort to the known modular equation, and to an expression of M
which I obtain by means thereof.
The modular equation is
(1 - M«) (i-v»)-(i- uvy = 0,
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS,
which may also be written
(d - m') (u - v') + luv (1 - uvf (1 - uv + mV)2 = 0,
as can be at once verified ; but it also follows from Cauchy's identity
{x + yj-x'-f^ Ixy (x + y) {a? + xy+ff.
149
We then have
Moreover
^,_ ia-v^)vF'v
n (l-u^)uF'u'
uF'u = -2iP(l-'^) + uv (1 - uvy
{I -uvy
l-v?
u(v — w') ;
and similarly
whence
F'v=^^v(u-v^),
1_ _-7u (v - u^)
M^
V u — v'
Writing this under the form
1 _ -luv (v - vT) (u - v'') _ 49t«2 (1 - uvy (l-uv + u^v'-y
M*~ x^ (u-vy ' ~ (m - ■y')2 '
I find, as will appear, that the root must be taken with the sign — , and that we
thus have t> = —
7u (l-uv)(l-uv + mV)
M'
u — v'
, whence also M=
V (1 — uv) (1 —uv + mV)
v — u'
44. Recurring now to the fundamental equations for the septic transformation, the
coefficients are a, ^, y, 8, and we have
a = l, 2^ = «v(i-g^
so that the coefficients are all given in terms of v, M. The unused equations are
u' (207 + 2a/S + ^) =^ (y' + 2yB + 2^88),
«-» (y + 2;37 + 2a8 + 2^8) = v" (2a7 + 2^7 + 2aS + ^%
which, substituting therein for a, /8, 7, 8 the foregoing values, give two equations;
from these, eliminating M, we should obtain the modular equation, and then M in
terms of u, v.
150 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
Substituting in the first instance for a, B their values, the equations are
u' (2/9 + 27 + /3") = v^ J-y= + 2 - (/9 + 7) •
7= + 2/97 + (2 + 2/3) ^' = wV K + 2/37 + 2 ^- + M .
The first of these is
4 (1 -Mi;) (2/9 + 27)+ 4/3= - 4-„7»= 0,
viz. this is
2
or observing that in this equation the coefficient of -v^ is
(1 - uhf) {2 - 2uv + 2mV - 1 - mV},
= (1 - ttV) (1 - uvy, = (1 - Mt>)' (1 + uv),
the equation becomes
(l-i;«)^,+ ^(l-M«)»(H-«t;) + l-w»-4(l-w)(l+^') = 0.
45. This should be satisfied identically by the foregoing value of -r^; viz. it should
be satisfied on writing therein
1 _ lu v — v?
M^~ V u — v''
1 7m (1 - uv) (1 - uv + uV) .
M~ u-v' '
that is, we should have
fly
-7-(v-u')(l-'if)- Uu{l-uvy{l + u>v')
+ (u - v') |l - w» - 4 (1 - uv) (\ + "')l = 0,
where observe that the — sign of the second temi is the sign of the foregoing value
of -jj^; so that the identity being verified, it follows that the correct sign has been
attributed to the value of -i>.
M
46. Multiplying by v, the equation is
- 7 (1 - M« - \^uv) il-'if')- l^uv (1 - uv)* (1 + mV)
+ {1 - ?;« - 1 - mi;} {- 8 (1 - uv) + 1 - m«} + 4 (1 - uv) (v - u?) (u - v') = 0,
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 151
viz. this is
- 7 (1 - u^) (1 - 1;8) + 7 (1 - uv) (1-v^)- Uuv (1 - iivy (1 + uV)
+ il-u«){l-v')-8(l-tiv)(l-if)+ 8 {1-uvy
- 1 (1 - uv) (1 - u^) +4 (1 - lov) (v - v7) (u - v') = 0.
In the second column the coefficient of 1 — uv is 2 — u' — v^, viz. this is
= (1 - u^) (l-if)+l- (uvf, or it is = (1 - uvf +1- {imf.
Reducing also the other two columns by means of the modular equation, the equation
thus becomes
- 6 (1 - uvf - (1 - uv) {(1 - uvf + 1 - {uvf] - I4euv (1 - uvf (1 + uV)
+ 8 {l—uvf
- 28uv {l-uvf{l-w) + uhi'Y = 0.
This is in fact an identity; to show it, writing for convenience 6 in place of uv,
and observing that the terms
-(\-d){i-e') + s{i-ey,
= (i-ey{s-{\ + e+e^ + 6''+e' + e' + e»+e')}
are
= (l-e/(7 + 6^ + 5^ + 4^=+3^^ + 20' + (9«),
the whole equation divides by (1 — 6y ; or thi'owing out this factor, it is
-Qii-ey-ii-ey+i + ^6 + 0$^ + 4^' + u* + 2^» + ^
-ue(i-e){\ + e')-'2.se{i-e+ej = o.
The first line is =14^(3-50 + 6^-3^' + ^); whence, throwing out the factor
140, the equation is
3 - -r>e + 6(9^ - 30^ + 0^ - (1 - 61) (1 + 00 - 2 (1 - 0 + &")',
that is,
( 1 _ ^ + ^s) (3 _ 20 + 0^ - (1 - 0=i) ( 1 - 0 + 0") - 2 ( 1 - 0 + (9»)» = 0 ;
or throwing out the factor 1 - 0 + 0», the equation is
(3 - 20 + 0»)- (1 - 0") - 2 (1 -0+ 0") = 0,
which is an identity.
The other equation is
7= + 2/87 + (2 + 2y8) ^' = mV (27 + 2/37 + 2 ^' + ;S=) ;
that is,
y + 2/87 - mV/3^ + 2 (1 + /3) (^ - 7M'u") - 2w»«^ = 0,
which might also be verified, but I have not done this.
152 A MEMOIR ON THE TBANSFORMATION OF ELLIPTIC FUNCTIONS. [578
47. The conclusion is
where
1 _ - 7m (1 - uv) (1 - uv + u^ri^)
M~ u-v'
and of course
1-y _ 1-x n -fix + ya^- SafV _
r+y ~l+x \1+ /3a; + 70? + Baf) '
but the resulting form may admit of simplification.
48. The endecadic transformation, m=11.
I have not completed the solution, but the results, so far as I have obtained
them, are interesting. The coefficients are a, ^, 7, 8, e, f ; and we have, as in general.
a = l . 2e=«v(^-!;;),
The unused equations then are
m" (2a7 + 2a/3 + ;S=) = v- (e= + 2e?+ 28^,
«« (y + 2ae + 2aS + 2^y + 2/8S) = v^ {iye + 27? + 2Se + 2/3? + P),
M-» (276 + 2af + 27S + 2/3e + 2/Sf + S^) = t)= (7= + 2ae + 2a? + 278 + 2jS€ + 2,SS),
tt-" (€-" + 27? + 2Se + 2S?) = If (2a7 + 2aS + 2/37 + /S') !
but I attend only to the first and the last, which, it will be observed, contain 7, S
linearly. If in the first instance we substitute only for a, ? their values, the equations
become
M")8(2+/3)-^e(e+2^*J) ^-w'^27 -vm».2S = 0,
«-"€' -''^^ +i---Jl+^)}.27+{---]+M-4.2S = 0;
say, for a moment, these are
4 + P . 27 + Q . 28 = 0,
£ + i2 . 27 + /S . 28 = 0,
giving
1 : 27 : 2B = PS-QR : Qfi-/S4 : RA-PB.
Here
P/S- QjB = — + €- m'V + m'- wV(l +/3)
= ^ l^!^ 4. ^„7j^ ^^ _ !f!!^ _ 2miv + 2m' - 2mV - (mV -^ - mvU .
578]
A MEMOIR ON THE TRANSFOEMATION OF ELLIPTIC FUNCTIONS.
153
where the terms containing ^ disappear of themselves, viz. this is
= i (— - 2w'V + 2«« - ?tv]
= -i — (^ + 2i)V-2t;!t-M*);
observe that the term in ( ), equated to zero, gives the modular equation for the
case n = 3. It thus appears that 7 and S are given as fractions, having in their
denominator this function v}-^1uv — 1uh? — tf.
49. To complete the calculation, we have
'6=
viz. multiplying by 8, and substituting for 2/3, 26 their values, this is
or, what is the same thing.
I
viz. the left-hand side is
J,a-^)+|(i-«^)-3(i-«')}{i+«'("-^;
or, say we have - — {QB- SA) = Tl, where
n= M-s-^^i-^)
+ jr»-"'(''"^'^^^^~'^^
+ ^ . 4mV + 1)^1 - 3m') - 4mV + 2m*
+ .-2i;*+6?^V(l -«') + «■* (-3 + 5^8);
C. IX.
20
154 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
wherefore the value of £7 is = \Tl -i- (v* + 2v^u' — 2vu — u*). Similarly, writing
+ ii> . 4-u'ifi + v*(3- v?) + 4mv -2whfi
+ v'{-5 + 3it«) + 6t>M. (1 - v?) + 2m",
we find
«•
28= J-n'-^(t;* + 2^t'-2vM-tt«);
in verification whereof observe that this being so, the first equation gives the identity
l(»-')(i+=)-'(»-?)(B+?)}<-+^*'-^'»-'*'>+"-"'=''-
50. The result is that, writing for the moment if + 2ifu^ — 2vu — m* = A, the values
of the coefiicients are
a, /3 , 7 > ^ . c , ? ,
and
1 - y _ 1 -X /l-0x + ya?-Ba^+ex*- ^afiV _
T-Ty "~ 1+x \1 + ^x + ^0^ + Sx^ + €X* + laf) '
the modular equation is known, and to complete the solution we require only an
expression for M in terms of u, v.
51. We may herein illustrate the following theorem, viz. we may simultaneously
change w, t), -^ , a : ^ : 7 : S : e : ?; into -, - , - -^, ? : e : S : 7 : /3 : a.
Thus making the change in the equation
we have
which is right.
So in the equation - = i -r- > if for a moment (11), (A) are what IT, A become, the
578] A MEMOIE ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 155
equation is | = ig), that is, 1 ^J^^, or (n) = l <^)n'; but obviously ^f=-^;
and the equation thus is (n) = -^— 11', or say 2*1^(11)=- 11'; that is,
w» * Jl/'^ ■ m' \u if) \ if)
which is right.
The general theory by q-transcendents. Art. Nos. 52 to 71.
52. I recur to the formula
1— y_ 1-^a: /a — ^x + ya?-\-.. ± o-?ii'"-"y
1 + 2/ ~ 1 + iT \a + /ya; + 7^^^ + . . + awii"-!'/ '
and seek to express the ratios a : /3 : ... : o- in terms of q. Writing with Jacobi
mK + m'iK'
<o =
n
we have in general
a + /9a; + 7a;» + ... +a^<"-" =a(l + — ^) fl + ^-)...fl + — ^^— -),
\ snc2o)/\ snc4a)/ \ 8nc(w — l)a)/
(snc = sin CO am ; viz. sne 2« =sn (/T— 2&)), &c.) ;
and the values of a, /3, ..., 6 which correspond to the moduli t;„, «i,...t;„, or say the
values (a„, /3o, ...,^o), (flj, /8i, ..., ^j), ..., (a„, /8„, ...,^„), are obtained by giving to w the
values
_2Z 2K + iK' ^K + iK' iK^
~ n ' n ' n '"" n '
viz. the cases Wo, q>„ correspond to Jacobi's first and second real transformations, and
the others to the imaginary transfonnations.
I remark that w = Wo gives for snc 2ga) an expression which is rational as regards
1
q, but eo=Q)n gives an expression involving q", the real nth root of q; the other values
1 1
Oh, Wj, ... give the like expressions, involving aq", a'^q", ... (a an imaginary nth root
of unity), the imaginary nth roots of q.
20—2
156 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
53. I consider first the expression
_ dn 2ga)o
snc 2gmo ' sn (K — 2go)o) ' en 2g(0i, '
2KP
Here, writing 2gcoo = — (f for Jacobi's a;, as a; is being used in a different sense),
TT
that is,
^ tr „ 2K 2g-Tr
Slrt 2irt'
(and thence e^= e " = a^, e^ = o"", if a = e " , an imaginary nth root of unity), we have
(Jacobi, p. 86, [Ges. Werke, t. i., p. 143])
1 , 2KP 2K^
ji — = dn — ^^ -T- en —
snc 2gra>„ ir ir
where
that is.
C 2etf (1 + ge"0 ■ . (1 + qe--^) . .
C
B
{(l+q)..] ^ ^^^'
_ 1 ^^o^ f^.. (l + a'^g)..(l+a"-''^g)..
8nc2^a)o 1+a^'-' ^^'' ' (! + a'*'?') • • (1 + a"^ S') • • '
where, for shortness, I write (1 +5'e'''0 ... to denote the infinite product
(1 + q&^)C\ + fe^) (1 + g»e^) ...,
and similarly (l+5»e»«)... to denote the infinite product (1 + g'e'*) (1 + g^e^) (1 + g'e'*) . . . ,
and the like for the terms in e"^: the notation, accompanied by its explanation, is
quite intelligible, and it would be difficult to make one which would be at the same
time complete and not cumbrous. Then attributing to g the values 1, 2, ...,^(71 — 1),
and forming the symmetric functions of these expressions, we have the values of - , - , &c.,
or a being put = 1, say the values of /9, 7, . . . , o-.
54. I stop to notice a verification afforded by the value of /80. Putting m = 0,
that is, 2 = 0, we have
1 _ 2ag
snc 2gm^ ~ 1 + a^ '
and thence
1 1 . —
we have 2/9o = ju — 1 ; and putting as above m = 0, the value of jr is = (— ) * w ;
whence
fa a= a" a*<"~" 1
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 157
a theorem relating to the imaginary ?ith roots of unity, n an odd prime. In particular,
n = 3, —4 = 4 \- \, at once verified by a^ + a + 1 = 0 ;
w = 5, 4 = 4 J + z i, verified by a' — 1 = 0,
(1 + a^ 1 + a^j •'
viz. the theorem is also true for the real root a = 1 ; in fact, the term in { } is
{a (1 + aO + a" (l+a^)t-^ (1+0^(1+0^), that is, (a + 1 + a^ + a^)-^ (1 + a^ + a^ + a), =1;
which may be verified by means of a*+a' + a^ + a' + a^+a + l = 0; and so on.
.55. I further remark that we have
-L = (_-)J(»-i) [ sn2tOo.sn4wo..sn(w-l)Q>o y
Mo |snc 2a)„. snc 4a)o. . snc (n — l)ft)oJ '
But Jacobi (p. 86, [Lc.]),
sn 2cr«o = sn — ? ,
^ AK e^-1 (l-q^e^^)..(l-q^e-^)..
~ iri e'i (1 - qe^) . . (1 - qe-^) . . '
where (p. 89, [Ic, p. 146])
that is,
mzgio-j q. ^^^ (1 -a^i/^). . (1 -a"-^?). . •
Hence
sn 2^«o _ a^w - 1 1 - g^ g" . . l + a^q.. 1 - a""^ g" . . 1 + a"-^ q . . .
snc 2ga>o ~ i"(a^Tl) iTc^' 7. l^a^q . . 1 + a»-«fi' g» . , 1 - a'^a q,,'
and giving to g the values 1, 2, ...,^(n — 1), and multiplying the several expressions,
we have the value of ir? , viz. this is
Mo'
Mo ^ ' \i'{oL''-'+m^^''
where R{q) denotes the product of the several factors which contain q.
66. The (v") of the denominator gives a factor i""', =(-) '^ , which destroys the
n-l
factor (— ) 2 . We have then a factor
n CZ^)"' which is = (-)i<»-') n.
158 A MEMOIK ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
In fact, n = 3, this is
viz. the numerator is a — 2a^ + l, =— 3a', and the denominator is (— o)*, =a'.
So n = 5, the formula is
or
a' — 4a* + 6a<-4a + l _
a» + 2a«+l •
viz. this is 5 (1 + a' + 2o*) — (1 — 4a — 40" + a' + 6a*) = 0, which is right ; and so in other
cases.
We thus have
which, on putting therein u = 0, that is, q = 0, gives, as it should do, ^^ = (— )i('»-i» n.
57. As regards the expression of R(q), observe that, giving to g its different
values, the factors l-a^q'' and 1 — a"~^ q^ are all the factors other than l—q" of 1 — q^,
and so as to the other pairs of factors; viz. we have
R(„) = fi-g"*-- 1 + g"-- L+9' • • 1-g ••V
-"-Vtf^ Vl-3' .. i + 5 .. 1 + 3^.. 1-5"../ '
viz. this is
that is,
agreeing with a former result.
58. We have of course the identity 2/3,, = ^ —1; that is,
^''l + a.^J '^'^'■(l+a'^q^)..(l+a"-^q').. ^^ <f>'{q)
{g=l, 2, ..^(n — 1)), which, putting therein 5 = 0, is an identity before referred to; a
form perhaps more convenient is obtained by dividing each side by /* (q).
59. I notice further that we have
Vo = m" {snc 26)o snc 4a)„ . . . snc (n — 1 ) w,] ;
the term in { } is
2a?/ ^'i\l^ai'q)..{l + a»-'!>q) ..'
1 -\- a^ 1 + a^
where we have 11 = (_)i(n»-i). For example, n = 3, the term is =-1;
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
n = b, it is
159
(H-a=)(l + a*) 1+a^ + a^ + a
. =-1;
w = 7, it is
a . a" ' a'
(1 + g") (1 + a*) (1 + ol') _l+a + a2 + a' + a< + a» + 2a[<'
a. a'', a*
a?
= 1;
and so on. The term in question thus is
that is.
^_\i(n«-l) ^ /--n+l ((j\ l+g'^-.l + g ••
This has to be multiplied by it", = (v'2)" g-^y^ (g), and we thus obtain
l,„ = (_)i(n-.)V2jV(9").
agreeing with a former result.
We have in what precedes a complete g-transcendental solution for the trans-
fffrmatio prima ; viz. the original modulus k^ (= m') being given as a function of q,
then, as well the new modulus \oH=V) and the multiplier M^, as also the several
functions which enter into the expression
ffl- ^ ] (i ^ \\
\ snc 2(ao/ V sac {n — V)u) J
)l
l—y_l—x
\ snc 2(Bo/ ■ ■ ■ \' ' snc (n — 1) Wo.
are all of them expressed as functions of q.
60. I consider in like manner the expression
1 1 _ dn 2g(0n
snc 2ga>n sn (^K — 2ga)^ ' en 2gmn '
Here, writing 2^(»„= ^ (f instead of Jacobi's x as before), that is,
TT
TT iK' _gmK
and thence
we have
nZ '
1 , 2.K^ 2K^
s = an — ^ ^ en — ^
snc zgwn v ir
=/'(?)
2g" (1+g »).. (1+g ")..^
1+5" (1+2 ")•• (1+? ")..
160 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
where the notations are as follows:
(1 + 5 »)., is the infinite product (1+g '')(l+9 ")(H-g ")••.
and
«+«? 2+?? 4+?? 6+^
(l + g»).. is the infinite product (1 + 2 '*)(l + 5' ")(l+3 ")..;
and the like as to the expressions with exponents containing — ^ .
it
And then attributing to g the values 1, 2, ..,^(« — 1), and forming the symmetric
functions of these expressions, we have the values of - , -,..,-; or a being put = 1,
say the values of )8, y, ... , a.
It is easy to see, and I do not stop to prove that, if instead of w = a>„ we have
1
(0=0)1, ft>2. •••> or cB„_i, we simply multiply 5" by an imaginary nth root of unity; that is,
1
we replace the real nth root 5" by an imaginary nth root of q.
In the case w = 0, that is, g = 0, we have „ = 0, and thence /8 = 0 ; and the
^ snc zga>n
like for the values Wi, <>)«,..., cb„_i : the equation 2/3 = tj^— 1 gives consequently for ^, n
values each =1, agreeing with the multiplier equation.
61. We have for Mn the formula
— = ^_\i(n-a) [ 8n2a>„sn4a)„...Bn(n-l)a)n j"
^n (snc 2o)„ snc 4ft)„ . . . snc (n — 1) q)„J '
and, as before.
2+^. ., 2-^
2iq» {l-q »).. (I-5 »).,
^ 2+?? 1+?? 2-^ 1-^
hence
8n2ff<»„^ g" -1 (l-g"'»).. (1+g"").. (1-g" ").. (1+g' »)^
8nc2ga)„"" ?? ' 4+^ 1+*? «-*? i-B? '
^" i(g» + l) (l+g"^").. (1-g »).. (1+g »)..(l-g «)..
and we thence derive the value of ^ ; viz. observing that we have in the denominator
(i*)iin^»^ _ ^_)}(n-i) vsrhich destroys this factor in the expression of ^ , this is
1-g" {l-q '')-.(l + g '').-(l-g ")-.(l+g ")..|
^ 2+2? i+a? 2-2? i_^
,l+g»(l+g »)..(l-g^»)..(l + g ")..(l-9 »)
578] A AIEJIOIE OX THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 161
Now, giving to g its values, it is easy to see that we have
2
?? .2+?? 2-?? r'l _ o»^
n(i-5»)(i-3 -)..{i-q ").-=;i_y;;.
1 2 4 6 1_
where (1— g").. denotes (1 — g'")(l — 5'")(1 — g").., viz. it is the same function of j" that
(1 — 5=) . . is of 3 ; also
1
118 5 1
where (I+9").. denotes (1 + g»)(l +g'")(l 4-9").. , viz. it is the same function of 9"
that (1 + g) . . is of g ; and the like as to the denominator factors : we thus have
± _ jg-g")-- (1 + ?")•• a+g°)-- (i-g)-.r
itfn" ^ . ._ . .. ^. .^ t
'n
viz. this is
or, we have
\{\-ct)..{\+q )..(l+g»)..(l-g«)..
f(l-gj)-- a+g^)..l . |(l-g')..(l+g)..^'
l(l + g»).-(l -?")••' l(l + 30--(l-9)..
1 ^
agreeing with a former result.
We have
that is.
^^ = </,= (g»)-<^'(g).
2^»=i-i'
lsnc2(B„ snc 4a>„ snc(n— l)ft)„j </>''(?) '
a result which, substituting on the left-hand side the foregoing values of the several
functions, must be identically true.
62. We have also
ii„ = M» [snc 2(B„ snc 46d„ . . . snc {n — 1) «d„},
where the term in { } is
«? i+U iJt
_ n/-- (n^ (^+g"> (1+g ")■•(!+? ")•• .
2g» (l + g »)..(H-g ")..
0 . 1 71^ ^1
or, observing that the sum of the exponents - is -{1 + 2.. + ^(?i — 1)} = -^ - , this is
(v'2)»-igr8» (1 +?»)..(! + ?»)..
C. IX. 21
162 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
1
or, the last factor being / (5") -5-/(5'), the expression is
?+/
n
or, multiplying by m», ={'^Yq^f^(c[), we have
_ JL I
agreeing with a former result.
We have in what precedes the complete 5-transcendental solution for the trans-
formatio secunda; viz. the original modulus k(=u*) being given as a function of q,
then, as well the new modulus X„ (=««*) and the multiplier Jlf„, as also the several
functions which enter into the formula
1 — y _ 1 — a;
\ snc 2<»„/ v. snc (ra — 1) <»„/
(l.H_^)...(l^ - ) '
\\ snc 2a)„' \ snc (n — 1) qj„/,
1^
are all expressed in terms of q. The expressions all contain g", and by substituting
for this an imaginary nth root of q, we have the formulae belonging to the several
(n — 1) imaginary transformations.
63. As an illustration of the formulas for the transformatio secunda I write n = 7 ;
1
and putting for greater convenience q = r'', that is, r = g', then we have
—6 =^p{r')A, -\- = 2f-{r'')B, — ^=2/=(»-0C,
snc 2w7 J "- ' ' snc i(l)^ J ^ ' ' sue g^,^ y \ / >
where
5.19... 9.23.
A —r
B = r'.
2.16.
..12.26..'
3.17.
11.25..
4.18.
10.24,.'
1.15..
13.27..
6.20.. 8.22..*
where the numerator of A denotes (1 -f- r") (1 -(- r") . . (1 -I- r*) (1 -I- r") . . , and so in other
cases, the difference of the exponents being always =14. And we have, as mentioned,
the identical equation
/'(rO(^-fJB-HCr) = i{|J-l}.
The values of the several expressions up to r*" are as follows: Mr J. W. L. Glaisher
kindly performed for me the greater part of the calculation.
578]
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
163
Ind.
of r
B
Sum
Multiplied by
0
0
0
+
1
1
+ 1
+ 1
+ 1
+
4
2
+ 1
+ 1
+ 1
+
4
3
- 1
+ 1
0
0
0
4
+ 1
+ 1
+ 1
+
4
5
+ 1
+ 1
+ 2
+ 2
+
8
6
+ 1
- 1
0
0
0
7
- 1
- 1
- 1
—
4
8
- 1
- 1
- 3
—
12
9
+ 1
- 1
- 1
- 1
- 3
—
12
10
+ 2
+ 1
- 1
+ 2
+ 2
+
8
11
- 1
- 1
— 2
- 4
_
16
12
— 2
- 1
- 1
- 4
- 8
_
32
13
+ 2
+ 2
+ 2
+
8
14
+ 2
- 1
+ 1
+ 3
+
12
15
+ 1
- 1
+ 1
+ 1
+ 8
+
32
16
— 2
+ 2
+ 2
+ 2
+ 9
+
36
17
- 2
- 2
+ 2
- 2
- 6
—
24
18
+ 1
+ 1
+ 2
+ 4
+ 13
+
52
19
+ 2
+ 2
+ 2
+ 6
+ 24
+
96
20
- 3
+ 1
_ 2
- 6
—
24
21
- 2
+ 2
- 1
- 1
- 8
_
32
22
- 2
+ 1 i
_ 2
- 3
- 20
—
80
23
+ 2
- 4
- 3
- 5
- 24
—
96
24
+ 3
+ 3
- 4
+ 2
+ 16
+
64
25
- 1
- 4
- 5
- 33
—
132
26
- 4
- 3
- 3
-10
- 62
—
248
27
- 2
+ 5
- 1
+ 2
+ 16
+
64
28
+ 4
- 3
+ 1
+ 2
+ 19
+
76
29
+ 5
- 1
+ 3
+ 7
+ 46
+
184
30
- 3
+ 6
+ 5
+ 8
+ 56
+
224
31
- 7
- 6
+ 7
- 6
- 40
—
160
32
+ 1
+ 1
+ 7
+ 9
+ 77
+
308
33
+ 9
+ 5
+ 4
+ 18
+ 144
+
576
34
+ 3
- 8
+ 1
- 4
- 38
—
152
35
- 9
+ 5
- 1
- 5
- 42
—
168
36
- 7
+ 2
- 5
- 10
- 99
—
396
37
+ 7
- 9
- 9
-11
-122
—
488
38
+ 11
+ 10
- 11
+ 10
+ 88
+
352
39
- 4
- 3
- 10
- 17
-168
—
672
40
- 13
- 8
— 7
- 28
- 310
—
1240
41
- 2
+ 13
- 3
+ 8
+ 82
+
328
42
+ 13
- 8
+ 3
+ 8
+ 88
+
352
43
+ 8
- 3
+ 9
+ 14
+ 204
+
816
44
-11
+ 14
+ 14
+ 17
+ 252
+
1008
45
- 14
- 14
+ 16
- 12
- 182
—
728
46
+ 5
+ 4
+ 15
+ 24
+ 344
+
1376
47
+ 17
+ 11
+ 12
+ 40
+ 632
+
2528
48
+ 3
-20
+ 5
- 12
- 168
—
672
49
- 17
+ 13
- 5
- 9
-175
—
700
50
-13
+ 5
-14
- 22
-401
■
1604
21—2
164 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS, [578
64. Afl already mentioned, the foregoing expressions of the coefficients in terms
of q may be applied to the determination of the coefficients as rational functions
of u, V.
Representing by 6 any one of the coefficients a, y9, 7, ..., <r, consider the sum
/ a positive integer, and the summation extending as before to the «+ 1 values of
V, and corresponding values of -. This is a rational function of u, and it is also
integral. As to this observe that the function, if not integral, must become infinite
either for m = 0 (this would mean that the expression contained a term or terms
Au~') or for some finite value of u. But the function can only become infinite by
d 1
reason of some term or terms of Si/ - becoming infinite ; viz. some term ;r —
a " snc zgm
must become infinite; or attending to the equation
v = u^ {snc 2(u snc 4a) ... snc {n — 1) <aj,
it can only happen if w = 0, or if v= 00; and from the modular equation it appears
that if t) = 00 , then also « = 00 : the expression in question can therefore only become
8 7
infinite if m = 0, or if u= co . Now m = 0 gives the ratios - , -,..., each of them a
a a
determinate function of n, that is finite ; and gives also t; = 0, so that the expression
does not become infinite for m = 0 ; hence it does not become infinite either for w = 0
or for any finite value of u; wherefore it is integral. The like reasoning applies to
a
the sum St)"-^-; viz. this is a rational function of u: and it is quasi-integral, viz.
there are no terms having a denominator other than a power of u, the highest
denominator being nV<f; viz. the expression contains negative and positive integer
powers of u, the lowest power (highest negative power) being — ^.
65. It is to be observed, further, that writing the expression in the form
vA + Sfvf-,
(where S' refers to the values d,, Vi,...,v„ of the modulus), and considering the several
quantities as expressed in terms of q, then in the sum S' every term involving a
h
fractional power 5" acquires by the summation the coefficient (1 +a+ a'-(-... + a"~0. ^^^
therefore disappears; there remains only the radicality g* occurring in the expressions
of the v's ; and if nf= /u (mod. 8), fi=0, or a positive integer less than 8, then the
n
form of the expression is 5* into a rational function of q. Hence this, being a
rational and integral function of u, must be of the form
ilM" + jBu>'+« + Cm^+" + &c.
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 165
66. We have thus in general
and in like manner
Si/- = AiO^ + ^w+s + &c. :
a.
Sv--f- = A'u-^ +B'ir'^+^ +&C.
a
We may in these expressions find a limit to the number of terms, by means of the
before-mentioned theorem that we may simultaneously interchange -at, v; a, /3, . . . , p, a-
11 0
into - , - ; a, p,... , fi, a. Starting from the expression of Sv^ - , let <p be the corre-
spending coeflScient to 0 ; viz. in the series a, ^,.., 0,.. , <f),.., p, a, let <^ be as removed
from o- as 0 is from o; then the equation becomes
Svnf ^^Au-" + Bu-"-^ + &c.,
where - = - - = — - — : the equation thus is
o- a o- m" o ^
So'-/* = ^w»-'' +Bu"~i'--^ +&C.;
a
and by what precedes the series on the right-hand side can contain no negative power
higher than -57y_i, ; tliat is, the series of coefficients A, B, G, ... goes on to a certain
point only, the subsequent coefficients all of them vanishing.
In like manner from the equation for Sv~^ we have
/Sfi/+» ^ = il'« (»+"•'' -I- £'«<"+"/-« + &c.,
a
where the indices must be positive ; viz. the series of coefficients A', B', . . goes on
to a certain point only, the subsequent coefficients all of them vanishing.
67. The like theory applies to the expression -jg. V/e have, putting as before
n/= ft, (mod. 8),
/SV \-. = AiO-- + Bu'--^^+...,
M
Sv-f ^, = A'u-^ + B'u-'^-^' +...
M
and we find a limit to the number of terms by the consideration that we may simul-
taneously change u, v, -jr? into -, -, ^^; the equations thus become
M
166 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
[where, if /= or < 4, there must be on the right-hand side no negative power of u ;
but if/>4, then the highest negative power must be 7735^^), and
8v^^^ = A'u'^^* + RW^-* + ...,
where on the right-hand side there must be no negative power of u.
68. It is to be remarked that /3, p being always given linearly in terms of ^
it is the same thing whether we seek in this manner for the values of 0, p or for
that of T^; but the latter course is practically more convenient. Thus in the cases
n = 5, M = 7 we require only the value of ^.
In the case n = 11, where the coeflBcients are a, /3, 7, 8, e, f, it has been seen that
y, 8 are given as cubic functions of -rj^: seeking for them directly, their values would
(if the process be practicable) be obtained in a better form, viz. instead of the
denominator (F'vf there would be only the denominator F'(v).
69. I consider for -j^ the cases « = 3 and 5 :
M
n = 3,f=0, 1, 2, 3. then fi = 0, 3, 6, 1 ;
and we write down the equations
1 V*
viz. if we had in the first instance assumed S -jrj.= A+Bu^ + .. , this would have given
V*
S -jrf= Au* + Bu~* + .,, whence B and the succeeding coefficients all vanish; and so in
other cases. We have here only the coefficients A, A'; and these can be obtained
without the aid of the 5-formulae by the consideration that for m = 1 the corresponding
values of v, 1^ are
M
V =1, -1, -1, -1,
^=3, —1, —1, —1,
)78] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 167
whence A = 0, A' =6; or we have the equations
giving as before
M ' M ' M ' M
(2u» + Sif'it - m) -^ = 3 (v^= + 2m»u + 1) w,
1 2m'
reducible by means of the modular equation to ^f = 1 H ■ •
•' ^ M V
70. n = 5. Corresponding bo /= 0, 1, 2, 3, 4, 5, we have /i = 0, 5, 2, 7, 4, 1, and
we find
1 V*
^M^-^' gluing 'S^]0= =^<
8^ = 0, „ S^ =0,
S^ = A'u\ „ S^ =A'u\
S^ = A"u + £"< „ Sv-^ I. = A'^i" + B"u-\
M M
But for M = 1 the corresponding values of v, j^ are
« =1, -1. -1, -1, -1, -1,
^=5, 1, 1, 1, 1, 1;
whence ^ = ^' = 10, A" + B" = 0, or say the value of S^ is =A"u(l-u').
The value of A" is found very easily by the g-formulae, viz. neglecting higher
powers of q, we have
hence
STf=%+S'i. =5qH-^)' = A"qi'^;
M M^ M
that is, A" = 20, and the equations are
whence
F'v.^= 20«(1-M»)
-10m* (/§?;„-?;)
- lOw^ (Sv^ViVi — vSVoVi + v^Svo — if)
- 10 (SvoV,v^ViVt - vSvoViViVi + vfSv^ViV., - VSv^Vj + v*8v^ - v'),
168 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
where Sv^, &c. are the coeflScients of the equation
tf + iv^ii' + 5vhi^ — 5vHt* — 4im — m' = 0,
yvs.
SVo, VoVi, VoViV,, VoViVtV,, V„ViViV,V,
are
- 4m', + 5m», 0 , -ou*. 4m;
or the equation is
F'v.^= 20«(1-M«)
- lOw* (- 4m' - v)
-10m»( _ 5M»t; - 4«'b» - v>)
-10 ( 4m +5u*v - 5v'u'' — Wu' - if),
or, say
where
^F'v -^ = 5 {u' + 4^m' + Qvhi^ + 4dV + vu* - 2m (1 - m')j,
\;F'v =3 v'> + 10i;^» + IOw'm' -5vm*-2u.
Hence also, reducing by the modular equation,
^F'v j^=bu[ifu + 4n^* + 6v^u' + 2^ (1 + m«) + m»},
the one of which forms is as convenient as the other.
71. Making the change u, v, „ into v, —u, —5M, we have
- ^F'u . 5if = 5 {- m' + 'k^u* - 6vhi? + 4j;V -v*u-2v{l- ifi)} ;
and comparing with the equation
5jl/._ (l-^)vF'v
~ {l-u^)uF'u'
we obtain
V (1 -if) _ -2v(l-if)-v*ii + Wu' - 6vhi' + 4!ifu* - m'
M (1 - m8) ~ - 2m (1 - M») + u*v + 4mV + 6mV + 4mV + if '
Writing for a moment ilf = m* + 6wV + v*, N = u- + v-, this is
v{l-if) _-2v(l-if)-uM + iifu^N
~ m(1-m«) ~ -2ull-u«) + vM + 4n)hi'N'
that is,
-4m»(1 -M«) (1 - 1^) - {m'(1 - u^)-v^(l - if)] M+4,ifu» {m" (l-if) + if(l- m»)} N=0.
But we have
u^(l-u'>)-v\l-if) = (u--v-){l-u'-u'v''-u*if-whf-if],
u" (1 - if) + if(l -u')='(u^ + v^) {I -uhfiu* - vN' + if)}.
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 169
Hence, replacing M, N by their values, this is
— 4mv (1 — m') (1 — v^)
— («» - v") (1 - M« - u'v'' - wV - Jt V - v') (u* + 6mV + V*)
+ 4m^ (u^ + v^y {I -wViu*- uhf + v*)] = 0 ;
viz. writing u- — v- = A, uv=B, this is
-A-B{\-A*- 4^2^ - 2B* + B^]
— A[\-A*- SA^B' - SB*} (A' + SB')
+ 45' (4= + 45=) jl - A'& - B*} = 0,
that is.
- 45 {(1 -A*- 4,A'B' -2B* + &)-B' (A' + 45=) (1 - A^B' - B*)}
- A (l-A*-oA^B'-SB*){A^ + 8B') = 0;
VIZ.
-iB {1-A*-5A'B'-S&){1-B*)
- A (l-A^-oA'B'-SB^yiA' + SB'y^O;
or throwing out the factor —{1—A* — 5A-B' — SB*), this is
A (A^ + SB') + 45(1 - 5*) = 0,
the modular equation, which is right.
The four forms of the modular equation, and the curves represented thereby.
' Art. Nos. 72 to 79.
72. The modular equation for any value of n has the property that it may be
represented as an equation of the same order (=?!+l, when n is prime) between
u, v. or between u\ v': or between u*, v*: or between «', v^. As to this, remark that
in general an equation (m, v, 1)"* = 0 of the order m gives rise to an equation
(m", v', 1)*"* = 0 of the oitier 2m between u', v" ; viz. the required equation is
(u, V, 1)"'(«, -■;;, 1)"'(-M, V, 1)'"(-M, -V, 1)"' = 0,
where the left-hand side is a rational function of iv', v' of the form (^t^ v", 1)^; or
again starting from a given equation (u, v, w)'" = 0, and transforming by the equations
X : y : z = u^ : ifi : ttfl, the curve in (a;, y, z) is of the order 2m ; in fact, the inter-
sections of the curve by the arbitrary line a^ + hy + cz = 0 are given by the equations
(u, V, w)™ = 0, aw' -1- 6w= + ctif = 0, and the number of them is thus = 2m. Moreover, by
the general theory of rational transformation, the new curve of the order 2m has the
same deficiency as the original curve of the order m. The transformed curve in
^> y, ■^i = wS i;", wl' may in particular cases reduce itself to a curve of the order m
twice repeated; but it is important to observe that here, taking the single curve of
the order m as the transformed curve, this has no longer the same deficiency as the
original curve ; and in particular the curves represented by the modular equation in
its four several forms, writing therein successively u, v ; u', v' ; u*, v* ; «', v^, = x, y,
are not curves of the same deficiency.
73. The question may be looked at as follows: the quantities which enter
rationally into the elliptic-function formulae are ftf, V = m', i^; it a, modular equation
(m, vy = 0 led to the transformed equation « v^f' = 0, then to a given value of m'
C. IX. 22
170 A MEMOIR ON THE TEANSFORMATION OF ELLIPTIC FUNCTIONS. [578
would correspond 8 values of u, therefore 8v values of v, giving the same number,
8v, values of ifi; that is, the values of v^ corresponding to a given value of li* would
group themselves in eights corresponding to the 8 values of u. There is, in fact, no
such grouping ; the equations are (w, v)" = 0, (u", «*)" = 0 ; . to a given value of m"
correspond 8 values of u, and therefore 8v values of v, but these give in eights the
same value of if, so that the number of values of if is = v.
74. I consider the case n = S: here, writing x, y for u, v, we have here the sextic
curve
I. y*-x*+2xi/(xY-^) = 0;
and it is easy to see that the remaining forms wherein x, y denote «', if; vf, i^; and
«', if respectively, are derived herefrom as follows; viz.
II. {y'-afy-'kxyixy-iy^O, that is,
y* + Qx'f + af- 4iry (afy' + 1) = 0 ;
III. (y" +exy + afy-l6xy(xy + iy=0, that is,
y* + 6afy^ + af- ixy {4!afy' - Saf - Sy' + *) = 0 ;
IV. {y' + 6xy + off - 16xy (ixy -Sx-Sy + 4:^ = 0, that is,
y* - 7e2ofy' -^af- 4,xy {64^afy' - 96afy - 96xy^ + SSaf + .33/ - 96a; - 96y + 64} = 0,
where it may be noticed that the process is not again repeatable so as to obtain a
sextic equation between x, y standing for m", v^' respectively.
The curve I. has a dp (fleflecnode) at the origin, viz. the branches are given by
y* — 2a; = 0, — of — 2y = 0; and it has 2 cusps at infinity, on the axes a; = 0, y = 0
respectively ; viz. the infinite branches are given hy y + 2af = 0, —x+ 2?/' = 0 respect-
ively. These same singularities present themselves in the other curves.
The curve II. has the four dps (af — y'^^O, xy —\= 0), that is,
{x = y=\), {x = y = -\), {x = i, y = -i), (x = -i, y = i).
Corresponding hereto we have in the curve III. the 2 dps (a; = y=l, x = y= — l), and
in the curve IV. the dp {x = y = l).
The curve III. has besides the 4 dps y^ + 6xy + af = 0, xy + 1=0, that is,
(1 + \/2, 1 - \^), (1 - V2, 1 + V2), (- 1 - V2, - 1 + V2), (- 1 + v'2, - 1 - V2);
and con-esponding hereto in the curve IV. we have the 2 dps
(3 + 2V2, 3-2\/2), (3-2V2, 3-I-2V2).
The curve IV. has besides the 4 dps (y" + 6xy + af = 0, 4a;y — 3a; — 3y + 4 = 0), or
say (2a; - 1) (2y - f ) + ^ = 0, 2 (a; + 1)^ + 2 (y + f )" - -1^ = 0. Hence the 4 curves have respect-
ively the dps and deficiency following: —
dps. dps. Def.
2. 1 = 3, 7,
2, 1, 4 = 7, 3,
2, 1. 2, 4 = 9, 1,
2,1,1,2,4 = 10, 0;
viz. the curve IV. representing the equation between w' and w" is a unicursal sextic.
578]
A MEMOIR OX THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
171
It may be noticed that, except the fleflecnode at the origin and the cusps at
infinity, the dps in question are all acnodes (conjugate points).
75. The foregoing equations may be exhibited in the square diagrams : —
L II.
y* f if y I y* f y'' y \
a*
+ 1
-4
+ 6
-4
+ 1
1+2 0 -2-1 ={y + \f{y-\);
1 -4 +6-4 +1 ={y-\)*,
a*
a?
a?
y* f
III.
y* f
+ f
- 16
+ 12
+ 6
+ 12
-16
+ 1
1 _ 4 +6-4 +1 =(y-lY;
IV.
y' y
+ 1
-256
+ 384
- 132
+ 384
- 762
+ 384
-132
+ 384
-256
+ 1
1 - 4 +
4 +1 =(y-\y;
where the subscript line, showiug in each case what the equation becomes on writing
therein x = \, serves as a verification of the numerical values.
The curve IV. being unicursal, the coordinates may be expressed rationally in
terms of a parameter; in fact, we have
^ a^(2 + a) ^ a (2 + gy
* l+2a ' ^~(l + 2a)='
These values give
\&xy =16a*(2 + a)« -=-(l + 2a)'',
4 + 4a;y - 3a; - Sy = (4, 8, 12, 32, 50, 32, 12, 8, 4][1, a)' +- (1 + 2a)^
af + Qxy + y-' = 4a=(2 + a)'(4, 8, 12, 32, 50, 32, 12, 8, 45l, a)8-f-(l + 2a)^
and the equation of the curve is thus verified.
22—2
172
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
[578
76. Considering in like manner the modular equation for the quintic trans-
formation, we derive the four forms as follows: —
I. a?jf+ba?y^{sc'-f) + ^ayy{\-a*y*) = Q\
IL {a* - y + .5an/ (a; - y)Y -\&xy{\- sd'fy = 0, that is,
af + IBaiy + Uary + y'- 2xy{8-5x' + lOie'f - 5y*+8a^y*) = 0 ;
ni. (a^ + I5afy + 1 oivy' ■\- fY - 4>xy {8 - daf + lOxy - 5y» + Sx'yy = 0, that is,
of + 655a^'' + 65oa^y* + y* - 6Wx'y' - 640x'y*
+ iey{- 256 + 320ar' + 32(y - 70a;* - 660a^f - 70y* + SiOx'f + S20a^y* - 256xy) = 0 ;
IV. (x' + eSSx'y + 655a;y» + y» - 64.0a;y - 640ar^')'
-xy(- 256 + 320a; + 320y - lOaf - 660a^ - lOy" + 320ar'y + S20xy^ - 2o6a^y'y = 0 :
or, expanding the two terms in the last equa:tion separately, this is
= 0.
xy
- 65536
^y
+ 163840
^f
+ 163840
s(?y
- 138240
a?f
+ 409600
- 542720
xy^
- 138240
^
1280
+ 44800
^y'
- 838400
+ 631040
xy
- 838400
+ 631040
xy*
1280
+ 44800
of
+ 1
^y
+ 1310
- 4900
^y"
+ 430335
- 297200
^f
+ 1677252
- 986072
a=y
+ 430335
- 297200
xr/^
+ 1310
- 4900
/
1
3?f
1280
+ 44800
aY
- 838400
+ 631040
a?y^
- 838400
+ 631040
«y
1280
+ 44800
off
- 138240
aV
+ 409600
- 542720
a^y^
- 138240
«»/
+ 163840
a-V
+ 163840
^!f
- 65536
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 173
77. The square diagrams are: —
I.
■^f if y* if y^ y ^
II.
*•
-1
a?
+ 4
1
1
a*
-5
n?
a?
+ 5
a^
-4
1
+ 1
y'
y^
y"
?/
f
y
1
+ 1
-16
+ 10
+ 15
-20
+ 15
+ 10
-16
+ 1
1+4+5 0 -5-4-1
1-6+15-20+15-6+1
= (2/-i)«;
III.
6 +1 =(y-l)«;
»•
y"
y*
y
f
y
1
x«
1
+ 1
ar>
- 65536
+ 163840
- 138240
+ 43520
- 3590
ar*
+ 163840
- 133120
- 207360
+ 133135
+ 43520
ar^
- 138240
- 207360
+ 691180
- 207360
- 138240
x'
+ 43520
+ 133135
- 207360
- 133120
+ 163840
X
- 3590
+ 43520
- 138240
+ 163840
- 65536
1
+ 1
+ 1
-6
+ 15
20
+ 15
6 +1 =(2/-l)'
174
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
[578
where the subscript line, showing in each case what the equation becomes on writing
therein x = l, serves as a verification of the numerical values.
78. The curve I. has at the origin a dp in the nature of a fleflecnode, viz.
the two branches are given by af+4iy = 0, — y°+4a; = 0 respectively ; and there are
two singular points at infinity on the two axes respectively, viz. the infinite branche.s
are given by —y— ^af = 0, a; — 4^' = 0 respectively. Writing the first of these in the
form —yz^ — 4^ = 0, we see that the point at infinity on the axis x — 0 (i.e. the point
z=0, x = Q) is =6 dps; and similarly writing for the other branch ««*- 4^ = 0, the
point at infinity on the axis y = 0 (i.e. the point z = 0, y = 0) is =6
Moreover, as remarked to me by Professor H. J. S. Smith, the curve has 8 other
dps; viz. writing m to denote an eighth root of —1, (<u' + l=0), then a dp is « = <»,
y=w\
To verify this, observe that these values give
Qa? --
= + 6
+ 20a^3/=
-20
-lOxy
-10
+ 43/
+ 4
-20ir*2/»
+ 20
-20
+ 4
+ 20
- Qy> =+6
+ \Qa*y - 10
-20a;»y»
+ 4a;
-^Oa*y*
or the derived functions each vanish. Thus I. has in all 1 + 12+8, =21 dps.
In II. we have in like manner 1+12 + 4, =17 dps; viz. instead of the 8 dps,
we have the 4 dps x= or, y = to", {m'^ + 1=0), or, what is the same thing, a; = w,
y = — CO, where w* + 1 = 0. But we have besides the 12 dps given by
a;^-f + 5xy(x-y)=0, l-xh/^ = 0,
viz. we have in all 1+12 + 4+12, =29 dps.
In IIL we thence have 1 + 12 + 2 + 6, =21 dps; and, besides, the 12 dps given by
a^ + 15ay'y + 15xy' + f = 0, 8 - 5a? + 10xy - 5y' + 8aff = 0,
in all 1+12 + 2 + 6 + 12, =33 dps.
And in IV. we thence have 1 + 12 + 1 + 3 + 6, =23 dps; and, besides, the 12 dps
given by
a? + 655^^^ + 655xy^ + y* - GiOxy - 640ar'/ = 0,
- 256 + 320a; + 320,y - 70a;' - 660a;y - TO^'' + S20afij + 320xy' - 256x'y- = 0,
(these curves intersect in 16 points, 4 of them at infinity, in pairs on the lines
a; = 0, y =0 respectively ; and the intersections at infinity being excluded, there remain
16-4, =12 intersections); there are thus in all 1 + 12 + 1+3 + 6 + 12, =35 dps.
* These resnlts follow from the general formulae in the paper " On the Higher Singularities of Plane
Curves," Camb. and Dubl. Math. Joitm. t. vii. (1866), pp. 212 — 223, [374]; but they are at once seen to
be true from the consideration that the curve yz*-x^ = 0, which has only the singularity in question, is
unicarsal ; the singularity is thus =6 dps.
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 175
Arranging the results in a tabular form and adding the values of the deficiency,
we have
I.
dps.
1 + 12 + 8
dps.
= 21,
Def.
= 15.
II.
1+12 + 4+12
29,
7,
III.
1 + 12 + 2+ 6 + 12
33,
3,
IV.
1 + 12 + 1+ 3+ 6 + 12
35,
1,
80 that the curve IV. is a curve of deficiency 1, or bicursal curve. It appears by
Jacobi's investigation for the quintic transformation {Fund. Nov. pp. 26 — 28, [Ges. Werke,
t. I., pp. 77 — 79]) that we can in fact express x, y, that is, u^, if, rationally in terms
of the parameters a, yS connected by the equation
a^ = 2;S(l + a + ;8),
which is that of a general cubic (deficiency = 1); in fact, we have
2 — g _ v* o _ ■""
that is,
where a, /9 satisfy the relation just referred to. The actual verification of the equation
IV. by means of these values would be a work of some labour.
79. In the general case p an odd prime, then in I. we have at the origin one dp
(in the nature of a fleflecnode) and at infinity two singular points each =^(p — l)(jj — 2) dps.
I infer, from a result obtained by Professor Smith, that there are besides (p — 1)(^ — 3)
dps ; but I have not investigated the nature of these. And the Table of dps and
deficiency then is
I. l+(p-l)(p-2)+ {p-l)(p-S)
II. l+(p-l)(p-2) + i(p-lXp-S) + i(p'-l)
III. l+(p-l){p-2) + \ip-l)(p-S) + iip^-l) + i(P'-^)
IV. l + {p-l)(p-2) + i(p-l){p-S) + i(p^-l) + i{p^-l) + ^ip''-l)
viz. his values of the deficiencies being aa in the last column, the
dps must be as in the last but one column.
dps.
2p''-
-Ip + Q,
Def.
4p-
-5,
2p^-
-5p + 4^,
2p-
■3,
2j9»-
- 4p + 3,
p-
-2.
2j^-
-iP + ^.
ip-
■f;
total number of
176 [579
579.
ADDKESS DELIVEKED BY [PROFESSOR CAYLEY AS] THE
PRESIDENT [OF THE ROYAL ASTRONOMICAL SOCIETY]
ON PRESENTING THE GOLD MEDAL OF THE SOCIETY TO
PROFESSOR SIMON NEWCOMB.
[From the Monthly Notices of the Royal Astronomical Society, vol. xxxiv. (1873 — 1874),
pp. 224—233.]
The Council have awarded the medal to Professor Simon Newcomb for his
Researches on the Orbits of Neptune and Uranus, and for his other contributions to
mathematical astronomy. And upon me, as President, the duty has devolved of explaining
to you the grounds of their decision.
I think it right to remark that it appears to me that, in the award of their
highest honour, the Council of a Society are not bound to institute a comparison
between heterogeneous branches of a science, or classes of research — to weigh, for
instance, mathematical against observational astronomy or astronomical physics ; or, in
the several branches respectively, the happy idea which originates a theory against the
patience and the skilled labour which develops and carry it out ; and still less to decide
between the merits of different workers in the science. It is enough that the different
branches of a science coming before them in different years, the medal should in
every case be bestowed as a recognition of high merit in some important branch of
the science.
Before speaking of the Tables, I will notice some of Professor Newcomb's other
works.
Memoir " On the secular Variations and mutual Relations of the Orbits of the
Asteroids," Mem. American Academy, vol. V. (1860), pp. 124 — 152. The object is to
examine those circumstances of the forms, positions, variations, and general relations of
the asteroid orbits which may serve as a test, complete or imperfect, of any hypothesis
respecting the cause fi'om which they originated, or the reason why they are in a
579] ADDRESS DELIVERED BY THE PRESIDENT. 177
group by themselves. Every a posteriori test is founded on the supposition, that the
hypothesis necessarily or probably implies that certain conditions must be satisfied by
the asteroids or their orbits, viz. in the one case the conditions are those which follow
necessarily and immediately from the hypothesis itself, in the other case those which
are deducible from it by the principle of random distribution. The two principal
hypotheses are that of Olbers, where the asteroids are supposed to be the fragments
of a shattered single planet : and the hypothesis that they were formed by the breaking
up of a ring of nebulous matter. On the first hypothesis the orbits of all the
asteroids once intersected in a common point ; the second affords no conclusion equally
susceptible of an a posteriori test.
But for a rigorous or probable test of either hypothesis, what is needed is rigorous
expressions in terms of the time for the eccentricity, inclination, and longitudes of
perihelion and node of each of the asteroids considered, or, what is the same thing,
the computation of the secular variations of the quantities h, I, p, q, which replace
these elements. The investigation is applied to those asteroids the elements of which
were determined with sufficient accuracy, and the eccentricities and inclinations of
which were sufficiently small (limit taken is 11°). And the backbone of the memoir
is the investigation of the h, I, p, q, for twenty-five asteroids included between the
numbers (1) and (40). In this calculation, as was clearly necessary, the action of the
asteroids on the larger planets and on each other was neglected; the expressions for
the h, I, p, q, of the larger pljinets are regarded as given — they are, in fact, taken
from Le Verrier (as calculated by him before the discovery of Neptune, but afterwards
partially extended to that planet). The effect is that the differential coefficients -5-, &c.
ctt
are given each of them as a sum of sines or cosines of arguments varying with the
time ; and thus, although the calculation is sufficiently laborious, the process is not one
of the extreme labour and difficulty which it is in the case of the larger planets.
The resulting table of the h, I, p, q, of the twenty-five asteroids has, of course, a
value quite independent of the theoretical part of the memoir. Of this it is sufficient
to say here that the conclusion is on the whole against Olbers's hypothesis. The
subject is resumed, and more fully examined in a paper in the Astronomische Nachrichten,
t. LVIII.
"Investigation of the Distance of the Sun and of the Elements which depend
upon it, from the Observations of Mars made during the Opposition of 1862, and
from other Sources," Washington Observations for 1865, Appendix II., pp. 1 — 29. The
chief part of this valuable Memoir is occupied with a determination of the solar
parallax by the discussion of the observations of Mars made in 1862 on the plan of
Winnecke: three partial discussions had previously appeared, but these having been by
comparisons of pairs of observations, one in each hemisphere, many observations in one
hemisphere were lost by want of a corresponding observation in the other hemisphere ;
and out of a total of nearly 300 observations, only 12.5 were utilised. The idea is,
the perturbations of the Earth and Mars being perfectly known for the period under
consideration, every observation of the planet would lead rigorously to an equation of
condition between its parallax, the six elements of its orbit, and the six elements of
c. IX. 23
178 ADDRESS DELIVERED BY THE PRESIDENT ON PRESENTING THE [579
the Earth's orbit — thus 13 or more observations, when compared with any theory,
should suffice to correct the errors of that theory. But the observations extending
only over a short interval, say one month, the coefficients would be so minute as to
give no trustworthy value of the corrections; the equations only suffice to determine
& few functions of the elements which, being determined, the equations will be satisfied
by widely differing values of the elements, if only these values are such as to give
to the functions their right values. And by fixing a priori the entire number of
functions in question, and using them in place of the elements of the Elarth and
Mars, the equations will be practically as rigorous as if all the 13 unknown quantities
had been introduced. By such considerations as these, each observation is made to
give a relation between only 3 unknown quantities, the correction of the Sun's parallax
being one of them.
The principle appears to be one of extended application, in regard to the proper
mode of dealing with the constantly recuning problem of the determination of a set
of corrections from a large number of linear equations; and it is used by the author
in regard to the equations which present themselves in his theories of Neptune and
Uranus.
Returning to the Mars observations, these were made at six Northern and three
Southern Observatories, the total number being 1 .54 Northern, and 143 Southern, together
297 observations. There was the difficulty of reducing to a concordant system the
observations at the different Observatories, since (the whole number of comparison stars
not being observed on each night) the adopted mean position of each of them was
not unimportant. But this being carefully discussed and allowed for, the observations,
extending from August 21 to November 3, 1862, ai-e divided into five groups, and
from these is deduced a correction to the provisional value 8""9 of the parallax. The
author then reproduces or discusses other determinations, from micrometric observations
of Mars, the parallactic inequality of the Moon, the lunar equation of the Earth, the
transit of 1769, and Foucault's experiment on Light — the last result, as not a strictly
astronomical one, and with no means of assigning its probable error, is left out of
consideration — and the combination of the remaining ones gives the author's concluded
value of the parallax ; from which other astronomical constants are deduced.
" On the Right Ascensions of the Equatoreal Fundamental Stare and the Correct-
ions necessary to reduce the Right Ascensions of different Catalogues to a mean
homogeneous System," Washington Observations for 1870, Appendix III., pp. 1 — 73.
This important Memoir is referred to in the Council Report for 1873. The object
is to do for the right ascensions of the equatoreal and zodiacal Stai-s what had been
done by Auwers for the declinations, namely, to furnish the data necessary to reduce
the principal original catalogues of stai's to a homogeneous system by freeing them of
their systematic differences. The results are contained in two tables of corrections (aa
depending on the R.A. and N.P.D. respectively) to the several catalogues ; and in a
table of concluded mean right ascensions for the beginning of each fifth Besselian year.
I
579] GOLD MEDAL OF THE SOCIETY TO PROFESSOR SIMON NEWCOMB. 179
1750 to 1900, of 32 fundamental Stars, and of periodic terms in the right ascensions
of Sirius and Procyon.
The evil of systematic differences between the observations of different Observatories
of course presents itself in every case where such observations have to be combined :
for instance, in the just-mentioned determination of the solar parallax by the observ-
ations of Mars ; and in the making of a set of planetary tables : and all that tends
to remove or diminish it is most important to the progress of Astronomy. I cannot
help thinking that there should be some confederation of Observatories, or Central
calculating Board, for publishing the lunar and planetary observations, &c., reduced to
a concordant system. It seems hard upon the maker of a set of planetary tables that
he should not at least have, ready to hand for comparison with his theory, a single
and entire series of the observations of the planet.
" Thdorie des Perturbations de la Lune, qui sont dues a Taction des Planetes,"
Liouville, t. XVI. (1871), pp. 1 — 45. This is a very important theoretical Memoir on the
disturbed motion of three bodies : a problem which, so far as I am aware, has not
hitherto been considered at all. I have elsewhere remarked that the so-called "Problem
of Three Bodies," as usually treated is not really this problem at all, but a different
and more simple one — that of disturbed elliptic motion. Thus, in the planetary theory,
each planet is considered as moving in an ellipse, and as disturbed by the action of
forces represented by means of a disturbing function peculiar to the planet in question.
An approach is made to the problem of three bodies when, as in memoirs by Hamilton
and Jacobi, the (say) two planets are replaced by two fictitious bodies, and instead of
a disturbing function peculiar to each planet, the motion of the system is made to
depend on a single disturbing function. And there are memoirs by Jacobi, Bertrand,
and Bour, which do relate to the proper problem of three bodies, viz. to their undisturbed
motion. But in the present Memoir, Professor Newcomb starts from this problem as
if it were actually solved, viz. he takes the coordinates of the three bodies (Sun, Earth,
and Moon) as given in terms of the time and of 18 constants of integration *. And
then considering the system as acted upon by the attraction of a planet, represented
by means of a disturbing function, he applies to the system of the three bodies the
method of the variation of the elements. The six elements which determine the motion
of the centre of gravity of the system are left out of consideration ; there remain to
be considered 12 elements only ; six of these are eo, t,,, 6^, €„', tto, 6o (initial mean
longitudes and longitudes of pericentre and node) : but the other six k,, k„, &c., are
functions the invention of which is a leading step in the theory, and it is in fact by
means of them that the investigation is brought to a successful conclusion : the
expressions of the last-mentioned six functions can, it is stated, be formed with facility
by means of the developments (obtainable from the lunar theory) of the rectangular
* Of coarse the expreasiona actually used muat be approximationa : the centre of gravity of the Earth
and Moon ia regarded aa moving round the Snn in an ellipse affected by a secular motion of perihelion
(ultimately neglected) ; and the coordinatea of the Moon in regard to the Earth are considered to be given
by Delaunay's Lnnar Theory. The centre of gravity of the whole system (in the undisturbed motion) moves
oniformly in a right line, viz. the coordinates are a + a't, b + b't, c + c't; and we have thus the whole number
6 + 0 + 6, =18, of arbitrary constants.
23—2
180 ADDRESS DELIVERED BY THE PRESIDENT ON PRESENTING THE [579
coordinates x, y, z, as periodic functions of the time. With these twelve elements, the
expressions for the valuations a^ume the canonical form
dk, _ dR deo _ dit .
d^~d^' di~~dk/
The concluding part of the Memoir contains approximate calculations which seems
to show that the whole process is a very practicable one : but the author remarks that
it is only doing justice to Delaunay to say that, starting from his (Delaunay's) final
<lifferential equations, and regarding the planet as adding new terms to the disturbing
function, there would be obtained equations of the same degree of rigour as those of
his own Memoir.
Everything in the Lunar Theory is laborious, and it is impossible to form an
opinion as to the comparative facility of methods; but irrespectively of the possible
applications of the method, the Memoir is, from the boldness of the conception and
beauty of the results, a very remarkable one, and constitutes an important addition to
Theoretical Dynamics *.
I come now to the planets Neptune and Urantis: it is well-known how, historically,
the two are connected. The increasing and systematic inaccuracies of Bouvard's Tables
of Uranus were found to be such as could be accounted for by the existence of an
exterior disturbing planet ; and it was thus that the planet Neptune was discovered by
Adams and Le Verrier before it was seen in the telescope, in September 1846. It was
afterwards ascertained that the planet had been seen twice by Lalande, in May 179.5.
The theory of Neptune was investigated by Peirce and Walker: viz. Walker, by means
of the observations of 1795, and those of 1846 — 47, and using Peii-ce's formulae for the
perturbations produced by Jupiter, Saturn, and Uranus, determined successfully two sets
of elliptic elements of the planet. The values first obtained showed that it was
necessary to revise the perturbation-theory, which Peirce accordingly did, and with the
new perturbations and revised normal places, the second set of elements (Walker's
Elliptic Elements II.) was computed. With these elements and perturbations there was
obtained for the planet from the time of its discovery a continuous ephemeris, published
in the Smithsonian Contributions, Gould's Astronomical Journal, and since 1852 in the
Am^can Ephemeris and the Nautical Almanac. The theory was next considered by
Kowalski in a work published at Kasan in the year 1855. The long period inequalities
are dealt \vith by him in a manner different from that adopted by Peirce, so that
the two theories are not directly comparable, but Professor Newcomb, by a comparison
of the ephemerides with observation, arrives at the conclusion that the theory of
Kowalski (although derived from observations up to 1853, when the planet had moved
through an arc of 16°) was on the whole no nearer the truth than that of Walker;
* Since the above was written, Professor Newcomb has coramnuicated to me some very interesting details
as to the extent to which he has carried his computations, and in particular he mentions that, considering
the action of each planet from Mercury to Saturn, he has (in regard to the terms the coefficients of which
might become large by integration) estimated the probable limiting value of more than fifty such terms of
period from a few years to several thousands without finding any which could become sensible, except the
term leading to Hansen's first inequality produced by Venus.
579] GOLD MEDAL OF THE SOCIETY TO PROFESSOK SIMON NEWCOMB. 181
he observed, however, that this failure is accounted for by an accidental mistake in
the computation of the perturbations of the radius vector by Jupiter.
Professor Newcomb's theory of Neptune is published in the Smithsonian Contributions
under the title " An Investigation of the Orbit of Neptune, with General Tables of its
Motion," (accepted for publication, May 1865). The errors of the published ephemerides
were increasing rapidly ; in 1863 Walker's was in error by 33", and Kowalski's by
22"; both might be in error by 5' before the end of the century. The time was come
when (the planet having moved through nearly 40°) the orbit could be determined
with some degree of accuracy. The general objects of the work are stated to be :
(1) To determine the elements of the orbit of Neptune with as much exactness
as a series of observations extending through an arc of 40° would admit of
(2) To inquire whether the mass of Uranus can be concluded from the motion
of Neptune.
(3) To inquire whether these motions indicate the action of an extra-Neptunian
planet, or throw any light on the question of the existence of such planet.
(4) To construct general tables and formulae, by which the theoretical place of
Neptune may be found at any time, and more particularly between the years 1600 and
2000.
The formation of the tables of a planet may, I think, be considered as the
culminating achievement of Astronomy : the need and possibility of the improvement
and approximate perfection of the tables advance simultaneously with the progress of
practical astronomy, and the accumulation of accurate observations; and the difficulty
and labour increase with the degree of perfection aimed at. The leading steps of the
process are in each case the same, and it is well-known what these are ; but it will
be convenient to speak of them in order, with reference to the present tables : they
are first to decide on the form of the formulae, whether the perturbations shall be
applied to the elements or the coordinates — or partly to the elements and partly to
the coordinates; and as to other collateral matters. These are questions to be decided
in each case, in part by reference to the numerical values (in particular, the ratios
and approach to commensurability of the mean motions), in part by the degree of
accuracy aimed at, or which is attainable — the tables may be intended to hold good
for a few centuries, or for a much longer period. The general theory as regards these
several forms ought, I think, to be developed to such an extent, that it should be
possible to select, according to the circumstances, between two or three ready-made
theories; and that the substitution therein of the adopted numerical values should be
a mere mechanical operation ; but in the planetary theory in its present state, this is
very far from being the case, and there is always a large amount of delicate theoretical
investigation to be gone through in the selection of the form and development of the
algebraical formulae which serve as the basis of the tables. In Prof Newcomb's theory
the perturbations are applied to the elements; in particular, it was determined that
the long inequality arising from the near approach of the mean motion of Uranus to
twice that of Neptune (period about 4,300 years), should be developed as a perturbation,
not of the coordinates, but of the elements. And it was best, (as for a theory designed
182 ADDRESS DELIVERED BY THE PRESIDENT ON PRESENTING THE [579
to remain of the highest degree of exactness for only a few centuries) to take not
the mean values of the elements, but their values at a particular epoch during the
period for which the theory is intended to be used. The adopted provisional elements
of Neptune, and the elements of the disturbing planets, are accordingly not mean
values, but values affected by secular and long inequalities, representing the actual
values at the present time. Secondly, the form being decided on and the formulae
obtained, the numerical values of the adopted provisional elements of the planet, and
of the elements of the disturbing planets and their masses, have to be substituted, so
as to obtain the actual formulae serving for the calculation of a provisional ephemeris;
and such ephemeris, first of heliocentric, and then of geocentric positions, has to be
computed for the period over which the observations extend. Thirdly, the ephemeris,
computed as above, has to be compared with the observed positions ; viz. in the present
case these are, Lalande's two observations of 179.5, and the modern observations at the
Observatories of Greenwich, Cambridge, Paris, Washington, Hamburg, and Albany,
extending over different periods from 1846 to 1864 : these are discussed in reference
to their systematic differences, and they are then corrected accordingly, so as to reduce
the several series of observations to a concordant system. In this way is formed a
series of 71 observed longitudes and latitudes (1795, and 1846 to 1864); the comparison
of these with the computed values shows the errors of the provisional ephemeris.
FouHhly, the errors of the provisional elements have to be corrected by means of the
last-mentioned series of errors : as regards the longitudes, the comparison gives a series
of equations between he, Bn, Sh, Bk, and /j, (correction to the assumed mass of Uranus).
The discussion of the equations shows that no reliable value of fj, can be obtained
from them ; it indeed appears that, if Uranus had been unknown, its existence could
scarcely have been detected from all the observations hitherto made of Neptune (far
less is there any indication to be as yet obtained as to the existence of a trans-
Neptuniau planet): hence, finally, fj. is taken =0, and the equations used for the
determination of the remaining corrections. As regaixis the latitudes, the compai'ison
gives a series of equations serving for the determination of the values of Bp and Bq.
And applying the corrections to the provisional elements, the author obtains his con-
cluded elements ; viz. as already mentioned, these are the values, as affected by the
long inequality, belonging to the epoch 1850. Fifthly, the tables are computed from
the concluded elements, and the perturbations of the provisional theory.
After the elements of Neptune were ascertained, the question of its action on
Uranus was considered by Peirce in a paper in the Proc. American Acad., vol. i.
(1848), pp. 334 — 337. This contains the results of a complete computation of the
general perturbations of Uranus by Neptune in longitude and radius vector, but without
any details of the investigation, or statement of the methods employed : it is accompanied
by a comparison of the calculated and observed longitudes of Uranus (with three
different masses of Neptune) for years at intervals from 1690 to 1845, and for one of
these masses the residuals are so small that it appears that, using these perturbations
by Neptune and Le Verrier's perturbations by Jupiter and Saturn, there existed a theory
of Uranus from which quite accurate tables might have been constructed. But this
was never done. The ephemeris of Uranus in the American Ephemeris was intended
579] GOLD MEDAL OF THE SOCIETY TO PROFESSOR SIMON NEWCOMB. 183
to be founded on the theory, but the proper definitive elements do not seem to have
been adopted : and in the Nautical Almanac for the years up to 1876, Bouvard's Tables
of Uranus were still employed ; for the year 1877 the ephemeris is derived from
heliocentric places communicated by Prof. Newcomb.
An extended investigation of the subject was made by SafFord, but only a brief
general description of his results is published, Monthly Notices, R.A.S., vol. xxil. (1862).
The effect of Neptune was here computed by mechanical quadratures; and corrections
were obtained for the mass of Neptune and elements of Uranus.
Professor Newcomb's Tables of Uranus have only recently appeared. They are
published in the Smithsonian Contributions under the title "An Investigation of the
Orbit of Uranus, with General Tables of its Motion," (accepted for publication Februar)',
1873), forming a volume of about 300 pages. The work was undertaken as far back
as 1859, but the labour devoted to it at first amounted to little more than tentative
efforts to obtain numerical data of sufficient accuracy to serve as a basis of the theory,
and to decide on a satisfactory way of computing the general perturbations. First, the
elements of Neptune had to be corrected, and this led to the foregoing investigation
of that planet : it then appeared that the received elements of Uranus also differed
too widely from the truth to serve as the basis of the work, and they were provisionally
corrected by a series of heliocentric longitudes, derived from observations extending from
1781 to 1861. Finally, it was found that the adopted method of computing the
perturbations, that of the " variation of the elements," was practically inapplicable to
the computation of the more difficult terms, viz. those of the second order in regard
to the disturbing force. While entertaining a high opinion of Hansen's method as at
once general, practicable, and fully developed, the author conceived that it was on the
whole preferable to express the perturbations directly in terms of the time, owing to
the ease with which the results of different investigations could be compared, and
corrections to the theory introduced ; and under these circumstances he worked out the
method described in the first chapter of his treatise, not closely examining how much
it contained that was essentially new. With these improved elements and methods the
work was recommenced in 1868 ; the investigation has occupied him during the sub-
sequent five years : and, though aided by computers, every part of the work has been
done under his immediate direction, and as nearly as possible in the same way as if
he had done it himself: a result in some cases obtained only by an amount of labour
approximating to that saved by the employment of the computer.
The leading steps of the investigation correspond to those for Neptune : there is,
first, the theoretical investigation already referred to ; secondly, the formation of the
provisional theory with assumed elements ; thirdly, the comparison with observation ;
and here the observations are the accidental ones previous to the discovery of Uranus
as a planet by Herschel in 1781, and the subsequent systematic ones of twelve
Observatories, extending over intervals during periods from 1781 to 1872; all which
have to be freed from systematic differences, and reduced to a concordant system as
before: the operation is facilitated by the existence, since 1830, of ephemerides com-
puted from Bouvard's Tables serving as an intermediate term for the comparison of
184 ADDRESS DELIVERED BY THE PRESIDENT. [579
the observations with the provisional theory. Fourthly, the correction of the elements
of the provisional theory, viz. the equations for the comparison of the longitudes give
Se, Sn, hh, Bk, and a correction to the assumed mass of Neptune, which mass is thus
brought out = ^g^^. And the equations for the comparison of latitudes give Bp, 8q ;
there is thus obtained a con-ected set of elements (Newcomb's Elements IV.), being
for the year 1850, the elements as affected with the long inequality ; these are the
elements upon which the Tables are founded. But it is theoretically interesting to
have the absolute mean values of the elements, and the author accordingly obtains
these (his Elements V.) together with the corrections corresponding to a varied mass
of Neptune, ( that is, the terms in /j, coiTesponding to a mass 0700 ) > ^^ remarks
that, admitting the mass of Neptune to be uncertain by about one-liftieth of its value,
the mean longitude of the perihelion of Uranus is from this cause uncertain by more
than two minutes, the mean longitude of the planet by nearly a minute, and the
mean motion by nearly two seconds in a century. Fifthly, the formation of the tables,
based on the Elements IV. ; the tables calculated with these elements are intended
to hold good for the period between the years 1000 and 2200; but by aid of the
Elements V. they may be made applicable for a more extended period.
In what precedes I have endeavoured to give you an account of Professor Newcomb's
writings: they exhibit all of them a combination, on the one hand, of mathematical
skill and power, and on the other hand of good hard work — devoted to the furtherance
of Astronomical Science. The Memoir on the Lunar Theory contains the successful
development of a highly original idea, and cannot but be regarded as a great step in
advance in the method of the variation of the elements and in theoretical dynamics
generally ; the two sets of planetary tables are works of immense labour, embodying
results only attainable by the exercise of such labour under the guidance of profound
mathematical skill — and which are needs in the present state of Astronomy. I trust
that imperfectly as my task is accomplished, I shall have satisfied you that we have
done well in the award of our medal.
The President then, delivering the medal to the Foreign Secretary, addressed him in
the following terms :
Mr Huggins — I request that you will have the goodness to transmit to Professor
Newcomb this medal, as an expression of the opinion of the Society of the excellence
and importance of what he has accomplished ; and to assure him at the same time
of our best wishes for his health and happiness, and for the long and successful
continuation of his career as a worker in our science.
580]
185
580.
ON THE NUMBER OF DISTINCT TERMS IN A SYMMETRICAL OR
PARTIALLY SYMMETRICAL DETERMINANT.
[From the Monthly Notices of the Royal Astrcmomical Society, vol. xxxiv. (1873 — 1874),
pp. 303—307, and p. 335.]
The determination of a set of unknown quantities by the method of least squares
is effected by means of formulae depending on symmetrical or partially symmetrical
determinants ; and it is interesting to have an expression for the number of distinct
tei-ms in such a determinant.
The tenns of a determinant are represented as duads, and the determinant itself
as a bicolumn; viz. we write, for instance,
aa \
bb
\PP
>99'
to represent the determinants
aa, ab, ap, aq'
ba, bb, bp', bq
pa, pb, pp', pq
qa, qb, qp' , qq'
This being so if the duads are such that in general rs = sr, then the determinant
is wholly or partially symmetrical ; viz. the determinant just written down, for which
the bicolumn contains such symbols as pp' and qq', (each letter p, q,... being distinct
Iaa ^
bb
cc
is wholly sjrmmetrical. A determinant for which the bicolumn has m rows aa, bb, &c.,
and n rows pp', qq', &c. is called a determinant (m, n); and the number of distinct
terms in the developed expression of the determinant is taken to be ^ {m, n) ; the
problem is to find the number of distinct terms ^ (m, n).
C. IX. 24
186
ON THE NUMBER OF DISTINCT TERMS IN A SYMMETRICAL OR
[580
Consider a determinant (m, n) where n is not = 0 ; for instance, the determinant
above written down, which is (2, 2); this contains terms multiplied by qa, qb, qp', qq'
I bb
respectively : where, disregarding signs, the whole factor multiplied by ja is J ap'
[pq'
which is a determinant (1, 2), and similarly the whole factor multiplied by qb is a
aa
bb
determinant (1, 2). But the whole factor multiplied by qp' is the determinant ■
pq
which is a determinant (2, 1), and the whole factor multiplied by qq' is also a determ-
inant (2, 1).
Hence, observing that qa, qb, qp', qq' are distinct terms occurring onli/ in the last
line of the determinant, the number of distinct terms is equal to the sum of the
numbers of distinct terms in the several component parts, or we have
<^(2, 2)=2,|,(1, 2) + 2<^(2, 1);
and so in general :
<}> (m, n) = m<f) {m — 1, n) + n<f> (m, n—1).
Consider next a completely symmetrical determinant (m, 0); for instance (4, 0), the
determinant
na, ah, ac, ad
ba, bb, be, bd
ca, cb, cc, cd j
da, db, dc, dd
aa\, =
bb
cc
dd
r aa\
is J hh I
We have first the terms containing dd; the whole factor is i M k which is a
I cc /
determinant (3, 0) ; secondly, the terms containing ad . da, or the like combinations,
bd.db or cd.dc: the whole factor multiplied hy ad . da is •[ >, which is a determ-
inant (2, 0) ; thirdly, the terms containing ad.db + bd . da, = 2ad .bd; or the like
combinations '2ad . cd or 2bd . cd : the whole factor multiplying the term 2ad . bd is
fee 1 . .
-I >, which is a determinant (1, 1). Hence observing that ad, bd, cd, =da, db, dc,
[ba)
and dd are terms occurring only in the last line and column of the original determinant,
it is clear that the number of distinct terms in the original determinant is equal to
the sum of the numbers of distinct terms in the component parts, or that we have
^(4, 0) = <^(3, 0) + 3<^(2, 0) + 3(^(1, 1); and so in general:
(^ (m, 0) = (^ (m - 1, 0) -I- mcf, (m - 2, 0) + "^ ' ^ ^ (f> (m - 3, 1).
580] PARTIALLY SYMMETRICAL DETERMINANT. 187
The two equations of differences, together with the initial values <^ (0, 0) = 1,
0(1, 0) = <f>(0, 1) = 1, 0(2, O) = 0(l, 1) = 0(1, 2) = 2, enable the calculation of the
successive values of (f> (vi, n) : viz. arranging these in the order
0(0, 0),
0(1, 0), 0(0, 1),
0(2, 0), 0(1, 1), 0(0, 2),
0(3, 0), &c., &c.,
we calculate simultaneously the lines 0(m, 0), 0(m, 1); and thence successively the
remaining lines 0 {m, 2), 0 (m, 3), &c. : the values up to »>i + w = 6 being in fact
1.
1, 1,
2, 2, 2,
5, 6, 6, 6,
17, 23, 24, 24, 24,
73, 109, 118, 120, 120, 120,
388, 618, 690, 714, 720, 720, 720:
where the process for the first two lines is
5=2 + 2.H-.l, 6 = ^. 2+2,
17- 5 + 3- 2 + 3. 2, 23 = 3. 6+ 5,
73 = 17 + 4. 5 + 6. 6, 109 = 4. 23 + 17,
388 = 73 + 5 . 17 + . 23, 618 = 5 . 109 f 23,
the larger figures being those of the two lines, and the smaller ones numerical
multipliers. And then for the third line, fourth line, &c., we have
6=1. 2 + 2. 2, 120 = 2. 24 + 3. 24,
24=2, 6 + 2. 6, 714 = 3.120 + 4.118,
118 = 3. 24 + 2. 23,
690 = 4.118 + 2.109,
and so on.
This is, in fact, the easiest way of obtaining the actual numerical values ; but we
may obtain an analytical formula. Considering the two equations
0(ot, l) = 7?i0(m— 1, l) + 0(m, 0),
0(m, O) = 0(m-1, O) + m0(OT-2, 0)+ — ^ 0(m-3, 1);
24—2
188 ON THE NUMBER OF DISTINCT TERMS IN A SYMMETRICAL OR [580
and using the first of these to eliminate the term <f>{m — 3, 1) and resulting terms
^(m — 4, 1), &c. which present themselves in the second equation, this, after a succession
of reductions, becomes
<f>{m, 0)= <l>(m-l, 0)
+ {m-l)<f>(m-2, 0)
m , m — 1 , , , - „,
+ 2 l«/'(»*-3. 0)
+ (m-3)^(m-4, 0)
+ (to-3)...3.2</)(1, 0)
+ (m-3)...3.2.1 };
or, observing that the last term (ni — 3) ... 3 . 2. 1 is, in fact, =(m- 3) ... 3. 2. l(/)(0, 0),
this may be written :
2<f>{m, 0)-<f>(m-l, 0)-(m-l)<^(m-2, 0)= ^(m-1, 0)
+ (m-l)<f> (m - 2, 0)
+ (vi - 1) {m -2)<f> (m - 3, 0)
+ (m-l)..3.2.1^( 0,0).
And hence assuming
« = <^(0, 0) + |<^(l, 0) + j^<^(2, 0)+...+j-^^^<^(m, 0)+...,
we find at once
„ du u
2 -; u — a:u =
ax \ — X
that is,
2^" = d.fl+..+ ^-^V
XI, \ 1 — x)
or integrating and determining the constant so that u shall become =1 for a; = 0, we
have
wherefore we have
^ (m, 0) = 1 . 2 ... m coeift. «"* in . .
\l-x
580]
PARTIALLY SYMMETRICAL DETERMINANT.
189
Developing as far as a", the numerical process is
1 i ^V 7S7
2 i ^V Trir Wttt
i 8 "57 TuU
1
^7
A
3 5
T"as
l
■3^
17
73
1
1
i
3
A
■^84
T^h
TinrsiT
1
i
s
1^
i\%
m
m.
a i TS ITS' T^S SoflO
33 9 7 25
8 Ifl ^7 T35 T75?
105
63
Mi
1
1
1
tf
iJ
"Mr
t¥tt
X by 1
1
2
6
24
120
720
388,
112 5
agreeing with the former values.
The expression of <f>(m, 0) once found, it is easy thence to obtain
</) (m, 1) = 1 . 2 . . . . m coefift. «'" in
<^ (m, 2)= 1 . 2 .... m coefft. a;" in
<^ (m, 3) = 1 . 2 .... TO coefft. x"" in
and so on, the law being obvious.
[Addition, p. 335.] The generating function
2 . 3ei*+i*'
(1 - ^)*
«,
1 + !<]«+ ... +U„
1.2... n
+ ..., =
Vl -a;'
190 ON THE NUMBER OF TERMS IN SOME DETERMINANTS. [580
was obtained as the solution of the differential equation
Writing this in the form
2(l-x)^ = u{2-a^),
we at once obtain for «„ the equation of differences,
«„ = ««„_,- i (n - 1) (n - 2) i/„_3;
and it thus appears that the values of «„ (number of distinct terms in a symmetrical
determination of the order n) can be calculated the one from the other by the process
n = l,
1 = .. 1.
= 2,
2 = a. 1,
= 3,
5 = 3. 2- ,.1,
= 4,
17 = 4. 5- 3.1,
= 5,
73 = 5.17- 6.2,
= 6,
388 = 6.73- 10.5,
&c.
which is one of extreme facility.
581]
191
581.
ON A THEOREM IN ELLIPTIC MOTION.
[From the Monthly Notices of the Royal Astronomical Society, vol. xxxv. (1874 — 1875),
pp. 337—339.]
Let a body move through kpocentre between two opposite points of its orbit, say
from the point P, eccentric anomaly u, to the point P', eccentric anomaly u', where
u, u' are each positive, m < tt, u' > tt. Taking the origin at the focus, and the axis
of a; in the direction through apocentre, then —
Coordinates of P are x = a(— cos u + e), y = a'/l —^ sin u,
„ P' „ x — a{— cos u' + e), y = a'J\—^ sin m' ;
whence, expressing that the points P, P' are in a line with the focus,
sin v! (— cos « 4- e) — sin u (— cos n! + e) = 0,
that is,
sin (m' — m) = e (sin ?t' — sin w),
which is negative, viz. it' — w is >it.
192 ON A THEOREM IN ELLIPTIC MOTION. [581
The. time of passage from P to P' is
nt = (m' — e sin w') — (u — e sin u),
= m' — i( — e (sin u' — sin w),
= u' — u — sin (m' — u),
which, u' — u being greater than tt and — sin (u' — «) positive, is greater than ir ; viz.
the time of passage is greater than one-half the periodic time. Of course, if P and P'
are at pericentre and apocentre, the time of passage is equal one-half the periodic time.
The time of passage from P" to P through the pericentre is
nt = 2'rr — («' — w) + sin («' — u),
which is
= 27r — («' - m) — sin {2ir — (u' — u)],
where 27r — (u' — u), = a suppose, is an angle < tt. Writing, then
nt = a — sin a,
and comparing with the known expression for the time in the case of a body falling
directly towards the centre of force, we see that the time of passage from P' to P
through the pericentre, is equal to the time of falling directly towards the same centre
of force from rest at the distance 2a to the distance a (1 + cos a), where, as above
a = 27r — (m' — m), u' — u being the difference of the eccentric anomalies at the two
TT
opposite points P, P'. If o = tt, the times of passage are each = - , that is, one-half
the periodic time.
The foregoing equation sin («' — i(,) = e (sin u' - sin u) gives obviously
cos ^ {u' — u) = e cos ^ (m' -(- w) ;
that is,
1 + tan i u tan ^ m' = e (1 — tan ^ a tan \u'),
or,
1 / 1 — e
— tan i u tan isu = z ;
(in the figure tan^M is positive, tan ^ it' negative); and we thence obtain further
sin ^ (it' — m) = cos ^ u' cos \ u (tan i u' — tan ^ u),
sin ^ (m' -I- m) = cos ^ «.' cos ^ m (tan ^ w' -I- tan J m),
2e
cos J (m' — m) = cos i m' cos ^ tt . r— - ,
2
cos i (it' 4- «) = cos i w' cos A it . = ;
I + e
581] ON A THEOREM IN ELLIPTIC MOTION. 193
and thence also
cos U + COS U' = 2 COS ^ (u' + U) COS ^ (m' - u),
» 1 ' 0 1 8e
= cos- i u cos^ * M . 7^ r» •
^ ^ (1 + ey
But we have
Q 2
1 + COS (m' - m) = 2 COS= ^ (m' - m) = cos" ^ m' C0S= I M . -^j r^ ,
or, comparing with the last equation,
1 + COS (m' — m) = e (cos u + cos u),
or, what is the same thing,
1 - cos (u' - m) = (1 — e cos u') + (1 — e cos u) ;
and in like manner,
Q
1 + cos («' + m) = 2 cos= J (m' + m) = cos'' J m' . cos=i^ M -Tj r-;
(I T fi)
or, comparing with the same equation,
1 + cos (m' + m) = -i(cos u + cos w') :
which are formulae corresponding with the original equation
sin (m' — u) = e (sin u' — sin «).
I
c. IX. 25
194 [582
S
582.
NOTE ON THE THEORY OF PRECESSION AND NUTATION.
[From the Monthly Notices of the Royal Astronomical Society, vol. xxxv. (1874 — 1875),
pp. 340—343.]
We have in the dynamical theory of Precession and Nutation (see Bessel's Funda-
menta (1818), p. 126),
cf^+(B-A)qr=LS{x'y-ccy')dm'(^^^-^-),
A^^ + (C-B)rp = LS(y'z-yz')dm'[^^-^),
Bf^+{A-C)pq = LS {z'x - zx') dm' (1 - i) ,
where L is the mass of the Sun or Moon, x, y, z the coordinates of its centre referred
to the centre of the Earth as origin,
r = '^a^ + y^ + z',
the distance of its centre, and
A = 'J(x- xj + (y- yj + {z- /)>,
the distance of its centre from an element dm', coordinates {x', y', z') of the Earth's
mass, the sum or integral S being extended to the whole mass of the Earth — I have
written dm, r for Bessel's dm, r-^ — , we have
A'' = r»- 2 {xx + yy' + «y) + a;'" + i/'» + /»;
and thence
^. - ^ = ^ (^ + yy' + "O - f ^ ((«* + y' + ^') (^'' + y'' + «'')- 5 (aa/ + y/ + «/)»} + etc.
582] NOTE ON THE THEORY OF PRECESSION AND NUTATION. 195
The principal term is the first one,
- {xx' + yy' + zz') ;
but Bessel takes account also of the second term,
- f ^ {(^ + y' + «') (^'' + y'^ + ^'') - 5 {xd + yy + Z2;)%
viz. considering the Earth as a solid of revolution (as to density as well as exterior
form), he obtains in regard to it the following terms of sin <» -^ and -r- respectively ;
or 1
^ ^ . 2 (C - .4) Z (5 sin= S - 1) cos S sin a,
or -1
-^ ^.2(C-^)Z(58in^S-l)cosScosa,
WDGrC
2 (0 - 4) Z = S (3/* - 5/u.') 27rp E'di? d/i,
K being in fact a numerical quantity, relating to the Earth only, and the value of
which is by pendulum observations ultimately found to be =0'13603.
Writing, for shortness, ?j
(ar' + 2^» + ^=) (»'» + 2/'= + /^) - 5 (xx' + yy' + ^^')' == ^.
then the foregoing terms of sin w -^ and -^ depend, as regards their form, on the
theorem that for any solid of revolution (about the axis of z) we have
8 {x'y - xy') ndm, S (^z - yz') D.dm', S (z'x - zx') ndm'
= 0,
^yix' + y' + z'- 5z') S [3 (a;'^ + y'= + z'') - 5/^] z'dm',
- i a; (a,-» + y» + 2" - 5z^) S [3 (x'^ + y'» + /") - 5/=] z'dm',
respectively : viz. writing x' + y'' + z'^ — B?, and z' = Rft, also a^ + y= + 2- = r= and
x-=r cos 8 cos a., y = r cos 8 sin a, ^ = r sin 8, the values would be
0,
i r» cos 8 sin a (1 - 5 sin" 8) -S (3 - 5jU=) ixRHm',
- i r» cos 8 sin a (1 - 5 sin" 8) S (3 - S/i") iJiR^dm',
which are of the form in question.
The verification is easy: the solid being one of revolution about the axis of z,
any integral such as Sa/i^^dm' or Sx'y'z'dm' which contains an odd power of af or of
y is =0; while such integrals as Sx'^z'dm', Sy'^zdm! are equal to each other, or, what
is the same thing, each = ^ S {x^ + y'^) z'dm. That we have 8 {x'y - xy') fldm' = 0 is
25—2
196 NOTE ON THE THEORY OF PRECESSION AND NUTATION. [582
at once seen to be true ; considering the next integral S (y'z - yz') ildm', the terms of
{y'z — zy') il which lead to non-evanescent integrals are
-yg'.(ai' + y' + 2^){a^'+y'^ + z''),
-5y'z.2yzy'z,
+ hyz' . (a^a;'» + yV' + ^V") ;
giving in the integral the several terms
- y (a;^ + y» + «») fif (ar'» + y'» + /=>) z!dm,
- 10ya» . i <Sf (a;'' + y'' + /» - ^'») /dm'.
^.hy(a? + y-'^z^-z-').\S (a;'> + y'^ + z'^ - z'') z'dm',
+ yz^Sz''dm',
viz. collecting, the value is
(- 1 + 1 =) H'^ + f^ ^') yS (a;'» + y'' + z") z'dm',
(- f =) - ^ («" + 2/' + 2') ytz'^dm,
(- f - 5 =) - ¥ y^'S (x'' + y'' + /») z'dm',
(+f + 5 + 5=) + ^ yz^Sz'^dm' ;
which is
= iy(a^ + y' + z'- 5z') S [3 (x'' + y'^ + z'^) - 5/=] z'dm' ;
and similarly the last term is
= -i^x{a? + f- + z''- 52^) S [3 {x'^ + 2/'= + z"') - 5z''] z'dm',
which completes the proof.
583] 197
583.
ON SPHEROIDAL TRIGONOMETRY.
[From the Monthly Notices of the Royal Astrmwmical Society, vol. xxxvii. (1876 — 1877),
p. 92.]
The fundamental formulae of Spheroidal Trigonometry are those which belong to
a right-angled triangle PSS^, where P is the pole, PS, PSo arcs of meridian, and
SS„ a geodesic line cutting the meridian PS at a given angle, and the meridian
PSd at right angles. We consider a spheiical triangle PSS^,
Sides PS, PSo, SS, = y, y„ s.
Angles So, S, P =90°, 6, I,
where 7 is the reduced colatitude of the point S on the spheroid (and thence also
70 the reduced colatitude of S,,) and 6 the azimuth of the geodesic SSo, or angle at
which this cuts the meridian SP; and then if S be the length of the geodesic SS^
measured as a circular arc, radius = Earth's equatoreal radius, and L be the angle
SPSo, S, L differ from the corresponding spherical quantities s, I by terms involving
the excentricity of the spheroid, viz. calling this e and writing
. _ e cos 7o
Vl - e= sin" 7o '
then (see Hansen's "Geodatische Untersuchungen," Abh, der K. Sachs. Gesell., t. viii.
(186.5) pp. 1-5 and 23, but using the foregoing notation) we have, to terms of the
sixth order in e,
+ (i^+ ^ ^ + T^A')8in2s
+ inh^ + jAi^) sin 4s
+ Wnf^ sin 6s;
and
L =i-ie»sin7o{(l-ii»+ie^- if¥ + ie^)s
-(A*'+ 3V^)8in2s
+ ^^ sin 4s},
which are the formulae in question.
198 ■ [584
584.
ADDITION TO PROF. R. S. BALL'S PAPER, "NOTE ON A TRANS-
FORMATION OF LAGRANGE'S EQUATIONS OF MOTION IN
GENERALISED COORDINATES, WHICH IS CONVENIENT IN
PHYSICAL ASTRONOMY."
[From the Monthly Notices of the Royal Astronomical Society, vol. xxxvii. (1876 — 1877),
pp. 269—271.]
The formulae may be established in a somewhat different way, as follows: —
Consider the masses M^, M^,
Let Xi, Fi, Z^ be the coordinates (in reference to a fixed origin and axes) of
the C.G. of M^ ;
0^1, 2/i> •2'i the coordinates (in reference to a parallel set of axes through the C.G.
of ilfi) of an element in^ of the mass il/i, and similarly for the masses ikT,, ...; the
coordinates (X,, Fj, Z^, (X,, Fj, Z^), ... all belonging to the same origin and axes;
7 XT
And let Xi, &c. denote the derived functions ^-' , «&c.
at
We have
r= s^i, [(Z, + w,y + (F, + y,y + (Z, + £,y]
+ s ^ [(Z"j + d;,y + (Y, + y,y + (z, + z,y]
or since SmiXi = 0, &c., and therefore also SmiXi = 0, &c., this is
T= iilf, (Xi» + F,= + ir,») +S^m,{d:,' + y,' + z,')
+ ^M, {i,' + t,' + Z^) + 8 \m., (x^ + y^ + i,»)
584] ADDITION TO PKOF. R. S. BALL's PAPER.
Write u, V, w for the coordinates of the c.G. of the whole system: then
199
and thence
M,Z, +M,Z, +..
MX + MX + ..
MX + Mj, +..
and thence
T - ^ (Mj + M, + ...){iV + v' + 1^")
1
= {M, + M,...)v,
= (i¥i + ilf,...)w;
= (Mi + M^...)u,
= iM^ + M^...)v,
= (M, + M^...)w;
r^^-^^- IM,M, [(X, -x,y+(Y,- Y,y + (z, - z,y]}
M,
or, representing the function ,on the right-hand side by T', this is
T^ i {M, + M,+ ...)0i= + V, + w,) + T'..., = T, + r.
Suppose the positions are determined by means of the 6n coordinates ((q)) ; the
equations of motion are each of them of the form
d dT,_dT, d dT' _dr^_dV
dt ' dq dq dt ' dq dq dq '
But these admit of further reduction ; the part in T„ depends upon three terms,
such as
d / . du\ . du _du du . /d dil du\
dt\ dq) dq' ~ dt dq \dt dq dq)'
But we have u a function of {{q)), and thence
du _du d du du _d du du
dq~ dq' dt dq dq' ~ dt dq dq' ~ '
or the term is simply
The equation thus becomes
du du
dt dq'
(M +M \ (— du dvdv dw dtb\ d dT _ dT
^ ' ^"'\dtdq~dtTqlll4)'^dt~d4~~d^'
dy
dq '
200 ADDITION TO PROF. R 8. BALL's PAPER. [584
Suppose now that T, V are functions of 6n — 3 out of the 6n coordinates ((q)),
and of the differential coefficients q of the same 6n — 3 coordinates, but are independent
of the remaining three coordinates and of their differential coefficients ; then, first, if
q denotes any one of the three coordinates, the equation becomes
du du dv dv dw dw _
dt dq dt dq di dq~ '
or, better,
du du dv dv . dw diu _
dt dq dt dq dt dq ~ '
and the three equations of this form give
du _^ di) _- dw _ ^
~dt~ • di~ ' W '
viz. these are the equations for the conservation of the motion of the centre of
gravity.
And this being so, then, if q now denotes any one of the 6?i — 3 coordinates,
each of the remaining equations assumes the form
d dr_dT^^_dV
dt ' dq dq ~ dq'
viz. we have thus 6» — 3 equations for the relative motion of the bodies of the system.
585] 201
585.
A NEW THEOREM ON THE EQUILIBRIUM OF FOUR FORCES
ACTING ON A SOLID BODY.
[From the Philosophical Magazine, vol. xxxi. (1866), pp. 78, 79 ; Camb. Phil. Soc. Proc.
vol. I. (1866), p. 23.5.]
Defining the " moment of two lines " as the product of the shortest distance of
the two lines into the sine of their inclination, then, if four forces acting along the
lines 1, 2, 3, 4 respectively are in equilibrium, the lines must, as is known (Mobius),
be four generating lines of an hyperboloid ; and if 12 denote the moment of the lines
1 and 2, and similarly 13 the moment of the lines 1 and 3, &c., the forces are as
V23 . 34 . 42 : V34.41.13 : V41 . 12 . 24 : Vl2.23.31.
Calling the four forces Pi, Pj, P3, P4, it follows as a corollary that we have
P,Pj . 12 = 12 . 34 V13.42 . Vi4723 = P,P, . 34 ;
viz. the product of any two of the forces into the moment of the lines along which
they act is equal to the product of the other two forces into the moment of the lines
along which they act, — which is equivalent to Chasles's theorem, that, representing a
force by a finite line of proportional magnitude, then in whatever way a system of
forces is resolved into two forces, the volume of the tetrahedron formed by joining the
extremities of the two representative lines is constant.
c. IX. 26
202 [586
586.
ON THE MATHEMATICAL THEORY OF ISOMERS.
[From the Philosophical Magazine, vol. XLVii. (1874), pp. 444 — 446.]
I CONSIDER a "diagram," viz. a set of points H, 0, N, C, &c. (any number of
each), connected by links into a single assemblage under the condition that through
each H there passes not more than one link, through each 0 not more than two
links, through each JV" not more than three links, through each C not more than four
links. Of course through every point there passes at least one link, or the points
would not be connected into a single assemblage.
In such a diagram each point having its full number of links is saturate, or
nilvalent: in particular, each point H is saturate. A point not having its full number
of links is univalent, bivalent, or trivalent, according as it wants one, two, or three
of its full number of links. If every point is saturate the diagram is saturate, or
nilvalent ; or, say, it is a " plerogram " ; but if the diagram is susceptible of n more
links, then it is w-valent ; viz. the valency of the diagram is the sum of the valencies
of the component points.
Since each H is connected by a single link (and therefore to a point 0, C, &c.
as the case may be, but not to another point H), we may without breaking up the
diagram remove all the points H with the links belonging to them, and thus obtain
a diagram without any points H : such a diagram may be termed a " kenogram " : the
valency is obviously that of the original diagram plus the number of removed H's.
If from a kenogram, we remove every point 0, C, &c. connected with the rest of
the diagram by a single link only (each with the link belonging to it), and so on
indefinitely as long as the process is practicable, we arrive at last at a diagram in
which every point 0, C, &c. is connected with the rest of the diagram by two links
at least : this may be called a " mere kenogram."
586]
ON THE MATHEMATICAL THEORY OF ISOMERS.
208
Each or any point of a mere kenogram may be made the origin of a "rami-
fication " ; viz. we have here links branching out from the original point, and then
again from the derived points, and so on any number of times, and never again
uniting. We can thus from the mere kenogram obtain (in an infinite variety of ways)
a diagram. The diagi-am completely determines the mere kenogram ; and consequently
two diagrams cannot be identical unless they have the same mere kenogram. Observe
that the mere kenogram may evanesce altogether ; viz. this will be the case if the
diagram or kenogi-am is a simple ramification.
A ramification of n points C is (2n + 2)-valent : in fact, this is so in the most
simple case m = 1 ; and admitting it to be true for any value of n, it is at once seen
to be true for the next succeeding value. But no kenogram of points C is so much
as (2w + 2)-valent ; for instance, 3 points C linked into a triangle, instead of being
8-valent are only 6-valent. We have therefore plerograms of n points C and 2n + 2
points H, say plerograms (^-"+=; and in any such plerogram the kenogram is of
necessity a ramification of n points C ; viz. the different cases of such ramifications are *
n = 1. n = 2.
« *
« = 3.
(«)
(a)
(a)
= 4.
n = o.
w= 6.
(a)
03)
(y)
(a)
w
(y)
(5)
where the mathematical question of the determination of such forms belongs to the
class of questions considered in my paper " On the Theory of the Analytical Forms
called Trees," Phil. Mag. voL Xlii. (1857), [203], and vol. xviii. (1859), [247], and in
some papers on Partition.s in the same Journal.
* The distinction in the diagrams of asterisks and dots is to he in the first instance disregarded ; it is
made in reference to what follows, the explanation as to the allotrious points.
26—2
204 ON THE MATHEMATICAL THEORY OP ISOMERS. [586
The different forms of univalent diagrams 0"H^^^ are obtained from the same
ramifications by adding to each of them all but one of the 2w + 2 points H; that is,
by adding to each point C except one its full number of points H, and to the
excepted point one less than the full number of points H. The excepted point C
must therefore be univalent at least; viz. it cannot be a saturate point, which presents
itself for example in the diagrams n = 5 (7) and n = 6 (8). And in order to count the
number of distinct forms (for the diagrams C"if*"+'), we must in each of the above
ramifications consider what is the number of distinct classes into which the points
group themselves, or, say, the number of " allotrious " points. For instance, in the
ramification n = 3 there are two classes only ; viz. a point is either terminal or medial ;
or, say, the number of allotrious points is = 2 : this is shown in the diagrams by
means of the asterisks ; so that in each case the points which may be considered
allotrious are represented by asterisks, and the number of asterisks is equal to the
number of allotrious points.
Thus, number of univalent diagrams (7"!?^+' :
n = \, 1
71=2, 1
n = 3 2
n = 4, (a) 2; (/3) 2 ; together 4
71 = 5, (a)3; (,8)4; (7)!; „ 8
n=Q, (a)3; (/8) 5 ; (7)2; (S) 3 ; „ 13
where it will be observed that, n=o (7), and ?i = 6 (S), the numbers of allotrious points
are 2 and 4 respectively ; but since in each of these cases one point is saturate, they
give only the numbers 1 and 3 respectively. It might be mathematically possible to
obtain a general solution ; but there would be little use in this ; and for even the
next succeeding case, No. of bivalent diagrams C^H^; the extreme complexity of the
question would, it is probable, prevent the attainment of a general solution.
Passing to the chemical signification of the formulae, and instead of the radicals
Qnjjm+i considering the corresponding alcohols (7".ff^+^ OH, then, n = 1, 2, 3, 4, the
numbers of known alcohols are 1, 1, 2, 4, agreeing with the foregoing theoretic number
(see Schorlemmer's Carbon Compounds, 1874); but n=,Ji^\ie number of known alcohols
is =2, instead of the foregoing theoretic number 8. It is, of course, no objection to
the theory that the number of theoretic forms should exceed the number of known
compounds ; the missing ones may be simply unknown ; or they may be only capable
of existing under conceivable, but unattained, physical conditions (for instance, of
temperature) ; and if defect from the theoretic number of compounds can be thus
accounted for, the theory holds good without modification. But it is also possible that
the diagrams, in order that they may represent chemical compounds, may be subject
to some as yet undetermined conditions ; viz. in this case the theory would stand good
as far as it goes, but would require modification.
587] 205
587.
A SMITH'S PKIZE DISSERTATION.
[From the Messenger of Mathematics, vol. ill. (1874), pp. 1 — 4.]
Write a dissertation:
On the general equation of virtual velocities.
Discuss the principles of Lagrange's proof of it and employ it [the general equation']
to demonstrate the Parallelogram of Forces.
Imagine a system of particles connected with each other in any manner and
subject to any geometrical conditions, for instance, two particles may be such that their
distance is invariable, a particle may be restricted to move on a given surface, &c.
And let each particle be acted upon by a force [this includes the case of several
forces acting on the same particle, since we have only to imagine coincident particles
each acted upon by a single force]. Imagine that the system has given to it any
indefinitely small displacement consistent with the mutual connexions and geometrical
conditions ; and suppose that for any particular particle the force acting on it is P,
and the displacement in the direction of the force (that is, the actual displacement
multiplied into the cosine of the angle included between its direction and that of
the force P) is =8p. Then Bp is called the virtual velocity of the particle, and the
principle of virtual velocities asserts that the sum of the products PBp, taken for all
the particles of the system, and for any displacement consistent as above, is =0; say
that we have
tPSp = 0.
This is also the general equation of virtual velocities : as to the mode of using
it, observe that the displacements Sp are not arbitrary quantities, but are in virtue of
the mutual connexions and other geometrical conditions connected together by certain
linear relations ; or, what is the same thing, they are linear functions of certain inde-
pendent arbitrary quantities 8m. Substituting for Bp their expressions in terms of Bu
206
A SMITH S PRIZE DISSERTATION.
[587
we have 2PSp = 2 UBu, where the several expressions U are each of them a linear
function of the forces P, and where on the right hand 2 refers to the several
quantities Su ; and the resulting equation is 2 UBu = 0 ; viz. since the quantities Bu are
independent, the equation divides itself into a set of equations f/j = 0, Ui = 0,... which
are the equations of equilibrium of the system.
Lagrange imagines the forces produced by means of a weight W at the extremity
of a string passing over a set of pulleys, as shown in the figure, viz. assuming the
forces commensurable and equal to mW, nW, &c., we must have »?i strings at A,
A'
"^
W
n strings at B, and so ou. Suppose any indefinitely small displacement given to the
system ; each string at A is shortened by Bj), or the m strings at A by mSp ; and the
like for the other particles at B, &c. ; hence, if mBp + nSq+ ..., = ^ (PBp + QBq + . . .),
be positive, the weight W will descend through the space
^(PBp + QSq +...).
Now, in order that the system may be in equilibrium, W must be in its lowest
position ; or, what is the same thing, if there is any displacement allowing W to
descend, W will descend, causing such displacement, and the original position is not a
position of equilibrium. That is, if the system be in equilibrium, the sum XPBp cannot
be positive.
But it cannot be negative ; since, if for any particular values of Bp the sum 2P8p
is negative, then reversing the directions of the several displacements, that is, giving
to the several displacements Bp the same values with opposite signs, then the sura
2P8p will be positive ; and we assume that it is possible thus to reverse the directions
of the several displacements. Hence, if the system be in a position of equilibrium,
we cannot have IPBp either positive or negative ; that is, we obtain as the condition
of equilibrium 2PSp = 0.
The above is Lagrange's reasoning, and it seems completely unobjectionable. As
regards the reversal of the directions of the displacements, observe that we consider
587] A smith's prize dissertation. 207
such conditions as a condition that the particle shall be always in a given plane, but
exclude the condition that the particle shall lie on a given plane, i.e. that it shall
be at liberty to move in one direction (but not in the opposite direction) off from
the plane. But the pulley-proof is equally applicable to a case of this kind. Thus,
imagine a particle resting on a horizontal plane, and let z be measured vertically
downwards, x and y horizontally. Suppose the particle acted on by the forces X, Y, Z,
and replacing these by a weight W as above, the condition of equilibrium is, that
XSx + YBy + Zhz
shall not be positive. We may have hx and hy, each positive or negative ; whence
the conditions X = 0 and F=0. But Bz is negative; hence the required condition is
satisfied if only Z is positive ; that is, if the vertical force acts downwards. Clearly
this is right, for if it acted upwards it would lift the particle from the plane. The
case considered by Lagrange is where the particle is always in the plane ; here hz = 0,
and there is no condition as to the force Z.
The only omission in Lagrange's proof is, that he does not expressly consider the
case of unstable equilibrium, where the weight TF is at a position, not of minimum,
but of maximum altitude. In such a case, however, the sum 2PSp is still = 0, taking
account (as the proof does) of the displacements considered as infinitesimals of the first
order; although taking account of higher powers, the sum SPSp would have a positive
value. An explanation as to this point might properly have been added to make the
proof "refutation-tight," but the proof is not really in defect.
P.S. Lagrange excludes tacitly, not expressly, the case where the direction of a
displacement is not reversible; he observes that the various displacements Sp, when
not arbitrary, are connected only by linear equations ; and " par consequent les valeurs
de toutes ces qnantites seront toujours telles qu'elles pourront changer de signe a la
fois." The point was brought out more fully by Ostrogradsky, but I think there is
no ground for the view that it was not brought out with sufiicient clearness by Lagrange
himself
Parallelogram of forces.
Let P, Q, R be the forces, a, /3, 7 their inclinations to any line ; then taking Ss
the displacement in the direction of this line, the displacements in the directions of
the forces are Bs cos a, Bs cos /3, Bs cos 7, and the equation 'EPBp = 0 assumes the form
(P cos a -f Q cos /3 + iJ cos 7) Bs = 0,
that is, we have
P cos a -(- Q cos /9 -I- i2 cos 7 = 0,
viz. this equation holds whatever be the fixed line to which the forces are refen-ed.
It is easy to see that, supposing it to hold in regard to any two lines, it will hold
generally, and that the relation in question is thus equivalent to two independent con-
ditions; and forming these we may obtain from them the theorem of the parallelogram
of forces.
208 A smith's prize dissertation. [587
But to obtain this more directly, take A, B, C for the angles between the forces
Q and R, R and P, P and Q respectively, then A + B + C = 2v, and thence
a = o,
y=a + C + A=a+2Tr-B,
whence writing o = j7r, or taking the line of displacement at right angles to the
force P, we have
and the equation becomes OP — Q sin (7 + iJ sin i? = 0, that is, Q : P = sin £ : sin (7 ; and
similarly R : P = smC : sin^, that is,
P : Q : R = sinA : sinP : sinC,
equations which in fact express that each force is equal and opposite to the diagonal
of the parallelogram formed by the other two forces.
588] 209
588.
PROBLEM.
[From the Messenger of Mathematics, vol. ill. (1874), pp. 50 — 52.]
It is required to place two given tetrahedra in perspective ; or, what is the same
thing, the tetrahedra being ABGD, A'B'G'D' respectively, to place these so that the
lines A A', BB', CC", DD' may meet in a point 0.
The following considerations present themselves in regard to the solution of this
problem. Take the tetrahedron ABGD to be given in position, and the point 0 at
pleasure ; then drawing the lines OA, OB, OG, OD, we may in a determinate number
of ways (viz. in 16 different ways) place the tetrahedron A'B'G'D' in such manner
that the summits A', E, C" shall be in the lines OA, OB, OG respectively. But the
summit D' will then not be in general in the line OD; and in order that it may
be so, a two-fold condition must be satisfied by the point 0 ; viz. the locus of this
point must be a certain curve in apace.
Or again, we may look at the question thus: we have to place a point 0 in
relation to the tetrahedron ABGD, and a point 0' in relation to the tetrahedron
A'B'G'D', in such manner that the edges of the first tetrahedron subtend at 0 the
same angles that the edges of the second tetrahedron subtend at 0'; for this being
done, then considering 0' as rigidly connected with A'B'G'D', we may move the figure
O'A'FG'D' so that 0' shall coincide with 0, and the lines O'A', O'B', O'G', O'D' with
OA, OB, 00, OD respectively. Take a, b, c, f, g, h, for the sides of the tetrahedron
ABGD (BG, GA, AB, AD, BD, GD = a, b, c, f, g, h respectively), and take also x, y, z, w
for the distances OA, OB, OG, OD respectively ; and let a, b', c', /', g', h', x', y', z', w
have the like significations in regard to the tetrahedron A'B'G'D' and the point 0', and
write
ytj^z^-a* a'+aj'-fr" x^ + y'^-c'' a^ + w^-f' y^+itf-g^ z^ + w'-h'
2yz ' 2zx 2xy ' 2xw ' 2yw ' 2zw '
= A, B. C, F, G, H,
c. IX. ' 27
210 PKOBLEM. [588
respectively; and the like as regards the accented letters. Then A, B, C, F, G, H
are the cosines of the angles which the edges of the tetrahedron ABCD subtend
at 0; they are consequently the cosines of the six sides of the spherical quadrangle
obtained by the projection of ABCD on a sphere centre 0 ; and they are therefore not
independent, but are connected by a single equation ; substituting for A, B, G, F, G, H
their values, we have a relation between a, b, c, f, g, h, x, y, z, w; viz. this is the
relation which connects the ten distances of the five points in space 0, A, B, C, D
(and which relation was originally obtained by Carnot in this very manner). There is
of course the like relation between the accented letters.
The conditions as to the two tetrahedra are
A=A', B = F, C=C', F=F', G = G', H^H',
which, attending to the relations just referred to and therefore regarding w a& a, given
function of x, y, z, and w' as a given function of x', y', z', are equivalent to five
equations (or rather to a five-fold relation); the elimination of x', y', z' from the five-
fold relation gives therefore a two-fold relation between x, y, z, that is, between the
distances OA, OB, OC; or the locus of 0 is as before a curve in space.
The conditions may be written :
y'2 + z'^ _ 2Ay'z' = a% x' + w'^ - IFx'w =f'\
/2^.a;'2-2fi/a;'=6'^ y''+w'^-1Gy'w'==g'\
a;'» + y'i - 2Gx'y' = c'^ , z'"- + w'» - ^Hz'w' = A'» ;
whence eliminating x', \J, z , w, and in the result regarding A, B, C, F, G, H as given
functions of x, y, z, w, we have between x, y, z, and w a three-fold relation determining
w as a function of x, y, z, and establishing besides a two-fold relation between x, y, z.
As a particular case : One of the tetrahedra may degenerate into a plane quadrangle,
and we have then the problem : a given plane quadrangle ABCD being assumed to
be the perspective representation of a given tetrahedron A'B'CU, it is required to
determine the positions in space of this tetrahedron and of the point of sight 0.
A generalisation of the original problem is as follows : determine the two-fold
relation which must subsist between the 4x6, =24 coordinates of four lines, in order
that it may be possible to place in the tetrad of lines a given tetrahedron ; that is,
to place in the four lines respectively the four summits of the given tetrahedron. It
may be remarked that coasidering three of the four lines as given, say these lines are
the loci of the summits A, B, C respectively, we can in 16 different ways place in these
lines respectively the three summits, and for each of these there are two positions of
the summit D ; there are consequently 32 positions of D ; and the two-fold relation,
considered as a relation between the six coordinates of the remaining line, must in
efifect express that this line passes through some one of the 32 points.
589] 211
589.
ON RESIDUATION IN REGARD TO A CUBIC CURVE.
[From the Messenger of Mathematics, vol. iii. (1874), pp. 62 — 6.5.]
The following investigation of Prof Sylvester's theory of Residuation may be
compared with that given in Salmon's Higher Plane Curves, 2nd Edition (1873), pp.
133—137 :
If the intersections of a cubic curve U3 with any other curve V^ are divided in
any manner into two systems of points, then each of these systems is said to be the
residue of the other; and, in like manner, if starting with a given system of points
on a cubic curve we di-aw through them a curve of any order F„, then the remaining
intersections of this curve with the cubic constitute a residue of the original system of
points.
If the number of points in the original system is = 3/;, then the number of
points in the residual system is =^q; and if we again take the residue, and so on
indefinitely, the number of points in each residue will be = 0 (Mod. 3) ; viz. we can
never in this way arrive at a single point. But if the number of points in the original
system be 3p + 1, then that in the residual system will be 3^' T 1 ; and we may in
an infinity of different ways arrive at a residue consisting of a single point ; or say
at a "residual point," viz. after an odd number of steps if the original number of
points is =3/; — 1, but after an even number of steps if the original number of points
is = 3/j 4- 1. But starting from a given system of points on a given cubic curve, the
residual point, however it is anived at, will be one and the same point ; this is
Prof Sylvester's theorem of the residuation of a cubic curve. For instance, starting
with two given points on the cubic curve, the line joining these meets the curve in
a third point, which is the residual point ; any other process leading to a residual
point must lead to the same point. Thus if through the 2 points we draw a conic,
meeting the cubic besides in 4 points; through these a conic meeting the cubic besides
27—2
212 ON RESIDUATION IN REGARD TO A CUBIC CURVE. [589
in 2 points ; and through this a line meeting the cubic besides in 1 point ; this
will be the before-mentioned residual point.
The general proof is such as in the following example :
Take on the cubic U3 a system of Sk — 2 points, say the points a : through these
a curve Vt, besides meeting the cubic in 3k — SK + 2 points /8 : and through these a
curve Pt-^+i, besides meeting the cubic in a point C. And again through the 5k — 2
points a a curve TFf , besides meeting the cubic in 3A;' — 3« + 2 points /9' : and through
these a curve Qf-.+i, besides meeting the cubic in a single point; this will be the
point C.
The proof consists in showing that we have a curve ^t+fc'_»_, such that
■^k+k'-K-i Us = Qe-K+i Vk + Pk-K+\ Wjf.
For this observe that
Qi-_,+, meets TTf in 3k' — 3/e + 2 points /3' and besides in k'^ — k'{K + 2) + 3k - 2 points e ;
Pk-K+i meets Fj in 3^ — 3« + 2 points /S and besides in k''' — k(K + 2) + 3K—2 points 6 ;
Pk-K+\, Qt-K+i meet in (k — K+l){k' — k + 1) points G ;
Ft, IFf meet in 3/e — 2 points a and kk^—SK + 2 points a;
Qk-k+1 Vt and Pk-^+i W^ meet in
kk'-k{K-l)-k'{K-l)+ {k- ly points G
3k' -3/C + 2 „ /3'
k'" -k'(K + 2) + 3K-2 „ e'
3k -3k + 2 „ /9
k' - k (k + 2) + 3« - 2 „ €
kk' -3«-|-2 „ a
3k -2 „ a
(k + k'f - {2k -2){k + k') + (k- ly
= {k+k'-K + iy points.
Every (k + k' — k + l)thic through
^{k + k'-K + l)(k + k'-K + i)-l
of these points passes through all.
Now ^i+f_»_2 may be drawn to pass through
^{k + k'-K-2){k+k' -K + l)
of the points a.
589] ON RESIDUATION IN REGARD TO A CUBIC CURVE. 218
Hence Ajc+k-k-^'U^ is a {k + k' — k + \)ih.\c through
\{k + k'-K-2){k + k'-K + \)
= ^ (A; + ky -\{2k + 1) {k + k') + h{K' + K-2) points a
3«-2
n
a
Zh
-3/e + 2
»
y3
w
-3/«:+2
JJ
/S'
i (A + A;')' + (- « + I) (A; + A;') + ^^^ - §« + 1
= H(A; + A;')' + (A; + A;') (- 2« + 5) + (« - 1) (« + 4) - 2)
= J (^ + ^'' - " + 1) (^ + ^■' - « + •!•) - 1
of the points in question; and therefore through all. Whence
Also U, meets Qe-K+i T^t in 3 (A; + A;' — k + 1) of the {k + k' — K + iy points, viz. these
are
3« — 2 points a,
H 3^• - 3« + 2 „ /3,
3A;'-3« + 2 „ /3',
1 „ c,
and Ji+f_,_2 meets Qi_.+i Fj in (A; + A;' - « — 2) (^- + A;' — « + 1),
that is, in
(k + ky + {k + k'){-2K-l) + K-+K-2
(k + k' — K +iy points,
kk' + {k + k'){- K + l) + K^- 2k points C
k'' -k'(ic + 2) +3ic-2 „ e'
k^ -k {k + 2) +3k-2 „ e
kk' -Sk + 2 „ a
{k + k'y + (A; + A;') (- 2« - 1) + k' + « - 2 points.
Hence U, passes through 1 of the points C, that is, through an intersection of Qi^-n+i
and Pi_,+,, that is, Qt-K+i and Pt_,+i intersect f/3 in a common point C; which was
the theorem to be proved.
In the particular case 3k — 2=10, ^•=A;'=4, the theorem is, given on a cubic
10 points, if through these we draw a quartic meeting the cubic besides in 2 points;
of the
viz. these are
214 ON RESIDUATION IN REGARD TO A CUBIC CURVE. [589
and through these a line meeting the cubic besides in a point C; then this is a
fixed point, independent of the particular quartic. And the proof is as follows : we have
U a cubic through 10 points a ;
V a quartic through tlie 10 points, and besides meeting the cubic in 2 points /8;
W a quartic through the 10 points, and besides meeting the cubic in 2 points ff ;
P the line joining the two points /3, and besides meeting V in two points e;
Q the line joining the two points yS', and besides meeting W in two points e' ;
P, Q meet in the point C;
U, V meet in the 10 points a, and besides in 6 points a ;
A a conic through 5 of the points a.
Then quintics QV, PW meet in the 10 points a, 2 points /3, 2 points e, 2 points /3',
2 points e', 6 points a and 1 point C. Every quintic through 19 of these passes
through the 25. But we have AU, a, quintic through 5 points a, and the 10 points a,
2 points /8 and 2 points ff ; hence A U passes through all the remaining points, or we
have
AU=QV + PW,
P passes through /9 , /? , e , e , C,
Q „ ^', /3', e' , e' , C,
e , e , /3 , yS , 6 points a, 10 points a,
e , € , /3', /3' , 6 points a, 10 points a,
e , e , e' , e' , 6 points a,
/8, 0, 13', ^', C,
V
F
A
U
or, what is the same thing,
A, P intersect in e , e ,
A, Q „ € , e ,
A, V „ e , e , 6 points a,
A, W „ e , € , 6 points a,
U, P „ /3, ^, C,
U, Q „ ^', ^'. C.
U, V „ /3, yS , 10 points a,
U, W „ /3', ;S', 10 points a.
In particular U, P, Q intersect in the point C; that is, C as given by the inter-
section of U by the line P; and as given by the intersection of U by the line Q;
is one and the same point.
590] 215
590.
ADDITION TO PROF. HALL'S PAPER "ON THE MOTION OF A
PARTICLE TOWARD AN ATTRACTING CENTRE AT WHICH
THE FORCE IS INFINITE."
[From the Messenger of Mathematics, vol. ill. (1874), pp. 149 — 152.]
I DO not in the passage referred to* expressly profess to interpret Newton's idea.
After referring to his investigation I say, " The method has the advantage of explaining
the paradoxical result which presents itself in the case force « (dist.)"'-', and in some
other cases where the force becomes infinite. According to theory the velocity becomes
infinite at the centre, but the direction of the motion is there abruptly reversed, so
that the body in its motion does not pass through the centre, but on arriving there
forthwith returns towards its original position; of course such a motion cannot occur
in nature, where neither a force nor a velocity is actually infinite;" viz. while assuming
that the analysis gives a motion as just described, or in Prof Hall's figure, a recipro-
cating motion between A and C, I expressly state that the motion is not one that
can occur in nature; in fact, my view is that the question (which, to render it precise,
I state as follows: "What happens in nature when the moving point arrives at C")
presupposes what is inconceivable. But I consider that the analysis gives a motion
as above, viz. that it gives x, t each as a one-valued function of a parameter <f>, such
that this parameter 0 increasing continuously, we have for the moving point a con-
tinuous series of positions corresponding to the motion in question, gives in fact the
equations x= a(l — cos ^) and — -^ = ^ — sin <f).
In explanation and justification of the assumption, it is interesting to show how
the solution just referred U> can be obtained from the equation of motion tt^ = — — j .
etc 3)
without (in the process) the extraction of the square root of the two sides of an
[• By Professor Hall in his paper (p. 144, I.e.) quoted in the title. The passage is an extract from the
British Association Report (1862) On the progress of the solution of certain special problems of dynamics,
p. 186; [298], Coll. Math. Papers, vol. iv. p. 51S.]
216 ON THE MOTION OF A PARTICLE TOWARD AN ATTRACTING [590
equation. Taking x as the independent vaiiable and writing for a moment -j- :* t',
QfX
dH „ ,
■7-3 = t , the equation is
and if we herein assume a; = a (1 — cos <j>) and transform to <^ as the independent
variable, it becomes
a' sin' <^ f 1 dH _ cos(f> dt] _ /jl
^dtV [a sin <j> d^ ~ asm^ d^j ~ a» (1 - cos (f>y '
[d^J
or, what is the same thing,
• . d /dt\ , (dt\ IX 1 fdty
'''' '^ d<f, W - "^^^ "^ l#.) = ^ (1-COS0)' te j ■
a differential equation of the first order for the determination of ^ as a function
of <^. Since a is a constant of integration of the original equation, a particular
integral only is required, but it is as well to obtain the general integral. For this
purpose assume
dt a*
a»
d^-M'^^-''"^^'
then, omitting from each side of the equation the factor -j^. , the equation becomes
2 sin ^ + -7-7 (1 — cos <^)[ — cos ^ . ^ (1 — cos ^) = (1 — cos <f>) z',
viz. the left-hand side being (1 — cos <^) f ^H- -^j-sin <^j, the whole equation contains the
factor (1 — cos <f)), and omitting this, the equation becomes
^ + ^-sm<^ = ^;
or, what is the same thing,
dz _ d(f)
^ — z sin <^ '
The integral of this is
z^—\
log — — = 2 log A; + 2 log tan ^</> ;
z
or, what is the same thing,
^^=i»tanH<^,
z
where k is the constant of integration.
[In explanation of this constant k, observe that the equation gives
1
z =
^(l-k't&n^^)'
590] CENTRE AT WHICH THE FOECE IS INFINITE. 217
and that we thence have
that is,
or, since
this is
dt _ a^ 1 — cos <f)
34 ~ VOT) V(l - y" tan= \(j)) '
dt _ V(/0 Bm(f> 1 _ gi tan ^<f)
dx ai 1-COS0 V(l-^^tan2J</,)' ~ VO^ V(l -^-'tan^^^)'
X'
tan4<^ = 2—
dt _ a* »J{x)
dx V(m) V(2a - a; - te) '
or, what is the same thing,
v(i+W
dt _ VVl+ArV V(«)
viz. we in effect have i-ITl^ ^^ ^ constant of integration in place of the original
constant a.]
Recurring to the general solution
'^ ^^=A^tanH0,
we may take 2=1, as a particular solution answering to the value ^' = 0 of the
constant ; and we then have
dt a* ,, ,.
viz. reckoning t from the epoch for which <^ is = 0, we thus have
a*
which, combined with the assumed equation
a; = a (1 — cos (f>),
gives the foregoing solution.
I quite admit that, considering (with Prof Hall) the attracted particle as split
into two equal particles placed at equal distances above and below the centre C, the
motion when the distances become infinitesimal is a motion not as above, but back-
wards and forwards along the entire line AB; but it remains to be seen whether at
the limit this can be brought out as an analytical solution of the differential equation
d^x iL
-5T^ = — — . Possibly this may be done, and I remark as an objection, not to the fore-
going as an admissible solution of the problem but to its generality as the only
solution, that, in writing a; = a (1 — cos <^) and assuming that <^ is real, I in effect
assume that x is always positive. But the burthen of the proof is with Prof Hall,
to show that there is an analytical solution in which x acquires negative values.
C. IX. 28
218 [591
591.
A SMITH'S PRIZE PAPER AND DISSERTATION; SOLUTIONS
AND REMARKS.
[From the Messenger of Mathematics, vol. iii. (1S74), pp. 165 — 183, vol. iv. (1875),
pp. 6-8.]
1. Find the triangular numbers which are also square.
The " mise en equatimi" is immediate; we have to find n, m such that
^(n + 1) = m';
or, what is the same thing,
(2n + l)=-8m= = l.
Observing that this is satisfied by n = m = l, that is, 2k + 1=3, 2m = 2, we have the
general solution given by
2n + 1 + 2m ^(2) = {3 + 2 >J{2)\p,
where p is any positive integer ; viz. 2n + 1, 2m being rational, this implies
2n, + 1 - 2m V(2) = {3 - 2 V(2)p,
and thence the equation in question. The successive powers
3 + 2 ^f{2), 17 + 12 V(2), 99 + 70 ^/{2), &c.,
give the solutions
w, m= 1, 1 , 8, 6 , 48, 35 , &c,;
viz. the square triangular numbers are
l^ = i 1-2 ; 6^ = i 8-9 ; 35=, = J 49-50, &c.
591] A smith's pkize paper and dissertation. 219
2, Show how to express any symmetrical function of the roots of an equation in
terms of the coefficients. What objection is there to the method tuhich employs the sums
of the powers of the roots ?
The ordinary method is that referred to, employing the sums of the powers of
the roots; but it is a very bad one. In fact, writing
a;» - ia;"-! + ca;"-^ - &c., = (a; - a) (a; - /9) (« - 7) . . . = 0,
leading to
S, = b,
S, = b- - 2c,
S, = b'-Sbc + Sd,
then if the method were employed throughout, we should have for instance to find
Sa^y, that is, d, from the formula
6Sa0y= S,»= b'
-SS,S, -36(6=- 2c)
+ 2 ^3 +2{¥-Sbc + M)
= 6d, which is right,
but the process introduces terms 6' and be each of a higher order than d (reckoning
the order of each coefficient as unity), with numerical coefficients which destroy each
other. And, so again, Sa'0 would be calculated from the formula
Sa»/3= SA= b(b'-2c)
- S, -(b'-Sbc + Sd)
= be — 3d, which is right,
but there is here also a term 6* of a higher order, with numerical coefficients which
destroy each other. And the order in which the several expressions are derived the
one from the other is a non-natural one ; S3 is required for the determination of
*SV/3, whereas (as will be seen) it is properly Sit'fi which leads to the value of S^.
The true method is as follows: we have
Sa = b, Sa/3 = c, Sa^y = d,8ic.,
and we thence derive the sets of equations
b =
Sa;
c =
Sa^.
6» =
Sa' +2Sa0;
d =
Sa^y,
bc =
Sa'$ + 3Safiy,
6' =Sa' + BSci'^ + 6Sa^y;
28—2
220 A smith's pkize paper akd dissertation ; [591
viz. we thus have 1 equation to give Sa; 2 equations to give 8a^ and So'; 3 equations
to give Sa^y, /Sa'/S, Soi' ; and so on. And taking for instance the third set of equations,
the first equation gives Saffy, the second then gives Sa^ff, and the third then gives
/So*, viz. we have
Safiy = d,
Sa^ff =bc-Sd,
Sa' =6'-3(6c-3d)-6d,
= b'~3bc + 3d.
Of course the process for the formation of the successive sets of equations would
require further explanation and development.
3. Oiven a point P in the inteiior of an ellipsoid, show tlmt it is possible to
determine an exterior point Q such that for every chord RS through P, the relation
QR : QS = PR : PS may hold good.
There is no difficulty in the analytical solution and in showing thereby that the
point Q is determined as the intersection of the polar plane of P by the perpend-
icular let fall from P on this plane. But a simple and elegant geometrical solution
was given in the Examination. Constructing Q as above, let the chord RS meet the
polar plane of P in Z; then the polar plane of Z passes through P, that is, the
line ZP is harmonically divided in R, S, or we have
ZR : ZS = PR : PS.
Again ZQP being a right angle, the sphere on ZP as diameter will pass through Q ; and
jR, S being points on the diameter, and Z, Q points on the surface, ZR : ZS = QR : QS ;
whence the required relation QR : QS — PR : PS.
4. Find the number of regions into which infinite space is divided by n planes.
The number ^ (?i' + 5w + 6) is a known result, but not a generally known one, and
I intended the question as a problem ; I do not think it is a difficult one.
Consider the analogous problem for lines in a plane : the first line divides the
plane into 2 regions.
The second line is by the first divided into 2 parts, and therefore adds 2 regions.
The third line is by the other two divided into 3 parts, and therefore adds
3 regions ; and so on.
That is, the number of regions for
1 line is = 2 =2 regions,
2 lines =2 + 2 =4 „
3 lines =2 + 2 + 3 =7 „
n lines = 2 + 2 + 3 + ... + « = J^(n'' + n + 2) „
591] SOLUTIONS AND REMAEKS. 221
In exactly the same way for the problem in space :
The first plane divides space into 2 regions.
The second plane is by the first plane divided into 2 regions, and therefore add&
2 regions.
The third plane is by the other two planes divided into 4 regions, and therefore
adds 4 regions.
The fourth plane is by the other three planes divided into 7 regions, and there-
fore adds 7 regions : and so on.
That is the number of regions for
1 plane is = 2 =2 regions
2 planes =2 + 2 = 4 „
3 planes =2 + 2 + 4 =8„
4 planes =2 + 2 + 4 + 7 =15
n planes = 2 + 2 + 4 + 7 + . . . + ^ («" - ,i + 2) = ^ (n' + 5w + 6),
where, for effecting the summation, observe that the series is
= 2 + {1 + 1 + 1 ... (?i - 1) terms}
+ {l + 3 + 6...+in(w-l)},
= 2 + (?i — 1) + ^ (n + 1) n (n — 1), = as above.
5. In the theory of Elliptic Functions, explain and connect together the notations
F(0), am M (sinamw, cosaraw, Aamtt), illustrating them by reference to the circular
functions*.
What is asked for is an explanation of the fundamental notations of Elliptic
Functions. To a student acquainted with the subject, the only difficulty is to say
enough to bring the meaning fully out, and not to say more than enough.
Defining F(x) by the equation
dx
Fix) = f
(viz. the integral is taken from 0 up to the indefinite value x), then the fundamental
property of elliptic functions (derived from consideration of the differential equation
dx dy ^ Q.
VRl - ^) (1 - A:=^^)! Vi(l - f) (1 - kY)] '
consists herein, that the functional relation
F{x) + F{y)^F{z)
* It would have been better in the question to have written P(x) instead of F(e).
222 A smith's prize paper and dissertation ; [591
is equivalent to an algebraic equation between the arguments x, y, z. F(x) as defined
by the foregoing equation is properly an inverse function ; this at once appears from
a particular case, viz. writing k = 0, F{x) = siu~^x, and the theory of the function F{x)
in the general case corresponds to what the theory of circular functions would be, if
writing F{x) to denote sin~'a;, we were to work with the equation
F{x)-^F{y) = F{z)
as equivalent to the algebraical equations (one a transformation of the other)
z = xs/{\-f) + y^{\-a?),
V(l - z') = V(l - ^) V(i - y») - ^.
But in the actual theory of circular functions, we introduce the direct symbols sin,
cos; writing F(x) = 0, that is, a; = sin^, '.J(l—x') = eosd, and similarly F(y) ==<!>, that is,
y = sin ^ and \/(l — y") = cos <f>, then the equation
F{x) + F(y)^Fiz)
becomes F{z) = 6 + (j>, that is, z = sm(d + (f)), ^(1 — z') = cos (d + <f>), and the other two
equations become
sin (^ + <^) = sin 0 cos <^ + sin (^ cos 0,
cos (0 + <f>) = cos 0 cos (^ — sin ^ sin cf),
viz. these are the addition-equations for the functions sin and cos.
In passing from the original notation F(x) to the notation amw, we make the
like step of passing from an inverse to a set of direct functions ; first modifying the
meaning of F, so as to denote by F{0) what was originally J^(sin^), we have as the
new definition
d0 f d0
^^^^~lo^/a-lc'sm-'0) I,
VCl - i^' sin'^ 0) j„A((9)'
(if as usual A^ denotes V(l - ^ sin' ^)), and this being so, the relation F (0) + F (<f>) = F (fi)
is equivalent to a relation between the sine, cosine, and A of 0, <f>, ft. Writing then
F(0) = u, and considering this equation as determining 0 as a. function of u, 0 — a.mu,
we have sin ^ = sin . am m, cos 0 = cos . am t<, and A^=A.amtt, and similarly F(<f))=v,
<j) = a.Tnv, &c., then the equation F (0) + F {<!>) = F (fi) becomes F(iJ,) — u + v, that is,
fi = am (u + v); and the algebraic relation in its various forms gives the values of
sin . am (m + 1;), cos . am (« + v), A.am(«-|-i;) in terms of the like functions of u, v
respectively, viz. it is the addition-theorem for the function am.
Observe that am u is considered as a certain function of u, sin . am u is its sine,
coe . am u its cosine, and '
A . am u = V(l — ^ sin^ . am «),
a function analogous to a cosine. But making only a slight change in the point of
view, we have sinam m, a certain function of u, and
cosam « {= V(l - sinam' u)}, Aam u {= V(l — ^'° sinam* w)},
591] SOLUTIONS AND REMARKS. 223
two allied functions, viz. sinam u is analogous to a sine, and the other two functions
to cosines ; the algebraical equations give the sinam, cosam, and Aam of u -j-v in
terms of the like functions of u and v respectively, viz. they constitute the addition-
theorem for these functions.
6. Find the differential equation satisfied by a hypergeometric series, and express
by means of such series the coefficients of the expansion of (1 — 2acos ^ + a^)~" according
to multiple cosines of 0.
I understand the expression " hypergeometric series " in the restricted sense in
which it signifies the series
r./ o N -. a-/3 a(a + l)B(B+l) „ ,
1.7 1.27(7+1)
I find it was understood in the more general sense of a series
M = tto + a^x + a^- + . . . + «„«" + . . . ,
where the coefficient a„+i is given in terms of the preceding one a,j by an equation
of the form a„+i = <^(n) . a„. In this latter sense, but supposing for greater simplicity,
that 0 (n) is a rational and integral function of n, the solution is as follows : we operate
on the series with the symbol ^[x-j-\\ viz. a; -7- is regarded as a single symbol of
dxl ' ' dx
operation; x j- .x^ — na?^, ('^V-j x^=n-af^, «Sic. ; thus a; -7- is, as regards «", = «, and
therefore ^ ix -y-] = ^ {n). We thence have
4>U -T-] u = 4) {0) a^ + <i){l)a^x + <f>{2)a^ ... + ^(n) anof" + ...
= ai + a.:X + a^ ... + a„+i «" + . . . ,
and consequently
x<^ix-j-\u = u—a^,
which is the required dififerential equation. This is equivalent to the process given
in Boole, only he writes x = e*, in order to reduce a; t- to a mere differentiation -jn ■
■f ' dx dd
I regard this introduction of a new variable 6 as most unfortwmte; the effect is
entirely to conceal the real nature of the operation ; the notion of a; -7- as a single
symbol of operation is quite as simple as that of -j^; and by means of it we retain
the original variable.
The process is substantially the same when <f) (n) is a rational fraction, but I give
the investigation directly for the hypergeometric series in the restricted sense, viz.
writing u for the series F{a, /9, 7, x), we find
d \ / d ^\ d / d
(^i-^")(^i + ^)" = ^d^(^d^+'y-0^;
224 A smith's prize paper and dissertation; [591
or, what is the same thing,
as at once appears by writing the general term successively under the two forms
g.g + l ...g + w-1 ./3.y3+l.../8 + n-l
1.2 ... n 7.7 + 1 ... 7 + n — 1
and
a.a + l...a+M./3./8 + l .../3 + n
a^,
1.2 ... n + 1 .7 .7 + 1 ... 7 +n
The diflferential equation may also be written
«»+'.
('>^-^)^ + {a + ^+^)'^-y]^ + ^^y-o.
Take next the function
(1 - 2a cos 0 + a")-",
= {l-a(.+ l) + 4~"
= \(l—aa;)(l-a-]i , if a; + - = 2 cos ^,
f, w n n.n+1 „ n.n + 1 n,n+l ,n + 2 ^ ] f 1\, „ /,\
+ ^l+T^:2-«^ + -i:2 0:3— «' + ...}«(- +-)(=2a cos ^)
+ |l.^^-^f|^+&c. la^(a^ + ^)(=2a^cos2^),
&c. &c.
where the second term contains the factor - a, the third the factor —^ — ^ a", and
so on. Throwing these out, the remaining factors are each of them a hypergeometric
series, viz. representing the whole expression by
A^ + 2.4, cos e + 24j cos 16 + &c.,
we have
A^ = F{n, n, 1, a"),
A,==^aF{n, n + l, 2, a'),
and generally
. n.n + l ...n + r—1 ,„, , „,
Ar = ^V-H — a''F(n, n + r, r + I, a^).
1 . Z ... r
591] SOLUTIONS AND KEMAKKS. 225
1^
7. The function e '•"-"■'^ has been suggested as an exception to the theorem that if
a function and all its differential coefficients vanish for a given value of the variable,
then the function is identically =0; discuss the question as regards the precise meaning
of the theorem, and validity of the exception.
The suggestion was made by Sir W. R. Hamilton ; the following remarks arise
in regard to it :
1
The function e <*-«)' is a function which in a certain sense satisfies the condition
that for a given value (= a) of the variable, the function and all its differential
coeflScients vanish ; viz. each differential coefficient is of the form Xe <*"")*, where X
is a finite series of negative powers of x — a; if then x=a±r, where r is real and
positive, and if r continually diminishes to zero, then {x — a)", remaining always real
and positive, continually diminishes to zero, that is, — ^ remaining always real and
\3C ^~ Cur
1
negative continually increases to — x , and e <*-»)' remaining always real and positive
continually diminishes to zero. And, moreover, {X containing only a finite series of
negative powers of x — a) the expression Xe <*-'»)' will in like manner, remaining always
real, continually approximate to zero. But assume x = a-\-r (cos ^ + i sin 6), r real and
positive, 6 real ; then {x — ays=r' (cos 29 + i sin 26), and if cos 29 be positive, then the
real part of (x — a)', being always positive, continually diminishes to zero, and the like
conclusions follow. If however cos 29 be negative, then the real part of (x — a)^ is
negative, and the real part of — , ^ is positive, and as r diminishes continually
1
approximates to + oo ; so far from e (^'-"■y continually approximating to zero, it is in
general an imaginary quantity continually approximating to infinity; and the like is
the case with its successive differential coefficients; the conclusion is, it is not true
1
simpliciter that the function e <*-»)', or any one of its successive differential coefficients,
vanishes for the value a of the variable.
Generally, if a real or imaginary quantity a + 0i is represented by the point whose
rectangular coordinates are a, /3 ; say if the value a of the variable x is represented
by the point P, and any other value a + h + ki, by the point Q (h, k being therefore
the coordinates of Q measured from the origin P), then a function F{x) which as Q
(no matter in what direction) approaches and ultimately coincides with P, tends to
become and becomes ultimately = 0, may be said to vanish simpliciter for the value a
of the variable ; but if this is only the case when Q approaches P in a certain
direction or within certain limits of direction, the function not becoming zero when
Q approaches in a different direction, then the function may be said to vanish sub
modo for the value a of the variable.
Taking the theorem to mean "If for a given value a of the variable, a function
and its differential coefficients vanish suh modo, the function is identically = 0," the
C. IX. 29
226 A smith's prize paper and dissertation; [591
instance of the function e (*""'' shows that the theorem is certainly not true ; but
taking the theorem to mean "If for a given value a of the variable, the function
and its diflferential coeflScients vanish simpliciter, then the function is identically = 0 " ;
the instance does not apply to it, and the truth of the theorem remains an open
■question.
The above view is consistent with a theorem obtained by Cauchy and others,
defining within what limits of h the expansion by Taylor's theorem of the function
F(a + h) is applicable, viz. a and h being in general imaginary as above, if the
function (or ? the function and its successive difiFerential coefficients) is (or are) finite
and continuous so long as the distance PQ does not exceed a certain real and positive
value p, then the expansion is applicable for any point Q, whose distance PQ does not
exceed this value p: but it ceases to be applicable for a point Q, the distance of
which is equal to or exceeds p. In the case of a function such as e ^"'"f, dis-
continuity arises at the point P, that is, for the value p = 0, and according to the
theorem in question, the expansion is not applicable for any value of p however small.
I wish to remark on a view which appears to me to be founded on a radical
1
misconception of the notion of convergence. Writing F(x) = e (*-«)*, consider the series
F(a) + F' (a)^ + F" (a) ~^ + 8zc. ...
1
Then admitting that the exponential e (*-«)" becomes =0 for x = a, the successive
functions F{a), F'(a), F"{a},... are each = 0 as containing this exponential: but inas-
much as the successive differentiations introduce negative powers of x — a, each successive
function is regarded as an infinitesimal of a lower order than those which precede it;
say F{a) being =0'', the successive terms are multiples 0'', O**""', 0''"', 0""*, &c. respect-
ively; where however fj, is infinite, so that the several exponents /i, /i — 3, /* — 6, &c.,
however far the series is continued, remain all of them positive. This being so, it is
said that the series F{a)-\-F' {a)^-\-kiC, as being really of the form 0''-l-0''-» + 0''-* + ...
is divergent, and for this reason fails to give a correct value of F{a-{-h). I appre-
hend that the notion of divergence is a strictly numerical one; a series of numbers
a + b + c+ d + ... is divergent when the successive sums a, a+b, a-\-b + c, a + b + c + d,
&c., are numbers not continually tending to a determinate limit. In the actual case
the series is 0+0 + 0-1-0 + ..., viz. each term is by hypothesis an absolute zero ; the
successive sums 0, 0 + 0, 0 + 0 + 0, . . . are each = 0, and we cannot, by the process of
numerical summation, make the sum of the series to be anything else than 0. If it
could, there would be an end of all numerical equality between infinite series; for
taking any convergent series a + b + c + d+ ..., ifO means 0, this is the same thing
as the series, also a convergent one,
(o + 0) + (6 + 0) + (c + 0) + &c..
591] SOLUTIONS AND REMARKS. 227
and their difference 0 + 0 + 0 + ... must be =0. I regard the view as a mere failure
to reconcile the equation
F(a + h) = Fa + ^F' (a) + S^.,
with the supposed fact in regard to the function e (^-'»)'..
8. Find the value of the definite integrals
je~'^'dx, I sin a;^ da?, jcosafdx,
the limits being in each case qo , — oo . Examine whether the last two integrals can be
found by a process such as Laplace's (depending on a double integral) for the first
integral.
Laplace's process for the integral je~''dx is as follows: write i(= je~'' dx, then also
u—\e^dy, and thence
u" = [je-"='+y'> dxdy,
which, considering x, y as rectangular coordinates and substituting for them the polar
coordinates r, 6, becomes
M» = jTe-rVdrd0;
and then considering the double integral as extending over the infinite plane, and
taking the limits to be r = 0 to r= oo , 9 = 0 to 0 = 27r, we obtain
w' = (-ie-'T27r, =i.27r, =7r,
that is,
u= le-''dx = \/{7r).
There is an assumption the validity of which requires examination. We have u the
limit of the integral I er^ dx, as a approaches to oo ; and this being so, we have
«' the limit of
f° f" e-^^'^y'^ dxdy,
J —a J — a
viz. M* is the integral of e"'*""*^ taken over a square, the side of which is 2o, a being
ultimately infinite. But making the transformation to polar coordinates, and integi-ating
as above, we in fact take the integral over a circle radius =/3, /3 being ultimately
infinite. And we assume that the two values are equal; or, generally, that taking
the integral over an area bounded by a curve which is such that the distance of
every point from the origin is ultimately infinite, the value of the integral is inde-
pendent of the form of the curve.
29—2
228 A smith's prize paper and dissertation; [591
This is really the case under the following conditions: 1°. For a curve of a given
form, the integral tends to a fixed limit, as the size is continually increased. 2?. The
quantity under the integral sign is always of the same sign (say always positive);
{the last condition is sufficient, but not necessary). For, to fix the ideas, let the curves
be as before the square and the circle : take a square ; surrounding this, a circle ; and
surrounding the circle, a square. Imagine the two squares and the circle continually to
increase in magnitude; the integral over the smaller square and that over the larger
square, each tend to the same fixed limit; consequently the integral over the area
enclosed between the two squares tends to the limit zero; and d fortiori the integral
over the area enclosed between the circle and either of the two squares tends to the
limit zero; that is, the integral over the square, and that over the circle, tend to the
same limit. In the case under consideration, the function e~<*'+!''' is always positive;
and the integral lier'^^^ dxdy, taken over the circle, tends (as in effect shown above)
to the limit ir: hence the process is a legitimate one.
But endeavour to apply it to the other two integrals; write
w= Isina;'^^; ! v=lcoaafdx
= J8my'di/, =jcosy^dy,
then
I /sin (af + y') dxdy = luv, \ /cos (d' + y'') dxdy = v^ — u\
where the double integrals on the left-hand side really denote integrals taken over a
square and are not equal to the like integrals taken over a circle. This appears
A posteriori if we only assume that the integrals u, v have determinate values; for
taking the integrals over a circle they would be
//
sin „ , T/i
r.rdrdO,
cos
and would involve the indeterminate functions x ; that is, if it were allowable to
cos
take the integrals over a circle, we should have 2w« and i;' — v? indeterminate instead
of determinate.
A process of finding them is as follows: in the equation /e~*'da! = v'('t), sub-
stituting in the first instance oc \/(a) for x, a real and positive, we have
J V(a)
and if it be assumed that this equation extends to the case where a = a + ^i, Hie
591] SOLUTIONS AND KEMAEKS. 229
real part a real and positive*; or, what is the same thing, a = p{cos 0 +ism0), p real
and positive, 0 between the limits 0 and ^ir, then we have
L-p(o<«fl+.-8in«, ^^^ V(^) (cos |(9 _ i sin i0),
or, separating the real and imaginary parts and taking p = 1, we have
L-^co8 e cos (a? sin 0) dx = V(7r) cos ^0,
L-a^ooss sin (a? sin d) d« = V(7r) sin ^^.
Admitting these formulae to be true in general, there is still considerable difficulty in
seeing that they hold good in the limiting case 0 = ^tt. But assuming that they do,
the formulae then become
h^'^-'^M. /-«'*«-^>.
V(2)' J V(2)"
which are the values of the integrals in question,
9. Considering in a solid body a system of two, three, four, five, or six lines, deter-
mine in each case the relations between the lines in order that it may be possible to
find along them forces to hold the body in equilibrium.
If there are two lines, the condition obviously is that these must be one and the
same line.
If three lines, then these must lie in a plane, and meet in a point.
The conditions in the other cases ought to be in the text-books; they in fact
are not, and I assumed that they would not be known, and considered the question
as a problem ; it is, in regard to the cases of four and five lines, a very easy problem
when the solution is seen.
In the case of four lines; imagine in the solid body an axis meeting any three
of the lines, and let this axis be fixed; the condition of equilibrium about this axis
is that the fourth line shall meet the axis. The required condition therefore is that
every line meeting three of the four lines shall meet the fourth line; or, what is the
same thing, the four lines must be generators (of the same kind) of a skew hyperboloid.
In the case of five lines, taking any four of them, we have two lines (tractors)
each meeting the four lines ; and taking either of the two lines as an axis, then for
equilibrium the fifth line must also meet this axis; the required relations therefore
are that the fifth line shall meet each of the two lines which meet the other four
lines ; or, what is the same thing, that there shall be two lines each meeting the
five given lines.
* The equation is clearly not true unless this is so: for a being negative, then in virtue of the factor
«"°^, the exponential, instead of decreasing will increase, and ultimately become infinite as x increases to ±oo
230 A smith's prize paper and dissertation ; [591
The case of six lines is one the answer to which could not have been discovered
in an examination; the relations in fact are that the six lines shall form an involution;
viz. this is a system such that taking five of the lines as given, then if the sixth
line is taken to pass through a given point it may be any line whatever in a
determinate plane through this point ; or, what is the same thing, if the sixth line
is taken to be in a given plane, it may be any line whatever through a determinate
point in this plane. But in a particular case, the answer is easy ; suppose five of
the six given lines to be met by a single line, then the sixth line may be any line
whatever meeting this single line.
10. If X, F, Z, ... are the roots of the equation
(1, P. Q,...)(c, 1)» = 0,
show that the differential equation obtained by the elimination of c is ^X'Y'Z' = 0, where
^ denotes the product of the squared differences of the roots X, Y, Z,..., and X', Y', Z',...
are the derived functions of these roots ; and connect this result with the theory of
singular solutions.
We have identically
(1, P, Q...)(c, l)» = (c-Z)(c-r)(c-^...;
the original equation and its derived equation
(0, F, Q',...)(c. 1)» = 0
(the latter of them of degree n — 1) may therefore be written
(c-X)(c-Y)(c-Z)...=0,
X'{c-Y){c-Z)... + Y'(c-X)(c-Z)...+&c. = 0.
To eliminate c, we have in the nilfactum of the second equation to substitute succes-
sively the values c = X, c=Y, &c., multiply the several functions together and equate
the result to zero; the factors are evidently
X'(X-Y){X-Z)..., Y'(Y-X)(Y-Z)..., &c..
where each difference occurs twice, e.g. X—Y under the two forms X—T suid Y—X
respectively; the result thus is
X'Y'Z' ...{X-Y)'(X-Zy(Y-Z)\.. = 0;
that is,
f.X'F'Z'...=0.
Thus in particular in the case of a quadric equation
(1, P, Q) (c. 1)', = (c - Z) (c - F), = 0,
the differential equation is
(Z-F)'Z'F' = 0;
591] SOLUTIONS AND REMARKS, 231
viz. since X+Y = -P, and XY=Q, this is
(P'-4Q)Z'F' = 0,
and writing also
X J{-P+V(i^-4Q)}. Y=-UP-^/(P'-m>
we find
,_ f (PP'-^QJ]
the differential equation thus is
(P,.,«{^..(i^^j=„.
The application to the theory of singular solutions is that, in the case where the
function (1, P, Q...)(c, 1)" breaks up into rational factors c — X, c—Y,..., the factor
^={X —Yy(X — Zy ... divides out and should be rejected from the differential equation,
which in its true form is X'Y'Z' ... = 0; viz. this is what we obtain immediately,
considering the given integral equation as meaning the system of curves c — X = 0,
c— F=0, ..., and there is not really any singular solution; whereas in the case where
the factors are not rational, the factor in question, when the product X'Y'Z' ... is
expressed in terms of the coefficients P, Q, ..., and their derived coefficients does not
divide out from the equation; and in this case, equated to zero, it gives a proper
singular solution of the equation.
11. In the theory of elliptic motion, v denoting the mean anomaly and e the eccen-
1 + e , .
tricity, if m' he an angle such that tan \v = tan \m', find in terms of e, m' the
mean anomaly m.
Taking as usual u for the eccentric anomaly, to commence the solution write down
tan ^ = a/ (fir;) *^" ^^
1 + e.
1-
tan ^i',
that is,
tan ^u = aJ(j^^ *^ ^''
and u being given hereby as a function of w', we have by substitution in the equation
m = u— e sin u, to find m as a function of m'.
A creditable approximate solution would be m = m' +0.e, viz. this would be to
show that neglecting terms in e", &c., we have m = m'. In fact, taking e small, we have
tan ^M = (1 + e) tan ^m',
and thence if m = m' + a;, we have
tan ^m' + ^x Bed' ^' = (1 + e) tan |m',
232 A smith's prize paper and dissertation ; [591
that is,
a = 2e cos' ^m' tan Jm' = e sin m' ; m = m' + e sin m',
and
TO = m' + e sin m'
— esin(m'+ ...)
= m' + 0 . e.
The complete solution would be obtained by expanding u in tenns of e, m' from the
equation t&n^=./l:z jtanjm' (which is of the form tan ^w = n tan ^m', giving for
u a known series = m' + multiple sines of m), and then observing that the same
equation leads to
V(l - c=) sin m'
sin tt = — ^ J- ,
1—e cos m
we have
e V(l — e*) sin m'
m = series —
1—e cos m'
where the second term has also to be expanded in a series of multiple sines of m' ;
which can be done without difficulty.
12. If (u, v) are given f Mictions of the coordinates (x, y), neither of them a
maximum or a minimum at a given point 0; and if through 0 we draw Ox' in the
direction in which v is constant and u increases, and Oy' in the direction in which u is
constant and v increases ; then the rotation {through an angle not greater than tt), from
Oaf to Oy' is in the same direction with that from Ox to Oy, or in the contrary
,. .. ,. du dv du dv .
direction, according as j- t — TT T' ^ positive or negative.
The theorem has not, so far as I am aware, been noticed, and it seems to be
one of some importance; there is no difficulty in it, but the answer requires some
care in writing out; of course where the whole question is one of sign and direction,
the omission to state that a subsidiary quantity is positive may render an answer
worthless.
It depends on the following lemma: Consider the triangle OX'Y', such that Ox,
Oy being any rectangular axes through the origin 0, the coordinates of X' are h, k,
and those of Y' are hi, kii then considering the area as positive, the double area is
= + (hki — hjc), viz. the sign is + or — according as the rotation from OX' to 0 Y'
(through an angle less than tt) is in the same direction yrith that from Ox to Oy,
or in the contrary direction ; or, what is the same thing, hki — hJc is in the first case
positive and in the second case negative.
To show this, suppose for a moment that the lines OX', OY' are each of them
in the quadrant xOy, say in the first quadrant, the inclination of OY' to Ox exceeding
k k
that of OX' to Ox; then h, k, h^, ki are all positive, and ■r->T, that is, hki — hik is +,
591] SOLUTIONS AND REMARKS. 233
and the rotation from OX' to OY' is in the same direction as that from Ox to Oy;
or the lemma holds good. Now OX' remaining fixed, let OY' revolve in the direction
k
Ox to Oy; so long as OY' remains in the first quadrant, r- continues to increase,
k k
and we have always j^>t, and hki—hjc = + ; when OY' comes into the second quadrant
(h, k being always positive), h^ is negative and ^i positive, consequently hk^ — hjc is
the sum of two positive terms, and therefore = + ; as OF' continues to revolve and
k k
passes into the third quadrant, we have hi, ki each negative, but r<T, and therefore
liki — hjc still = + ; when, however, OY' comes into the position opposite to OX', then
k k
j^ = T, and hki—hjc is =0; and when OY', continuing in the third quadrant, has
k k
passed the position in question, we have ^>t, and therefore hkj, — hjc = — , but now
the angle X'OY' measured in the original direction has become >w, and the rotation
OX' to OY' through an angle less than ir will be in the opposite direction, that is,
in the direction opposite to that from Ox to Oy; and, similarly, when OY' passes
into the fourth quadrant, and until, passing into the first quadrant, it approaches the
position OX', the sign of hki — hJc will be — , and the rotation will be in the direction
contrary to that from Ox to Oy. The lenima is thus true for any position of OX'
in the first quadrant ; and the like reasoning would show that it is true for any
position of OX' in the second, the third, or the fourth quadrant; hence the lemma
is true generally.
This being so, taking a new origin, let the coordinates of 0 be x, y; and drawing
through 0 the axes Oaf, Oy' as directed, let X' be the point belonging to the values
M + S«, V of (w, v), and Y' the point belonging to the values u, v + hv of (w, v) ; taking
hu positive, X' will be on Ox' in the direction 0 to x, and similarly taking bv
positive, Y' will be on 0/ in the direction 0 to y'. Taking as before (A, k) for the
coordinates of X', and {h^, k^ for the coordinates of Y', these coordinates being measured
from the point 0 as origin, we have
-, du , du ,
6Ji = T- A + -;- A,
ax dy
- dv , dv J
0 = J- ft + J- A,
ax dy
, -.■ c ^ T <^^* d'" du dv , „ dv ^ J., dv ~.
whence, writing for a moment J = ^r i s — r • we have Jh = + -j-bu, Jk = — =- ou.
° dx dy dy ax dy dx
And in like manner
- du , du ,
^ = drx^^^dy^^'
5 dv , dv J
dx dy
whence
,, du J, ,, du ^
JK = --^yhv, Jh = ^Bv;
c. IX. 30
234 A smith's prize paper and dissertation ; [591
and hence
J-'(^A.-M)=(^ ^-| J)s«St,, =JSuBv,
that is,
hki — hik= -J- BuSv,
and Su, Bv being as above each of them positive, J has the same sign as ^^i - hik.
But the rotation from OX' to OY' is in the same direction as that from Ox to Oy,
or in the contrary direction, according as hk^ — Jhk is + or — , that is, according as
r du dv du dv . , i ■ i • .i. .l
•/, =j--5 T-j-,is + or-; which is the theorem m question.
13. Write a dissertation on:
The theory and constructions of Perspective.
In Perspective we represent an object in space by means of its central projection
upon a plane : viz. any point Pi* of the object is represented by P", the intersection
with the plane of projection of the line -DiPj from the centre of projection (or say
the eye) Di to the point Pi; and considering any line or curve in the object, this is
represented by the line or curve which is the locus of the points P', the projections
of the corresponding points Pi of the line or the curve in the object.
The fimdamental construction in perspective is derived from the following con-
siderations : viz, considering through Pi (fig. 1) a line meeting the plane of projection
in Q, and drawing parallel thereto through D^ a line to meet the plane of projection
in M and joining the points M, Q, then the lines D^M, MQ, QPi are in a plane;
that is, the plane through A and the line P^Q meets the plane of projection in MQ;
Fig. 1.
and consequently the projection P' of any point P, in the line P^Q lies in the line
QM; and not only so, but considering only the points Pj of this line which lie
behind the plane of projection (A being considered as in front of it), the projections
of all these points lie on the terminated line MQ; viz. Q is the projection of the
point Q, and M the projection of the point at infinity on the line QPi ; or, if we
please, the finite line QM is the projection of the line QPiOO .
* The subscript unity is used to denote a point not in the plane of projection, considered as a point
out of this plane ; a point in the plane of projection, used in the constructions of perspective as a con-
ventional representation of a point Pj, will be denoted by the same letter P without the subscript unity.
And the like as regards D, and D.
591] SOLUTIONS AND REMABKS. 235
If we consider a set of lines parallel to PiQ, these all give rise to the same
point M, and thus their projections MQ all pass through this point M, which is said
to be the " vanishing point " of the system of parallel lines. Again, if we consider
any two or more lines through Pj, to each of these there correspond different points
M and Q, and, therefore, a different line MQ, but these all intersect in a common point
P' which is the projection of Pj. If the lines are all in one and the same plane
through Pi, then the locus of the points Q is a line, the intersection of this plane
with the plane of projection, say the " trace " line ; and the locus of the points M
is a parallel line, the intersection of the parallel plane through Di with the plane of
projection ; say this is the " vanishing line " for the plane in question.
A construction in perspective presupposes a conventional representation on the
plane of projection (or say on the paper) as well of the position of the eye as of the
object to be projected. If for simplicity we suppose the object to be a figure in one
plane, then this plane intersects the paper in a trace line, and we may imagine the
plane made to rotate about the trace line until it comes to coincide with the paper,
and we have thus the plane object conventionally represented on the paper. Similarly
considering the parallel plane through the eye Dj, and regarding D, as a point of
this plane, the plane meets the paper in the vanishing line, and we may imagine the
plane made to rotate (in the direction opposite to that of the first rotation) until it
comes to coincide with the paper, bringing the point Dj to coincide with a point D
of the paper. We have thus khe " point of distance " D, being a conventional repre-
sentation on the paper of the position of the eye Dj-, but which point D has, observe,
a different position for different directions of the plane of the object.
To fix the ideas, suppose the plane of projection to be vertical, and the plane of
the object to be a horizontal plane situate below the eye. The trace line will be
represented by a horizontal line HH' (fig. 2), and the object by a figure in the plane
of the paper below the line HH' such that, bending this portion of the paper back-
wards through a right angle round HH', the figure would be brought to coincide with
the object*. The vanishing line will be a horizontal line KK' above HH', and the
• It ia assnmed in the text, that the figure on the paper is equal in magnitude to the object ; but
practically the figure ia drawn on a reduced scale, the distance between the lines KK', HH', and the distance
DS (representing respectively the distance between the parallel planes, and the distance of the eye from the
plane of projection) being drawn on the same reduced scale.
30—2
236 A smith's pbize paper and dissertation. [591
eye will be represented by a point D above KK', in suchwise that, bending the upper
part of the paper round KK' forwards through a right angle, the point D would come
to coincide with the position D^ of the eye. This being so, taking any line PQ in
the representation of the object, we draw through D the parallel line DM, and then
joining the points M and Q, we have MQ as the perspective representation of the
line QPoo , which represents a line QPi* of the object. And drawing through P any
number of lines, each of these gives a point Q and a point M, but the lines MQ
all meet in a common point P', which is the perspective representation of the point
P; which point P" may, it is clear, be obtained as the intersection with any one line
MQ of the line DP drawn to join P with the point of distance D. The plane of
the object has for convenience been taken to be horizontal ; but its position may be
any whatever, and in particular the construction is equally applicable in the case where
the plane is vertictil.
In the case of an object not in one plane, any point Q, of the object may be
determined by means of its projection by a vertical line upon a given horizontal plane,
say this is Pj, and of its altitude Q^Pi above this plane. We in fact determine the
object by means of its groundplan, and of the altitudes of the several points thereof
It is easy, from the foregoing principles, to see that, drawing through P the vertical
line PQ equal to the altitude, and joining the points Q, D, then the vertical line
through P' meets this line QD in a point Q', which will be the perspective repre-
sentation of Qi. We have thus a construction applicable to any solid figure whatever.
592] 237
592.
ON THE MERCATOR'S-PROJECTION OF A SKEW HYPERBOLOID
OF REVOLUTION.
[From the Messenger of Mathematics, vol. iv. (1875), pp. 17 — 20.]
In a note " On the Mercator's-proj action of a surface of revolution " read before the
British Association, [555, (5)], I remarked that the surface might be, by its meridians
and parallels, divided into infinitesimal squares ; and that these would be on the map
represented by two systems of parallel lines at right angles to ea<;h other, dividing the
map into infinitesimal squares ; and that, by taking the squares not infinitesimal but
small, for instance, by considering the meridians at intervals of 10° or 5°, we might
approximately construct a Mercator's-projection of the surface. But it is worth while,
for the skew hyperboloid of revolution, to develop analytically the ordinary accurate
solution.
Taking the equation of the surface to be
a? + y^ z-' _
{or, if as usual a^ + d^-= a'e°, then a? + y^ — {e^ — 1) z' = a^], and writing x = r cos 6, y = r sin 6,
the meridians corresponding to the several longitudes 6 are in the map represented
by the parallel lines X = a0, and the parallels corresponding to the several values of
z are in the map represented by a set of parallel lines Z=/(z), the form of the
function being so determined that the infinitesimal rectangles on the map are similar
to those on the surface. The required relation is readily found to be
J z^ + d'
where the integral is taken from the value z = 0.
238 ON THE mercator's-projection of a skew [592
The substitution which first presents itself is to write herein z = -;— r „. tan d> ;
or, what is the same thing,
z = a(e — J tan <^,
where observe that a(e — ) is the distance between a focus and its corresponding
directrix. The equation of the surface is satisfied by writing therein n/(af + y^) = aaec-\Jr,
2 = c tan yjr, and yjr as thus defined is the " parametric latitude " ; hence the foregoing
angle <^ is a deduced latitude connected with the parametric latitude ijr by the equation
c &
tan0 = — ; T^tani/r, = ___ tan i/r.
/ 1
ale —
V e
The resulting formula in terms of ^ is
Jocos^(c=' + a''co8''</))'
or, if we write herein f = tan ^<^, the formula becomes
^ c»(n-f)2+a=(i-r'y"
viz. the function under the integral sign is rational. The expression is, however, com-
plicated, and a more simple formula is obtained by using instead of <f> the parametric
latitude i/r ; viz. we have 5 = ctan'^, and thence
^^rV(a'sin'V. + c')
] cos i/r ^
or, putting herein
. , c u
8in'Jr = jy:. ^,
^ a i/(l- w») '
and therefore
and
a'sin'ilr + c= = - ,, cosy}rd-dr = j,
r l-u"' ^ ^ a (1 _„»)••
the formula becomes
«•_-,_/■ ^w
^^"'Jo(l- M') {a' - (a» + (f) «»} •
or, what is the same thing,
viz. we thus have
""^^~^^J„(l-M')(l-e»u')'
the logarithms being hyperbolic.
592] HYPERBOLOID OF REVOLUTION. 239
As already mentioned, u is connected with the parametric latitude i/r by the
equation
. , c u u>J(^ — l)
^ a V(l-M')' VCl-M") '
that is,
8in'^ = V'(e^ — l)tan^, if u = Biap,
or conversely
sini^
u =
V(e=-H-sin^-«|r)'
so that the point passing to infinity along the branch of the hyperbola, or ifr passing
from 0 to 90°, u passes from 0 to - ; and for u = - the value of Z becomes, as it
should do, infinite. The value of « in terms of m is
(e^-l)M , z
z — ^7\ T-s^ , or conversely u = ,. „ „ — ^^ ,
and we have, moreover,
u= V^, =-sin<^, =(£is before) --
It will be recollected that, in the Mercator's-projection of the sphere, the longitude and
latitude being 0, ^, the values of X, Z are
X = a0, Z= log tan ( J + J<^J ,
the logarithm being hyperbolic.
In the case of the rectangular hyperbola a = c, =1 suppose,
e = V(2), « = tan-«/r, m= ,Q^^°g^a . w =8in^, if sin-f = tan^;
whence
tan (45° -i;,) tan (22°30' - ^j>)
^- ^'^ • ' tan (45° + hp)~^ "^^^ tan (22°30' + ^p) '
the first term being of course
= A. itan (45° — ^/)), or — ^ . / tan (45° + ^p).
Transforming to ordinary logarithms, this is
Z = ^/pyi^e [- V(2) log tan (45° + ip) + {log tan (22°30' + ip) - log tan (22°30' - i^)j],
say this is
^= ,,J, (-A+B),
V(2)loge'
h
240 mercatoe's-projection of a skew hyperboloid of revolution. [592
4 = V(2)logtan(45° + Jp).
B = log tan (22° 30' + Jp) - log tan (22° 30' - ^p).
Taking ■^ as the argument, I tabulate z, = tani/r, and Z. <J{2)loge, = — A+B, as shown
in the annexed table : the last column of which gives, therefore, the positions of the
several parallels of 5°, 10°, ...,85°; the interval of 5° between two meridians is, on the
same scale,
V(2) log e . ^ = (1-4.142136) (-4342945) (-0872665), = -05360 ;
viz. this is nearly equal to the arc of meridian 0° to 5°, and the table shows that
the arcs 0° — 5°, 5° — 10°, &c. continually increase as in a Mercator's-projection of the
sphere, but more rapidly ; there is, however, nothing in this comparison, since the
determination of latitude on the hyperboloid by the equation 2 = tani/r is altogether
arbitrary.
f
« = tan^
p
A
B
-A+B
0*
0-
0°
0-
0-
0-
5
■08749
4° 58' 51"
•05348
•10719
•05370
10
•17632
9 51 0
•10611
•21439
•10827
15
•26795
14 30 40
•15724
•32174
•16450
20
■36397
18 53 0
-20619
•42943
•22324
25
•46631
22 54 30
-25342
•53780
•28539
30
•57735
26 34 0
•29557
•64758
-35201
35
•70020
29 50 20
•33538
•75959
•42421
40
•83910
32 44 0
•37170
•87506
•50337
45
1^00000
35 15 50
•40446
•99554
•59108
50
M9175
37 27 20
•43360
1^12355
•68995
55
1^42815
39 19 20
•45913
1^26151
-80238
60
r73205
40 53 40
•48117
r41445
•93328
65
214450
42 11 10
•49971
1 •58840
1 •08869
70
2-74747
43 13 10
■51478
179504
1^28026
75
3^73205
44 0 30
•52646
2^05524
1-52877
80
5^67128
44 33 40
•53469
2^41347
1-87877
85
1143005
44 53 30
•53973
302355
2^48381
90
00
45° 0' 0"
•54133
00
00
593] 241
593.
A SHEEPSHANKS' PROBLEM (1866).
[From the Messenger of Mathematics, vol. iv. (1875), pp. 34 — 36.]
Apply the formulce of elliptic motion to determine the motion of a body let fall
from the top of a tower at the equator.
The earth is regarded as rotating with the angular velocity w round a fixed axis,
80 that the body is in fact projected from the apocentre with an angular velocity = w ;
and we write a for the equatorial radius, /9 for the height of the tower; then g
denoting the force of gravity, and fi, h, n, a, e, 6, as in the theory of elliptic motion,
we have
^= n-a' =goi',
A = (a + /3)2 w = naV(l - e^),
o + /3 = a (1 + e) ;
whence
(a + /S)«a)» = 5ra»a(l-e='),
(a + /3) = a (1 + e ).
(« + /3)»a)' aa)7, , /3\
where -^ = ratio of centrifugal force to gravity.
1 g \ o-i
80 that 1 — e is small ;
whence
?F7 '
= «(!-«') = (a+/3)(l-e)
1 — e cos d' 1 — e cos 0 '
c. IX. 31
L
242 A sheepshanks' problem (1866). [593
Suppose
r = a,
l_eco8^ = (l-e)('l+^)=l-e + ^(l-e),
\ (x/ a
which is nearly
= l-e,
that is, tf is small, and therefore approximately
or
^ = Ml-i.
a e
we then have
i^dd = hat, or d< = -t— = 7 — — ^rx — 7^ 5^ at)
«2(l-ecos^)«"
■«(l-e + ie^)»
1
d^
(o\ 1 — e /
that is,
Integrating, we have
where tat = earth's rotation in time t, =<j> suppose ; therefore
hence, if d be as above, the angle described in falling to the surface,
i-e a '
593]
Writinj
y herein
A SHKEPSHANKS' PROBLEM
1—e , am'
e 9
243
this is
VIZ,
whence
or say
a(g-<^)_2V(2) //^\ /W|
/3 3
where a (6 — (f)) is the distance at which the body falls from the foot of the tower.
Substituting for . /( — J its value, =jj, we have
/ ■
31—2
244 [594
594.
ON A DIFFERENTIAL EQUATION IN THE THEORY OF ELLIPTIC
FUNCTIONS.
[From the Messenger of Mathematics, vol. iv. (1875), pp. 69, 70.]
The follo\ving equation presented itself to me in connexion with the cubic trans-
formation :
.dQ
'dk
Q'-Q(k + ly3 = 3(l-k')\
Writing as usual k = u*, I was aware that a solution was
Q = ^, + 2uv.
where u, v are connected by the modular equation
u*-v*+ 2uv (1 - itV) = 0 ;
but it was no easy matter to verify that the differential equation was satisfied. After
a different solution, it occurred to me to obtain the relation between (Q, u); or, what
is the same thing, (Q, k), viz. eliminating v, we find
or say
Q
whence also
^{Q'-6Q'-S) = *(k + ly
I
594] ON A DIFFERENTIAL EQUATION IN THE THEORY OF ELLIPTIC FUNCTIONS. 245
that is,
^(Q-1)'(Q + 3) = 4JVA; + ^}',
and
and thence
iQ + mg-s) fk-i\\
(Q-inQ+s)~\k+ij '
viz. the value of Q thus determined must satisfy the differential equation. This is
easily verified, for, in virtue of the assumed integral, we have
that is,
or finally
Q._3-J(Q*-6Q=-3) = 3(1-A^)^;
Q'-10(?+9 = -12(l-i=)^,
(Q'-l)(Q'-9) = -12(l-A:=)^,
an equation which is at once obtained by differentiating logarithmically the former
result, and we have thus the verification of the solution. This is, however, a particular
integral only ; and it appears doubtful whether there exists a general integral of an
algebraical form.
246 [595
595.
ON A SENATE-HOUSE PROBLEM.
[From the Messenger of Mathematics, vol. iv. (1875), pp. 75 — 78.]
The following was given [5 Jan., 1874,] as a problem of elementary algebra :
" Solve the equations
u (2a — x) = x (2a — y) = y (2a — z)'=z (2a — m) = h",
and prove that unless 6' = 2a', a;=y = z=u, but that if 6" = 2a'', the equations are not
independent."
This is really a very remarkable theorem in regard to the intersections of a
certain set of four quadric surfaces in four-dimensional space; viz. slightly altering the
notation, we may write the equations in the form
x{2e-y) = md'...(l2),
y(2e- ^) = m^...(23),
z{2d-w) = m0'...{Si),
w{2e- x) = md'...{4!l),
where, regarding (x, y, z, w, 6) as coordinates in four-dimensional space, each equation
represents a quadric surface. I remark that in such a space we have the notions, point-
system, curve, subsurface, surface, according as the number of equations is 4, 3, 2, or 1.
Four quadric surfaces intersect in general in 16 points. But for the system in
question {m being arbitrary), the common intersection consists of two lines and the two
points
x = y=z=w = d[\± V(l — w)} ;
and in the case where m = 2, then the intersection consists of two lines and a certain
unicursal quartic curve.
595] ON A SENATE-HOUSE PROBLEM. 247
To obtain these results, I consider the four points
^ = 0, x=0, y=0, 2=0, ...123,
e = 0, y = 0, z=0, w = 0, ... 234,
^ = 0, z =0, w = 0, « = 0, ... 341,
^ = 0, w=0, x=0, y = 0, ...412:
the two points
and the six lines
x = y = z = w=e{l±^(l-m)}, ... PQ:
(9 = 0, a; = 0, y=0, ... 12,
0 = 0, y = 0. z=0, ... 23,
0 = 0, z =0, w = 0, ... 34,
5 = 0, ^ = 0, a;=0, ...41,
0 = 0, x=0, ^=0, ... 13,
0 = 0, y=0, w = 0, ... 24,
being the edges of a tetrahedron, the vertices of which are the four points, viz. the
point 123 is the intersection of the lines 12, 13, 23, and so for the other points.
The surfaces contain the several lines, viz.
the surface 12 contains (12)=, 13, 14, 23, 24,
23 „ (23)», 12, 24, 13, 34,
34 „ {U)\ 13, 23, 14, 24,
41 „ (4in 24, 34, 12, 13,
■where (12)' denotes that 12 is a double line on the surface, and so in other cases.
And it thus appears that the surfaces pass all four of them through the lines 13, 24,
so that these lines are a part of the common intersection. To obtain the residual
intersection, observe that the equations give
x = 20 — m
whence
w 20-y'
ofl ^ ^0^
z = 20-m — = „~^ ,
y 20— w
(25-3,) (2^ -"^ = m^.
(25-w)(25-— ) = m5^
248 ON A SENATE-HOUSE PROBLEM. [595
or omitting from each equation the factor B, the equations become
(2d - y) (2w - md) = mdvi,
(26 - w) (2y - mO) = mdy ,
that is,
(4 - 2m) dw - 2md* - 2yw + md (y + w) = 0,
{^-2m)dy -2mff'-2yiv->rm0{y + w) = O.
Whence, m not being = 2, we have y = w, and then
w^-26w+me' = 0,
or, what is the same thing,
eta '>^^
2ff — w= ,
w
giving x=y = z = iu = 6 {1± \/(l — m)], viz. the surfaces each pass through the points
P, Q. As regards the omitted factor 6, it is to be observed that, writing in the
equations of the four surfaces 0=0, the equations become xy = 0, yz== 0, zw = 0, wx = 0,
satisfied by x = 0, z=0, or by y = 0, w=0, we have thus (0 = 0, x = 0, z = 0) and
{0 = 0, y = 0, w= 0), viz. the before-mentioned lines 13 and 24.
In the case m = 2, we have between y, w the single equation
yw - d (y + w) + 2^ = 0,
giving
0{w-20)
y w-0 '
and thence
_20{w- 6)
z =
w-0'
VJ
or, writing for convenience « = ^ . then the equations are
w
y_a-2
z_ -2
d~a-2'
a; 2 (3-1)
0~ a
595]
or, what is the same thing,
ON A SENATE-HOUSE PROBLEM.
x= 2(a-l)=(a-2)(l-^)
y : a ... (a-2)=(l-^)
-2a(a-l) ... (l-^J
a»(a-l)(a-2)
a(a-l)(a-2) (l - ^) .
249
: w
: e
where, for the sake of homogeneity, I have introduced the factors f 1 j and ( 1
a
X / \ CO ,
viz. we have x, y, z, w, 6 proportional to quartic functions of the arbitrary parameter
a, or the curve is a unicursal quartic. Writing in the equations a=0, 1, 2, oo successively,
we see that this quartic curve passes through the four points 123, 234, 341, 412 (inter-
secting at these points the lines 13 and 24 respectively); and writing also a=\±i we
see that the curve passes through the points P, Q, the coordinates of which now are
x = yr=LZ = w = (\ ±i)d.
It should admit of being proved by general considerations that, in 4-dimensional
geometry when 4 quadric surfaces partially intersect in two lines, the residual inter-
section consists of 2 points ; and that, when they intersect in the two lines and in a
unicursal quartic met twice by each of the lines, there is no residual intersection — but
this theory has not yet been developed.
C. IX.
32
250
r596
596.
NOTE ON A THEOREM OF JACOBI'S FOR THE TRANSFORM-
ATION OF A DOUBLE INTEGRAL.
[From the Messenger of Mathematics, vol. iv. (1875), pp. 92 — 94.]
Jacobi, in the Memoir "De Transforraatione Integralis Duplicis..." &c., Crelle, t. viii.
(1832) pp. 253—279 and 321—357, [Ges. Werke, t. in., pp. 91—158], after establishing
a theorem which includes the addition-theorem of elliptic functions, viz. this last is " the
differential equation
dr, de
^{0'" cos= V +0"' sin» v-G')'^ s/iG'" cos= 6 + G"" sin» d-G^)'
has for its complete integral
G+G' cos »? cos ^ + G" sin i; sin 5 = 0,"
{observe, as to the integral being complete, that the differential equation contains only
the constant G"" — G'^-r- (G" — G"'), whereas the integral equation contains the two con-
stants G' -r-G and G" -r- G], obtains a corresponding theorem for double integrals ; viz.
this, in the corresponding special case, is as follows : If the variables (^, ■^) and
(?;, 6) are connected by the two equations
= 0,
+ a' cos <^ . cos rf
+ a" sin <f> cos i/r . sin t] cos 6
+ a'" sin <|) sin 1^ . sin tj sin 6
and if putting for shortness
/3 =0,
-J- ^ cos (^ . cos »?
+ ^" sin (^ cos ^ . sin r) cos 6
■\- ff" sin ^ sin •^ . sin ij sin 6
a"^"'-a"'/3"=/, oiff -a' ^ = a,
a"'/3' -aT=9' ^^" -a"/9 = 6,
a' ^' -a"^' =h, ajS"'-a"'/8 = c,
(whence af+bg + ch'=0);
596] ON JACOBl'S TRANSFORMATION OF A DOUBLE INTEGRAL. 251
R' = /- (sin (f) COS •^)* (sin <f) sin yfr)-
+ g"^ (sin ^ cos i/r)" (cos ^J
+ A" (cos i^)= (sin ^ cos r/r)!
— a' (cos ^)2
— ¥ (sin (^ cos i/ry
— c' (sin (^ sin ■v^)^
/S2= /"(sin 17 cos 0)2 (sin »; sin ^)»
+ 5f=' (sin 1) sin 0)= (cos t;)^
+ h? (cos 17)" (sin 17 cos 0)2
— a' (cos i)f
— 6' (sin 1) cos 0)-
— c" (sin 17 sin 0)=,
. sin «^ (^^ d'</r _ sin 77 di? d0
R 8 •
then we have
And it may be added that the integral equations are, so to speak, a complete
integral of the differential relation; viz. in virtue of the identity af->rhg-\-ch = <i, the
differential relation contains really only four constants ; the integral relations contain
the six constants a : a' : a" : a'" and y8 : /3' : /3" : y9"', or we have two constants
introduced by the integration.
The best form of statement is, in the first theorem, to write x, y for cos 77, sin77, {a?+y^= 1),
f, 77 for COS0, sin0, (^- + 772=1), and similarly in the second theorem to introduce the
variables x, y, z connected by a^ + y^ + z- = \, and ^, 77, f connected by ^ + 7]^ + i^'^=\ ;
then in the first theorem ^77, d9 represent elements of circular arc, and in the second
theorem sin ^ d<^ dyjr and sin 77 ^17 dd represent elements of spherical surface, and the
theorems are:
I. If {z, y) are coordinates of a point on the circle a?-\-y'^ = \, and (f, 77) coordinates
of a point on the circle ^■ + rf = \, and if ds, dcr are the corresponding circular elements,
then
da da
s/{ouc'+by^-c) •s/{a^ + bv^-c)'
has for its complete integral
ax^ + byt) — c = 0.
32—2
252 ON JACOBl'S TRANSFORMATION OF A DOUBLE INTEGRAL. [596
II. If (x, y, z) are coordinates of a point on the sphere a;' + y° + 2' = l, and
(f> Vt ?) coordinates of a point on the sphere f + »?' + ?' = l; and if ds, da are the
corresponding spherical elements, and
/37'-/9'7=/> aSf-a'B = a,
7a' - 7'a = g, /8S' - /S'S = 6,
a/3'-a'y3=A, yS'-y'B=c,
(whence af+bg + ch = 0);
and for shortness
then the differential relation
ds _ da-
has for its complete integral the system
ax^ + ^yrj + yz^+B =0,
a'x^ + ^yv + yz^+B' = 0,
where by complete integral is meant a system of two equations containing two arbitrary
constants.
597] 253
597.
ON A DIFFERENTIAL EQUATION IN THE THEORY OF ELLIPTIC
FUNCTIONS.
[From the Messenger of Mathematics, vol. iv. (1875), pp. 110 — 113.]
The diiferential equation
(2._Q(,H.1)_3 = 3(1-Z.=)f,
considered ante, p. 69, [594, this volume, p. 244], belongs to a class of equations trans-
formable into linear equations of the second order, and consequently is such that,
knowing a particular solution, we can obtain the general solution.
In fact, assuming
the equation becomes
= 3(I-J.){3(l-«=.)i^+64|-3(l-i.)ig},
1 /dz\'
viz. omitting the terms in -j ( tt ) which destroy each other, and dividing by 3(1- k^).
this is
1 (P£
or finally
^^^ "Ut^ k dk i-/fc=^-"-
254 ON A DIFFERENTIAL EQUATION [597
But knowing a particular value of Q we have
. = exp.|-ijj^},
a particular value of z, and thence in the ordinary manner the general value of z,
giving the general value of Q.
The solution given in my former paper may be exhibited in a more simple form
by introducing, instea(i of k, the variable a connected with it by the equation
/fc2 = --^ — --J. We have in fact, Fundamenta Nova, p. 25, [Jacobi's Qes. Werke, t. I.,
p. 76],
8 3 2 + a ,.,
■2 + a
viz. these expressions of u, v in terms of the parameter a, are equivalent to, and
replace, the modular equation m* — ir" + 2uv (1 — mV) = 0. We thence obtain
(l + 2a)^ ' u^ a?{\ + 2af'
that is.
uv
~^^°'V(l^)' u'~W)\\T+2a)'
Q'-Q
7^
and the particular solution, Q = — , + 2mj;, becomes
Introducing into the differential equation a in place of k, this is found to be
But from this form it at once appears that it is convenient in place of a to introduce
the new variable /3, = a + - ; the equation thus becomes
satisfied by Q = V(5 + 2/3) ; or, what is the same thing, writing 5 + 2/3 = 7", that is,
/8 = — ^ + 7^, the equation becomes
4Q« + ^(3 + 67-^-7*)-12=-('/-l)('/-9)^.
satisfied by Q = 7.
597] IN THE THEORY OF ELLIPTIC FUNCTIONS. 255
Writing here
Q = H7-1)(7-9)J|.
we have for z the equation
satisfied by
, 7' - 9\i
=(^D-
[In fact, this value gives
^=(y-9)i(y-l)-i,
^ = (-12y + 57y + 36)(r-9)-i(7=-l)-*
which verify the equation as they should do.]
Representing for a moment the differential equation \)yA^-\-B^-\-Cz = () and
d'f ay '
putting ^1= y^Tzr-^) > then assuming z = z-^ ly^7> we find
that is,
VIZ.
or
whence, integrating
that is,
\ dy 2 dzi B ^
—-\ — — H — =0
y d"^ Zi d'f A '
1 dy 2d2, .sy- 14^ + 3
y &Y'^z,dy'^{rf-l)(rf-9)-^'
y drf Zj dy y — 1 7^ — 9
logFa' + 37-ilog^-§log^=0,
y = e
3\«
V
zi'Ky- 1/ V7-3/'
V7-3.7+3/ W-1/ W-3/
(7+1X7+ ay
■ (7-3)'
256 A DIFFERENTIAL EQUATION IN THE THEORY OF ELLIPTIC FUNCTIONS. [597
Hence, the general value of « is
the constants of integration being K and 70, or, what is the same thing,
the corresponding value of Q being
Q = i(7^-l)(7-9)i|,
which contains the single arbitrary constant ^ ; when this vanishes, we have the fore-
going particular solution Q = y.
I recall that the expression of 7 is
7 = V(5 + 2y3), =^{5 + 2(a+i)}, = J-^ V{(2 + a) (1 + 2a)},
where a is connected with k by the relation
a°(2 + a)
" ~ l+2a ■
598] 257
598.
NOTE ON A PROCESS OF INTEGRATION.
[From the Messenger of Mathematics, vol. iv. (1875), pp. 149, 150.]
I HAD occasion to consider the integral
r-g r^-' dr
H Jo {r» + e«)i'+«'
where e is small in regard to R and q is negative. The integral is finite when e = 0,
and it might be imagined that it could be expanded in positive powers of e ; and,
assuming it to be thus expansible, that the process would simply be to expand under
the integral sign in ascending powers of e, and integrate each term separately, so that
the series would be in integer powers of e".
Take two particular cases. First, let
8 = 2, ? = -f;
the integral is
rB fR
I r -Jir" + d") dr = \ dr{r' + ^eV -^e^r-- -{■ ...)
Jo Jo
viz. the integral is not thus obtainable : the series is right as far as it goes, but the
true expansion contains a term in e'; and the failure of the series to give the true
expansion is indicated by the appearance of infinite coefficients. In fact, the indefinite
integral is §(r' + e')*; taking this between the limits, it is
Again, let s = 1, q== — 2; the integral is
I (f + ^)*dr=\ (r»+feV+|e^r-'+..,)
J 0 J 0
c. IX. 38
258 NOTE ON A PROCESS OF INTEGRATION. [598
viz. the integral is not thus obtainable : the series is right as far as it goes, but the
true expansion contains a term as e*loge, and the failure is indicated by the infinite
coefficients. In fact, the indefinite integral is
(ir* + |eV) V(^ + C) + fe* log {r + V(r' + e»)},
which between the limits is
(ii? + f e^i^) V(ii^ + 6») + f e* log ^^±^^^f^±^\
0
= ifl*+fe='i?'+ ... -fe'loge.
In the general case, the term causing the failure is Ke~^ when q is fi-actional, and
Ke~^\oge when q is integral. As a step towards determining the entire expansion, I
notice that, writing a; = — — — or r = eari{l—x)i, the value of the integral is
= ^er^j afl-^ (1 - «;)*»-• dx.
where
Y- ^
599]
259
599.
A SMITH'S PRIZE DISSERTATION.
[From the Messenger of Mathematics, vol. iv. (1875), pp. 157 — 160.]
Wbite a dissertation on Bernoulli s Numbers and their use in Analysis.
The function -^ — :. +^^ is an even function of t, as appears by expressing it in
the form
y + l _ ^ ^ e*' + e-i'
and its value for f = 0 being obviously =1, we may write
^j+i* = l + Aj^2-5,j-/3-^+&c.;
or, what is the same thing,
where the several coeflScients B^, B„ B,, &c., are, as is at once seen, rational fractions,
and, as it may be shown, are all of them positive. These numerical coefficients
Bi, B^, JBj, &c., are called Bernoulli's numbers.
There is no difficulty in calculating directly the first few terms; viz. we have
+ tHi + it + ^t'+...)
=i+t{-i)+tH-i+i =+^)+t^-^j+i-^=o)
VIZ.
33—2
260 A smith's prize dissertation. [599
which is therefore
and consequently
and so a few more terms might have been found.
But a more convenient method is to express the numbers in terms of the
differences of 0™ by means of a general formula for the expansion of a function of e*,
viz. this is
<^(e«) = <^(l + A)e'-»,
where
and the ^ (1 + A) is to be applied to the terms 0°, 0', 0', 0', &c. We have thus
t ^log(e')
e*-l e«-l
log(l + A)
~ A ^
_log(l + A)[ t t' t-^-^ f- I
We have, as may be at once verified,
log(l+^) log(l + A)
A " -^' A " - 5'
and by what precedes, since the coefficient of every higher odd power of t vanishes,
log(l±A) _
and then, by comparing the even powers of t,
that is.
(_)«-.5„ = ^M(i±A)o«_
(-)"-■ ^„= (1 - i A + i A-^ ... + 2^ A=») 0-
the series for " ^ being stopped at this point since A'"+'0*" = 0, &c. For instance,
in the case w = l, we have
5, = (1-^A + JA=)0== 0'^
-i(i'-o»)
= — I + §, = ^ as above.
599]
A SMITHS PRIZE DISSERTATION.
261
The formula shows, not only that 5„ is a rational fraction but that its denominator
is at most = least common multiple of the numbers 2, 3, . . . , 2n + 1 ; the actual
denominator of the fraction in its least terras is, however, much less than this, there
being as to its value a theorem known as Staudt's theorem. It does not obviously show
that the Numbers are positive, or afford any indication of the rate of increase of the
successive terms of the series.
These last requirements are satisfied by an expression for £„ as the sum of an
infinite numerical series, which expression is obtained by means of the function cot 6,
as follows :
We have
gje ^ g-it
g^^j + H -i*eit_e-J«-l+-^'i.2 "'n.2.3.4
B,
+ &C.,
or, writing herein t = 2i0 {i = \/(— 1) as usual), this is
2'^
2*0*
But we have
^cot^ = l-5,j-2-£.j-2-3-4-
&c.
log8ind = log^ + log(l--) + log(l-2^) + ...,
and thence, by difiPerentiation,
+ ..1
0 cot 5 = 1 -
26^
0 \
Hence
that is.
B„
2d*
•K*
%
1
+ 2i +
&c.
- &c.
2»n
2 (
I^-
1
2'"
....]
1
.2...
2w
}
2(1.2.. 2n)/l 1 \
" (27rr Vl"* 2"'^"7'
showing first, that £„ is positive, and next, that it rapidly increases with w, viz. n being
large, we have
2(1.2...2n)
(27rf
5„ =
or, inBtead of 1.2...2« writing its approximate value v'(27r) . (2n)™+4e~"', this is
5„ = 4V(n7r)(jJ
262 A smith's prize dissertation. [599
The result may of course be considered from the opposite point of view, as giving a
determination of the sum n^ + oST"'"^* ••• ^° terms of Bernoulli's Numbers, assumed
to be known, viz. we thus have
l«,-i-2«.^-" 2(1.2...2n) "■
For instance, n = l,
l+l^ =i27ry_ ^7r«
l»-^2» 2.1.2'*' 6'
and this is one and a good instance of the use of Bernoulli's Numbers in Analysis.
Another and very important one is in the summation of a series, or say in the
detennination of Smx, =W(, + t<j + ... +Mi-i ; viz. starting from
e«-l t ^ ' 1.2 1.2.-3.4
and writing herein t = dx, and therefore
111
= — or S,
e'_i e^-l' A
and applying each side to a function % of x, we have
r 7? • 7?
or taking the two sides each between the integer limits a, x,
dxU:, - i K - Ua) + y\ (<^xM*f - l~2\~^ (a^*""*)' + • • • .
where if Ux is a rational and integral function the series on the right-hand side is
finite. If for instance % = x, the equation is
a + {a+l)... + {x-\) = ii{af-a')-\{x-a),
viz.
{1 + 2 ... +(a;-l)l -11+2 ... +{a-l)]=^{a? -n)-^{a?-a),
which is right.
Applying the formula to the function log x, we deduce theorems as to the F-function ;
and it is also interesting to apply it to - .
OS
The above is given as a specimen of what might be expected in an examination :
I remark as faults the omission to make it clear that B^ is a rational fraction ; and
the giving the series-formula as a formula for the convenient calculation of 5„. The
omission to give the first-mentioned straightforward process of development strikes me
as curious.
600] 263
600.
THEOREM ON THE nth ROOTS OF UNITY.
[From the Messenger of Mathematics, vol. iv. (1875), p. 171.]
If n be an odd prime, and a an imaginary nth root of unity, then
for instance,
verified at once by means of the equation 1 + o + a^ = 0 :
„=5, 4 = 4(j-^,+ j|^),
where the term in ( ) is
a(H-g*) + a'(l 4-a')
(1 + a=) (1 + a*) '
that is,
a + 1 + a' + a*
l+a'+a^ + a'
and so in other cases.
= 1:
264 [601
601.
NOTE ON THE CASSINIAN.
[From the Messenger of Mathematics, vol. iv. (1875), pp. 187, 188.]
A Symmetrical bicircular quai-tic has in general on the axis two nodofoci and four
ordinary foci; viz. joining a nodofocus with either of the circular points at infinity,
the joining line is a tangent to the curve at the circular point (and, this being a
node of the curve, the tangent has there a three-pointic intersection) : and joining an
ordinary focus with either of the circular points at infinity, the joining line is at some
other point a tangent to the curve, viz. an ordinary tangent of two-pointic intersection.
In the case of the Cassinian, each circular point at infinity is a fleflecnode (node with
an inflexion on each branch); of the four ordinary foci on the axis, one coincides with
one nodofocus, another with the other nodofocus, and there remain only two ordinary
foci on the axis ; the so-called foci of the Cassinian are in fact the nodofoci, viz. each
of these points is by what precedes a nodofocus plus an ordinary focus, and the line
from either of these points to a circulai- point at infinity, qud, tangent at a fleflecnode,
has there a four-pointic intersection with the curve.
The analytical proof is very easy ; writing the equation under the homogeneous form
{{x - azj + y'} {{x + azf + if\ -(^z*=0,
then the so-called foci are the points (x =:az, y = 0), {x = — az, y = 0); at either of
these, say the first of them, the line drawn to one of the circular points at infinity
is x = az + iy, and substituting this value in the equation of the curve we obtain
z^ = 0, viz. the line is a tangent of four-pointic intersection ; this implies that there
is an inflexion at the point of contact on the branch touched by the line x = az + iy ;
and there is similarly an inflexion at the point of contact on the branch touched by
the line x = — az + iy; viz. the circular point x = iy, z=0 is a fleflecnode; and similarly
the circular point x=—iy, z = 0, is also a fleflecnode.
601] NOTE ON THE CASSINIAN. 265
To verify that there are on the axis only two ordinary foci, we write in the
equation x = az + iy, and determine a by the condition that the resulting equation for
y (which equation, by reason that the circular point z=Q, x = iy, is a node, will be
a quadric equation only) shall have two equal roots; the equation is in fact
{(a - afz^ + 2 (a - a) iyz] {(a + af ^= - 2 (a + a) iyz\ - c=2^ = 0,
viz. throwing out the factor z^, this is
(a' - a») {(a -a)z + 2iy] {(a + a)z + 2iy] - c*z" = 0,
or, what is the same thing, it is
(a? - a") {{az + 2iyy - a'^z-] - c'z'' = 0,
viz. it is
{2iy + azy-(a^ + ^^z^- = 0.
The condition in order that this may have equal roots is
hence a has only the two values iA/la' -J, viz. there are only two ordinary foci.
c. rx. 34
266 [602
602.
ON THE POTENTIALS OF POLYGONS AND POLYHEDRA.
[From the Proceedings of the London Mathematical Society, vol. vi. (1874 — 1875),
pp. 20—34. Read December 10, 1874.]
The problem of the attraction of polyhedra is treated of by Mehler, Crelle, t. LXVi.
pp. 375 — 381 (1866) ; but the results here obtained are exhibited under forms, which
are very different from his and which give rise to further developments of the theory.
General Formtdce for the Potentials of a Cone and a Shell.
1. The law of attraction is taken to be according to the inverse square of the
distance ; and I commence \vith the general case of a cone standing upon any portion
of a surface S as its base, and attracting a point at its vertex, the cone being con-
sidered as a mass of density unity.
2. Considering, in the first instance, an element of mass, the position of which
is determined by its distance r from the vertex (or origin) and by two angulai'
coordinates defining the position of the radius vector r, then the element is =r^dr da
(where dw is the element of solid angle, or surface of the unit-sphere), and the con-e-
sponding element of potential is - r' dr dco, =rdr dm ; whence
V = irdr dco,
which, integrating from ?• = 0 to r= its value at the surface, is
= ^jr^da>,
where r now denotes the radius vector at a point of the surface, being, therefore, a
given function of the two angular coordinates : and the remaining (double) integration
602] ON THE POTENTIALS OF POLYGONS AND POLYHEDRA. 267
is to be extended to all values of the angular coordinates belonging to a position of
r within the conical surface which is the other boundary of the attracting mass, or
say over the spherical aperture of the cone.
3. If the value of the radius vector at the surface is taken to be mr (m a
constant), then we have obviously
'=WJ
r'da
and hence also, writing m + dm instead of m, we obtain, for the potential of the
portion of the shell Ipng between the similar and similarly situated surfaces 2 and 2',
belonging to the parameters m and m + dm respectively, the value
V= m dm \ r^do) ;
this is = 2 — into the potential of the cone ; and we thus see that it is the same
m ^
problem to determine the potential of the cone, and that of the subtended portion of
the indefinitely thin shell included between the two surfaces.
4. The same result may be arrived at as follows: the element of solid angle day
determines on the surface an element of surface dS, and if dv be the corresponding
normal thickness of the shell, then the element of mass is = dv dS, and the element
of potential is = — dc rfS (mr being, as before, the radius vector at the surface).
Take a the complement of the inclination of the radius vector to the tangent plane —
that is, a the inclination of the radius vector to the normal, or, what is the same
thing, to the perpendicular from the origin on the tangent plane (whence, also, if mp
be the length of this perpendicular, then p=r cos a). The shell-thickness in the direction
of the radius vector is = r dm, or we have dv = r dm cos a ; the element of potential
is therefore = — cos a d2. But d(o being the spherical aperture of the cone subtending
the element dS, the perpendicular section at the distance vir is = mhMco ; we have
therefore dS = mV da> ; and hence the element of potential is = m din . ?-^ do), or
cos a ^
the potential of the subtended portion of the shell is as before, = m dm I r^ dm.
5. It may be added that, integrating between the values m, n (m > n), we obtain
\ {vv' — 71^)17^ day for the potential of the shell-portion included between the surfaces
mr, nr; and if n = 0, then, as before, the potential of the cone is =^m''lr^da).
34—2
268 ON THE POTENTIALS OF POLYGONS AND POLYHEDRA. [602
Cone on a plane base, and plane figure,
6. Suppose that the surface 2 is a plane; the surface %' is, of course, a parallel
plane. Taking here mp for the perpendicular distance of the plane 2 from the origin,
then, if 8 be the infinitesimal distance of the two planes from each other, we have
B = p dm, that is, dm = - ; the potential of the cone is, as before, = ^m" I r" da, and
that of the plane figure, thickness B, is = — I r'do).
7. Taking, for greater convenience, m = 1, we have
Potential of cone = i / ^ dm,
B
B f
Do. of plane figure =- j r' da,
where p is now the perpendicular distance of the plane from the vertex ; or if, as
regards the plane figure, the infinitesimal thickness B is taken as unity, then
Potential of plane figure = - I r^ da.
In each case r is the value of the i-adius vector corresponding to a point of the plane
figure which is the base of the cone, and the integration extends over the spherical
aperture of the cone.
8. If the position of the radius vector is determined by the usual angular
coordinates, 0 its inclination to the axis of z, and <f> its azimuth from the plane of
zx — viz. if we have
x = rsmff cos <f),
y = r sin ^ sin (j>,
z =rcos0;
then, as is well-known, da = sin 0 d0 d<f>, and the integral I r' da is = I r* sin ^ d^ d<}>.
Taking the inclination of p to the axes to be a, /3, 7 respectively, the equation
of the plane which is the base of the cone is
X cos a + y cos ^ + z cos 7 = p ;
viz. we have
r [(cos a cos 0 + cos /3 sin <^) sin ^ + cos 7 cos d]=p;
that is,
P
r =
(cos a cos <^ + cos /9 sin <f)) sin 0 + cos 7 cos 0 '
602] ON THE POTENTIALS OF POLYGONS AND POLYHEDKA. 269
and the integral i r^da is therefore
jf sin Oddd<f>
^ J [(cos a cos <j) + cos /3 sin (p) sin 6 + cos 7 cos 6]^ '
and, in particular, if p coincide with the axis of 2, so that the equation of the plane
is z = p, then the integral is
_^^^smeded<i)
cos'^ 6
-f\'-
9. The integi-ation in regard to 6 can be at once performed ; viz. in the latter
case we have I — ^5" = ^®^^! ^'^^ ^^ the former case, writing, as we may do,
(cos a cos ^ + cos yS sin ^) sin ^ + cos 7 cos ^ = if cos {6 — iV),
then
/• sin Ode ^ ^^ rsin(d-N + N)de
j [(cos a cos ^ + cos y8 sin ^) sin ^ + cos 7 cos ^j- i/^/ cos''(^ — iV)
1 r .^ f sin (d-N')dd . --/• d0 1
= W r' '^ I cosM^-iV^) + '"^ ^i cos(^-iyr)J
1 H
= ^ [cos JVsec (^ - iV) + sin iVlog tan {i-n- + i (^ - N)}].
Case of a Polyhedron or a Polygon.
10. Consider now the pyramid, vertex the origin 0, standing on a polygonal base.
Letting fall from the vertex a perpendicular OM on the base of the pyramid, and
drawing planes through OM and the several vertices of the polygon, we thus divide
the pyramid into triangular pyramids ; viz. AB being any side of the polygon, a com-
ponent pyramid (or tetrahedron) will be OMAB, vertex 0 and base MAB, where MO
is a perpendicular at M to the triangular base MAB. And drawing through MO a
plane at right angles to AB, meeting it in D (viz. MD is the perpendicular from M
on the base .4J^ of the triangle), we divide the triangular pyramid into two pyramids
OMAD, OMBD, each having for its base a right-angled triangle ; viz. the vertex is 0,
the base is the triangle ADM (or, as the case may be, BDM) right-angled at D, and
OM is a perpendicular at the vertex M to the plane of the triangle. It is to be
observed that, in speaking of the original p3Tamid as thus divided, we mean that the
pyramid is the sum of the component pyramids taken each with the proper sign, -I- or — ,
as the case may be.
11. In the case of a polyhedron, this is in the like sense divisible into pyramids
having for the common vertex the origin or point 0, and standing on the several
faces respectively ; hence the polyhedron i.s ultimately divisible into triangular pyramids
such as OADM, where ADM is a triangle right-angled at D, and where OM is a
perpendicular at M to the plane of the triangle. Hence the potential of the polyhedron
270
ON THE POTENTIALS OF POLYGONS AND POLYHEDRA.
[602
in regard to the point 0 depends upon that of the pyramid OADM ; and (what is the
same thing) the potential of any plane polygon in regard to the point 0 depends upon
that of the right-angled triangle ADM, situate as above in regard to the point 0. I
y
take OM = h, MD =/, DA = g ; viz. supposing, as we may do, that the plane of the
triangle is parallel to that of xy, the point M on the axis of z, and the side MD
parallel to the axis of x, then /, g, h will be the coordinates of the point A.
Formuke for component triangular Pyramid, and Triangle.
12. Writing, as above, a; = r sin ^cos <(>, y =r sin 6 sin <^, ^ = rcos d, and observing that
h is the perpendicular distance originally called p, we have, for the potential of the
pyramid,
= ^h'jd<t> (sec 6),
where, <f) being regarded as a given angle, the integral expression sec 0 must be taken
from ^ = 0 to the value of 9 corresponding to a point in the side AD. For any
f
such point we have /=rsin^cos^, h = r cos 6, that is, 4 = tan ^ cos ^, or the required
f
value of ^ is = tan~' i • > ^°d consequently that of sec 6 is
^/
l+Ii^^-Ti. =j-^— iV/» + A'cos'^.
h^cos'<f> hcos<f> •' ^
or, as this may also be written,
hence
= i V/» + /*«+/> tan' <^;
602] ON THE POTENTIALS OF POLYGONS AND POLYHEDRA. 271
13. The first term of the integi-al, writing therein for a moment tan (^ = «, is
[ if + h' +fx>) dx
j(l+ar')\/7>TF+7V'
Hence, replacing x by its value, we have
V=\h\^ tan-' ^^^ _^ ^,*^^t tan' ./. "^-^^"^ ^"^'^^^ "^ "^ '^•^'' "^ ^'' "^-^^ ^""^ '^^ " ^"^1 '
to be taken from 0 = 0 to the value of <^ corresponding to the point A ; viz. we
have here /= r sin Q cos <^, ^ = r sin ^ sin <f>, h = r cos ^, and thence tan <^ = ? or y tan (f>=g',
whence, writing for shortness, s = '//^+g^+h^ (viz. s denotes the distance OA), we have
F = iA |a tan-> ^ +/log /"^^ - A tan-' fl ;
or, observing that
8 + 9^ (g + g)'
« - 5' V/^ + A» '
this is
F=iA|Atan-'^ + i/log*i^-Atan-'4,
for the potential of the pjTamid OMDA in regard to the point 0 ; by omitting the
factor J A, we have
V=h tan-' ^ + i/log ^-ii' - A tan"' 2
for the potential of the triangle MDA. The expression tan-' denotes, here and else-
where, an arc included b
as the tangent is + or — ,
where, an arc included between the limits — „ , + o '• it is therefore + or — according
Formula for rectangular Pyramid, and Rectangle.
14. Completing the rectangle MDAE, the potential of the triangle AME is
obtained by interchanging the letters g and /; viz. we have
7= A tan-' -^ + iff log ^^- A tan-'^
gs '^ ^8-f g
for the potential of the triangle ME A.
272 ON THE POTENTIALS OF POLYGONS AND POLYHEDRA. [602
The sum of the two gives the potential of the rectangle MDAE\ viz. for this
rectangle, we have
But we have
tan-$ + tan-^+taa-^ = ^;
js gs hs 2
for the function on the left hand is
, fs as hs
1
s^ s^ s'
viz. the denominator being 1 —-^ ^ , = 0, the tangent of the arc is oo , and the
component arcs being each positive and less than -x , the arc in question can only be
= „- . We have consequently
F=-Atan-^ + i/log^^ + i5rlogf^^
for the potential of the rectangle MDAE. And, multiplying this by ^h, we have
for the potential of the rectangular pyramid, vertex 0 and base MDAE.
Formula for the Cuboid.
15. Completing the rectangular parallelepiped, or, say for shortness, the "cuboid,"
the sides whereof are (/, g, h) ; this breaks up into three pyramids, standing on the
rectangles fg, gh, and hf respectively; and the potentials for the last two pyramids
are at once obtained from the last-mentioned expression of V by mere cyclical inter-
changes of the letters. Adding the three expressions, we obtain
1^= i^/^ log j^+ i A/ log ^"^ + i/^ log '^; - i/nan-^ - isr' tan-' ^{- ^A' tan-'l
for the potential of the cuboid.
Group of Results, for Point, Line, Rectangle, and Cuboid.
16. It is convenient to prefix two results, that for the potential of the point A
(mass taken to be unity), and that for the potential of the line AE (density taken to
602] ON THE POTENTIALS OF POLYGONS AND POLYHEDRA, 273
be unity, or mass of an element of length dw, taken to be =dx). We have, the
attracted point being always at 0,
Potential of point A =-, (s = V/^ + g^+h\ as before),
s + f
Potential of line AE = i log -^.,
Potential of rectangle MDAE = y log *— 4-+ if log ^-^ - h tan"'-^ ,
S — J s — o its
Potential of cuboid =igh log ^v.+ i A/log ^ + i/^f log — r
S — T S ~~ Qf o ft
-i^'tan-f-i^nan-^-iAHan-l,
which functions may be called A (f, g, h), B{f, g, h), C(f, g, h), and D{f, g, h)
respectively. It is to be observed that f, g, h are taken to be each of them positive,
and that s denotes in every case the positive value of ^f^ + g^+h?; for a symmetrically
situated body, corresponding to negative values of each or any of these quantities, the
potential has in each case its original value, without change of sign. But 5 is an odd
function as regards /, C an odd function as regards f or g, D an odd function as regards
/, g, or h; for example, C {—/, g, ± h) and C (/, —g, ±h) are each = — C{f, g, h), and
therefore of course C (— /, —g, ±k) = C (/, g, h).
Extension to case where the attracted point has an arbitrary position.
17. The attracted point has thus far been considered as in a definite position in
regard to the attracting mass ; but it is easy to pass to the general case of any
relative position whatever. Thus, for a line AB, if M be the foot of the perpendicular
let fall from the point 0, and if, to fix the ideas, the order of succession of the three
points is A, B, M, then, with respect to the point 0,
Une AB = line AM - line BM.
A B M
Taking the y- and ^^-coordinates to be b, c, the ^-coordinates for the points A, B, M to be
x„, Xi, a respectively, and in the figure a>Xi, x,>Xt„ then a—x^, a — Xi are each of
them positive, a — x„ being the greater, the potential of the line AM is = B{a — x„, b, c),
that of BM is =B{a — Xj, b, c), and the potential of the whole line is
= B(a — Xa, b, c) — B(a — Xj, b, c);
viz. this formula is proved for the case where M is situate as in the figure. But
supposing that A and B retain their relative position (viz. Xi > «„), then the formula
holds good for any other position of M; thus, if M be between the points A, B —
viz. if the order ia A, M, B — then
line AB = line AM + line BM,
c. IX. 85
274 ON THE POTENTIALS OF POLYGONS AND POLYHEDRA. [602
and potential is
= B(a-Xo, b, c) + B(xi — a, b, c),
where the second term is = — £(o — a;,, b, c); and so, if the order is M, A, B, then
line AB — line BM — line AM,
and the potential is B{xi — a, b, c) — B {x„ — a, b, c), which is
= — B{a — Xi, b, c)+B{a — x„, b, c).
18. Similarly for a rectangle ABCD, if M, the foot of the perpendicular from the
point 0, has the position shown in the figure, then
rectangle AD— rectangle MG
— rectangle MA
— rectangle MD
+ rectangle MB,
M
B
C D
where 0 is a point on the perpendicular at the common vertex M of the four rectangles;
and the resulting expression for the rectangle AD will apply to any position of the
point M.
19. And in like manner for a cuboid; taking the point 0 in any determinate
position, the cuboid may be decomposed into eight cuboids (each with the sign + or —
as the case may be) having the point 0 for a common vertex; and the resulting
expression for the potential will apply to any position whatever of the point M.
20. The results may be collected and exhibited as follows: — the coordinates of
the attracted point are a, b, c; and it is assumed that a;j>a!o, yi>yo> Zi>^(» (viz. for
X the order is + oo , Xi, x^, — oo , and so for y and z respectively).
Potential of point {x, y, z) is = A(a — x , b — y , c — z);
Potential of line («„ y, z), (x„, y, z) is = B {a — x^, b — y , c — z)
-B(a-Xu b-y , c- z);
Potential of rectangle (x^, y^, z), (x^, y,, z) is = G (a — x^, b — y„, c—z)
(a^, yo, z), (a^o. yo> ^) -(7(a-a;„, t-y,, c-z)
— G (a — x-i, b — yo, c — z)
+ G(a-Xu b-yi, c-z);
602] ON THE POTENTIALS OF POLYGONS AND POLYHEDRA. 275
Potential of cuboid (x^, y,, z^, («„, y,, ^i) is = D(a — Xo, h — y„, c — Zo)
(a!i, yo. «i), (a?o. yo, .Si) -D{a-a^, b-y^, c-z„)
(ai, yu Zo), (a^o, yi. 2^o) - D (a - «„, 6 - yi, c - ^o)
(a?!, yo, .^o), («o. yo, -^o) +D{a-xi, 6-yi, c-^o)
-i)(a-a;„, 6 - yo, c-Zi)
+ D{a-Xi, b-yo, c-Zi)
+ D{a-x^, b-yu c-0,)
-D{a-x-„ b-yu c-Zi).
21. These are connected together as follows, viz. : —
Potential of line = I dx Potential of point,
J Xo
fVi
Potential of rectangle = I dy Potential of line,
•'Wo
Potential of cuboid = I dz Potential of rectangle,
equations which are in fact of the form
B {x, y, z)=ldxA (x, y, z),
C (x, y, z)=jdyB (x, y, z),
D (x, y, z)=jdzC (x, y, z).
Differential properties of the fwnctions A, B, C, D.
22. These relations, with other allied ones, may be verified as follows. Writing
r = 'Jo? + ^+1^, the fundamental forms are
. r -k- X J ^ ,yz
log , and tan~' ^- .
" r — X rx
We have dxr = - , &c., and thence
r
^ X -, X
, , r+x r r 1 1 2
°r — x r + x r — x r r r
y y
^ ° r — X r + x r — x' r\r + x r — x) ' r(r^ — ai')'
&c. &c. &c.
d,tan-y? = "^^^''"'^-^ = "y" ^' + '^ ,
* rx r^x' + y'z' r 1^3? + f^'
35—2
276 ON THE POTENTIALS OF POLYGONS AND POLYHEDRA. [602
or, since
r» + a* = (r*-y') + ('^-'2') and T^ar" + y»^> = (r» - y') (r" - ^»),
this is
r [r' - y' r* - i^J '
xy
, ^ ,yz '' r xz i^ — ifl
" rx r^x' + y^z" r r»a? + i/»^'
which, the denominator being, as before, (r* — y') (r' — z'), is
a;^ 1
It is now easy to form the following results: —
23. First,
it = A (x, y, z) = - (symmetrical),
dzU =- :^, &c.,
and thence
24. Secondly,
then
J , 3a:» 1 , , , 3«y -
(4= + d„» + d/) w = 0.
u = B{x, y, ^) = ^log (symmetrical as to y, z);
7* '~~ £C
d^u = - (= ^ (^, y, z)). dyu = ^r^^rz^) > ^°-'
dx^ = -^' djyu = -^, dyd,u = ^ ^^/_ ^^ + ^-^r^y . fee-.
, _ — a; 2a^' a;y' „
and thence
/ 7 „ 7 „ 7 „x « 2a; 2a; x
25. Thirdly,
u = C{x, y, z) - \y log ^^-^-^ + ^a; log --^ - z tan"' ^ (symmetrical as to x, y)
dyu = ^ log^ (= ■S(a'. y. ^)) :
602] ON THE POTENTIALS OF POLYGONS AND POLYHEDRA. 277
in verification whereof, observe that the remaining terms are
_ xy^ X xz^ '>^ — y^
r\ r^-x' r^-x'J'
which is =0;
, _ xyz xyz acyz r^ — a^ + r^—y^ _^xy
^" ~~r(r^-a^)~r(r»-2/»)'*' r^ (f - a?) {r' - y'^)' ^'
= -tan-'^,
zr
r(r'-j^)'
_ ^ r^-a? + r^-y^ _ ocy / 1 1 N
' V(r'-ar')(r^-y=)' ~ V [r' - a^ r^ -y") '
and thence
26. Fourthly,
M=i)(a;. y, z)=^yz\og^^—^ + ^zx\og^ + ^xylog^;^
- i ar" tan-' — - i y" tan"' — - ^ «" tan"* ^ (symmetrical),
d,M = i y log ^— ^ + i « log ™^ - ^ tan-' ^ = ^(^' y- ^)'
d> = -tan-'^;
zr
and thence
(dx' + cL' + 40 M = - tan-' ^ - tan-' — - tan-' ^
yz zx xy xyz
, xr yr zr r^
= -tan-' ^:5 i i— ;
a? y^ z^
^2 ^ ^
viz. the denominator being = 0, the arc is + ^ , or we have
mm M_
the value being — - \i x, y, z are all three of them, or only one, positive ; but + -^
if they are all three of them, or only one, negative.
278 ON THE POTENTIALS OF POLYGONS AND POLYHEDKA. [602
Application to the Potentials of the Point, the Line, the Rectangle, and the Cuboid.
27. Take now V to denote in succession the foregoing expressions of the potential
of a point, a line, a rectangle, or a cuboid, at the point (a, 6, c). In the first three
cases respectively, each of the component terms is reduced to zero by the operator
da* + di* + dc'; and we have, therefore,
(da' + di^ + dc')V=0,
which is as it should be. But in the case of the cuboid, each of the eight com-
ponent terms is by the operator reduced to + ^ , and we have therefore
(da^ + db'' + dc')V=-Z
^(^I)
2 denoting the sum of eight terms, the ± denoting + or — , accoi-ding to the sign
of the terra in the formula (viz. in four cases this is +, and in four cases it is — ),
and the + „ denoting the value ^ with its proper sign depending on the signs of the
quantities (a— a^o. ^ — yo> c — z„), &c., as explained in the preceding Number.
Suppose for a moment «>«!, b>yi, oz^, or the attracted point in one of the
regions exterior to the cuboid ; then + ^ will in each case be = — - , and the sign
+, being + for four of the terms and — for the four remaining terms, the sum is =0.
And similarly, in all cases where the attracted point is exterior to the cuboid, the
sum of the eight terms is =0. But when the attracted point is interior, that is,
when a>Xa<Xi, h>y^<yi, c>z„<Zi, then it is found that, for the four terms which
have the sign + , the value of + -^ is = — ^ ; and for the four terms which have the
sign — , its value is = + — ; whence, in the sum, each term is = — ^ , or the value is
= — 47r. Hence, in the case of the cuboid, we have
(da" + dt^ + do') F= 0 or - iv,
according as the attracted point is external or internal.
Verification in regard to the Rectangle.
28. I start from the formula
V= G{a -Xo, b — yo, c)
-G(a-a;u b-y„, c)
-C{a-x^, b-yu c)
+ C(a-Xi, b-yi, c),
602] ON THE POTENTIALS OF POLYGONS AND POLYHEDRA. 279
where, as before, Xi >«„, y^>y^. V is here a function of (a, b, c), satisfying the partial
differential equation
and (as is easily verified) vanishing when any one of the variables a, b, c becomes
infinite; it does not become infinite for any finite values of a or b, or any positive
value of c. Hence, by a theorem of Green's *, there exists on the plane z = 0 a dis-
tribution of matter giving rise to the potential V; and not only so, but the density
at any point (x, y) of the plane is given by the formula
_ _ J_ (dW\
f~ 2ir\dcJc=o'
where W is what V becomes on writing therein x, y in place of a, b, and c = 0 is
regarded as an indefinitely small positive quantity.
We have
deC (x, y, c) = - tan-> — , where r = Vx" + t/' + c'.
And hence
d.W=^ - tan- (^-^„)(y-y,)_^
c\/{x-Xoy + iy-yoy + c^
+ tan- — (^-^.Xy-yo) —
+ tan"
— tan"
c^ix-x,y+(y-yoy + c^
(x-x,)(y-y,)
c^(x-x,y + {y-y,y + c'
, (x-xi){y-yi)
c^{x-x,y + {y-y,y + (?'
Putting c = 0, as above, each arc is = ^ or — „ . according as the fraction under
the tan~' is positive or negative — that is, according as the numerator is positive or
negative. Suppose for a moment x>Xi, y>yi, viz. the point {x, y) is here in a
region exterior to the rectangle (a;,, y^, {x^, y^, (xo, yi), {xo, y,,)- the value of dcW is
*fr *fr *ir tt
= — -n'^a + a~a' =0; and similarly, for every other position of the point (x, y)
dV dV
* The theorem in question ia a particular case of Green's, iirf>=--j .— ("Essay on the Application
of Mathematical Analysis to the Theories of Electricity and Magnetism" (1828), see p. 31 of the Collected
Works) ; viz. the surface is here a plane, and 1'= V. And it is also a particular case of the formula
»'= , P' ("Memoir on the Determination of the Exterior and Interior Attraction of Ellipsoids
of Variable Densities" (1835), see p. 199 of the Collected Works); viz. « is taken =2; and Green's extra-
spatial coordinate u then becomes the coordinate z of ordinary tri-dimensional space.
280 ON THE POTENTIALS OF POLYGONS AND POLYHEDEA. [602
exterior to the rectangle, the value is = 0. But for a point interior to the rectangle,
we have x<a>i>Xo, y<yi>yo, and in this case the value is
ir
Hence
<-lH-l)-l-'-
p, =--^(dcW)c=a, is =0 or 1,
according as the point is exterior or interior to the rectangle, viz. the distribution
producing the potential in question is a uniform distribution (density unity) over the
rectangle, which is as it should be.
Potential of a Cuboidal Surface.
29. The preceding formulae lead to the expression of the potential of a cuboidal
surface (viz. the surface composed of the six faces of a cuboid, each of them being
considered as a plate of the same uniform density) upon a point a, b, c. Writing, for
convenience,
Eif, g, h) = ^(g + h)log'^+Hh+f)hg'^+Hf+9)H{^)
-/tan-' $ - a tan- ^- h tan""-^ ,
fi gs hs'
where each term is supposed to have (compounded with its expressed sign) a sign +,
as follows: viz. in axiy fg term (i/log ~, ^gr log — -^, or A,tan-'-^), this sign +
\ s — g s —J hs/
is + if / and g are both positive or both negative, but is — if / and g are the
one of them positive and the other negative ; and the like as to the gh terms and
the hf terms respectively. And this being so, the expression for the potential (applying
as well to an interior as to an exterior point) is
F= E(a-Xo, b-yo, c-z„)
+ E(a-Xi, b-yo, c-z^)
+ E{a-x„, b-yi, c — z„)
+ E(a-Xi, b-yu c-2„)
+ E{a-Xo, b-y„, c-z,)
+ E(a-Xi, b-y„, c-Zi)
+ E(a-Xo, b-yu c-Zi)
+ E(a-Xu b-yu c-z^).
It is, in fact, easy to verify that the final result, interpreted as above, represents
the sum of the six positive values, which are the values of the potential for the six
faces of the cuboid respectively.
603] 281
603.
ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE.
[From the Proceedings of the London Mathematical Society, vol. vi. (1874 — 1875),
pp. 38 — 58. Read January 14, 1875.]
The Potential of the Ellipse.
1. I CONSIDER the potential of an ellipse (or say an elliptic plate of uniform
density); viz. this is
J 'J(a-xy + (b-yy + c!''
the limits being given by the equation i^j + ^ = 1-
J O
Writing herein a; = 7n/"cos m, y = m^ sin m, we have dxdy = fg mdmdu; and consequently
mdmdu
'=^^/v^
V(a — mf cos uf +(b — jng sin uy + d'
where the integrations are to be taken from m = 0 to m = 1, and from m = 0 to m = 27r.
2. It is to be remarked that, by first performing the integration in regard to m,
we may reduce the potential to the form j du.F, where F is an algebraic function of
cosu, sinw; and that the result so obtained, although in the general case too complex
to be manageable, is a useful one in the case f=g, where the ellipse becomes a
circle. The case of the circle will be treated of separately, but in the general case
it will be sufficient to show that the integral is of the form in question.
c, IX. 36
282 ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. [603
3. To accomplish this, writing
A = a^ + ¥ + c^,
B = q^cos u + bg sm u,
C =/» cos' u + g" sin' u,
then the integral in regard to m is
which is
f mdm *
J '^A-2£m + Gm''
= i V^ - 2Bm + Cm^ + —-^ log |f7m - 5 + VC V2-2£m+Cm'l .
Taken between the limits 0 and 1, this is
= ^(V^-25 + 0-V^)4-^log{ _^^^^^ [;
and we have therefore
F=/,/c^4(VXir2^0-VZ)+/,/d./f°-;;-^^-7)iog^
J ^ J (/^ cos' w + 5'' sm' m)»
where, for greater clearness, the value of the coefficient — ^ of the logarithmic term
has been written at full length.
4. But this coefficient admits of algebraic integration, viz. we have
. f , a/'cos M + ft^f sin M _ a^f sinw — 6/"cosm
•' (/'cos''M + 5''sin*tt)* (/'cos'M + gr^sin'w)*'
hence, integrating the second term by parts, we have
V=fgjdu ^{s/A-2B + C-'/A]
agsinu — bfcosu
(pcos^'u + g^sin^uy
logT
-/
du
ag sin u — bf cos u f
( poos' u + g' Bin' v)i ' T'
T
where the second term, taken between the limits u = 0, u = 2ir, is = 0 ; and ^ being
an algebraic function of sin u, cos u, the potential is expressed in the form in question.
5. But we may, by means of a transformation upon u (that made use of in
Gauss' Memoir* on the attraction of an elliptic ring), transform the expression so as
• [Ge». Werhe, t. in., pp. 333—355; in particular, I.e., p. 338].
603]
ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE.
283
to obtain the integral in regard to m under a much more simple form. We, in fact,
assume
o + a' cos r + a" sin T
COSM =
8mM =
7 + 7 COS Th- 7" sin 2"
/3 + y8'cosr+,8"sinr
7 +7' COS T +7" sin 2"
where the nine coefficients are such that identically
(a + a'cos2'+a"sinT)' + (/3 + /S'cos2'+/3"sinT)2-(7 + 7'cosr+7"sinr)» = cos=2'+sin''T-l,
(this of course renders the two equations consistent) ; and also that
(a - m/cos uf + Q}- mg sin «y + c= = .-— -^ ^\ „ . ^,, {G + G' co8= T+ G" sin= T).
^ •' ' ^ ^ (7 + 7 cos jf + 7 sm 2 f
This last condition gives, for the determination of the coefficients G, G', G", the identity
or, what is the same thing, G, — G', — G" are the roots of the equation
g' h' c= _ 1 = o
e^rm^f'^ d + my^ d
This equation has one positive root, which may be taken to be G, and two negative
roots, which will then be — C, — G" ; viz. G, G', G" are thus all positive ; and G
denotes the positive root of the last-mentioned equation.
6. We have
du =
and thence
dT
{G + G'coaT+G"smTf'
V=f9\mdmj^^-^^
dT
' (G + G' COS' T+G" sin' T)i'
the integral in regard to T being taken from 0 to 2v, or, what is the same thing,
we may multiply by 4 and take the integral only from 0 to ^ ; viz. we thus have
dT
V=^fgjmdmfj^^-^^r^^
'T + G"aiji'T)i'
where the integral in regard to T can be at once reduced to the standard form of
an elliptic function, or it might be calculated by Gauss' method of the arithmetico-
geometrical mean.
7. But, for the present purpose, a further reduction is required. Writing
t = G+(G + G')coV'T,
36—2
284 ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. [603
we have
sin' T
t + G'=(G + 0') ^
sin'T"
t + 0" = (G + G' cos" T+G" sin'' T)J^7r:
sin' i
whence
^t-6.t+G'.t + G" = {G + G')(G + G'cos^T+G"sm^T)i^^^;
moreover
dt = -2(G + G')^^dT.
^ ' sin' 1
Hence
di -MT
•Jt-G.t^G'.t + G" {<?+<?' cos» T + G" 8in» Tji '
It
and, observing that to the limits 0, _ of T correspond the limits oo, G of f, we
thence obtain
■'^i Jo'Jt-G.t + G'.t+G"
or, what is the same thing,
dt
"""^
iy» t + my
where G denotes, as before, the positive root of the equation
h^ r"
e + m'f" e+my e
8. Writing for f, mH, and for G, m'ff, the formula becomes
V=2fgjmdmf /' a' 6' Cx '
where (? now denotes the positive root of the equation
a* h' c» , „
603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 285
Thus G is a function of m; but it is to be remarked that the integration in respect
to m can be perfoimed through the integral sign / dt in precisely the same way as
Jo
if O were constant, and that we, in fact, have
F= 2fg [II dt ^m^ - ^-^, - ^, - f ^JJ^^^T^] .
where the function of m is to be taken between the limits 0 and 1. The reason is that,
diflferentiating this last integral in respect to m, the term depending on the variation
of the limit G is
V
o' &» c' 1 dO
""" G+f^ G + g^ G^G.G^f\G+g^dm'
which is =0 in virtue of the equation which defines G; hence the whole result is
the term arising from the variation of m in so far as it appears explicitly.
9. Proceeding next to take the function of m between the two limits : for m = 0
we have G=<X! , and the integral vanishes ; for wi = 1 we have G the positive root
of the equation
K a* ^_ ^_-. _(.
e+p'^e + g"'^ e '
or, using 0 to denote the positive root of this equation, the value is G=6; we thus
finally obtain
F.2/,/>/-4-.-r^-fvrT?7
'Jt.t+f\t + g^
as thp expression for the potential of the ellipse semiaxes (/, ^r) on the point (a, h, c).
Case where the Attracted Paint is on the Focal Hyperbola.
10. The result becomes very simple when the attracted point is in the focal
a? &
hyperbola of the ellipse, viz. when we have 6 = 0 and -j-^. 1 — '2 = !• '^^^ function
J *7 *J
a^ 6= c» . ,
1 - ^ -., - 7 ; - T 18 here
<+/' t + f t
p-f ^ f' + t t
_ t + g^ (t_'LfI\
t{t+p)V g'J-
286 ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. [603
Hence also 0 = —■ ; introducing this value, the function in question becomes
and we have
V=P'9[
' dt"^^+Uz.^ 1
-2/,/;
le t.t+/'-'
which, writing t = a^ + 0, becomes
= ^ (^d+p tan-' ^^ - ^e tan-i "^X
= 27r|:(V^Tr-V^);
or, substituting for 6 its value -~ , this is
7 = 27r(Vc» + 5i'-c),
which is, in fact, the potential of the circle a^ + y* = ^r' on the axial point (0, 0, c) ;
and, observing that the value is independent of /, we have at once the theorem that,
considering / as variable, and taking the attracted point at the constant altitude c in
the focal hyperbola ^i 5 — 'a = ^' t^® potential is the same, whatever is the value
J i/ %/
of the semi-axis major f of the ellipse.
11. A point in the focal hyperbola determines, with the ellipse, a right circular
cone having for its axis the tangent to the hyperbola; viz. the tangent in question
is equally inclined to the two lines joining the point with the foci of the hyperbola,
or with the extremities of the major axis of the ellipse. Taking d for the inclination of
the tangent to either of these lines, viz. 6 is the semi-aperture of the cone, and 7
for the inclination of the tangent to the axis of z, then it is easy to show that
t~i : cos 7
Vcos' 7 — sin' d
and we thence have
W cos» 7 - sm' 0 I
603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 287
viz. the ellipse is here considered as the section of a right cone of semi-aperture 6,
the perpendicular distance from the vertex being = c, and the inclination of this
distance to the axis of the cone being =7; and this being so, the potential is then
IT
expressed by the last preceding equation. It will be observed that, when 7 = — — ^, the
section becomes a parabola, and the potential is infinite ; for any larger value of 7,
the section is a hyperbola, and the formula ceases to be applicable.
12. 1 origrDally obtained the result by thus considering the ellipse as the section
of a right cone. Consider for a moment, in the case of any cone whatever, the plate
included between the plane, perpendicular distance from the vertex =c, and the con-
secutive parallel plane, distance =c + dc. Let dS denote an element of the first plane,
r its distance from the vertex, and r + dr the distance produced to meet the second
plane ; also let da denote the subtended solid angle. We have dl,dc = r^ dr dm, or,
dc dr . \ \ \
since — = — , we obtain d2 = - r* dco, or - d% = - r'' dm ; wherefore the potential of the
C T C T C
plane section is F = - / j^ dm, where r denotes the value at a point of the plane
section, and the integration extends over the spherical aperture of the cone.
13. Let the position of r be determined by means of its inclination 6 to the
axis of the cone, and the azimuth (/> of the plane through r and the axis of the cone ;
viz. taking the axis of the cone for the axis of z, suppose, as usual, x = r sin 6 cos ^,
y = r sm 6 Bin <l>, z = r cos d. We have then, as usual, dm = sin Q ddd^\ and if the
equation of the plane be x cos a + y cos yS -I- ^ cos 7 = 0, then the value of r is obtained
from the equation
r {(cos a cos ^ -|- cos y8 sin ^) sin 6 -f- cos 7 cos 6\ = c;
80 that we have for the potential
■^ _ f sin 0 dd d(f>
J {(cos a cos d) + cos /S sin <^) sin
{(cos a cos <]) + cos /S sin <^) sin 6 + cos 7 cos ^j'' '
where the integration is extended over the whole spherical aperture of the cone ; viz.
in the case of a right cone of semi-aperture 6, the limits are from 6 = 0 to 0=6 and
from 0 = 0 to <^ = 27r.
14. Write
(cos o cos 0 + cos /3 sin </>) sin 6 + cos 'ycos6 = M cos (6 — N),
where M, N are given functions of <t> ; then we have
^_ f d4> f sin 6 d0
j M'j COBH0-N)
and the ^-integral is
■ [sin (0-N) cos iV+ cos (0 - N) sin N]dd
P
co»'{0-N)
= cos iVsec (6-N) + sin iVlog tan {^tt + ^ (0 - iV)j,
whence
and
288 ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. [603
which between the limits is
= oos N {aec(e -N)-aec N} + 8m N {\ogta.n[\ir + ^(0 - N)]-logtm(iTr - ^N)},
0 now denoting the semi-aperture of the right cone. And we have
^ = c/ ^, |co8 i^r (_^1-^^ - ^) + sin i\r [log tan (iTT + i(^ - iyr)} - log tan (i,r - i^^^
We may mthout loss of generality write cos /3 = 0, and therefore cos a = sin 7, where 7
now is the inclination of the perpendicular on the plane to the axis of the cone. We
thus have
cos 7 cos ^ + sin 7 cos ^ sin 0 = Jlf cos (0 — N),
that is,
cos 7 = if cos N,
sin 7 cos (j) = M sin N ;
tan N — tan 7 cos <f> or N = tan~' (tan 7 cos </>),
if2 = cos'' 7 + sin' 7 cos' <^ = 1 - sin' 7 sin' <j>,
cosN _ 1
cos (iV — 0) cos ^ + sin 5 tan 7 cos <f> '
15. We have, therefore,
V=o( J^^-( -^ 1]
J 1— sin' 7 sin' (j) \cos ^ + sin ^ tan 7 cos <f> I
■^ ° /(l-6b'7s5^* ^^°^ *^° [i'r + i^ - i tan-' (tan 7 cos ^)]
— log tan Wit — J tan"' (tan 7 cos (^)]}.
But
f d<^ cos ^ _ sin <^
^ (1 - sin' 7 sin' ^)* (1 - sin' 7 sin' ^)i '
hence the second line is
'^ ^"^^ ^ (l-sin'°7 8in''^)* ^'°^ *^° ^^"^ + i^ " i *an~' (t^n 7 cos (^)]
— log tan [J7r — ^ tan"* (tan 7 cos <f>)]}
-c^y jd<l> (i_3i^?^tin'<^)i ^ !^^S ^'^^ t*'" + *^ - * *^" ('^^ 'y "''' '^^^
— log tan [Jtt — J tan-' (tan 7 cos <f>)]\.
But, restoring for a moment N in place of tan-' (tan 7 cos <f>), we have
d , ,, 1/1 »7\ dN 1 sin 7 cos 7 sin <^ 1
^^logtan(iu + i&-i\0 = -^ cos(i\r_^)=l_sin'7sin'.^ cos(i\r-^)'
'^ 1 + /I AA _ ^^ ^ _ sin 7 cos 7 sin <^ 1
^logtanUTT-iV) -'d^^^N- -l-sin'7sin'"^H3^-
603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 289
And then, in place of j^ — ^r- i^) writing
^ cos{N-6) cosN *
( ^ iV
\cos 0 + Siin0 tan y cos <A / '
cos 7 Vl - sin" 7 sin" ^ \cos d + siin0 tan 7 cos (f>
the expression in question becomes
" ^ '^ (1 -sin' 7 sin' ^* '^''^ tan [iTT + i^ - i tan- (tan 7 cos i>)]
— log tan [^TT — ^ tan~' (tan 7 cos <^)]}
_ c f d,* sin'7sin'</> / 1 ^\
J ^ 1 — sin" 7 sin' ^ Vcos ^ + sin 0 tan 7 cos <^ / "
And we have
^ = (1 -'in'V^hi^^)^ i'"g tan [Itt + i^ - ^ tan-' (tan 7 cos ,f>)]
- log tan [i^ - i tan- (tan 7 cos ^)]} + c/d<^ (cos g + sin Ln 7 cos^ " ^ ^ '
16. The integral is here
(cos 7 (coS-^ cos 7 — sin 0 sin 7 cos <^)
cos' ^ cos' 7 — sin' 6 sin' 7 cos' (jt
d<t>
-1
= cos' 7 cos ^ I — Tfl — 5 •,/),, rx
' J cos' ^ cos' 7 — sin' 0 sin' 7 cos' <p
• a i cos (bdd> f J.
— cos 7 sm 7 sm r — -^ : „l, . „ — — —, — ( aq>
' ' ; co8'0cos'7-8in'^8in'7cos'<^ J ^
_ cos 7 i. -1 cos ^ cos 7 tan 0
V cos' 7 — sin' 6 vcos' 7 — sin' 0
cos
7 , , sin 0 sin 7 sin <i ,
-!—_ tan"' . — 9,
Vcos' 7 - sin' 0 Vcos' 7 - sin' 0
as may be immediately verified.
Hence
Tr c sin 7 sin A ,, r, 1/11. , /. .^t
V = ^ [log tan ii'T + k^ ~ k tan"' (tan 7 cos 0)J
Vl — sin' 7 sin' (^
— log tan [^TT — J tan— (tan 7 cos <^)])
c cos 7 , , cos 0 cos 7 tan 0
+ -p=4== tan— — p====4=^=r
vcos' 5 — sin' 7 vcos' p— sin' 7
C cos 7 X -1 sin 0 sin 7 sin 0
— Y~ -=• tan . . ^^
vcos' 0 — sin' 7 V cos' 0 — sin' 7
— C(^,
which is to be taken between the limits 0 and 2ir; or, what is the same thing, the
integral may be taken between the limits 0, tt, and multiplied by 2. But as ^ passes
C. IX. 37
290 ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. [603
from 0 to TT, the arc of the form tan"' (A tan <f>) passes through the values 0, -, — ^ , 0,
but the other arc of the form tan"' (5 sin 0) through the values 0, ^ , „-, 0; the first
arc gives therefore a term tt, the second arc a terra 0, and the final result is
Vvcos»7-sin'^ /
which is right.
The Potential of the Circle.
17. In the case of the circle we have g=f; the terms containing a*, 6* unite
throughout into a single term containing a* + bf, and there is obviously no loss of
generality in assuming 6 = 0, and so reducing this to a* ; viz. we take the axis of x
to pass through the projection of the attracted point, the coordinates of this point
being therefore (a, 0, c). We in fact consider the potential
=/
dxdy
V(a - a;)' + y« + c"
over the circle a^+y^ =/' ; or, writing x = m/cos 4>< V = mf sin (/>, we have dxdy =f-))idmd<f>,
and therefore
V — /"a f ^ '''" ^^
" ' ^/a^ + d' + rn^p - 2TOa/cos ^ '
the integral being taken from m = 0 to m — l, and <^ = 0 to cf) = 27r.
Writing in the general formula g =f and 6 = 0, we have
^N^-tTp-1
{t+p)^t
where 6 denotes the positive root of the equation
V= 2/»|^
or, observing that
e+p e '
<+/' t~"'\0+f' t+/')^'^[e t)
= <*-^)|(d+/.)(e+/.) + 4
t.t+p
603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 291
we have also
F=2/=f ^-^ 7=^.
.'« t{t+f^)^t+p
18. The pi'esent particular case gives rise to some interesting investigations. We
may, in the first place, complete the process of first integrating directly in regard to m.
Writing
fr_ f [[{'fi/— a cos ^) + a cos ^] din d^
~ J ((»i7- a cos 4>y + a» sin'' ^ + c^j* '
the integral in regard to m is
= f { V(m/- a cos <f>y + a' sin' ^ + c^ + a cos (^ log [mf- a cos <^4 'J(mf- a cos <^f+ a? sin'' (f+d']}
to be taken from m = 0 to m = 1 ; and we thus obtain
r= fdc^ {Va' + c»+/''-2a/cos<^ - VoM- c^
+ acos<^[log(/- acofi<j) + Va» + c* +/' - 2a/cos cf)) - log (- acos<}> + -J a? + c")]}.
Writing for shortness Va' + c* +/' — So/" cos <^ = A, the second line of this is
a sin <^ [log (/— a cos <^ + A) - log (- a cos <^ + '^a- + c")]
- f rfrf) a? sin' <& {^ £^^ , I ,
.' lA (/- a cos ^ + A) - a cos ^+ va^ + c*'
and we thus have
F= a sin ^ (log (/— a cos (^ + A) — log (— a cos ^ + Va* + c')}
a' sin" <^ (/ + A) a= sin= </>
[d0 JA-Va»
+ rf<^ -^ A - Va» + c" ^-^ '— +
A (/ — a cos ^ + A) — a cos (^ + Va' + o") "
19. We have
/+A ^ (/+A)(/-acos^-A)
A (/- a cos </) + A) " A {(/- a cos <\>y - A'''} '
the numerator of which is /» - A" - a cos </> (/+ A),
=/' + A' + a cos (^ (/— a cos 0 - A) - 2a/ cos <^ + a' cos' <f),
= — c'' — a' sin' ^ + a cos <^ (/— a cos ^ - A),
and the denominator is = - A (c' + a' sin' <t>). The second line of V is thus
= f dd) Ia - Va' + c^- "'^^"'"^ 4- «° sin' <f> cos ^ /- a cos <jbj- A a' sin' (^ (Vg' + c-' + a cos <^)1
J \ A " A c» + a'8in''(^ c' + a'sin-(/) j'
37—2
292 ON THE POTENTIAL OF THE ELUP8E AND THE CIRCLE. [603
which is easily reduced to
r (c* +/' — of cos <f> _ d'a cos <f>{/- a cos <f>) c* Va' + C |
j""P| A (c^ + a»sin»<^)A c» + a" sin* </)] '
the last term of which is = — c tan~' ; and we thus have
F = a sin <^ {log (/— a cos i^ + A) — log ( — a cos </> + Va' + c*)} — c tan~'
c
fj (cr'+f' — af cos <f) Ca cos <^ (/— a cos ^)
+ )^9\ ^ (c» + a»8in»^)A
between the limits 0, 2t ; or, finally,
ir_ o— . a r'^.1 |C+/^-a/cos(^ c''acos<^(/-acos(^)] _
in partial verification whereof observe that for o = 0 we have A = v c* +/*, and the
value becomes
F=27r(Vc^+/=-c),
which, writing therein gr in place o{ /, agrees with a foregoing result.
20. The process applied to finding the Potential of the Ellipse is really applicable
step by step to the Circle ; but if we begin by assuming g =/, it presents itself
under a different and simplified form. Starting from
V=p fmdm I, '^^ ,
J J 's/a' + <f + rri'p - 2mafco8 ^
for convenience we assume
PQ = ma/,
thereby converting the radical into \fP' + Q* — 2PQ cos (f). Writing also
n=a* + c' + 711* f* + 2a'c' + 2mV/- - 2mW/», = (P= - Q»)^
and hence assuming P^ - Q" = VS, and combining with the foregoing equation
F' + Q' = a' + d' + m'f',
we have
P» = ^ (a» + c" + m"/' + Vn),
Q» = ^ («» + <f + m'p - Vli).
21. This being so, the transformation-equations to the new variable T are
, PcosT + Q , „ Pcos<f>-Q
cos 9 = L> , n m> whence cos T = t5 — tt — I >
. , v/nsinT _, v/fi sin <i
^ P -f- Q COS r P - Q cos ^ '
603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 293
and also
-J a = {P + Q cos T)(P-Q cos <f>), =r--(^.
We find moreover
,. x^dT </nd<f>
^~P+QcosT' P-Qcos<f>'
and
whence
and hence
d<f> dT
-J P^ + Q'-^PQ cos 4, 'JP^-Q'cos^T'
V=p fm dm f , ^'^ ,
■' J J ^/P'-Q'cos'T
where the limits of T are from 0 to 27r, or, what is the same thing, we may multiply
by 4, and take them to be 0, ^tt.
22. Assuming next
we have
t=P'- m'f^ + (P'-Q') cot^" T,
t-P^-^rr.p^iP^-Q'f^^,
t +m»/^=(P^-Q'cos=D^j^,
lence
'Jt-P' + m'f\ t-qr- + m?f\ t + m'f = (P» - Q«) ^^ "^ P^ - Q' cos' T ;
also
and consequently
dt=^-2(P'-Q')'^^dT-
dt - 2dT
^t-F" + my. < - Q» + m»/'. t + m'/' VP»- Q»co8» T
T=0 gives « = oo, and T=\ir gives t = P^-m?f', =G suppose; and we thus have
V=2f'[ m dm r , ^
J Jo-/t-P^+m'f'.t-Q' + m'/'.t + m''p
23. We have
(« - P» + my») (« - Q» + TO'/») = t' + (rn^f - a- - c^) t - m'c^p,
or, putting mH in the place of t, this is
= m" {m'i= + (my -a''-e)t- c-f%
294 ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. [603
or, what is the same thing,
=-''<'+/=) h'-«+7.-7};
whence, completing the substitution, we have
V=2f*[mdm( ^^ T=^- .
where the inferior limit 0, = - C, =--F' — f' is, in fact, the positive root of the
equation
24. We may hence integrate in regard to m, through the sign I dt, in the same
way as if ^ were constant ; viz. we have
dt
V=2p
IW'^'-ttp-
f t -Jtit+p).
where the function of m is to be taken between the limits 0, 1 : for m = 0, we have
5 = 00, and the function vanishes; hence, writing wi=l, we obtain
-w.V'-,i:^-'
dt
■+r- t -Jtit+f)'
where 8 now denotes the positive root of
1 ?!__£^ = o
e+f^ e •
25. But it is interesting to reverse the transformation, so as to bring the radical
back into its original form. For this purpose, taking now
and consequently
P' = i(a»+c'+/^ + Vi:i),
where
n = a* + c^ -j-/* + 2a»c» + 2c?p - 2a*f\
and writing
t = P'-/'+ (P' - Q') cot' T,
we first obtain
„_., r' ilcos^TdT
(P» - Q» cos' T -/' sin' T){P'-Q' cos' T)* '
603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 295
and then, writing
i^ - y COS ^
. ™ v/n sin d>
F— (4 cos <\>
we bring in the variable <^. But it is important to remark that this is not the
quantity which was, at the beginning of the investigation, represented by this letter,
and that it is not easy to see the connexion between the two quantities ^. We find
■ c? +/" cos'' ^ - 2a/cos <j)) (a= + c" +/' - 2a/ cos cj))^ '
26. To reduce this, write as before
A = Va^ + c» +/=> - 2a/ cos <f),
and also
ip=a- -^-c'- 2o/cos <f> +p cos" (f>,
so that the denominator in the integral is = 4>A'.
We have
(P-Q cos <f>y {P COB ^■^Qy= (A» - Q» sin= </>) ( A» - P* sin^" ^),
= A* - (a' + e +/') A= sin'' 4> + a?f sin* ^,
= A" (A'' - {c- +/») sin= <^\ - a^ sin'' <t> (A' -/' sin" </>),
= A' (A" - (c" +/=) sin" <^} - a" sin" </> . 4),
and hence
/•/"JA^-j^+/Vn^^]d^ fsin'.^rf^
*^~j *A ""-^ j A' '
the limits being always 0, 27r. But we have identically
d sin <^ _ cos <f> a/" sin" tf)
d^ ~A~ " "A A' •
and thence
r sin" if>d^ _ 1 /sin <]fr\ 1 f cos <f>d(f)
J A^ 7if\ ^')~afj A~'
where the term ( -i—j is to be taken between the limits, but for the present I retain
it as it stands. Moreover, A" = fJ>+/^siD"^, and consequently
A" - (c" +/*) sin" </) = O - c" sin" ^,
and we thus obtain the result
where the denominators under the integral signs are
A, = Va' + c" +/" - 2afcoB <f>, and *A, = (a" + c" - 2a/cos <j> + /" cos" <f>) A.
296 ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. [603
27. We may, by a transformation such as that for the change of parameter in an
elliptic integral of the third kind, make the denominators to be A and {c? + a' sin» </>) A ;
fiA
viz. for this purpose we assume A = tan ' -j , where B and A are functions of <^
such that we have identically ^'+ jff'A''=(c= + a'sin=^)(a'' + c»- 2rt/cos (/> +/'cos'^);
the values of B, A are found to be ccos</) and sin </)(a'' +0* — a/cos^), whence,
dividing each of these for greater convenience by sin <j>, we have
A = tan-'(,^^f t^ J.
\(r + c^ — aj cos <pj
so that, ^vriting now B, A = c cot ^ and a- + c- — a/cos <^ respectively, the value is
/BA\
A = tan-' f
where
A.. .-.(f),
sm" 9
and, as before, 4> = a^ + c° — 2afcoa (j> + p cos' if>, and also 11 = c^ + a' sin^ <f>. We have
dA _ (AB - A'B) A' + ^^jB (A')' f.,_dA \
dd> ~ (A' + 5-^A») A ' [ dd>' ^^-J '
d<f> (A" + B'^i") A ' V #
and then
AB' - A'B = -A-[ (- a' - C + a/ cos' A),
sin' (^ ^ J r
iAB (A=)' = "'''.'^"f^ (a' + c= - a/cos </.) a/sin <^,
a Sin' <p
and the numerator thus is
-^4j, {(- a' - C + a/cos» <^) (a- + c' +/= - 2a/ cos <}>)
+ af cos (^ (1 - cos' <f>) (a' + c' - o/cos <f))],
which is in fact
= -^-^j, {~ (C' + '*'■' "^^^^ *^) ("'^ + ''' +/■ ~ 2a/ cos (t>)
+ {afcos(j)-a- cos- </)) (a' + c' - 2a/ cos <f> + /' cos' <^)} ,
= -J^- {- UA' + (af cos 4>-a' cos' (f>) *1 ;
sin <p
or, what is the same thing,
= -r^ {- Ud) - n/' sin' d) + (af cos A - a- cos' <^) *j,
sin' ^ ' ^ 7- V .^
and the denominator, by what precedes, is
= ^.n*A.
sin' 9
603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 297
We thus have
1 dA _ _ 1 _ /' sin' <j) a/cos ^-a^ cos' <f)
c d4~~^ <I)A ITA •
whence, by integration,
1 _j/ ccot</)A \_ Cd(l> C{af cos (j) — a' cos- (l>)d^ ^^rsm'<j)d(p
c W+(^-afcos<f>)~~J A ■•"; ITA -^ J ^*A~ '
which is the required formula of transformation.
28. Multiplying by c', and subtracting from the value of V, we find
[((f +f^ — af COS <l))d(j} J, ^cos^if- acos^)d<f>
■*"] A ""'] (d' + a' sin' (f,) A '
which is to be taken between the limits 0 and 27r ; viz. we thus have
F = -2c7r+ 2 [' ('^ +/' ~ "Z*^"^ '^) ^'^ 2c'a r^os <^(/- a cos <^)d(^
.'o A Jo (c' -I- a''' sin'' (^) A
agreeing with a former result.
29. But this former result, previous to the final step of taking the integrals
between the limits, was
F = 2a sin ^ log f ^^^^^^^^1^±£.) - c tan- f ^«'+^^*)
\-acos<^ + Va''+c"' V c /
C(c' +/' — af cos ^)d<p „ f cos (f) { f — a cos (f)) d4)
viz. the integrals are the same, but the integrated terms are altogether different;
the explanation of course is that the <j>'s are different in the two formulae, which there-
fore do not correspond element by element but only in their ultimate value between
the limits.
30. In order to discuss numerically the Potential of the Circle,
/(t-6.t + '^pj dt
F=2/'j'^
this must be reduced to elliptic functions. Writing t = d+x', we have
a? '^a? + yS* dm
•^0 {of +6) (of + 0?)^'
c. IX. 38
298 ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. [603
if for shortness
The constants a, ^, 0 may be considered as replacing the original constants a, c, f;
viz. from the last two equations and the equation
. + '1 = 1,
we deduce
showing that a', ^, 6 are in order of decreasing magnitude ; viz. a- — /3-, ^— 0, a"— 6
are all positive. The formula may be written
or" (x'' + B') dx
iF=(a»-5)f -— -
(of +e)(x'+ a^) \/af + a' . ai' + ff''
which, in virtue of the identity
becomes
i F= (a^ - ^ ) r , ^
dx
-0 (^-0)['°
-'o (a^ + 0)
ViC^ + 0» . iB* + /S" '
31. Writing here a; = acotM, and therefore tir = — a cosec' « du, to the values
a; = 00 , 0 correspond m = 0, ^tt, and we have
J 0 Vo» COS* u + ^ sin" u{ a*cos'u + 0 sin" wj
-r, ^" ;.» f/^^-/> (.- .r)-inMr "'^^^-^> L___
^ 0 Vo»cos''M + /8'sin»M I o'-^ ^ ' 0.^-0 a" cos^ u + 1> sin:^ u
8^
Writing k'=l -, we have
and thence
Vo'cos'M + ^S^sin'M = a Vl-A»sin'M,
00^-0
iF=f — ^=.x
Jo V 1 - Ar" sin' u
a (1 - ^ sm= u) ^ -
>-l 1-,' i-(i-S»»-
603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 299
Q
viz. writing ?i = — 1 + — (so that n is negative and in absolute magnitude < 1), and
moreover ^ = c^k'' and 0 = (n + l)aP, this is
i K = , X Ul-K' sm^ it) a + - (w + 1) a a -. 1 ;
J 0 ^/1-k' sin'' u { ' n"- ' n 1 + n sm" u] '
viz. this is
= a -^^iK + «" ^iK — III («. ^)f
32. This may be further reduced by substituting for the complete function
Hi (n, k), its value ; viz. writing
e
= (-l+^) = -l+*'^sin»X,
that is, sin' X = -^ ! then, writing the value first in the form
a [eJc -{n + \)F,k- " + ^-^» + ^ [ri, («, k) - FJci ,
and observing that
n+ l.n + i" rTT / i\ IT71 i^* sin' \ cos' \ TTT / 7\ Tin
[n. {n, k) - FM = i_^,3.^,^ [n, {n,k)- FJc]
kf^ sin \ cos \
Vl - A'' sin'
we have
k'^ sin \ cos \
= I^TT + {F,k - E,k) F {¥, \) -FJc.E {k', X) \ ,
i' X (. '
iF= a \E,k - k'' sin' \ FJc - /'"''^T^ [Jtt + {F,k - EJc) F {k', \) -FJc.E (k', \)]l ,
{ vl — A;'sin'\ j
where
" ^ 0 ^ \ e)
ce-e+f\ *» = i-5. =1-
a'" e+f^ ' 0+f' '
or, what is the same thing,
6 being, it will be recollected, the positive root of
" , + ^ = 1.
33. Thus when in particular a = 0, we have ^ = c', and thence
a = v'c' +/', A = 0, i'=l, 8in\ =
38—2
300 ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. [603
whence
J F = i TT Vc* +/» { 1 - sin' X - sin \ (1 - sin \)},
= i TT Vc»+/» ( 1 - sin \), = i TT (^c'+'f^ - c),
or
7=27r(Vc»+/'-c).
which is right.
' f
34. If c = 0, a being >/ then 6 = a--f', k = -, \ = ^7r, a=a; so that, retaining
f
k as standing for its value - , we have
° a
\V=a{E,k-k'^FJc), or V ='ia{EJc-k''' FJi),
which may easily be verified.
If c = 0, a being <f, then, recurring to the original equation for the determination
of e, viz. {e+pye(."'~.^ + '^-l\=0, which for c = 0 becomes ^ ( ^ +/') (^ -«'+/') = 0,
we have (as the positive root of this equation) 6 = 0; whence a=f\ also, observing
that 1 — ^=1^. k = -., and sin'\ = — (where -^ is finite), =0, and retaining k
f> J J ^ + ^/
to denote its value =^, we obtain \V=fEJc, or V=^fEJc.
If a =f, then in each of the formulae k=\\ and since in the first formula
4
K*FJc, k nearly =1, is =A.'Mogr,, vanishing for ^=1 or k' = 0, we have F=4/&il, =4/
Section of Equipotential
sarfaces of a Circle.
It would be interesting to consider the value of the potential at different points
of the ellipse -w-. — „ +5 = 1 (^ constant, a, c current coordinates). For this purpose
writing a = V/' + 6 cos q, c = Vising, we should have a = ^f'^ + d (a constant), and
/cosg ,, . Vg+/'sin'g
46 , /sin g
sm \ = -; — = — ^-^r^ , cos \ =
^6 +/» sin' 5 •Je +/« sin' q
and then V through k, k' , \, is a given function of g.
603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 301
35. Suppose, to fix the ideas, /= 1, and consider the points (0, c) and (a, 0), which
have equal potentials. First, if a >f (that is, a > 1), then writing k = -, the relation is
and we have
27r (Vl + c= - c) = I (E,k - k'^F.k) ;
^1 30° = 1-68575, £•, 30° = 1-46746, - = 1-27324.
TT
Secondly, if a </ (that is, a < 1), then writing k = a, the relation is
27r(Vr+c2-c)=4^,Jfc.
(1) In particular a = ^, = sin 30°, this is
Vf+"d»-c = -^,30'' = -93421.
TT
(2) a = l, then
Vr+?-c = - = -63662.
TT
(3) a=2, k = i, = sin 30°,=^
Vl + c^ - c = - {^, (30°) - iF, (30°)| = -25866.
TT
But if a/1 + c" - c = m, then c = ^ ( m ) ; whence
m
a c
0
-0
4
-06810
I
-46709
2
1-80376
for the values of c, corresponding to the foregoing values of a.
302 [604
604.
DETERMINATION OF THE ATTRACTION OF AN ELLIPSOIDAL
SHELL ON AN EXTERIOR POINT.
[From the Proceedings of the London Mathematical Society, vol. vi. (1874 — 1875),
pp. 58—67. Read January 14, 1875.]
The shell in question is the indefinitely thin shell included between two con-
centric, similar, and similarly situated ellipsoidal surfaces, the density being uniform
and the attraction varying as the inverse square of the distance.
It was shown by Poisson that the attraction was in the direction of the axis of
the circumscribed cone, and expressible in finite terms; the theorem as to the
direction of the attraction was afterwards demonstrated geometrically by Steiner, Crelle,
t. XII. (1834), his method being to divide the shell into elements by means of conical
surfaces having their vertices at an interior point Q ; and the investigation was about
two years ago completed by Prof Adams, so as to obtain from it the finite expression
for the attraction of the shell. The process was explained in a lecture at which I was
present: I did not particularly attend to the details of it; and I now reproduce the
solution in my own form, stating, in the first place, the geometrical theorems on which
it depends.
Statement of the Geometrical Theorems.
1. We consider (see figure, p. 305) an ellipsoid, and two corresponding points, an
external point P, and an internal point Q; as will appear, the correspondence is not
a reciprocal one. The points are such that each of them is, in regard to the ellipsoid,
in the polar plane of the other; moreover I'Q is the perpendicular at P to the polar
plane of Q ; that is, Q being regarded as given, then P is determined as the foot of
the perpendicular let fall from Q upon its polai- plane ; to a given position of Q
there corresponds thus a single position of P. It follows that PQ is the normal at
604] DETERMINATION OF THE ATTRACTION OF AN ELLIPSOIDAL SHELL. 303
P to the confocal ellipsoid through this point ; that is, given the position of P, then
Q is the intersection of the polar plane of P by the normal at P to the confocal
ellipsoid. Analytically, to a given position of P, there correspond three positions of Q,
viz. these are the intersections of the polar plane of P by the normals at P to the
three confocal surfaces through this point, and the correspondence of the points P, Q
is a (1, 3) correspondence ; but the other two positions of Q are external to the
ellipsoid, and we are not concerned with them ; we determine Q as above by means
of the normal to the confocal ellipsoid.
2. If through the point Q we draw at pleasure a chord R'QR", and join the
extremities R, R" with P, then the line PQ bisects the angle R'PR"; whence also
PR:QR = PR" : QR", or writing QR', QR" = r, r" and PR', PR" = p', p", then ^' = C-
Putting each of these equal ratios = p , where H is a length depending on the
position of Q but independent of the direction of the chord R'QR", then R will be
a length depending on the direction of the chord, and if along the chord (say in
the sense Q to R') we measure off from Q a length QT, = R, thence the locus of
the extremity T of this line will be an ellipsoid, centre Q, similarly situate to the
given ellipsoid, say this is the "auxiliary ellipsoid."
Consider now the given ellipsoid and a concentric and similarly situated similar
ellipsoid, exterior to and indefinitely near it. To fix the ideas, let the semi-axes
of the given ellipsoid be nif, mg, mh, and those of the consecutive ellipsoid be
(m + dm)/, {m + dm) g, (m + dm) h. Producing the chord R'R" to meet the consecutive
ellipsoid in /S", S", then the radial thicknesses R'S', R"S" of the included shell will
be equal to each other, or say each =Adm, where A is a quantity dependent as
well on the position of the point Q as on the direction of the chord R'R" through
this point.
3. Let 2<^ denote the angle R'PR", or, what is the same thing, let <j> denote
either of the equal angles R'PQ, R'PQ; then, R, A being as above, it is found that
mR
Determination of the Attraction of the Shell,
4. We may now solve the attraction-problem. We consider the indefinitely thin
shell (density unity) included between the given ellipsoid and the consecutive ellipsoid,
and attracting the exterior point P. We determine the corresponding interior point
Q, and then dividing the shell into elements by means of indefinitely thin cones
having their vertices at Q, we consider in conjunction the elements determined by
any two opposite cones, say the two opposite cones, having for their axis the chord
R'QR" and a spherical aperture = dco. The shell-element at R' is
r''da> . RS' = r"'Ad(o dm ;
304 DETERMINATION OF THE ATTRACTION OF AN [604
its attraction on P is therefore
-y Xdw dm, = j=r-, A dm dm,
p 12
and the attractions in the du'ections QR' and PQ are this quantity multiplied by
sin ^ and cos <^ respectively.
5. But the shell-element at R" exerts upon P the same attraction ^^Kdmdm,
and the attractions in the directions QB!' and PQ are this quantity multiplied by
sin <^ and cos <^ respectively : hence the attractions in the directions QR, QR" exactly
counterbalance each other, and there remain only the two equal attractions in the
direction PQ; viz. this, for either of the elements in question, say for the element
at R', is
i?
= „ J A cos <f> dm da>,
or, substituting for cos ^ its value, = — r- , this is
A
m,dm,
R?dw.
Hence the whole attraction of the shell is in the direction PQ, its value being
mdm
fll^"--
over the whole solid angle at Q; and recollecting that R denotes the radius vector
in the auxiliary ellipsoid, we have the volume of this ellipsoid
= II j r^drdm = ^ll R' dm,
that is, 1 1 i? dft) = thrice the volume of the auxiliary ellipsoid, = 4eirF6H, if F, 0, H
are the semiaxes of the auxiliary ellipsoid. That is.
Attraction of shell = '"^ 47rJ;'<?.ff.
The problem is now solved ; but it remains to prove the geometrical theorems, and
to determine the values of the quantities fl, F, 0, H, which enter into the
expression for the attraction ; and we may also deduce the formula for the attractions
of a solid ellipsoid.
Proof of the Geometrical Theorems.
6. I take
of y' z'
for the equation of the ellipsoid; a, b, c for the coordinates of P; f, /;, ^ for those
of Q ; a, /9, 7 for the cosine-inclinations of the radius QR' to the axes. Hence, in
604j ELLIPSOIDAL SHELL ON AN EXTERIOR POINT. 305
the equation of the ellipsoid, substituting for x, y, z the values ^ + ra, 77 + r^, ^ + ry,
and writing for shortness
A-'^^+^ + t
£- rf ^
C = ^ + -^ + 1^ — m', (C being therefore negative),
•/ if
we have r', — r" as the roots of the equation
^r'' + 2£r- + C = 0:
viz.
25 , „ C
-J- = — r + r , -J = — r r ,
and thence
r' -^ ,r ^ ,r.+r'= -^ .
7. Suppose for a moment that the semidiameter parallel to B!K' is =m/o\ we
have evidently v'^=-7. And tihen, if in the central section through B!K' the conjugate
A
semidiameter is mw, the equation of the section referred to these conjugate axes
will be „+ „„=1, or say, v'' = wiV „a?, where « is the coordinate parallel to
R'R", so that, taking the coordinate to belong to the point R', we have
y = ^{r +r)= -^ .
For the exterior surface of the shell, m is to be changed into m + dm ; hence, y and m
alone varying, we have
ydy = mif dm, = m d/m -j ,
that is,
dy = mdm^^-^^,
C. IX. 39
306 DETEEMINATION OF THE ATTRACTION OF AN [604
viz. this is the value of the radial thickness R8' of the shell ; or, since the same
process applies to the point R", we have
1
RS'=R'S" = mdm
^/B'-AC'
7ti
or, calling this, as above, A dm, the value of A is =
8. The points P and Q are connected by the condition that, for every direction
whatever of the chord RR", we have
PR' : PR" = QR' : QR",
or, what is the same thing, that the line QP bisects the angle RPR'. Taking
PR = p, PR" = p", the condition is p : r = p" : r" ; and taking (a, h, c) as the
coordinates of the point P, we have
p'» = (f + /a - ay + (17 + r'/3 - 6)' + (f + r'7 - c)=
= <T"- + 2r'U+r'^,
if, for shortness, __
^ = (f_a)^ + (i,-6)» + (?-c)», (=Qn
fr=a(?-a) + /9(,;-6) + 7(r-c);
and similarly
p"'' = <7'-2r"U+r"\
The required condition therefore is
viz. this is
so that, omitting a factor, it becomes
that is,
«7=. ^+2^7 = 0, or t7 = °^,
which must be satisfied independently of the values of a, /3, 7.
9. Writing, for greater convenience, ^ = — ^, the equation is U= — 6B, viz.
substituting for U, B their values, this gives f - a + -^1 = 0, &c., or say,
«-f(i+7.).
604] ELLIPSOIDAL SHELL ON AN EXTERIOR POINT. 307
and the assumed relation -^ = — 0 is
viz. substituting for a, b, c the foregoing values, and omitting a factor 0, this is
or, writing for shortness
^+/r^j=-l/^+^"^+f^-"^^):
the equation is
0 = -n^C.
We thus see that, (f, rj, f) being given, 0, and therefore also (a, b, c), are uniquely
determined. It may be added that, writing (7 = — -^ , we have ff' = H^cr^, or say ila- = 0.
10. We have, moreover.
and
whence
c
5+TrT2="^''
^H-Z'^ + ^f'^ + A'
or, regarding (a, 6, c) as given, 0 is determined as a function of (a, 6, c) by this cubic-
equation; and 0 being (in accordance with the foregoing equation 0 = — D,-G) assumed
to be positive, we have 0 the positive root of this equation, and m' (0 +/'), inP (0 + g'^),
m?{0 + h?) as the squared semiaxes of the confocal ellipsoid through the point P. And
0 being known, f, t), f are, by the foregoing equations a = Ml +xi)> &c., determined
in terms of f, ?;, §"; that is, starting from the given external point P, we have the
internal point Q. And it appears that PQ is the normal at P to the confocal ellipsoid,
or, what is the same thing, the axis of the circumscribed cone, vertex P.
11. The foregoing equation
|! (^ +/') + 1 (^ +Sr») + 1^ (^ + A») = mS
considering a, b, c, and therefore 0, as given, shows further that the point Q is situate
^ v^ z^
on an ellipsoid which is the inverse of the confocal ellipsoid ^ — -^ + ^ — + -^ — r- = m-
o +j " + 5' p + A
a? y^ z"
in regard to the given ellipsoid ^ + ^+r^ = w^
J i/
39—2
308 DETERMINATION OF THE ATTRACTION OF AN [604
12. Expressing Q, in terms of a, b, c, we have
y»~ flf) J. i^M "^ /■/) J. «»\> "^ ,
c»
We have 0-"=^,, =C'n\ and
whence
i7.,({-a) + /S(,-t) + 7(f-c),
or, since
this is
A + 2B-, + c\=0,
r r^
I
r
"= if 1 _ 1 AC
This last equation may also be written
or, what is the same thing,
if for shortness
F' ~ il' f
G^ ~ W g' '
1 _ 1__C
H* n» A"'
a
viz. substituting herein for C its value — — , these equations give
V^+/»' V<?+^»' -^e+h"'
where il stands for its expression in terms of a, b, c.
604] ELLIPSOIDAL SHELL ON AN EXTERIOR POINT. 309
13. The expression for ^^ shows that R is the radius vector, cosine-inclinations
o, /9, 7, in an ellipsoid semi-axes F, G, H, which may be regarded as having its centre
at Q; viz. this is the "auxiliary ellipsoid." And this being so, we have
r r K
It appears from these equations that, drawing from Q parallel to PR" a line
QM, = n, and from its extremity M parallel to PQ a line to meet QR' in T, the
locus of T is the auxiliary ellipsoid.
14. By what precedes, the angles R'PQ, R"PQ are equal to each other, say each
is =^; the triangle R'PR" gives
p' + p"' -(y + rj
cos 2^ ■■
2p'p"
that is,
4pp
viz. this is 1^
= (|-l)(r-' + r")-4grV',
/I lUr' + r'J
~ \R? nV 4rV"
--AtR'. ^p .-^
= R'iB'-AC);
or say
cos <f> = R'^B'- AC;
a remarkable equation which may also be written
C08<^ = ^.H^'+n
if, as before, v is the semi-diameter parallel to R'R".
In virtue of the equation A = , which defines A, the equation becomes
mR
cos (^ = -^ ;
and we thus complete the demonstration of the several geometrical theorems upon
which the investigation was founded.
310 DETERMINATION OF THE ATTRACTION OF AN [604
Analytical Expressions for the Attraction of the Shell, and for the Resolved Attractions.
15. The attraction of the shell was shown to be
or, since the mass of the shell, the density being unity, is
-n^fgh . Snv'dm = 4m^ dm trfgh,
o
we have
Attraction -^ Mass = — ^rr ^r- ;
mil' fgh
which, by what precedes, is
^ n
m\/(f' + e)(g' + e){h' + dy
where
^^ ■*■/'«»;
o» (f'+dy^(cf^ + 0y (h^' + ey
6 being the positive root of
16. It is to be observed that the cosine-inclinations of the line PQ to the axes are
afl 611 cfi
f^ + d' g' + d' h' + d'
respectively; so that, considering, for instance, the attraction parallel to the axis of x,
we have
Kesolved Attraction -=- Mass =
m if' + 6) '/(f^ + 6) if + 6) ih' + 6) '
a? y- z'
Resolved Attractions of the Ellipsoid -k-^- „ + fi = 1-
J if
17. We may find the attraction of the solid ellipsoid
For this purpose, dividing it into shells, semi-axes mf, mg, mh, and (m + dm)f, (m + dm) g,
(m + dm) h respectively, we have for the shell in question
aft'
Resolved Attraction -r- Mass =
m (/» -I- 6) V(f' + 6) (^» + e) {h' + 6) '
604] ELLIPSOIDAL SHELL ON AN EXTERIOR POINT. 311
4nr
and the mass of the shell is -^fgh.Sm^dm, where the first factor is the mass of
o
the ellipsoid; whence
a . SmCl^ dm
Resolved Attraction -r Mass of Ellipsoid =
(r-+0)'^if' + e)(g^+0)ih^ + e)'
0 being here a function of m, and m extending from 0 to 1. But taking 0 as the
variable in place of m, the equation
6^
gives
— jY^d0 = 2m dm ; that is, Smfl^ dm = — fdft
Moreover m = 0 gives ^ = x , and m = \ gives 0 = its value as defined by the equation
a? 6' c' _
/» + ^ + ^ + ^ + Ai> + ^-l'
so that, reversing the sign, the limits are oa , 6; or, finally, writing under the integral
sign <^ in place of 6, the formula is
Resolved Attraction -r- Mass of Ellipsoid = M \ ^ — — ,
Ms (/^ + ,^) V(/= + 4,) {g- + 4,) (h' + 4)
which is a known formula.
312 [605
605.
NOTE ON A POINT IN THE THEORY OF ATTRACTION.
[From the Proceedings of the London Mathematical Society, vol. vi. (1874 — 1875),
pp. 79—81. Read February 11, 1875.]
Consider a mass of matter distributed in any manner on a surface, and attracting
points P, Q not on the surface. Consider a point Q accessible from P, viz. such that
we can pass continuously from P to Q without passing through the surface. (It is
hardly necessary to remark that, if for example the matter is distributed over a
hemisphere or segment of a closed surface, then by the surface we mean the hemisphere
or segment, not the whole closed surface.) The potential and its differential coefficients
ad infinitum, in regard to the coordinates of the attracted point, all vary continuously
as we pass from P to Q; and it follows that the potential is one and the same
analytical function of (a, b, c), the coordinates of the attracted point, for the whole
series of points accessible from the original point P; in particular, if the surface be
an unclosed surface, for instance a hemisphere or segment of a sphere, then every
point Q whatever not on the surface is accessible from P ; and the theorem is that
the potential is one and the same analytical function of (o, b, c), the coordinates of
the attracted point, for any position whatever of this point (not being a point on
the surface). But this seems to give rise to a difficulty. Consider the matter as
uniformly distributed over a closed surface, and divide the closed surface into two
segments: the potential of the whole shell is the sum of the potentials of the two
segments; and the potential of the first segment being always one and the same
function of (a, b, c), whatever may be the position of the attracted point, and
similarly the potential of the second segment being always one and the same function
of (a, 6, c), whatever may be the position of the attracted point; then the potential
of the whole shell is one and the same function of (a, b, c), whatever may be the
position of the attracted point. This we know is not the case for a uniform spherical
605] NOTE ON A POINT IN THE THEORY OF ATTRACTION. 313
shell ; for the potential is a different function for external and interior points, viz.
. . . M
for internal points it is a constant, =M-t- radius : for external points it is = -. — ,
if a, 6, c are the coordinates measured from the centre of the sphere.
The difficulty is rather apparent than reaL Reverting to the case of an unclosed
surface or segment, and considering the continuous curve from P to Q, let this be
completed by a curve from Q to P through the segment; viz. we thus have P, Q
points on a closed curve or circuit meeting the segment in a single point. To fix
the ideas, the circuit may be taken to be a plane curve, and the position of a point
on the circuit may be determined by means of its distance s from a fixed point on
the circuit. Considering this circuit as drawn on a cylinder, we may at each point
of the circuit measure off, say upwards, along the generating line of the cylinder, a
length or ordinate z, proportional to the potential of the point on the circuit, the
extremities of these distances forming a curve on the cylinder, say the potential curve.
We may draw a figure representing this curve only; the points P, Q being marked
i=m»
I
as if they were points on the curve (viz. at the upper instead of the lower extremities
of the corresponding ordinates z) : the generating lines of the cylinder, and the plane
section which is the circuit, not being shown in the figure. The potential curve is
then, as shown in the figure, a continuous curve, viz. we pass from P to Q in the
direction of the arrow, or along that part of the circuit which does not meet the
segment, a curve without any abrupt change in the value of the ordinate z or of
dz d^z
any of its differential coefficients, -5- , -^-^ , &c. ; but there is, corresponding to the
point where the circuit meets the surface, an abrupt change in the direction of the
dz
potential curve or value of the differential coefficient j- , viz. the point on the curve
is really a node, the two branches crossing at an angle, as shown by the dotted
lines, but without any potentials corresponding to these dotted lines.
In the case of two segments forming a closed surface, or say two segments forming
a complete spherical shell; then, if the points P, Q are one of them internal, the other
external, the circuit, assuming it to meet the first segment in one point only, will meet
the second segment in at least one point; the potential curves corresponding to the two
segments respectively will have each of them, at the point corresponding to the intersec-
tion of the circuit with the segment, a node ; and it hence appears how, in the potential
curve corresponding to the whole shell (for which curve the ordinate z is the sum
of the ordinates belonging to the two segments respectively), there will be a dis-
continuity of form corresponding to the passage from an exterior to an interior point.
C, IX. 40
314
NOTE ON A POINT IN THE THEORY OF ATTRACTION.
[605
This is best shown by the annexed figure, which represents a uniform spherical shell
made up of two segments, one of which is taken to be a small segment or disc
having the point A for its centre, the other the large segment B, which is the
remainder of the shell; the circuit is taken to be the right line ..PAQB.. through
Q A P
the centre of the sphere (viz. we may imagine the two extremities meeting at infinity,
or we may, outside the sphere, bend the line so as to unite the two extremities,
thus forming a closed curve). The curve (a) represents the potential curve for the
segment A, the curve {b) that for the segment B, these two curves having, as shown
by the dotted lines, nodes corresponding to the points A, B respectively (but these
dotted portions not indicating any potentials) ; and then, drawing at each point the
ordinate which is the sum of those for the curves (a), (b) respectively, we have the
discontinuous curve (c), composed of a horizontal portion and two hyperbolic branches,
which is the potential curve for the whole spherical shell
Practically the figure is constructed by drawing the curves (c), (a), and from
them deducing the curve (6). As regards the curve (a) it may be noticed that,
treating the segment (a) as a plane disc, the curve (a) is made up of portions of
two hyperbolas; viz. it breaks up into two curves, instead of being, as fissumed in
the discussion, a single curve ; this is a mere accident, not affecting the theory ;
and, in fact, taking the segment to be what it really is, the segment of a sphere
the potential curve does not thus break up.
606] 315
606.
ON THE EXPRESSION OF THE COORDINATES OF A POINT OF
A QUARTIC CURVE AS FUNCTIONS OF A PARAMETER.
[From the Proceedings of tJte London Mathematical Society, vol. vi. (1874 — 1875),
pp. 81—83. Read February 11, 1875.]
The present short Note is merely the development of a process of Prof. Sylvestei-'s.
It will be recollected that the general quartic curve has the deficiency 3 (or it is
4-eur8al) ; the question is therefore that of the determination of the subrational *
functions of a parameter which have to be considered in the theory of curves of the
deficiency 3.
Taking the origin at a point of the curve, the equation is
{x. yy + (x, yy + {x, yf + {x, y) = 0;
and writing herein y = \x, the equation, after throwing out the factor x, becomes
(1, \)*a^ + (l, \ya?-\-{\, Xfx-\-{\, \) = 0;
or, say
a.a^ + Zha? + 3ca; + d = 0,
where we write for shortness
a, b, c, d = (i, \)\ HI. >-n HI. ^f< (1. M;
viz. a, b, c, d stand for functions of X of the degiees 4, 3, 2, and 1 respectively.
The equation may be written
{ax + by -3(b!'-ac)(ax +b) + a^d- 2abc + 2b^ = 0;
* The expression " subrational " includes irrational, but it is more extensive ; ii Y, X are rational
functions, the same or different, of y, x respectively and }' is determined as a function of x by an equation
of the form Y=X, then y i» a. subrational function of x. The notion is due to Prof. Sylvester.
40—2
316 ON THE EXPRESSION OF THE COORDINATES OF A POINT OF A [606
viz. writing for a moment ax + b = 2 sW—ac . u, this is
4m» - 3u + IL-^ — = 0.
2(6»-ac)v6»-ac
HeDce, assuming
a'd - Sabc + 26»
— cos rf) = -. ,
2(6''-ac)v6^-ac
then we have 4rtt' — 3« — cos <f> = 0; consequently u has the three values cos ^<^, cos ^(^ + 27r),
cos ^ (0 — 27r), and we may regard cos J<^ as representing any one of these values.
We have thus ax + b = 2\/b'-accos^<f), and y = \x, giving x and y as functions
of X and (f>, that is, of X. But for their expression in this manner we introduce the
irrationality \b^ — ac, which is of the fonn V(l, \)', and the trisection or derivation
of cos^^ from a given value of cos^; viz. we have, as above, — cos^, a function of
X of the form
(1, X)'-(l, x)«V(Xli)''.
The equation for <j) may be expressed in the equivalent forms
. . a V- (a»d^ + 4ac» + 46'd - Gabcd - 36V)
sin A = ^^— 7-= ,
(b"--ac)'^b'-ac
^ _ g V- (a'cf + 4:0^ + 'ib'd - 6abcd - 36'c')
-tan<^- a'd-Sabc + 2b' '
and inasmuch as we have
(b' — ac) cos <p
we may, instead of
ax + b = 2'J¥ — ac cos J^,
write
, (a'rf - 3a6c + 26') cos 4 d>
(or — ac) cos 9
or, what is the same thing,
^ - (g'd - 3a6c + 26')
(6» - oc) (4 cos^ 0-3)"
The formulae may be simplified by introducing ji, a function of X, determined by
the equation
c/i" - 26/* + a = 0 ;
viz. this equation is
HI, X)»,*'-§(1, X)> + (1, X)« = 0,
80 that (X, (i) may be regarded as coordinates of a point on a nodal quartic curve,
or a quartic curve of the next inferior deficiency 2. And we then have
(c/i - 6) = V6^^^,
606] QUARTIC CURVE AS FUNCTIONS OF A PARAMETER. 317
and consequently
^ a?d - Zahc + 26'
— cos<p= — jTT 1\'. — ;
^ 2 {en- by '
viz. cos^ is given as a rational function of the coordinates (X, fi); there is, as before,
the trisection; and we then have
aas + b=2 {cfi — h) cos \^, y = \x,
giving X and y as functions of \, /i, <f); that is, ultimately, as functions of \. I have
not succeeded in obtaining in a good geometrical form the relation between the point
{x, y) on the given quartic and the point (\, ji) on the nodal quartic.
Reverting to the expression of tan <^, it may be remarked that a = 0 gives the
values of \ which correspond to the four points at infinity on the given quartic
curve ; a'd" + ^sac? + 46'd — &ahcd — 36V = 0, the values con-esponding to the ten tangents
from the origin ; and a'd — ^ahc + 26' = 0, the values corresponding to the nine lines
through the origin, which are each such that the origin is the centre of gravity of
the other three points on the line.
I take the opportunity of mentioning a mechanical construction of the Cartesian.
The equation r' = — .4 cos 6 —N represents a lima9on (which is derivable mechanically
from the circle r' = - A cos ff), and if we efifect the transformation r' = r-\ — , the new
r
JO
curve is r + - + ^ cos^+iV=0; that is, r» + r(^ cos0 + iV) + £ = O, which is, in fact,
the equation of a Cartesian. The assumed transformation r' = r H — can be effected
r
immediately by a Peaucellier cell.
318 [607
607.
A MEMOIR ON PREPOTENTIALS.
[From the Philosophical Transactions of the Royal Society of London, vol. CLXV. Part li.
(1875), pp. 675—774. Received April 8,— Read June 10, 1875.]
The present Memoir relates to multiple integrals expressed in terms of the (s+1)
ultimately disappearing variables (x, . . , z, w), and the same number of parameters
(a, . . , c, e) ; they are of the form
/
pdv
{(a -xy + .. + {c-zy + (e- «;)'}*«+« '
where p and dw depend only on the variables (x, ... z, w). Such an integral, in regard
to the index J« + q, is said to be " prepotential," and in the particular case ? = — ^
to be "potential."
I use throughout the language of hyper-tridimensional geometry: (x, ... z, w) and
(a, . . , c, e) are regarded as coordinates of points in (s + l)-dimensional space, the former
of them determining the position of an element p dnr of attracting matter, the latter
being the attracted point ; viz. we have a mass of matter = j P dia distributed in such
manner that, dm being the element of (s+1)- or lower-dimensional volume at the point
{x, . ., z, w), the corresponding density is p, . a given function of (x, . . , z, w), and that the
element of mass pdw exerts on the attracted point (a, .., c, e) a force inversely proportional
to the {s + 2q+ l)th power of the distance {(a -x)- + . .+(c-zy + ie- wy}K The integra-
tion is extended so as to include the whole attracting mass j pdm; and the integral
is then said to represent the Prepotential of the mass in regard to the point (a, . . , c, e).
In the particular case 8=2, q = — ^, the force is as the inverse square of the distance,
and the integral represents the Potential in the ordinary sense of the word.
The element of volume dur is usually either the element of solid (spatial or (s-|-l)-
dimensional) volume dx . . dzdw, or else the element of superficial (s-dimensional)
volume dS. In particular, when the surface (s-dimensional locus) is the (s-dimensional)-
607] A MEMOIR ON PREPOTENTIALS. 319
plane w=0, the superficial element dS is =dx ... dz. The cases of a less-than-s-dimen-
sional volume are in the present memoir considered only incidentally. It is scarcely
necessary to remark that the notion of density is dependent on the dimensionality of
the element of volume dtn : in passing from a spatial distribution, pdx. .. dz dw, to a
superficial distribution, pdS, we alter the signification of p. In fact, if, in order to
connect the two, we imagine the spatial distribution as made over an indefinitely thin
layer or stratum bounded by the surface, so that at any element dS of the surface
the normal thickness is dv, where dv is a function of the coordinates (x, . . , z, w) of the
element dS, the spatial element is = dv dS, and the element of mass pdx...dzdw is
= pdvdS; and then changing the signification of p, so as to denote by it the product
pdv, the expression for the element of mass becomes pdS, which is the formula in
the case of the superficial distribution.
The space or surface over which the distribution extends may be spoken of as the
material space or surface ; so that the density p is not = 0 for anj' finite portion of the
material space or surface ; and if the distribution be such that the density becomes = 0
for any point or locus of the material space or surface, then such point or locus,
considered as an infinitesimal portion of space or surface, may be excluded from and
regarded as not belonging to the material space or surface. It is allowable, and
frequently convenient, to regard p as a discontinuous function, having its proper value
within the material space or surface, and having its value = 0 beyond these limits ;
and this being so, the integrations may be regarded as extending as far as we please
beyond the material space or surface (but so always as to include the whole of the
material space or surface) — for instance, in the case of a spatial distribution, over the
whole (s + l)-dimensional space ; and in the case of a superficial distribution, over
the whole of the s-dimensional surface of which the material surface is a part.
In all cases of surface-integrals it is, unless the contrary is expressly stated,
assumed that the attracted point does not lie on the material surface; to make it
do so is, in fact, a particular supposition. As to solid integrals, the cases where the
attracted point is not, and is, in the material space may be regarded as cases of
coordinate generality ; or we may regard the latter one as the general case,
deducing the former one from it by supposing the density at the attracted point to
become =0.
The present memoir has chiefly reference to three principal cases, which I call
A, C, D, and a special case, B, included both under A and C: viz. these are: —
A. The prepotential-plane case; q general, but the surface is here the plane
w=0, so that the integral is
pdx ...dz
/
{{a-xy+...+{c- zj + e»ji«+« •
B. The potential-plane case ; q — —\, and the surface the plane w = 0, so that
the integral is
pdx ...dz
I
{(a-xy + ... + (c-zy + e'ji^i '
320 A MEMOIR ON PREPOTENTIALS. [607
C. The potential-surface case ; q = — ^, the surface arbitrary, so that the integral is
[ pdS
j\(a-xy+... + {c-zy + {e-wy\i-i'
D. The potential-solid case; 2 = — i, and the integral is
f pdx ...dz dw
J {{a - xy + ... + {c - zY + (e - wy}i^'
It is, in feet, only the prepotential-plane case which is connected with the partial
differential equation
W rfC de' e de) "'
considered in Green's memoii-* "On the Attractions of Ellipsoids" (1835), and called
here "the prepotential equation." For this equation is satisfied by the function
{a»-l-...-l-c»-l-e»}i»+9'
and therefore also by
1
and consequently by the integral
pdx ... dz
[ pdx...dz
J {{a- xy + ... + {c - zf + e^]!"^ ^ ^'
that is, by the prepotential-plane integral; but the equation is not satisfied by the value
1
{{a-xy+ ... + (c-zy + {e-v!y}i'+9'
nor, therefore, by the prepotential-solid, or general superficial, integral.
But if J = — i, then, instead of the prepotential equation, we have "the potential
equation "
and this is satisfied by
1
{a' -1-... + C' +«')*•-*•
and therefore also by
1
{(a-xy+... + (c-zy + {e-wy]i-^'
Hence it is satisfied by
r pdx ... dzdw ,_^
j {(o - a;)" -(-... + (c-«)» + (e -•»)»)»•-* ^ ''
' [Oreen't Mathematical Papers, pp. 186—222.]
607] A MEMOIR ON PREPOTENTIALS. 321
the potential-solid integral, provided that the point (a, .. , c, e) does not lie within the
material space : I would rather say that the integral does not satisfy the equation,
but of this more hereafter; and it is satisfied by
pdS
h
.(C),
^{{a-xf-¥ ... +(c-2)- + {e-wy}i'~i
the potential-surface integral. The poteutial-plane integral (B), as a particular case of
(C), of course also satisfies the equation.
Each of the four cases give rise to what may be called a distribution- theorem;
viz. given V a function of (a, . . , c, e) satisfying certain prescribed conditions, but
otherwise arbitrary, then the form of the theorem is that there exists and that we
can find an expression for p, the density or distribution of matter over the space or
surface to which the theorem relates, such that the corresponding integral V has ita
given value : viz. in A and B there exists such a distribution over the plane w = 0,
in C such a distribution over a given surface, and in D such a distribution in
space. The establishment, and exhibition in connexion with each other, of these four
distribution-theorems is the principal object of the present memoir; but the memoir
contains other investigations which have presented themselves to me in treating the
question. It is to be noticed that the theorem A belongs to Green, being in fact
the fundamental theorem of his memoir of 1835, already referred to. Theorem C, in
the particular case of tridimensional space, belongs also to him, being given in his
" Essay on the Application of Mathematical Analysis to the theories of Electricity and
Magnetism" (Nottingham, 1828*), being partially rediscovered by Gauss** in the year
1840; and theorem D, in the same case of tridimensional space, to Lejeune-Dirichlet:
see his memoir " Sur un moyen general de verifier I'expression du potentiel relatif a
una masse quelconque homogene ou hdterogfene," Grelle, t. xxxn. pp. 80—84 (1840). I
refer more particularly to these and other researches by Gauss, Jacobi, and others in
an Annex to the present memoir.
On the Prepotential Surface-integral. Art. Nos. 1 to 18.
1. In what immediately follows we require
dx ... dz
"=/,
limiting condition of + ... + z'' = R-, the prepotential of a uniform (s-coordinal) circular
diskf, radius R, in regard to a point (0, . . , 0, e) on the axis ; and in particular the
* [Also Crelle, t. xxxix., pp. 73 — 89, t. xliv., pp. 356 — 374, t. xlvii., pp. 161 — 221; Green's Matliematical
Papert, pp. 1 — 115.]
** ["AUgemeine Lelirsatze in Beziehung auf die im verkehrten Verhaltnisse des Quadrats der Entfernung
wirkenden Anziehnngs- und Abstossungskrafte," Ges. Werke, t. v., pp. 195 — 242.]
t It is to be throughout borne in mind that x, .. , z denotes a set of s coordinates, x, . . , z, w a set of
»+l coordinates; the adjective coordinal refers to the number of coordinates which enter into the equation;
thaB,a^ + ...+^ + w^=f^ is an (»-(-l)-coordinal sphere (observe that the surface of such a sphere is g-dimensional);
«*+... + «'=/', according as we tacitly associate with it the condition ■w = 0, or w arbitrary, is an s-coordinal
circle, or cylinder, the surface of such circle or cylinder being s-dimensional, but the circumference of the
circle (s-l)-dimensional; or if we attend only to the s-dimensional space constituted by the plane w = 0, the
locas may be considered as an »-coordinal sphere, its surface being (s - l)-dimen8ional.
C. IX. 41
822 A MEMOIR ON PREP0TENTIAL8. [607
value is required in the case where the distance e (taken to be always positive) is
indefinitely small in regard to the radius R.
Writing a; = rf , , . , z = rf, where the s new variables ^, . . , f are such that ^ + ... + ^=\,
the integral becomes
where dS is the element of surface of the s-dimensional unit-sphere ^ + . . . + f* = 1 ; the
integral I dS denotes the entire surface of this sphere, which (see Annex I.) is = p. .
The other factor,
.'o (r» + €^)i'+^ '
is the r-integral of Annex II.
2. We now consider the prepotential-surface integral
V=( P^
As already mentioned, it is only a particular case of this, the prepoteritial-plane integral,
which is specially discussed; but at present I consider the general case, for the purpose
of establishing a theorem in relation thereto. The surface (s-dimensional surface) S is
any given surface whatever.
Let the attracted point P be situate indefinitely near to the surface, on the
normal thereto at a point N, say the normal distance NP is = «* ; and let this point
N be taken at the centre of an indefinitely small circular (s-dimensional) disk or
segment (of the surface), the radius of which It, although indefinitely small, is in-
definitely large in comparison with the normal distance «. I proceed to determine
the prepotential of the disk ; for this purpose, transforming to new axes, the origin
being at N and the axes of x, .., z in the tangent-plane at N, then the coordinates
of the attracted point P will be (0, . . , 0, «), and the expression for the prepotential
of the disk will be
where the limits are given by {c' + ... + z^ < R-.
Suppose for a moment that the density at the point iV^ is = p, then the density
throughout the disk may be taken = p', and the integral becomes
„ _ , r dx ... dz
''J {a^+...+i^+H^]i'+9'
where instead of p' I write p ; viz. p now denotes the density at the point N.
Making this change, then (by what precedes) the value is
2(r^)' r/t 7-^'dr
~''r(^s)Jo{r» + «•]»•+«•
* « is positive; in afterwards writing « =0, we mean by 0 the limit of an indefinitely small positive
qaantity.
607] A MEMOIR ON PREPOTENTIALS. 323
q = Positive. Art. Nos. 3 to 7.
3. I consider first the case where q is positive. The value is here
~^ V (is) 2«^9 |r (is + q) Jo (1+ «)*'+»
or, since -^ is indefinitely small, the a;-integral maj' be neglected, and the value is
«=» ^ r (is + q) '
Observe that this value is independent of R, and that the expression is thus the
same as if (instead of the disk) we had taken the whole of the infinite tangent-plane,
the density at every point thereof being = p. It is proper to remark that the neglected
terms are of the orders
(PA)* To
so that the complete value multiplied by »"' is equal to the constant p pyr ^ + terms
of the orders l^j , (pj , &c.
i. Let us now consider the prepotential of the remaining portion of the surface ;
every part thereof is at a distance from P exceeding, in fact far exceeding, R; so
that imagining the whole mass i pdS to be collected at the distance R, the pre-
potential of the remaining portion of the surface is less than
jpdS
R'+^g '
viz. we have thus, in the case where the mass jpdS is finite, a superior limit to the
prepotential of the remaining portion of the surface. This will be indefinitely small
in comparison with the prepotential of the disk, provided only «^ is indefinitely small
compared with ii'+^, that is, « indefinitely small in comparison with R ^. The proof
assumes that the mass \pd8 is finite ; but considering the very rough manner in which
[pdS
the limit p^ was obtained, it can scarcely be doubted that, if not universally, at
least for very general laws of distribution, even when \pd& is infinite, the same thing
is true; viz. that by taking « sufficiently small in regard to R, we can make the
41—2
324 A MEMOIR ON PREPOTENTIALS, [607
prepotential of the remaining portion of the surface vanish in comparison with that
of the disk. But without entering into the question I assume that the prepotential
of the remaining portion does thus vanish ; the prepotential of the whole surface in
regard to the indefinitely near point P is thus equal to the prepotential of the disk ;
viz. its value is
- 1 „ (r^yrg
which, observe, is infinite for a point P on the surface.
5. Considering the prepotential V at an arbitrary point (a, . . , c, e) as a given
function of (a, . . , c, e) the coordinates of this point, and taking {x, .., z, w) for the
coordinates of the point N, which is, in fact, an arbitraiy point on the surface, then the
value of V at the point P indefinitely near to N will be = TT, if W denote the same
function of (x, .., z, w) that V is of (a, . . , c, e). The result just obtained is therefore
or, what is the same thing,
As to this, remark that V is not an arbitrary function of (a, . . , c, e) : non constat
that there is any distribution of matter, and still less that there is any distribution
of matter on the surface, which will produce at the point (a, . ., c, e), that is, at every
point whatever, a prepotential the value of which shall be a function assumed at
pleasure of the coordinates (a, .. , c, e). But suppose that V, the given function of
(a, . . , c, e), is such that there does exist a corresponding distribution of matter on the
surface, (viz. that V satisfies the conditions, whatever they are, required in order that
this may be the case), then the foregoing formtila determines the distribution, viz. it
gives the expression of p, that is, the density at any point of the surface.
6. The theorem may be presented in a somewhat diflferent form ; regarding the
prepotential as a function of the normal distance «, its derived function in regard
to u is
2q (r^y Tq
that is,
L_ 2(ri)^r(9 + i).
„2?+i P T{is + q) '
and we thus have
dw_ 1 2(r^yr{q+i)
ds ~ «»«+■'' Tiis + q) ' '' '''
or, what is the same thing,
p = — ^ . — ::^-,'--^ I »^
2(ri)'f(j+i) V d«;«-.'
607] A MEMOIR ON PREPOTENTIALS. 325
dW
where, however, W being given as a function of {x, .., z, w), the notation , requires
CtrW
explanation. Taking cos a, . . , cos y to be the inclinations of the normal at N, in the
direction NP in which the distance « is measured, to the positive parts of the axes
of {x, .., z), viz. these cosines denote the values of
dS dS
dx"" dz'
each taken with the same sign + or — , and divided by the square root of the sum
of the squares of the last-mentioned quantities, then the meaning is
dW dW dW
-i— = -^r- cos «+...+ -7— cos 7.
dn dx dz
7. The surface S may be the plane w = 0, viz. we have then the prepotential-
plane integral
V-i pdx...dz
J {{a-xy + ...+(c-zy + ^}i'-<' ^^''•
where e (like «) is positive. In afterwards writing e = 0, we mean by 0 the limit of
an indefinitely small positive quantity.
The foregoing distribution-formulae then become
'> = Wrf^^'^'- ^^>'
and
P 2{nyr{q+l)V deJe^o ^ ''
which will be used in the sequel.
It will be remembered that in the preceding investigation it has been assumed
that q is positive, the limiting case q = 0 being excludedf .
q = -J£. Art. Nos. 8 to 13,
8. I pass to the case q = —^, viz. we here have the potential-surface integral
J {{a- xy + ... + {c - zf + {e -wYji'-i ^^'
it will be seen that the results present themselves under a remarkably different form.
The potential of the disk is, as before,
2(r^y r r^'dr
P r^s j (r» + «»)*«-*'
t This is, as regards q, the case thronghout ; a limiting value, if not expressly stated to be included, is
always excluded.
326 A MEMOIE ON PREPOTENTIALS. [607
where p here denotes the density at the point N; and the value of the r-integral
= iJ (l + terms in ^, ^„ ...)-« j,^|iIL.
Observe that this is indefinitely small, and remains so for a point P on the surface;
the potential of the remaining portion of the surface (for a point P near to or on
the surface) is finite, that is, neither indefinitely large nor indefinitely small, and it
varies continuously as the attracted point passes through the disk (or aperture in the
material surface now under consideration); hence the potential of the whole surface
is finite for an attracted point P on the surface, and it varies continuously as P
passes through the surface.
It will be noticed that there is in this case a term in V independent of « ;
and it is on this account necessary, instead of the potential, to consider its derived
function in regard to «; viz. neglecting the indefinitely small terms which contain
powers of p, I write
dV ^_ 2 (riy+'
d«~ r(^-s-i)^-
The corresponding term arising from the potential of the other portion of the
surface, viz. the derived function of the potential in regard to «, is not indefinitely
small ; and calling it Q, the formula for the whole surface becomes
d« *^ r(i5-i)^-
9. I consider positions of the point P on the two opposite sides of the point N,
say at the normal distances «', «", these being positive distances measured in opposite
directions from the point N. The function V, which represents the potential of the
surface in regard to the point P, is or may be a different function of the coordinates
(a, . . , c, e) of the point P, according as the point is situate on the one side or the
other of the surface (as to this more presently). I represent it in the one case by
V, and in the other case by V" ; and in further explanation state that »' is measured
into the space to which V refers, a" into that to which V" refers ; and I say that
the formulae belonging to the two positions of the point P are
dw _ry _^<nr\
rf«' -^ r(js-l)'''
dW" _ 2 {Ti^r' ,
dn"-"^ VXhs-kV'
where, instead of V, V", I have written W, W", to denote that the coordinates, as
well of P* as of P", are taken to be the values {x, .. , z, w) which belong to the
point N. The symbols denote
dW dW . , dW
T-7 =^}— cosa + ... + — J— cosy,
da dx dz
dW" dW" „ , dW"
d7- = -d^^°^« +-+-d.^''^'y'
607] A MEMOIR ON PREPOTENTIALS. 327
where (cos a', . . , cos 7') and (cos a", . . , cos 7") are the cosine-inclinations of the normal
distances «', «" to the positive parts of the axes of {x, .., z); since these distances are
measured in opposite directions, we have cos o" = — cos a', . . , cos 7" = — cos 7'. If we
imagine a curve through N cutting the surface at right angles, or, what is the same
thing, an element of the curve coinciding in direction with the normal element P'NP",
and if s denote the distance of N from a fixed point of the curve, and for the point
P' if s become s -(- S's, while for the point P" it becomes s — h"s, or, what is the same
thing, if s increase in the direction of NF and decrease in that of NP", then if any
function 0 of the coordinates (a;, . . , z, w) of N be regarded as a function of s, we
have
ds ds' ' ds ds"'
10. In particular, let 0 denote the potential of the remaining portion of the
surface, that is, of the whole surface exclusive of the disk ; the curve last spoken of
is a cui"ve which does not pass through the material surface, viz. the portion to which
0 has reference : and there is no discontinuity in the value of 0 as we pass along
this curve through the point N. We have Q' = value of j 1 ^.t the point P', and
d%
Q" = value of -v->, at the point P" ; and the two points P', P" coming to coincide
together at the point N, we have then
d0
^ - ds"
rf0
~ ds'
^ ~ ds" •
d0
~ da'
dW dW
ds' ds
dW" dW"
ds" ds
dw de
ds ds
2(riy+'
r(is-i)'''
dW" _ d0
ds ds
2 ir^y+'
We have in like manner -j~r = ^^— , —^ „ = ;77~ i ^^^ *^® equation obtained
above may be written
in which form they show that as the attracted point passes through the surface from
the position P' on the one side to P" on the other, there is an abrupt change in
dW dV
the value of -r- , or say of , , the first derived function of the potential in regard
to the orthotomic arc a, that is, in the rate of increase of V in the passage of the
attracted point normally to the surface. It is obvious that, if the attracted point
traverses the surface obliquely instead of normally, viz. if the arc s cuts the surface
dV
obliquely, there is the like abrupt change in the value of t- .
828 A MEMOIR ON PREPOTENTIALS. [607
Reverting to the original form of the two equations, and attending to the relation
<y+Q" = 0, we obtain
dW' dW"_-ijr^y+'
d>i' ■•" ds" - r(i«-i)'''
or, what is the same thing,
11. I recall the signification of the symbols: — V, V" are the potentials, it may
be different functions of the cooixlinat.es (a, . . , c, e) of the attracted point, for positions
of this point on the two sides of the surface (as to this more presently) : and W, W"
are what V, V" respectively become when the coordinates (a, . ., c, e) are replaced by
(ar, . . , z, w), the coordinates of a point N on the surface. The explanation of the
dW' dW"
symbols -y-r , , „ is given a little above; p denotes the density at the point (x,..,z, w).
€f/ti uti
12. The like remarks arise as with regard to the former distribution theorem (A);
the functions V, V" cannot be assumed at pleasure ; non constat that there is any
distribution in space, and still less any distribution on the surface, which would give
such values to the potential of a point (a, . . , c, e) on the two sides of the surface
respectively ; but assuming that the functions V, V" are such that they do arise from
a distribution on the surface, or say that they satisfy all the conditions, whatever they
are, required in order that this may be so, then the formula determines the distri-
bution, viz. it gives the value of p, the density at a point (x, .. , z, w) of the surface.
13. In the case where the surface is the plane w = 0, viz. in the case of the
potential-plane integral,
Y-{ pdic.dz
j [{a-xy+...+{c-z)'+^]i^ ^ ''
(e assumed to be positive) ; then, since the conformation is symmetrical on the two sides
of the plane, V and V" are the same functions of {a, .., c, e), say they are each = V ;
W, W" are each of them the same function, say they are each = W, of (a;, . . , z, e)
that V is of (a, . . , c, e) ; the distribution-formula becomes
2(ri)«+' UcA-o ^ ^'
Kr^)*
viz. this is also what one of the prepotential-plane formulae becomes on writing therein
q = 0, or Negative. Art, Nos. 14 to 18.
14. Consider the case q = 0. The prepotential of the disk is
607] A MEMOIR ON PREPOTENTIALS. 329
to get rid of the constant term we must consider the derived function in regard to «,
viz. this is
2{r^y 1
-P
and we have thus for the whole surface
I'is ■
»
where Q, which relates to the remaining portion of the surface, is finite ; we have thence,
writing, as before, W in place of V,
dw__ 2(r^y
or say
r^s f dW\
15. Consider the case q negative, but —q<^. The prepotential of the disk is here
P r> 1-29 + *'' T(is + q)+-V
to get rid of the first term we must consider the derived function in regard to «,
viz. this is
whence, for the potential of the whole surface,
dF_ 2(riyr(9+2)
where Q, the part relating to the remaining portion of the surface, is finite. Multiplying
by B^+' (where the index 2q + l is positive), the term in Q disappears ; and writing,
as before, W in place of V, this is
dTf^_ 2(ri)V[M9+l)
dn '^ V\8 + q
or, say
'' 2(ri)'r(g + i) r d«/«=o'
viz, we thus see that the formula (A*) originally obtained for the case q positive
extends to the case 9 = 0, and q = — but — q<^', but, as already seen, it does not
extend to the limiting case q= — ^■
16. If 9 be negative and between —\ and —1, we have in like manner a formula
dv 2(ri)'r(g + i)
d7-*^ C V{^a + q)
C. IX. 42
330 A MEMOra ON PREPOTENTIALS. [607
but here, 2q + l being negative, the term «'«+■ Q does not disappear : the formula has
to be treated in the same way as for 5 = —^, and we arrive at
viz. the formula is of the same form as for the potential case q = — ^. Observe that
the formula does not hold good in the limiting case q = — 1.
17. We have, in fact, for q = — 1, the potential of the disk
whence
since, in the complete diflferential coefficient » + 2« log «, the term » vanishes in com-
parison with 2« log B. Then, proceeding as before, we find
dW 1 dW" _ -8(r|>
s'logB^ d«'"^8"iog«" d«" r(^s-i)^'
but I have not particularly examined this formula.
18. If q be negative and > — 1 (that is, —q> 1), then the prepotential for the
disk is
ns
-2q 1 -2^-2
and it would seem that, in order to obtain a result, it would be necessary to proceed
to a derived function higher than the first ; but I have not examined the case.
Continuity of the Prepotential-surface Integral. Art. Nos. 19 to 25.
19. I again consider the prepotential-surface integral
pdS
/
{(a- xf + ... -t- {c - zy + {e - wf\<"-^
in regard to a point (a,.., c, e) not on the surface ; q is either positive or negative,
as afterwards mentioned.
The integral or prepotential and all its derived functions, first, second, &c. ad
infinitum, in regard to each or all or any of the coordinates (a, . . , c, e), are all finite.
This is certainly the case when the mass I pdS is finite, and possibly in other cases
also; but to fix the ideas we may assume that the mass is finite. And the pre-
potential and its derived functions vary continuously with the position of the attracted
607] A MEMOIR ON PREPOTENTIALS. 331
point (a, . . , c, e), so long as this point in its course does not traverse the material
surface. For greater clearness we may consider the point as moving along a continuous
curve (one-dimensional locus), which curve, or the part of it under consideration, does
not meet the surface ; and the meaning is that the prepotential and each of its
derived functions vary continuously as the point (a, . . , c, e) passes continuously along
the curve.
20. Consider a "region," that is, a portion of space any point of which can be,
by a continuous curve not meeting the material surface, connected with any other
point of the region. It is a legitimate inference, from what just precedes, that the
prepotential is, for any point (a, . . , c, e) whatever within the region, one and the same
function of the coordinates (a, . . , c, e), viz. the theorem, rightly understood, is true ;
but the theorem gives rise to a difficulty, and needs explanation.
Consider, for instance, a closed surface made up of two segments, the attracting
matter being distributed in any manner over the whole surface (as a particular case
s + 1 = 3, a uniform spherical shell made up of two hemispheres) ; then, as regards
the first segment (now taken as the material surface), there is no division into regions,
but the whole of the (s + l)-dimensional space is one region; wherefore the prepotential
of the first segment is one and the same function of the coordinates (a, . . , c, e) of the
attracted point for any position whatever of this point. But in like manner the
prepotential of the second segment is one and the same function of the coordinates
(a, . . , c, e) for any position whatever of the attracted point. And the prepotential of
the whole surface, being the sum of the prepotentials of the two segments, is
consequently one and the same function of the coordinates (a, . . , c, e) of the attracted
point for any position whatever of this point ; viz. it is the same function for a
point in the region inside the closed surface and for a point in the outside region.
That this is not in general the case we know from the particular case, s + 1 = 3, of
a uniform spherical shell referred to above.
21. Consider in general an unclosed surface or segment, with matter distributed
over it in any manner; and imagine a closed curve or circuit cutting the segment
once ; and let the attracted point (a, . . , c, e) move continuously along the circuit. We
may consider the circuit as corresponding to (in ordinary tridimensional space) a plane
curve of equal periphery, the corresponding points on the circuit and the plane curve
being points at equal distances s along the curves from fixed points on the two
curves respectively; and then treating the plane curve as the base of a cylinder, we
may represent the potential a.s a length or ordinate, V = y, measured upwards from
the point on the plane curve along the generating line of the cylinder, in such wise
that the upper extremity of the length or ordinate y traces out on the cylinder a
curve, say the prepotential curve, which represents the march of the prepotential.
The attracted point may, for greater convenience, be represented as a point on the
prepotential curve, viz. by the upper instead of the lower extremity of the length or
ordinate y; and the ordinate, or height of this point above the base of the cylinder,
then represents the value of the prepotential. The before-mentioned continuity-theorem
is that the prepotential curve, corresponding to any portion (of the circuit) which
42—2
332 A MEMOra ON PREPOTENTIALS. [607
does not meet the material surface, is a continuous curve: viz. that there is no abi-upt
change of value either in the ordinate y {=V) of the prepotential curve, or in the
first or any other of the derived functions ^ , -j~ , &c. We have thus (in each of
the two figures) a continuous curve as we pass (in the direction of the arrow) from
a point P on one side of the segment to a point P" on the other side of the
segment; but this continuity does not exist in regard to the remaining part, from
P" to P", of the prepotential curve corresponding to the portion (of the circuit)
which traverses the material surface.
22. I consider first the case 5f = — J (see the left-hand figure) : the prepotential
is here a potential. At the point N, which corresponds to the passage through the
material surface, then, as was seen, the ordinate y (= the Potential F) remains finite
d'u
and continuous; but there is an abrupt change in the value of -v , that is, in the
direction of the curve : the point N is really a node with two branches crossing at
this point, as shown in the figure; but the dotted continuations have only an analytical
existence, and do not represent values of the potential. And by means of this branch-
to-branch discontinuity at the point N, we escape from the foregoing conclusion as to
the continuity of the potential on the passage of the attracted point through a closed
surface.
23. To show how this is, I will for greater clearness examine the case (s-t-l) = 3,
in ordinary tridimensional space, of the uniform spherical shell attracting according to
the inverse square of the distance ; instead of dividing the shell into hemispheres, I
divide it by a plane into any two segments (see the figure, wherein A, B represent
the centres of the two segments respectively, and where for graphical convenience the
segment A is taken to be small).
We may consider the attracted point as moving along the axis xx', viz. the two
extremities may be regarded as meeting at infinity, or we may outside the sphere
bend the line round, so as to produce a closed circuit. We are only concerned with
what happens at the intersections with the spherical surface. The ordinates represent
the potentials, viz. the curves are a, b, c for the segments A, B, and the whole
spherical surface respectively. Practically, we construct the curves c, a, and deduce the
curve h by taking for its ordinate the difference of the other two ordinates. The
curve c is, as we know, a discontinuous curve, composed of a horizontal line and two
hyperbolic branches ; the curve a can be laid down approximately by treating the
segment .4 as a plane circular disk ; it is of the form shown in the figure, having
a node at the point corresponding to A. (In the case where the segment A is
607]
A MEMOIR ON PREPOTENTIALS.
333
actually a plane disk, the curve is made up of portions of branches of two hyperbolas ;
but taking the segment A as being what it is, the segment of a spherical surface,
the curve is a single curve, having a node as mentioned above.) And from the
curves c and a, deducing the curve h, we see that this is a curve without any
discontinuity corresponding to the passage of the attracted point through A (but with
an abrupt change of direction or node corresponding to the passage through B). And
conversely, using the curves a, h to determine the curve c, we see how, on the passage
of the attracted point at A into the interior of the sphere, in consequence of the
branch-to-branch discontinuity of the curve a, the curve c, obtained by combination
of the two curves, undergoes a change of law, passing abruptly from a hyperbolic to
a rectilinear form, and how similarly on the passage of the attracted point at B from
the interior to the exterior of the sphere, in consequence of the branch-to-branch
discontinuity of the curve h, the curve c again undergoes a change of law, abruptly
reverting to the hyperbolic form.
24. In the case q positive, the prepotential curve is as shown by the right-hand
figure on p. 332, viz. the ordinate is here infinite at the point N corresponding to
the passage through the surface ; the value of the derived function changes between
+ infinity and — infinity ; and there is thus a discontinuity of value in the derived
function. It would seem that, when q is fractional, this occasions a change of law
on passage through the surface : but that there is no change of law when q is
integral.
In illusti-ation, consider the closed surface as made up of an infinitesimal circular
disk, as before, and of a residual portion; the potential of the disk at an indefinitely
near point is found as before, and the prepotential of the whole surface is
334 A MEMOIR ON PREPOTENTIALS. [607
where F,, the prepotential of the remaining portion of the surface, is a function which
varies (and its derived functions vary) continuously as the attracted point traverses
the disk. To fix the ideas, we may take the origin at the centre of the disk, and
the axis of e as coinciding with the normal, so that «, which is always positive, is
= ± e ; the expression for the prepotential at a point (a, . . , c, e) on the normal through
the centre of the disk is
viz. when q is fractional there is the discontinuity of law, inasmuch as the term
changes from ^ ^ to ^ — : but when q is integral this discontinuity disappears. The
(+ e)^ (— e) 9
like considerations, using of course the proper formula for the attraction of the disk,
would apply to the case ^ = 0 or negative.
25. Or again, we might use the formulae which belong to the case of a uniform
(s+ l)-coordinal spherical shell (see Annex No. III.), viz. we decompose the surface
as follows,
surface = disk + residue of surface ;
and then, considering a spherical shell touching the surface at the point in question
(so that the disk is, in fact, an element common to the surface and the spherical
shell), and being of a uniform density equal to that of the disk, we have
disk = spherical shell — residue of spherical shell ;
and consequently
surface = spherical shell — residue of spherical shell + residue of surface ;
and then, considering the attracted point as passing through the disk, it does not
pass through either of the two residues, and there is not any discontinuity, as regards
the prepotentials of these residues respectively; there is consequently, as regards the
prepotential of the surface, the same discontinuity that there is as regards the
prepotential of the spherical shell. But I do not further consider the question from
this point of view.
The Potential Solid Integral. Art. No. 26.
26. We have further to consider the prepotential (and in particular the potential)
of a material space ; to fix the ideas, consider for the moment the case of a
distribution over the space included within a closed surface, the exterior density being
zero, and the interior density being, supposed for the moment, constant; we consider
the discontinuity which takes place aa the attracting point passes from the exterior
space through the bounding surface into the interior material space. We may imagine
the interior space divided into indefinitely thin shells by a series of closed surfaces
similar, if we please, to the bounding surface ; and we may conceive the matter
included between any two consecutive surfaces as concentrated on the exterior of the
607] A MEMOIR ON PREPOTENTIALS. 335
two surfaces, so as to give rise to a series of consecutive material surfaces ; the
quantity of such matter is infinitesimal, and the density of each of the material surfaces
is therefore also infinitesimal. As the attracted point comes from the external space
to pass through the first of the material surfaces — suppose, to fix the ideas, it moves
continuously, along a curve the arc of which measured from a fixed point is = s — there
is in the value of V (or, as the case may be, in the values of its derived functions
dV
-J- , &c.) the discontinuity due to the passage through the material surface ; and the
like as the attracted point passes through the different material surfaces respectively.
Take the case of a potential, q = — -^', then, if the surface-density were finite, there
would be no finite change in the value of V, but there would be a finite change
dV
in the value of -j—; as it is, the changes are to be multiplied by the infinitesimal
density, say p, of the material surface; there is consequently no finite change in the
value of the first derived function ; but there is, or may be, a finite change in the
value of -V. and the higher derived functions. But there is in V an infinitesimal
change corresponding to the passage through the successive material surfaces respectively;
that is, as the attracted point enters into the material space, there is a change in
the law of V considered as a function of the coordinates (a, . . , c, e) of the attracted
point; but by what precedes this change of law takes place without any abrupt
change of value either of V or of its first derived function ; which derived function
may be considered as representing the derived function in regard to any one of the
coordinates a, .. ,c, e. The suppositions, that the density outside the bounding surface
was zero and inside it constant, were made for simplicity only, and were not essential ;
it is enough if the density, changing abruptly at the bounding surface, varies con-
tinuously in the material space within the bounding surface*. The conclusion is that
V, V" being the values at points within and without the bounding surface, V and
V" are in general different functions of the coordinates (a, . . , c, e) of the attracted
point ; but that at the surface we have not only V = V", but that the first derived
functions are also equal, viz. that we have
dT^dW^ dV' ^dV'^ dV'^dr^
da da '"' dc dc ' de de '
27. In the general case of a Potential, we have
y_f pdx ...dzdw
~J {(a-xy+ ... + (c-zy + {e- wy}i»-i'
If p does not vanish at the attracted point (a, . . , c, e), but has there a value p'
different from zero, we may consider the attracting (s + l)-dimensional mass as made
* It is, indeed, enough if the density varies continuously within the bounding surface in the neighbourhood
of the point of passage through the surface; but the condition may without loss of generality be stated as
in the text, it being understood that for each abrupt change of density within the bounding surface we must
consider the attracted point as passing through a new bounding surface, and have regard to the resulting
discontinuity.
336 A MEMOIR ON PREPOTENTIALS. [607
up of an indefinitely small sphere, i-adius e and density p, which includes within it
the attracted point, and of a remaining portion external to the attracted point.
Writing V to denote -3— „ + • • • + j"a + j^ > tben, as regards the potential of the sphere,
we have ^^= — fTTi r^p' (see Annex III. No. 67), and as regards the remaining
portion V F = 0 ; hence, as regards the whole attracting mass, V Y has the first-
mentioned value, that is, we have
where />' is the same function of the coordinates (a, . . , c, e) that p is of {x, . . ,z, w);
viz. the potential of an attracting mass distributed not on a surface, but over a
portion of space, does not satisfy the potential equation
fd^ dr d-
\da''^'"'^dc-'^ d^.
)v=o.
but it satisfies the foregoing equation, which only agrees with the potential equation
in regard to a point (a, . . , c, e) outside the material space, and for which, therefore,
p' is =0.
The equation may be written
'^ 4 (ri)»+' Ua' de ^de'J '
or, considering F as a given function of (a, . . , c, e), in general a discontinuous
function but subject to certain conditions as afterwards mentioned, and taking W the
same function of (x,.., z, w) that F is of (a, . . , c, e), then we have
4(r|)»+' V(^
viz. this equation determines p as a, function, in general a discontinuous function, of
(x,. . , z, w) such that the corresponding integral
~J{(a-
pdx ...dz dw
{{a-xf+...+{c- zf + (e - w)»|4»+«
may be the given function of the coordinates (a, . . , c, e). The equation is, in fact,
the distribution-theorem D.
28. It is to be observed that the given function of (a, . . , c, e) must satisfy
certain conditions as to value at infinity and continuity, but it is not (as in the
distribution-theorems A, B, and C it is) required to satisfy a partial differential
equation ; the function, except as regards the conditions as to value at infinity and
continuity, is absolutely arbitrary.
607] A MEMOIR ON PREPOTENTIALS. 337
The potential (assuming that the matter which gives rise to it lies wholly within
a finite closed surface) must vanish for points at an infinite distance : or, more
accurately, it must for indefinitely large values of a''+ ... -\- c^ + ^ be of the form,.
Constant -7- (a^ +...+ C-+ e^)i*~*. It may be a discontinuous function; for instance,
outside a given closed surface it may be one function, and inside the same surface a
different function of the coordinates (a, . . , c, e) ; viz. this may happen in consequence
of an abrupt change of the density of the attracting matter on the one and the
other side of the given closed surface, but not in any other manner; and, happening
in this manner, then V and V" being the values for points within and without the
surface respectively, it has been seen to be necessary that, at the surface, not only
_ _, , ^ , dV dV" dV dV" dV dV" „ , . , , ,, ,.,.
V =V , but also -J— = — ; — ,.., , = V , 5^ = , . Subiect to these conditions as
da da dc dc de de "^
to value at infinity and continuity, V may be any function whatever of the coordinates
(a, . . , c, e) ; and then taking W, the same function of (x, . . ,z, w), the foregoing
equation determines p, viz. determines it to be =0 for those parts^ of space which
do not belong to the material space, and to have its proper value as a function of
{x, .. ,z, w) for the remaining or material space.
The Prepotential-Plane Theorem A. Art. Nos. 29 to 36.
29. We have seen that, if there exists on the plane w = 0 a distribution of
matter producing at the point (a, . . , c, e) a given prepotential V — viz. V is to be
regarded as a given function of (a, . . , c, e) — , then the distribution or density p is
given by a determinate formula; but it was remarked that the prepotential V cannot
be a function assumed at pleasure : it must be a function satisfying certain conditions.
One of these is the condition of continuity ; the function V and all its derived
functions must vary continuously as we pass, without traversing the material plane,
from any given point to any other given point. But it is sufiicient to attend to
points on one side of the plane, say the upperside, or that for which e is positive ;
and since any such point is accessible from any other such point by a path which
does not meet the plane, it is sufficient to say that the function V must vary
continuously for a passage by such path from any such point to any such point ;
the function V must therefore be one and the same function (and that a continuous
one in value) for all values of the coordinates {a,.., c) and positive values of the
coordinate e.
If, moreover, we assume that the distribution which corresponds to the given
potential F is a distribution of a finite mass \ pdx ...dz over a finite portion of the
plane w = 0, viz. over a portion or area such that the distance of a point within the
area from a fixed point, or say from the origin (a, . . , c) = (0, . . , 0), is always finite ;
this being so, we have the further condition that the prepotential V must, for in-
definitely lai-ge values of all or any of the coordinates (a, , . , c, e), reduce itself to the
form
'' pdx ... dz] -=- (a= + ... + c= + e=)*'+?.
a^
c. rx. 43
338 A MEMOIR ON PREPOTENTIAIA [607
The assumptions upon which this last condition is obtained are perhaps unnecessary;
instead of the condition in the foregoing form we, in fact, use only the condition that
the prepotential vanishes for a point at infinity, that is, when all or any one or more
of the coordinates (a, ... c, e) are or is infinite.
Again, as we have seen, the prepotential V must satisfy the prepotential equation
(d^ ^ d" 2q + l d\^_
\da''^--'^dc''^dd''^ e rfej
These conditions satisfied, to the given prepotential V there corresponds, on the
plane w=0, a distribution given by the foregoing formula ; it will be a distribution
over a finite portion of the plane, as already mentioned.
30. The proof depends upon properties of the prepotential equation
or, what is the same thing,
dx\ dx ) '" dz\ dz j de\ de )
say, for shortness, D TT = 0.
Consider, in general, the integral
/,....,...e»«|(f)%....(f)%(f)].
taken over a closed surface S lying altogether on the positive side of the plane c = 0,
the function W being in the first instance arbitrary.
Writing the integral under the form
fj J J ( .^^.dW dW ^„^,dW dW ^ ^.^dW dW\
j \ dx da; dz dz de de J
we reduce the several terms by an integration by parts as follows: —
The term in '^^ is ={dy...dzdeW^'>^'^-\dx...dzdeW^ {^*^^S) '
= jdx deW^+''^J^-jd^...dzdeW^^ (^^'w)'
= jdx dzWe'i^'^^-jda:...dzdeW~(^+'^^y
dW
dz
607] A MEMOIR ON PREPOTENTIALS. 339
Write dS to denote an element of surface at the point {x, .. , z, e). Then taking
a, . . , 7, S to denote the inclinations of the interior normal at that point to the positive
axes of coordinates, we have
dy ... dzde= — dS cos a,
da; de = — dS cosy,
dx dz = — dS cos B;
and the first terms are together
f o„+, ^^r/dW , dW dW .\ j„
= _ Ig2?+i ^/ cos a 4-... + -^ cos 7 + -^ cos 8] d/S,
W here denoting the value at the surface, and the integration being extended over
the whole of the closed surface : this may also be written
where « denotes an element of the internal normal.
The second terms are together
==-jda;..dzd^w\^ {f^*'-Jx) + - +i {^^^''^'S)'^Te{^''^''^de)h~l'^--'^"^^°^-
We have consequently
= _ fe-2,+i W^J^dS-jdx ... dzdee^i^' WU W.
31. The second term vanishes if W satisfies the prepotential equation D TT = 0 ;
and this being so, if also W =0 for all points of the closed surface S, then the first
term also vanishes, and we therefore have
where the integration extends over the whole space included within the closed surface ;
whence, W being a real function,
dx "■• dz ' de '
for all points within the closed surface ; consequently, since W vanishes at the surface,
W=0 for all points within the closed surface.
32. Considering W as satisfying the equation D W = 0, we may imagine the closed
surface to become larger and larger, and ultimately infinite, at the same time flattening
43—2
340 A MEMOIR ON PREPOTENTIALS. [607
itself out into coincidence with the plane e = 0, so that it comes to include the whole
space above the plane e = 0 : say the surface breaks up into the surface positive infinity
and the infinite plane e = 0.
The integral 1 6^+' W ^ dS separates itself into two parts, the first relating to
the surface positive infinity, and vanishing if W = 0 at infinity (that is, if all or
any of the coordinates x, .. , z, e are infinite) ; the second, relating to the plane e = 0,
is I W{^'*'^—T-jdx...dz, W here denoting its value at the plane, that is, when e = 0,
And the integral being extended over the whole plane. The theorem thus becomes
/^...^*..»..{(i?)V....(<^?)V(«)'} = -/r(^.^J)^...^.
Hence also, if ir=0 at all points of the plane e = 0, the right-hand side vanishes,
and we have
/-•••— ■{(f)"--(fr-©]--
dW dW dW
Consequently , = 0, . . , -j— = 0, -j- = 0, for all points whatever of positive space ; and
therefore also W = 0 for all points whatever of positive space.
33. Take next U, W, each of them a function of (x, . ., z, e), and consider the
integral
fdx...dzde.^^^(f'^^...^fi'' + fm,
J Vote dx dz dz de de J
taken over the space within a closed surface S; treating this in a similar manner, we
find it to be
= _ fe^+i w'^—dS-jdx... dzde.e'^' WUU,
where the integration extends over the whole of the closed surface S ; and by parity
of reasoning it is also
= - fe°-«+' U^J^dS-jdx...dzde.^+' UOW,
^vith the same limits of integration ; that is, we have
j d^+'W^^ dS+j dx ...dzde.^+' WDU= j ^+' U^^ dS + j dx ... dzde.^>9+^ UnW,
which, if U and W each satisfy the prepotential equation, becomes
And if we now take the closed surface S to be the surface positive infinity, together
with the plane e = 0, then, provided only U and V vanish at infinity, for each integral
607] A MEMOIR ON PREPOTENTIALS. 341
the portion belonging to the surface positive infinity vanishes, and there remains only
the portion belonging to the plane e = 0 ; we have therefore
Je^+i W^da . . . dz=^je^+' ^^^'^ ■■•^^'
where the functions TJ, W have each of them the value belonging to the plane e = 0 :
viz. in U, W considered as given functions of («,.., z, e) we regard e as a positive
quantity ultimately put = 0 ; and where the integrations extend each of them over the
whole infinite plane.
34. Assume
{(a-a;f + ... +(c-^)'' + e=}i»+9'
an expression which, regarded as a function of {x, .., z, e), satisfies the prepotential
equation in regard to these variables, and which vanishes at infinity when all or any
of these coordinates (x, .. , z, e) are infinite.
We have
de {{a-xy + ... + {c-zf + e'}i'+i+^''
and we have consequently
(W -2as + q)^-' ^^_f(^.,dW\ dx...dz
where it will be recollected that e is ultimately = 0 ; to mark this, we may for W
write Wo.
Attend to the left-hand side ; take V„ the same function of a, . . ,c, e = 0, that W^
is of X,. . , z, e = 0 ; then, first writing the expression in the form
y r -2(^s + q)^9+'da;...dz
'j {(a-xY+...+(c-zy + e^\i»+9+^'
write x = a + e^,.. ,z = c + e^, the expression becomes
_y [-2{^s + q)e-'^Ke^d^...d^ __2as + o)v( ^^-^^
°i {e= (1 + ^ + ... + f^)jl«+?+> • ^ ^** ^ ^^ "^'V {1 + |2 + ..:+^]¥+q+^ '
where the integral i.s to be taken from — x to + oo for each of the new variables
Writing f = ra, .., 5'=r7, where 0" + ... +7^ = 1, we have d^ ... d^^r'-' drdS: also
f + . . . + f = ^. and the integral is
^ ^\}»+9+l >
342 A MEMOIR ON PREPOTENTIALS. [607
where I dS denotes the surface of the s-coordinal unit sphere a' + ... + y = l, and the
r-integral is to be taken from r=0 to »• = oo ; the values of the two factors thus are
fds-^^^ and [ ^"' '^^ _^r^^r(g + i)
j "^ r (is) • * ° j (1 + r^)»'+9+' - r (is + 3 + 1) •
Hence the expression in question is
and we have
/(
^^^dw\ dx...dz ^-2(ri)'r(g + i)
efe/o {(a -«;)= + ... + (c-^)-^ + ef '+« r(is + 3) "•
or, what is the same thing,
r -Tjis + q) / dW\
V = 2(ri)''r(g + i)r delr-"^'
' j [{a-xf-i- ...+(c-zf + ef'\^-^9 •
35. Take now V a function of (a, .. , c, e) satisfying the prepotential equation in
regard to these variables, always finite, and vanishing at infinity ; and let W be the
same function of («,.., z, e), W therefore satisfying the prepotential equation in regard
to the last-mentioned variables. Considei- the function
V-
2(ri)'r(g + i)r deJo*"-^^
V=i pdx...dz
{{a-xy + ...+{c-zy + e']i'+''
where the integral is taken over the infinite plane e = 0; then this function (F— the
integral) satisfies the prepotential equation (for each term separately satisfies it), is
always finite, and it vanishes at infinity. It also, as has just been seen, vanishes for
any point whatever of the plane e = 0. Consequently it vanishes for all points whatever
of positive space. Or, what is the same thing, if we write
< dx ... dz
\{a - xf + ... + {c-zy + e")*
where p is a function of (x, .., z), and the integral is taken over the whole infinite
plane, then if F is a function of (a, . . , c, e) satisfying the above conditions, there
exists a corresponding value of p ; viz. taking W the same function of («,.., z, e)
which F is of (a, . . , c, e), the value of p is
dW
where e is to be put =0 in the function e^*+' -^ . This is the prepotential-plane
theorem ; viz. taking for the prepotential in regard to a given point (a, . . , c, e) a
function of (a, . . , c, e) satisfying the prescribed conditions, but otherwise arbitrary,
there exists on the plane e = 0 a distribution p given by the last-mentioned formula.
607] A MEMOIR ON PREPOTENTIALS. 343
36. It is assumed in the proof that 2q + l is positive or zero ; viz. q is positive,
or if negative then — qlf' ^ ; the limiting case q= — ^ is included.
It is to be remarked that, by what precedes, if q be positive (but excluding the
case q = 0), the density p is given by the equivalent more simple formula
r(^s + q)
^~ {nyvq ^^ ^>''-
The foregoing proof is substantially that given in Green's memoir on the Attraction
of Ellipsoids ; it will be observed that the proof only imposes upon V the condition
of vanishing at infinity, without obliging it to assume for large values of (a, . . , c, e)
the form
M
{a» + ... + c» + e''p+9'
The Potential-surface Theorem C. Art. Nos. 37 to 42.
37. In the case q = -\', writing here V = j-j + • • ■ + j" 2 + Jl > ^® '^*^®' precisely
as in the general case,
(w^f^dS+jdw...dzde W^U=jU^dS + jdx...dzde CTVTF;
and if the functions U, W satisfy the equations W=0, VW = 0, then (subject to the
exception presently referred to) the second terms on the two sides respectively each of
them vanish.
But, instead of taking the surface to be the surface positive infinity together with
the plane e = 0, we now leave it an arbitrary closed surface, and for greater symmetry
of notation write w in place of e ; and we suppose that the functions U and W, or one
of them, may become infinite at points within the closed surface; then, on this last
accouDt, the second terms do not in every case vanish.
38. Suppose, for instance, that U at a point indefinitely near the point (a,. . , c, e)
within the surface becomes
\{x-ay + ...+{z-cy + (w-ey}i^i'
then if V be the value of W at the point (a, . . , c, e), we have
jdx...dzdwW^U= vldx...dzd2vVU;
and since VU=0, except at the point in question, the integral may be taken over any
portion of space surrounding this point, for instance, over the space included within the
344 A MEMOIR ON PREPOTENTIALS. [607
sphere, radius R, having the point (a, . . , c, e) for its centre ; or taking the origin at
this point, we have to find \ dx ... dzdw W, where
U=-, '
and the integration extends over the space within the sphere x'+ ... + z^ + vj' = Rr.
39. This may be accomplished most easily by means of a particular case of the
last-mentioned theorem; viz. writing W=l, we have
j-j^dS + jdx...dzdwVU = 0,
•dU
or the required value is = — I , dS over the surface of the last-mentioned sphere.
We have, if for a moment r- = a?+ ... +z^ + w^
dU _ /x d z d w d\ jj _ dll Ux d z d w d \ ) _ dU
dn \r dx '" r dz r dwj ' dr \\r dx '" r dz r dwj j ' dr '
that is, ,- = , = „ at the surface; and hence
dn r' E'
where I dS is the whole surface of the sphere x' + ... + z^ + 'u^ = if, viz. it is = iJ*,
multiplied by the surface of the unit-sphere x-+ ... + z'- + w-=l. This spherical surface,
say I dZ, is
_ 2(rj)»+' 4(r^)»+'
-rH« + i)' "(s-i)rHs-i)'
fdU 4 ( FA)*''''
and we have thus j j d8 = -^, ■,•., and consequently
J da r^(s- 1) ^ •'
/
dx ...dzdw^ U = —
4(r0
i«+i
40. Treating in like manner the case, where IF at a point indefinitely near the
point (a, . . , c, e) within the surface becomes
^ 1
{{x-&f+ ...■i-{z-cy+{w-eyii'-i'
and writing T to denote the same function of (a, . . , c, e) that U is of (x,.., z, w), we
have, instead of the foregoing, the more general theorem
I W*^ dS + [ dx ... dz dw WS7 U- ^ff^.V
J da J r(^s-i)
= [ U^dS+ fdx...dzdw U^W-:^jf^*^^ T,
J dn J r(^s-|) '
607] A MEMOIR ON PREP0TENTIAL8. 345
where, in the two solid integrals, we exclude from consideration the space in the imme-
diate neighbourhood of the two critical points (a, . . , c, e) and (a, . . , c, e) respectively.
Suppose that W is always finite within the surface, and that U is finite except at
the point (a,..,c, e): and moreover that U, W are such that VCr=0, WW=0; then
the equation becomes
In particular, this equation holds good if U is
1
{{a-a;y+ ... + {e- wy}i'-i '
41. Imagine now on the surface S a distribution pdS producing at a point
{a',..,c, e) within the surface a potential V, and at a point (a",..,c", e") without
the surface a potential V" ; where, by what precedes, V" is in general not the same
function of (a", . . , c", e") that V is of (a', . . , c', e').
It is further assumed that at a point (a, . . , c, e) on the surface we have V = V" :
that V, or any of its derived functions, are not infinite for any point (a', . . , c', e')
within the surface : '
that V", or any of its derived functions, are not infinite for any point (a", . . , c", e")
without the surface:
and that V" = 0 for any point at infinity.
Consider F' as a given function of (a, . . , c, e) ; and take W the same function
of (x,. . , z, w). Then if, as before,
u !
{(a-xy+ ... + (c-2y+(e-wy}*^'
we have
Similarly, considering V" as a given function of (a, . . , c, e), take W" the same
function of (x, . . , z, e). Then, by considering the space outside the surface S, or say
between this surface and infinity, and observing that U does not become infinite for
any point in this space, we have
rdW" ,„ f ^„ dU
ln^^,S.jwPs;
adding these two equations, we have
fjr/dW' dW'\ (( dU dU\ UJV^
C. IX. 44
346 A MEMOIR ON PREP0TENTIAL8. [607
But in this equation the {unctions W and W" each of them belong to a point
{x,.., z, w) on the surface, and we have at the surface W = W", — W suppose ; the
term on the right-hand side thus is \ ^ \-y~> + ~fi, ) dS, which vanishes in virtue of
-J-, + J-,, = 0 ; and the equation thus becomes
as an
that is, the point (a, . . , c, e) being interior, we have
r-r{^s-^)fdW' dW"\ ds
J 4 (ri)'+' V ds ^ ds" J[(a-xy + ... + (c-zy+(e- tvy]i^ ■
In exactly the same way, if (a, . . , c, e) be an exterior point, then we have
iu'^d8.jw'§dS.
adding, and omitting the terms which vanish,
( (dW dW"\ 4 (Fir-
that is,
J 4(ri)'« \ds' "^ ds" J{ia-x)
■^yu, - xf + ... + (c - zf + {e - w)»ji»-i "
42. Comparing the two results with
y^ f P<i^
J [{a-xy+... + ic-zy + ie- wy]
l»-i'
we see that, V and V" satisfying the foregoing conditions, there exists a distribution p
on the surface, producing the potentials V and V" at an interior point and an
exterior point respectively ; the value of p in fact being
„- r{hizA)(dW'.dW"\
P--T{rir'~\d7^~d7') ^^^•
where W, W" are respectively the same functions of (x,.., z, w) that V, V" are of
(a,.., c, e).
The Potential- solid Theorem D. Art. No. 43.
43. We have as before (No. 40),
f w~dS+ {dx...dzdw WV U- ^Z,^^^""' V
J ds J r(^s-i)
= ( U^dS+ldx...dzdwUVW-^JP^,T,
J ds J "(^5-^) '
607] A MEMOIR ON PREPOTENTIALS. 347
no term in T; and takine next U = — r- -. — -, .,,. . as before, we
where, assuming first that W is not infinite for any point {x,..,z, w) whatever, we have
J_
[{a-xf-\-... + {c-zf-it{e-wy\
have V CT = 0 ; the equation thus becomes
{w^dS-lu'^d8-4^^^^V= {dx...dzd'wUVW,
1 d» J ds r(^s-^) J
where W may be a discontinuous function of the coordinates (a?, . . , z, w), provided only
there is no abrupt change in the value either of W or of any of its first derived
dW dW dW
functions -rj— •• • •'j > j~ • ^i^- ^* ^^y be any function which can represent the
potential of a solid mass on an attracted point («,.., z, w) ; the resulting value of
V W is of course discontinuous. Taking, then, for the closed surface S the boundary
of infinite space, U and W each vanish at this boundary, and the equation becomes
-£r^,V= (dx...dzdwUVW-
viz. substituting for U its value, and comparing with
< dx ... dzdw
y._ I pax ... azaw
~ j f(a - xy+ ... +(c-zy + {e- wf]i'-i '
where the integral in the first instance extends to the whole of infinite space, but
the limits may be ultimately restricted by p being = 0, we see that the value of p is
W being the same function of (x,.., z, w) that V is of (a, . . , c, e) : which is the
theorem D.
Examples of the foregoing Theorems. Art. Nos. 44 to 50.
44. It will be remarked, as regards all the theorems, that we do not start with
known limits ; we start with V a function of (a, . . , c, e), the coordinates of the
attracted point, satisfying certain prescribed conditions, and we thence find p, a function
of the coordinates (x, . . , z) or {x,. . , z, "w), as the case may be, which function is
found to be = 0 for values of (x,. . , z) or {x,.. , z, w) lying beyond certain limits, and
to have a determinate non-evanescent value for values of {x,.., z) or {x,.., z, w) lying
within these limits; and we thus, as a result, obtain these limits for the limits of
the multiple integral V.
45. Thus in theorem A, in the example where the limiting equation is ultimately
found to be ar* + . . . + ^' =/', we start with V a certain function of a" + . . . + c"
(= «' suppose) and e', viz. F is a function of these quantities through 6, which
denotes the positive root of the equation
, + z = l.
44—2
348 A MEMOIR ON PREPOTENTIAL8. [607
the
value in fact being V = I i~«~* {t +/')"** dt, and the resulting value of p is found
to be = 0 for values of {x,.., z) for which a^ + . . . + ^:' >/'. Hence V denotes an
integral
r pdx ... dz
J {(a -a;y+'... + (c-zy + ef']i^9 '
the limiting equation being sd' + ...+z''=/^ : say this is the «-coordinal sphere.
And similarly, in the examples where the limiting equation is ultimately found to
be 7i + ••• +r-3 = 1, we start with V a certain function of a,..,c,e through 0 (or
directly and through 0), where 0 denotes the positive root of the equation
i+-+T^T^ + ^ = l-
p + e k' + o^ 0'
and the resulting value of p is found to be = 0 for values of {x,. ., z) for which
Hence V denotes an integral
pdx ... dz
/{(«-
a;)'+... + (c-z)» + e»)*'+9'
.a? z'
the limiting equation being ji+ •••+t, = ^ ■ say this is the s-coordinal ellipsoid. It is
clear that this includes the before-mentioned case of the s-coordinal sphere ; but, on
account of the more simple form of the ^-equation, it is worth while to work out
directly an example for the sphere.
46. Three examples are worked out in Annex IV. ; the results are as follows : —
First, 0 defined for the sphere as above ; q+l positive ;
ar+...+z'Y
over the sphere x''+ ... +y- =/'',
r
dx ...dz
1 \{a-xy+ ...+(c-zf + ^\i'+^
-'^t^[^V-/V'<-/-)-*
This is included in the next-mentioned example for the ellipsoid.
Secondly, 0 defined for the ellipsoid as above ; 9+1 positive ;
\9
F =
\{a-xy+... + {c-zy+^]^*9
607] A MEMOIR OX PREPOTENTIALS. 349
3? Z'
over the ellipsoid ^ + . . . + y^ = 1,
This result is included in the next-mentioned example ; but the proof for the
general value of m is not directly applicable to the value m = 0 for the case in
question.
Thirdly, 6 defined for the ellipsoid as above ; 5 + 1 positive ; m = 0 or positive,
and apparently in other cases,
over the ellipsoid as above,
_(riyr(i + g + m) .r(^_jl _^ iT r«-Ma + /-^-i (t+h^\\-idt
-t(^8+q)T{\+myJ-''U\ r + e - h'+e e) * i«+/ )...(«+ a )) *«f«-
And we have in Annex V. a fourth example ; here 6 and the ellipsoid are as
above: the result involves the Greenian functions.
47. We may in the foregoing results write e = 0; the results, — writing therein
« + 1 for 8, and in the new forms taking (a, . . , c, e) and (x,.., z, w) for the two
sets of coordinates respectively, also writing q — ^ for q — , would give integrals of the
form
pdx ...dz dw
[{a-xf+... + {c- zY + (e - M;)2ji»+»
for the (« + 1 )-coordinal sphere and ellipsoid a?-\- ...+z'^ + w- =/* and 7a + • • • + rs + Ts = 1 =
say these are prepotential-solid integrals ; and then, writing 5' = — |, we should obtain
potential-solid integrals, such as are also given by the theorem D. The change can
be made if necessary ; but it is more convenient to retain the results in their
original forms, as relating to the s-coordinal sphere and ellipsoid.
There are two cases, according as the attracted point (a, . . , c) is external or
internal.
k'
For the sphere: — For an external point K->f'; writing e=0, the equation -^^ — - = 1
has a positive root, viz. this is d=K''—/'; and 0 will have, or it may be replaced
by, this value /c' — f: for an internal point k'</^; as e approaches zero, the positive
root of the original equation gradually diminishes and becomes ultimately =0, viz. in
the formulae 6 is to be replaced by this value 0.
For the ellipsoid: — For an external point 72 + ••• + t-2> 1; writing e = 0, the equation
a' c'
2 — fi + ... + a — Ti = 1 has a positive root, and 6 will denote this positive root : for an
/i
350 A MEMOIR ON PREPOTENTIALS. [607
a-
intemal point 2=i+ ••• + /j < 1 ! ^ ^ approaches zero the positive root of the original
equation gradually diminishes and becomes ultimately = 0, viz. in the formulae 0 is
to be replaced by this value 0.
The resulting formulae for the sphere a^+...+^'=/- may be compared with
formulae for the spherical shell, Annex VI., and each set with formulae obtained by
direct integration in Annex III.
We may in any of the formulae write q = — i, and so obtain examples of theorem B.
48. As regards theorem C, we might in like manner obtain examples of potentials
relating to the surfaces of the (s+ l)-coordiiial sphere x' + ... +z'+to'=f', and
ellipsoid 7^+'--+tj+-t5=1) or say to spherical and ellipsoidal shells ; but I have
confined myself to the sphere. We have to assume values V and V" belonging to
the cases of an internal and an external point respectively, and thence to obtain a
value p, or distribution over the spherical surface, which shall produce these potentials
respectively. The result (see Annex VI.) is
h
dS
{{a - xy + ... + (c - zy + {e - wy]i'-i
over the surface of the (5 + l)-coordinal sphere xr + ... + z' + ii/'=f^,
and
2 (rAy+' f' 1
where «' = a' + . . . + c* + e". Observe that for the interior point the potential is a mere
constant multiple of/
The same Annex VI. contains the case of the s-coordinal cylinder 0^"+ ... +«*=/',
which is peculiar in that the cylinder is not a finite closed surface ; but the theorem
C is found to extend to it.
49. As regards theorem D, we might in like manner obtain potentials relating
to the (s + l)-coordinal sphere 3? ■>r . . . -V z"^ -k- v? = p and ellipsoid 7i+---+rj + ]Gi = l;
but I confine myself to the case of the sphere (see Annex VII.). We here assume
values 7' and V" belonging to an internal and an external point respectively, and
thence obtain a value p, or distribution over the whole (s+ l)-dimensional space,
which density is found to be =0 for points outside the sphere. The result obtained is
-^_ /■ dx...dzdw
j{(a-xy+ ... + {c-zy + (e-wy}i
607] A MEMOIR OX PREPOTENTIALS. 351
over the (s + l)-coordinal sphere af+ ... + z- + ttf' = f-,
~ fvi T\' Zi^ ^^^ ^^ exterior point «>/,
— {(i« + i)/' — (i* — i) '''} for an interior point k</,
where «' = a' + . . . + c- + e%
50. The remaining Annexes VIII. and IX. have no immediate reference to the
theorems A, B, C, D, which are the principal objects of the memoir. The subjects to
which they relate will be seen from the headings and introductory paragraphs.
Annex I. Surface and Volume of Sphere 0^+ ... + 2^ + w^ =f-. Art. Nos. 51 and 52.
51. We require in (s + l)-dimen8ional space, I dx ... dzdw, the volume of the
sphere af+ ... +z^ + tt)^ = f', and i dS, the surface of the same sphere.
H,
Writing a; = / Vf , ..,z =f Vf , w = fVa, we have
dx ... dzdw = 27+,/'"" V' ...?-*«-* d? ... rffdw,
with the limiting condition f+ ... + f+(»=l ; but in order to take account as well
of the negative as the positive values of x,. . ,z, w, we must multiply by 2*+'. The
value is therefore
= /'"" f ?"*••• ?"^ '""^ (^f ■•• <^? (^«.
extended to all positive values of ^, ..,f, to, such that f+... + f+«<l; and we obtain
this by a known theorem, viz.
Volume of (s+ l)-dimensional sphere =/*'^' p-rp-"^ — jt.
Writing x=f^,..,z=-f%, w=fu>, we obtain dS=f'd'%, where cS is the element of
surface of the unit-sphere ^+... + ?' + <b'=1; we have element of volume d^...d^do)
= r*drd^, where r is to be taken from 0 to I, and thence
jd^...d^dto = j^r'drjdX=j^^jdl,
that is,
/ci2=(«+l)fci^..c^?rf., =2a. + i)J^^=lg^^;
consequently I d(S' = surface of (s-l-l)-dimenfiional sphere =/* _, ,\ ^ , , .
352 A MEMOIR ON PREPOTENTIALS. [607
52. Writing « — 1 for s, we have
Volume of (s — IVdimensional sphere = /"* r> ,, ,x .
Surface of do. ^/.-i^O^^
which forms are sometimes convenient.
Writing in the first forms s + 1 = 3, or in the second forms s = 3, we find in
ordinar)' space
Volume of sphere =/»^^ =/\ "^ =-9-.
and
Surface of sphere =p~^ =/» ^=, = 47r/',
as they should be.
rB ,.8-1 ^j.
Annex II. ^.^e Integral I ^^^ . Art. Nos. 53 to 63.
53. The integral in question (which occurs ant^, No. 2) may also be considered
as arising from a prepotential integral in tridimensional space ; the prepotential of an
element of mass dm ia taken to be = 5-—- , where d is the distance of the element
from the attracted point P. Hence if the element of mass be an element of the plane
z = 0, coordinates (x, y), p being the density, and if the attracted point be situate in
the axis of ^ at a distance e from the origin, the prepotential is
V =
pdxdy
(ar' + y' + e^)*'^'
For convenience, it is assumed throughout that e is positive.
Suppose that the attracting body is a circular disk, radius R, having the origin
for its centre (viz. that bounded by the curve a? +y- = R') ; then writing x — r cos 0,
y=r sin d, we have
V— [ P^'^^^^
i(r=+e'')i^«'
which, if p is a function of r only, is
-^'^j(r^+e»)4.+«
and in particular, if p = r*"*, then the value is
607] A MEMOIR ON PREPOTENTIALS. 353
the integral in regard to r being taken from r = 0 to r = R. It is assumed that s — 1
is not negative, viz. it is positive or (it may be) zero. I consider the integral
/,
* r*~' dr
which I call the r-integral, more particularly in the case where e is small in com-
parison with R. It is to be observed that e not being = 0, and R being finite, the
integral contains no infinite element, and is therefore finite, whether q is positive,
negative, or zero.
54. Writing r = e Vt), the integral is
= ie-^j.
R'
the limits being — and 0.
(1 + i;)J'+« '
In the case where q is positive, this is
viz. the first term of this is
* rus + q)'
and the second term is a term expansible in a series containing the powers 2q, 2q + 2,
g2 . . 1
&c. of the small quantity ^ , as appears by effecting therein the substitution v = -;
viz. the value of the entire integral is by this means found to be
*^ \r(i8 + q) Jo (l + a;)W
•55. In the case where q ia =0, or negative, the formula fails by reason that the
element /^ r^^^^q of the integrals I , I becomes infinite for indefinitely large values
of V. Recurring to the original form I .— — .^^ , it is to be observed that the
integral has a finite value when e = 0 ; and it might therefore at first sight be
imagined that the factor (r* + e*)-**-' might be expanded in ascending powers of e*, and
the value of the integral consequently obtained as a series of positive powers of e*.
rR
But the series thus obtained is of the form e'* | r-^v-^-^ dr, where 2q being positive,
Jo
the exponent —2q — 2k—l is for a sufficiently small value of k at first positive, or if
negative less than — 1, and the value of the integral is finite ; but as k increases the
exponent becomes negative, and equal or greater than — 1, and the value of the
C. IX. 45
354 A MEMOIR ON PREPOTENTIALS. [607
integral is then infinite. The inference is that the series commences in the form
A + B(f'+ Ce*...: but that we come at last when q is fractional to a term of the form
Ke"*', and when q is = 0 or is integral, to a term of the form Ker^ log e ; the process
giving the coeflScients A, B, C, .., so long as the exponent of the corresponding term
e°, e*, e*, . . is less than — 2q (in particular q = 0, there is a term k log e, and the
expansion-process does not give any term of the result), and the failure of the series after
this point being indicated by the values of the subsequent coefficients coming out = oo .
56. In illustration, we may consider any of the cases in which the integral can
be obtained in finite terms. For instance,
Integi'al is |r(7^ + e=')*dr, =^(r^ + e»)*, from 0 to iJ,
viz. expanding in ascending powers of e, this is
or we have here a term in (?. And so,
s = \,q = -2,
Integral is |(»^ + e')*dr, = (i^ + fe") r 'Jr^e' + fe* log (r + Vr" + e"), from 0 to R,
viz. expanding in ascending powers of e, this is
= iii^ + f if'e' + . . . + f e* log - *
or we have here a term in e* log e.
•57. Returning to the form
«• i;i«-l dv
{l+v)
H+q '
1 —X 1
and writing herein v = , or, what is the same thing, x = i- , and for shortness
° X ° 1+v
^^V^^' = — ^' ^^^ ^^^"^ ^'
=ie-j;
afl-^ (1 - a;)*^' dx.
where observe that 9 - 1 is 0 or negative, but X being a positive quantity less than
1, the function ««-' (1 — a;)i»~' is finite for the whole extent of the integration.
Term is f«*log — , =|e* (log — +log 2), which, — being large, is reduced to fc*log
607] A MEMOIK ON PREPOTENTIALS. 355
58. If q = 0, this is
- 1 r l-{l-(l-a^)M<^
J X *^
dx
where observe that, in virtue of the change made from -(!—«)*»"' to - jl — (1 —«)*•""{
sc so
(a function which becomes infinite, to one which does not become infinite, for x = Q),
it has become allowable in place of I to write \ — \ .
When e is small, the integral which is the third term of the foregoing expression
is
obviously a quantity of the order e"; the first term is ^(log — •" ^^^g v ^ + "pa ) > which,
sglecting ter
R
neglecting terms in e*, is = J log — , and hence the approximate value of the r-integral
f* r'-^dr
IS
or, what is the same thing, it is
^1_(1_^
= logf-i/'d/j^""\
y
where the integral in this expression is a mere numerical constant, which, when Js — 1
is a positive integer, has the value
neglecting this in comparison with the logarithmic term, the approximate value is
69. I consider also the case q^= — \\ the integral is here
ie f ar»(l-a;)i»-'dfl;
= Ae f «-» (1 - !l - (1 - xf-'^X) dx
J X
= e (Z-i - 1) + ie f'a;-' {1 -(1 -«)»«-) dx;
45—2
356 A MEMOIR ON PREP0TENTIAL8. [607
and the first term of this being = Ve" + iZ* - e, this is consequently
= V:RM^ + h ef'x-i {1 - (1 - a:)*^'l dx-e(l+^j a;-» (1 - (1 - x)^'] dx^ .
As regards the second term of this, we have
-2arJ{l-(l-a;)*'-')+2(is-l) j x-i (I - a:)i<>-^ dx = j x'i {1 - (1 - a:)J*-'j ck;
or, taking each term between the limits 1, 0,
-2 + 2{^s-l)^^^^ = f\-i{l-{l-x)i-^]dx;
viz. this integral has the value
^ >•*"' dr
and the value of the r-integral / , ^\i«-i ^^ consequently
= Vi? + e» + ief x-i {1 - (1 - x)i^'} dx - e p^-f ^\. ,
which is of the form
ijjl + terms m ■^, ^^, ...[ -e j,^_^^;
say the approximate value is
E-e^i^
r(|s-i)'
r^ .
where the first term R is the term I dr, given by the expansion in ascending powers
Jo
of e*; the second term is the term in e~^. And observe that the term is the value of
^e\ x-i(l-x)i>-^dx,
Jo
calculated by means of the ordinary formula for a Eulerian integral (which formula,
on account of the negative exponent — |, is not really applicable, the value of the
integral being = oo ) on the assumption that the F of a negative q is interpreted in
accordance with the equation F (q + l) = qrq; viz. the value thus calculated is
on the assumption r^ = — ^r(— ^); and this agrees with the foregoing value.
60. It is now easy to see in general how the foregoing transformed value
^e~*» I afl~^ (1 — a;^' dx, where q is negative and fractional, gives at once the value of
607] A MEMOIB ON PREPOTENTIALS. 357
= p^,
a positive quantity less than 1 j ; the function to be integrated never becomes infinite.
Imagine for a moment an integral / af^ dx, where a is positive or negative. We may
conventionally write this = | of-dx — \ af- dx, understanding the first symbol to mean
^0 Jo
— , and the second to mean ; they of course properly mean — — and
l+a
; but the terms in 0'+", whether zero or infinite, destroy each other,
the original form j sifdai, in fact, showing that no such terms can appear in the
result.
In accordance with the convention, we write
f X^-' (1 - «)*•-' da;=l afl-'(l- a;)**-' dx-j a:9-> (1 - x)i'-' dx ;
Jx Jo Jo
and it follows that the term in e~^ is
^e-^ / a^' (1 - a;)**-' dx,
Jo
this last expression (wherein q, it will be remembered, is a negative fraction) being
understood according to the convention ; and so understanding it, the value of the
term is
* rus+qy
where the T of the negative q is to be interpreted in accordance with the equation
T(q+l) = qrq; viz. we have rq = -r(q + l),= . .r(q + 2), &c., so as to make
the argument of the T positive. Observe that under this convention we have
r^r (1 - g) = ^^: or the term is ie-^ . ^^ „.. ^ ^.^^ ,, r .
^ ^ sm qv * sm qv V (|s + q)r (1 —q)
61. An example in which J^s — 1 is integral will make the process clearer, and
will serve instead of a general proof. Suppose J = — |, ^s — 1 = 4, the expression
I a;"* (1 - «)* da; = / (x~^ - 4a;"* + 6a;* - ^x'^ + x^) dx
Jo Jo
is used, in accordance with the convention, to denote the value
— 7 2401 — 7'
= 7(-l-§ + A-i + M=7(-M-i + ^). =T5Tl'3727'=5.13.27-
in
358 A MEMOIR ON PREPOTENTIAL8. [607
But we have
T^aTq _ r5r(-f) ^ 24r(-|) ^ -7'
r(i»+9)~ r(6-f) v.y.j^.f.^r(-i) 5.13.27'
agreeing with the former value.
62. The case of a negative integer is more simple. To find the logarithmic term of
we have only to expand the factor (1 — a)**"' so as to obtain the term involving x~9.
We have thus the term
1 / Ii^\ R r e*
where log y = log { 1 + ^ ) . = 2 log — h 2 log a/ 1 + ps ! ^o \^^%, neglecting the terms
^ , &c., this is = 2 log — , and the term in question is
The general conclusion is that q being negative, the »'-integral
Jo (r= + e=)H+?
has for its value a series proceeding in powers of e", which series up to a certain point
is equal to the series obtained by expanding in ascending powers of c* and inte-
grating each term separately ; viz. the series to the point in question is
R-^_i8±9 iir^"- ^ .is+q.js + q + l R-^-*
-2q 1 -2q-2 '^ 1.2 ' _2g-4 ■^••■•
continued so long as the exponent of e is less than —2q: together with a term Ker*^
when q is fractional, and Ke~'^ log — when q is integral ; viz. q fractional, this term is
*^ r(i« + 3)' *' 8inq7rT{is + q)ril-q)'
and q integral, it is
= (-)q e-i? Til lo? -
607] A MEMOIR ON PREPOTENTIALS. 359
63. It has been tacitly assumed that ^s + q is positive ; but the formulae hold
good if ^s + q is =0 or negative. Suppose ^s + q is 0 or a negative integer, then
r (^s + q)= ao , and the special term involving e~^ or e~^ log e vanishes ; in fact, in
this case the r-integral is
_ ^y-i (^ 4. e2)-(i«+9) dr,
J n
where (r'+e*)" '*'"'"'' has for its value a finite series, and the integral is therefore equal
to a finite series A + B^ + Ce* + &c. If ^s + q be fractional, then the F of the negative
quantity ^s + q must be understood as above, or, what is the same thing, we may,
instead of F (^s + q), write
am(is + q)7rril-q-^s)'
thus, q being integral, the exceptional term is
_ , V, _^ r^ssia(is + q)7r.r{l-q-^s) R
"^ ^ (F^)'r(l-?) '''^e-
For instance, 8=1, q = —2, the term is
'I FisinJ-jTOFI ^.
or, since Ff = f.^F^, and F3 = 2, the term is +|e*log— , agreeing with a preceding
result.
Annex III. Prepotentials of Uniform Spherical Shell and Solid Sphere.
Art. Nos. 64 to 92.
64. The prepotentials in question depend ultimately upon two integrals, which
also arise, as will presently appear, from prepotential problems in two-dimensional space,
and which are for convenience termed the ring-integral and the disk-integral respect-
ively. The analytical investigation in regard to these, depending as it does on a
transformation of a function allied with the hypergeometric series, is I think interesting.
65. Consider first the prepotential of a uniform (s + 1 )-dimensional spherical shell.
This is
y^[ dS
} {{a- x)' + ... + (c - zf + (e - tvYli'-^i '
the equation of the .surface being 0^+ ...+z^ + w' =f^ ; and there are the two cases
of an internal point, a'' + ... + (f + ef' <f-, and an external point, a^ + ...+c^+ e^>f\
The value is a function of a:'+ ... +c- + e\ say this is = k'. Taking the axes so
that the coordinates of the attracted point are (0, . . , 0, k), the integral is
=/:
dS
ai'+ ... + z' + (k - wy\i'+i '
360 A MEMOIR ON PREPOTENTIALS. [607
where the equation of the surface is still ir»+ ... + ^■- + w»=/». Writing x=/^,..,z=JX,
w=f(Op where ^+ ... + f* + o)'= 1, we have dS^-' — ^-^^ — - , or the integral is
=/•/
d^...di;
»(/»-2«/a) + «")*•+«■
Assume ^ = px,. . ,^=pz, where ai^ + ... + z'^=l; then p' + a^ = 1. Moreover, df...df,
= ;j^*dpd2, where cTE is the element of surface of the s-dimensional unit-sphere
a? + ... + z^=l; or for p, substituting its value Vl — «', we have dp = -. ; and
Vl — 0)'
thence d^ ... d^= — (l — a>')^-^<oda>d1. The integral as regards p is from p = — 1 to
+ 1, or as regards w from 1 to —1; whence reversing the sign, the integral will be
irom » = — 1 to + 1 ; and the required integral is thus
_ !• {l-o,')i'-^dwdl ^fJ^^C (l-<»')*^'rfa>
where I dS is the surface of the s-dimensional unit-sphere (see Annex I.), = 1. -^ ;
J A ^s
and for greater convenience transforming the second factor by writing therein to = cos 6,
(riv
the required integral is = J, .^ ^ multiplied by
¥'l
8m'-'0d0
« (/» - 2«/cos e + /r')i'+9 '
(r*y
which last expression — including the factor 2/"*, but without the factor y.^^ — is the
ring-integral discussed in the present Annex. It may be remarked that the value can
be at once obtained in the particular cEise s = 2, which belongs to tridimensional space :
viz. we then have
F= 2,r/-» [' «^°^^^
io (/' - 2/c/cos 0 + «=)«+'
= ^^if' -W<^os0 + Kr'
which agrees with a result given, Mdcanique Cdleste, Book xii. Chap. II.
66. Consider next the prepotential of the uniform solid (s + l)-dimensional sphere,
dx ... dzdw
y. _ f dx ... dzdw
~J {(a-xy + ...+{c-zy + {e-'
7)»jl*+9 '
the equation of the surface being a^ + ... +z^+ ^u' =/" ; there are the two cases of an
internal point « </, and an external point « >/ («'+... + c* +6" = «' as before).
607] A MEMOIR ON PREPOTENTIALS. 361
Transforming so that the coordinates of the attracted point are 0, . . , 0, k, the
integial is
_ r dx ... dzdw
~J {a^ + ... +z^ + (K-wy]i'+9'
where the equation is still af+ ... +z' + w'' =/'■ Writing here x=r^,..,z = r^, where
f + ... + ^ = 1, we have dx ...dz =r^^drd^, where dS is an element of surface of the
«-dimensional unit-sphere ^+ ... + ^ = 1; the integral is therefore
_ r r^' dr dS dw
~i j{r» + (/e-w)»}i'+»'
where, as regards r and w, the integration extends over the circle 'r' + v/' =/'. The
value of the first factor (see Annex I.) is = 1,/^ ; writing y and x in place of
2(r'y
r and w respectively, the integral is = — pr^— multiplied by
r y»~' dx dy
J {{x-Ky + y'\^+9
over the circle a? + y" =/' ; viz. this last expression ( without the factor „ ,,^ j is the
disk-integral discussed in the present Annex.
67. We find, for the value in regard to an internal point «</,
which, in the particular case q = — ^, is
viz. the integral in f is here
or we have
It may be added that, in regard to an external point «>/, the value is
c. IX. 46
362 A MEMOIR ON PREP0TENTIAL8. [607
which, in the same case q = — ^, is
r(i«-i) •'"«-/'
where the ^-integral is
and the value of F is therefore
r(i«+^)«-'-
Recurring to the case of the internal point; then, writing ^='^+---+^ + (^'
and observing that V (/c») = 4 (^s + ^), we have
^{¥-\y
A. \
(in particular, for ordinary space s + l = 3, or the value is ,^ , =-47r, which is
right).
68. The integrals referred to as the ring-integral and the disk-integi-al arise also
from the following integrals in two-dimensional space, viz. these are
f y«-' d8 r y'-' dx dy
J {{x -Ky4- y^li'+i ' J {{x- kY + 2/^ji*+9 '
in the first of which dS denotes an element of arc of the circle af + y'-=/'', the
integration being extended over the whole circumference, and in the second the
integration extends over the circle a.^ + y'=f'; y^~^ is written for shortness instead of
(2/»)J(«-'>, viz. this is considered as always positive, whether y is positive or negative ;
it is moreover assumed that s — 1 is zero or positive.
Writing in the first integral x —fcos 6, y =/sin 6, the value is
(sin ey-^ dd
J J(/s-2k/cos 5 + «»)*•+«'
viz. this represents the prepotential of the circumference of the circle, density varying
as (sin 6y~^, in regard to a point x= k, y — O in the plane of the circle ; and similarly
the second integral represents the prepotential of the circular disk, density of the
element at the point («, y) = y-', in regard to the same point x = k, y = 0; it being
in each case assumed that the prepotential of an element of mass pd'or at a point
at distance d is = ^rr^ .
607] A MEMOIR ON PREPOTENTIALS. 363
69. In the case of the circumference, it is assumed that the attracted point is
not on the circumference, k not = /; and the function under the integral sign, and
therefore the integral itself, is in every case finite. In the case of the circle, if k
be an interior point, then if 2g — 1 be = 0 or positive, the element at the attracted
point becomes infinite; but to avoid this we consider, not the potential of the whole
circle, but the potential of the circle less an indefinitely small circle radius e having
the attracted point for its centre; which being so, the element under the integral
sign, and consequently the integral itself, remains finite.
It is to be remarked that the two integrals are connected with each other; viz.
the circle of the second integral being divided into rings by means of a system of
circles concentric with the bounding circle 3?-\-y^=f^, then the prepotential of each
ring or annulus is determined by an integral such as the first integral; or, analytically,
writing in the second integral a; = r cos ^, y =r sin 6, and therefore dxdy = rdr dd, the
second integral is
(sin ey-^ d0
= \r'dr\
(r» + K'-2/«rcos^)*»+9'
viz. the integral in regard to 9 is here the same function of r, k that the first
integral is of /, « ; and the integration in regard to r is of course to be taken
fi-om r=0 to r=f. But the ■^-integral is not, in its original form, such a function
of r as to render possible the integration in regard to r; and I, in fact, obtain the
second integral by a different and in some respects a better process.
70. Consider first the ring-integral which, writing therein as above x=fcos6,
y=/sinO, and multiplying by 2 in order that the integral, instead of being taken
from 0 to 27r, may be taken from 0 to ir, becomes
(sin ey-' de
(/«- 2/c/cos ^ + /«:'')**+« •
Write cosi^ = '\/«; then sinJ^ = Vl-a7, sin 5 = 2a;i (1 - a;)* ; dO = - x-'' {\ - x)-^ dx ;
cos 6 = — l-ir2x; 0 — 0 gives x=\, d = v gives x = 0, and the integral is
= 2— /'I
I a^'-^(l-x)i'-'dx
o[(f+>cT-i'cfx}i'+i'
2-1 f n a^<-i (1 _ a;)i»-i dx
Jo
if+KY+n}^ (l-«a;)i»+9 '
if for shortness u=-, ^>r,., so that obviously m < 1.
The integral in x is here an integral belonging to the general form
n (a, ^, 7, m) = f «•-" (1 - a;/-' (1 - ux)-y dx,
Jo
viz. we have
Ring-integral = (j^f^+,g n (^s, y, ^s + q, u).
46—2
364 A MEMOIR ON PREPOTENTIALS. [607
71. The general function 11 (a, /9, 7, u) is
Pff no
n («, ft, y, u) = ^~^'^F{a, 7, o + (8, u),
or, what is the same thing,
^(a, 0, 7, u) = rar^7-a) " <«■ Y - «. ^. «).
and consequently transformable by means of various theorems for the transformation
of the hypergeometric series, in particular, by the theorems
F{a, fi,y,u) = F(0, a,y,u),
F{a, /3, 7, u) = a-u)y—^F{y-a, 7 - /9, 7. «);
(1 _ Vl — uY . . 4 \/v
,^= ) , or, what is the same thing, u = =- , then
1 + V1-m/ ^ (1+V^)«
F{a, y3, 2/9, u) = {1 + -Jv)^ F (a, a-^ + ^v).
In verification, observe that if m = 1 then also v = 1, and that with these values,
calculating each side by means of the formulae
F(a a -V n- r7r(7-a-ff) rar(/3-7)
the resulting equation, F{a, /3, 2y3, l) = 2^«J'(a, a-/9 + |, /9+^, 1), becomes
r2;3r(/3-«) _ r(;8 + |)r(2ff-2a)
r(2/3-a)ry3 r(2/3- «) r(^ - a + 4)'
that is,
r2/3 ^2- r (2/3 -2a)
r/sr(/s+i) r(^-a)r(^-a+|)'
r 2a; FA
which is true, in virtue of the relation _ „ . — 2_ = 2^~K
ra:r(a; + ^)
72. The foregoing formulae, and in particular the formula which I have written
F(a, 0, 2/9, u) = (1 + '^vy* F (a, a-y8 + i, /9 + i, v), are taken from Rummer's Memoir,
"Ueber die hypergeometrische Reihe," Crelle, t. xv. (1836), viz. the formula in question
is, under a slightly different form, his formula (41), p. 76 ; the formula (43), p. 77,
is intended to be equivalent thereto; but there is an error of transcription, 2a— 2/9+1,
in place of /9 + 1, which makes the formula (43) erroneous.
It may be remarked as to the formulae generally that, although very probably
n (o, /3, 7, u) may denote a proper function of u, whatever be the values of the indices
(a, /9, 7), and the various transformation-theorems hold good accordingly (the F-function
of a negative argument being interpreted in the usual manner by means of the
equation Fa; = - F(l +«), = — r-TF(2 + a;) &c.), yet that the function 11 (a, /9, 7, u).
607] A MEMOIR ON PREPOTENTIALS. 365
used as denoting the definite integral / a^~' (1 — xf^^ (1 — ux)~y dx, has no meaning
jo
except in the case where a and ^ are each of them positive.
In what follows we obtain for the ring-integral and the disk-integral various
expressions in terms of Il-funetions, which are afterwards transformed into (-integrals
with a superior limit oo and inferior limit 0, or «- —f'^ ; but for values of the
variable index, q lying beyond certain limits, the indices a and y3, or one of them,
of the Il-function will become negative, viz. the integral represented by the Il-function,
or, what is the same thing, the (-integral, will cease to have a determinate value,
and at the same time, or usually so, the argument or arguments of one or more of
the F-functions will become negative. It is quite possible that in such cases the
results are not without meaning, and that an interpretation for them might be found;
but they have not any obvious interpretation, and we must in the first instance
consider them as inapplicable.
73. We require further properties of the Il-functions. Starting with the foregoing
equation
F(a, /9, 2/9, tt) = (l+V;)>«i'(a, a-0 + l /9 + i, v),
-each side may be expressed in a fourfold form : —
F{<x, /9, 2^, u)
^Fi^, a, 2/3, u)
= (l-uy-F(2^-a, yS. 2/9, u)
= {l-uy-'F(a, 2/9 -a, 20, u)
(1 + V»)^^(a, a-/9-|-i /9 + J, v)
^^(l+'/vy-Fici-^ + l 7, 0+1 v)
= (1 + Vw> (1 - v)«^-^ F{0 -a + ^, 20-a, 0 + ^, v)
= (l+VD)''(l-«)-*-^«i^(2/3-a, 0-a+l 0 + i, v),
where, instead of {1 + •^vy"{l-v)'^-^, it is proper to write (l + ^/vy* (l-'/iy^-^;
And then to each form applying the transformation
Tve have
^(0, 0, y, ")=rar('y-a)"^"' '^~''' ^' "^'
^^^ n(a, 2/3- a, 0, u)
Fa V {-20 -a)
T20
rySTyS
n (0, 0, a, u)
V 9.R
=^^-")'"'r(2;9-a)r«"^^^-"'"-^-">
366 A MEMOIR ON PREPOTENTIALS. [607
= <l-^^"^)^" r(«4 + ^!y(2^-a-)"("-^ + » 2^-«- «' ^)
= (1 + 'Tvr (1 - ^^)'^""r(J-ttt)ra n (^ - a + ^, a, 2/3 - a, v)
I select the second of the first four forms; equating it successively to each of the
second four forms, and attending to the relation p o^ = 2'"*^ TJ^, we find
n (y3, /3, a, «) = (! + Vi;)»« 2>-^ rar(^-! + i) n (a, ^ - a + i, a -/3 + i «)
= (1 + Vii)* (1 - Vv^-^ ^'"'^r(;3-t+\)ra n (;S - a + J, a, 2y8 - a, »)
= (l+V^)^(l-V;r---2-^j,^^^_[^^j.^^_^_^^^n(2/3-a, a-^ + i /3-a + i, .).
Putting herein /3 = is, a=iijS + q, the formulae become
n^s, is, is + g, «) = (! +^^r=«2-' j,^^-^^^ip^^ n (is + g, i-j, J + 9, v) (I.)
= (i+v^r'»2- __^iil__^na+9,i*-9,is+9,i;) (II.)
=(i+vj;)«(i-v^)-»'2-'j,^^-^^Ji^~^na-9,is+9,is-j,«) (III.)
= (1 + ^yy (1 _ ^/i;)-». 2-« j,^Ji^L___ U(y-q,^ + q.i-q,v) (IV.),
where observe that on the right-hand side the Il-functions in I. and IV. only differ
by the sign of q, and so also the Il-functions in II. and III. only differ by the sign
of q. We hence have
n(H H is-q, u) = (1 + ^/vy-^ 2-' . ^ ^^^ 5g) l\^ + g) " d' - g' i+9> *-?' «);
607] A MEMOIR ON PREPOTENTIALS.
and comparing with (IV.),
_j-j n(^s, ^5, \s-q, u).
74. The foregoing formula,
367
2^1/.
Ring-integral = ^y^ /).+,, n (|s, |s, ^s + q, u),
*Kf
i, gives, as well in the case of an exterior as an interior point, a
where m — , ^ . .
convergent series for the integral ; but this series proceeds according to the powers
4*/'
of ■ . "^ .^. We may obtain more convenient formulae applying to the cases of an
internal and an external point respectively.
75. For an internal point
int K </, Vl — M =-^ — '^ ,
and therefore v = -s; .
V
m -
where the Il-functions on the right-hand side are respectively
^ afl-i (I - a;y-9-' dx
Jo
JO
(/» - /«:»a;)i»+9
»a*^»+'(l -«)«-! da;
d<
the f-fonns being obtained by means of the transformation x =
gives
t
; viz. this
whence the results just written down.
368 A MEMOIR ON PREP0TENTIAL8. [607
We hence have
Ring-integral = ^j/_-^ i^^/) r^^ - ,) /J ^'^' (* +/' " *')-*'^' <« +/')--* *
= /• fTf ^rw+T) il '""* ^' "-^^ - '^^^""* ^' ^^'^~"'' ^'
As a verification write « = 0, the four integrals are
r t^'^^^dt _ rjis+^jK (i-g)
r ji+»:' dt ^ T{^ + q)T(\s-q)
r-^-^ r(ig-g)r(i + g).
Jo «+/')*'+*• "^ ra* + i)
hence from each of them
Ring-mtegral=^~^j,^^^^,
which is, in fact, the value obtained from
2*"' f' I 4«/' \
Ring-integral = ^^-p^-'p^ n (^is, |s, \s^q, (—~Jy)
on putting therein « = 0 ; viz. the value is
76. For an external point « >/, vl— m= — ^ and therefore v='^.
= C^^'("-r-r-<4-^^^W)" (» - *-'■ ^'-i")
Jo («' -/»«)*«+«
„23+l
607] A MEMOIR ON PREPOTENTIALS.
where the Il-functions on the right hand are respectively
Jo {K'-/'x)9+i
1 afl-i (1 - x)*»-9-i (iB
^ /•ta;-^(l-a;)i«+9-'<fo
fia^»-5+'(l_ar)?-ida;
36S>
= «--29+i
We have then
dt
Ring-integi.l = ^^^, j,^^^I^^^^-^JJ_^ri-n« V=--=)^--(^+/r«-* d<
= •^' r (^ -^j r a. + ,) /!,. ^--^ (^ -^/' - --)-^ (^ V-n--
Observe that in II. and III. the integrals, except as to the limits, are the same
as in the corresponding formuliE for the interior point.
If in the ^integrals we put t + K- —f in place of t, and ultimately suppose k
indefinitely large in comparison with f, they severally become
and they all four give
Ring-integral = ^^.j,^^— ^^.
which agrees with the value
when ^ is indefinitely large.
C. IX.
47
370 A MEMOIR ON PBEPOTENTIAL8. [607
77. We come now to the disk-integral,
y*~' dxdy
C y*~' dx
yS)»»+9 '
over the circle a^ + y^=f\ Writing x = K + pcoa<f>, y = p sin <fi, we have dx dy = p dp d(f>,
and the integral therefore is
rrsin*-' <}>dpd<f>
J J ^ '
where the integration in regard to p is performed at once ; viz. the integral is
= r^gf(p'-^)^i^'-'4>d<f>;
.2g
or multiplying by 2, in order that the integration may be taken only over the semi-
circle, y = positive, this is
= r^ j{p'-^) sin-' <f>d<l>,
the term {p'~'^) being taken between the proper limits.
78. Consider fii-st an interior point k </. As already mentioned, we exclude an
indefinitely small circle radius e, and the limits for p are from p = e to /> = its value at
the circumference; viz. if here x=fcos6, y=f sin 6, then we have /cos ^ = « -H p cos (^,
/sin ^ = p sin <f>, and consequently
p2 = K^ +/2 - 2«/cos 0,
8m<i=^smp, = , •'^ =11,
P VV+7' - 2ac/cob 0
and the integral therefore is
As regards the second term, this is = — j — I sin*"' <f> d(f>, from ^ = 0 to <^ = tt ; or,
what is the same thing, we may multiply by 2 and take the integral from <^ = 0 to </> = -5 •
607] A MEMOIR ON PREPOTENTIALS. 371
Writing then sin <^ = Va;, and consequently sin*~' (f) d<f> = ^*i'~' (1 — a))~i dx, the term is
= — -, =—1 — %x ■ the value of the disk-integral is
_ f-' r sin'-' e d^ ^ TjsT^
But we have
and thence
that is,
. , /sin 9 , fcos 6 — tc
sin d> ='' , cos <f) = ,
P P
, f if- K cos 6) dO _ f{f-Kco%e)de
or, what is the same thing,
^ i ((/' -'^) + (P + «' - 2«/co8 ^)} .
/= + /<:■' - 2«/cos e '
the expression for the disk-integral is therefore
J/»-' /•'sin'-' g ((/' - /f') -f- (/" +>c'- 2*;/ cos g)} dg e^ F^sT^
l/» -(-«'- 2«/cos 0JJ»+9 i-? r(|s-|-^)'
79. Writing as before cosi^ = Va;, sin ^^ = Vl +«, &c., and u=j — ~-^, this is
-qms+i)-
As a verification, observe that, if « = 0, each of the Il-functions becomes
= I '^'-' (1 - a;)i-' rfa;, = ^*ill? ;
Jo 'la
hence the whole first term is = ; — . -K^-, viz. this is -t— i~ ^, and
^-q Fa ' ^-qn^s + i)'
the complete value is
vanishing, as it should do, if /= e.
80, In the case of an exterior point k >/, the process is somewhat different ; but
the result is of a like form. We have
Disk-integral = —- f (/3,'-»» - p'-^') sin»-' 0 d(^,
47—2
/
372 A MEMOIR ON PREPOTENTIALS. [607
where pi refers to the point M' and p to the point M. Attending first to the integral
p^~^ 8m'~' <f> d<f>, and writing as before /cos ^ = « + p cos <^, y sin ^ =p sin <^, this is
_ r Bin^^ ed<f>
•' j{«»+/'-2/c/cos^ji'+«
^ f sin-' e {(/' - ^) + (p +K'- 2/c/cos d)} de
*-^ j (/» + /e»-2//«:cos^)i»+«
the inferior and the superior limits being here the values of 6 which correspond to the
points N, A respectively, say ^ + a, and ^ = 0 ; hence, reversing the sign and inter-
changing the two limits, the value of — I p'~^ sin*"' d d<f> is the above integral taken
from 0 to a. But similarly the value of + pi'~^ sin*~' 0 d^ is the same integral taken
from a to tt. For the two terms together, the value is the same integral from 0 to tt ;
viz. we thus find
T^- 1 • ^ 1 i/*"' /■' sin*"' ^ {- («' -/') + (/" + «' - 2«/cos 6)] dff
Disk-mtegral = f^ — * — \ ., ■' \ — ^-. ,,. . -^ '-^ — ;
r- ^icf
or, writing as before cos^6 = yx, &c., and u = -. — fT^, this is
81. As a verification, suppose that k is indefinitely large : we must recur to the
last preceding formula; the value is thus
/'
(i -9) «*+»»-'
" sin«-> d (- cos 0 + ^]
l-'f
cos 0
viz. this is
= (prgy^i+iFr //in- 0 {- cos 0+[l-(s+ 2q) COS' 0] {-| d0,
where the integral of the first term vanishes ; the value is thus
= {i-q)Kf+-i i„ '^"'"' ^ [1 - (« + 2g) cos= ^] d^,
607] A MEMOIR ON PREPOTENTIALS. 373
TT
where we may multiply by 2 and take the integral from 0 to - . Writing then
sin Q = Va;, the value is
= (F-"9V+^Jo
where the integral is
r xi'-i {l-(s+ 2q\ (1 - x)] (1 - «)-* dx,
a
and hence the value is
viz. this is = I ^' da; dy, over the circle a?-¥y'' =f^, as is easily verified.
82. Reverting to the interior point k <f.
Disk-integral
then reducing the expression in { J by the transformations for 11 {^s, |s, ^s + q, ii)
and the like transformations for n(^s, ^s, ^s + q—l, u), the term in { } may be ex-
pressed in the four forms: —
rj^r4__ if+.r^^ ^^,^jp,.^, ^y
2.-«-__Ii!-Ei (L+^Tl^ multiplied by
[(i-pn(i+^. i.-<?. i«+g. j,)+ ^j^ n(-K?. i«-9 + i. i«+?-i.p]>
n^ (f+J^TH/^J^ ,„,„iplied by
r{i-q)r(i8 + q) f^ ^
2-. - -J>Ei (/+^)'-;(/-^)'-'^ n^ultiplied by
374 A MEMOIR ON PREPOTENTIALS. [607
83. The first and tlie fourth of these are susceptible of a reduction which does not
appear to be applicable to the second and the third. Consider in general the function
(l-v)U(a, ^, 1-/3, «) + "^n(a-l. ^ + 1, -/3, v);
the second 11 -function is here
I x^~'^ (1 — ar . 1 — vxy dx ;
Jo
viz. this is
g*—\ J n fi
= J (1 — a; . 1 — vxf I ;»•-' -=- (1 — a; . 1 — vxf dx,
a— 1 a- 1 .'o ««
or, since the first term vanishes between the limits, this i.s
= -^, ( a^-> . (1 - « . 1 - vxf-' {l+v- 2vx) dx,
a — 1 .' 0
= -^Kl+^)n(a, /8, 1-/3, v)-2v r x^(l-x.l -vxy-'dx}.
a— I Jo
Hence the two Il-functions together are
= {l -v+l + v) a^-' (1 - a; . 1 - vxf-^ dx - 2 j vx. «*-' (1 - a; . 1 - vx)^-' dx,
Jo Jo
= 2 I of"-' (1 - xy-' (1 - vxf dx,
Jo
that is,
{l-v)U(<x, A 1-/9, v)+~^U(<x-l. /9+1, -0, v) = 2U(a. /S, -0, v).
We have therefore
(i-j;)n(i.+g, ^-g, ^+g.^)-f^^+^~^n(i. + g-i, f-g, -^ + g, ^
= 2n(i«+g, i-g, -J + g,p:
and from the same equation, written in the form
n(a-l. /9+1, -/S, v) + ^^(l-v)U(a, /3, 1-/S, t;) = 2^f ^HCa, /9, -/9, v),
we obtain
84. Hence the terms in [ ] in the first and the fourth expressions in No. 82 are
= ra. + g)r(i-g)- /'^w-» •"l^^ + '?-^-g--^ + g-7'J'
ra«-g+i)r(i+g) 7^=^^ n(^i.-g + i, -i + g, i-g.^j,
607] A MEMOIR ON PREPOTENTIALS. 375
respectively ; the corresponding values of the disk-integral are
which we may again verify by writing therein « = 0, viz. the Il-functions thus become
and consequently the integral is
85. But the forms nevertheless belong to a system of four. In the formulae
n (a, /3, 7, V)
n(7, a + ^-7, a, v)
= {l-vy-y U{^, a, a + /9-7, v)
= a-)'-r(J|^n(a+^-7. 7. A .),
writing a=ls+q, fi=^ — q, y = — ^ + q, we deduce
11(^8 + 5', i-9, -^•¥q, v)
= (1-")'-^ Tl{i-q, is + q, ^S-q+l,v)
= <^ - ^y-' r ail;Vi)yf-7? ,) n (i. - , + 1, - i 4- ,, i - ,, .) ; .
and the last-mentioned values of the disk-integral may thus be written in the four
forms :
rd-gmt + g) •^'"' n(i* + g, i-g, -l + g,^ -terming,
T(i + q)r%?-q + i) /'-" n(-Hg.i-g + i.i«+g.p- .. .
rW-q)T{i,S + q) V-j) n(i-?'i* + ?. i«-?+l./J - " .
r(^ + g)T(i.-g + i)(/-7J n(i«-g + i, -i + g, j-g.j,j- „ ;
376 A MEMOIR ON PRE POTENTIALS. [607
and since the last of these is in fact the second of the original forms, it is clear
that, if instead of the first we had taken the second of the original forms, we should
have obtained again the same system of four forms.
86. Writing as before x = - — .^ _ ^, &c., the forms are
^^1;^ — — (/=-/ry-'-» [<**+*-'(<+/=- /c')-*^«-' (<+/»)"**"« d«-termine,
87. The third of these possesses a remarkable property. Write mf instead of /,
and at the same time change t into niH: the integral becomes
r(f
^ttT^ x/'+' f tr^ [m-a +/-) - «'!-«+! a +/n-i»+?-' dt - term in e ;
and hence, writing inf=f+Sf or m = l + -^, and therefore m^ = l+2 ^-, the value is
Hence the term in 8/ is
= 8/ into expression j,^^?Ii^^IL_.^ /« J"<-,-i (<+/._ ^)-»-i (« +/.)-i^ dt,
where the factor which multiplies 8/" is, as it should be, the ring-integral ; it in fact
agrees with one of the expressions previously obtained for this integral.
88. Similarly for an exterior point of; starting in like manner from, Disk-
integral
and reducing in like manner, the terra in { ) may be expressed in the four forms
607] A MEMOIR ON PREPOTENTIALS. 377
2 r{^ + q)ri^s-q) /^««- multiplied by
89. For the reduction of the first' and the fourth of these, we have to consider
viz. this is
-(l-t;)n(a, A 1-/9, v) + '^U(ci-l, 0+1, - /3, v);
{-l+v+l+v)\ x<'-^{l - X .1 - vxy-i dx - 2 i vx.x'-^{l -x .l-vxf-^dx,
Jo .'0
= 2» I iC-^(l -x)(l-a. l-vx'f-^dx,
Jo
= 2vU{a, 0+1, -0 + 1, v);
that is,
-{l-v)U{a. 0, 1-0, v) + ^U(<x-l, 0 + 1, -0, v) = 2vU{a, 0 + 1, -0+1, v).
1 repeat, for comparison, the foregoing equation
+ {l-v)U(a, 0, 1-0, z,) + ?^n(a-l, 0 + 1, -0, i;) = 2n (a, 0, - 0. v);
by adding and subtracting these we obtain two new formulae ; for reduction of the
fourth formula, the equation may be written
-n(a-l,/9+l, -0,v) + {l-v)^n(a, 0, 1-0, v) = -2 ^vU(a, 0 + 1-0 + 1, v).
90. But it is suflScient to consider the first formula; the term in [ ] is
r(i8 + 5)r(i-9) [ K ) K^ ^\¥ + q.^ ?'* + ?■«.;'
and the corresponding value of the disk-ii\tegral is
c. IX. 48
378 A MEMOIR ON PREPOTENTIALS. [607
which we may again verify by taking therein k indefinitely lai'ge; viz. the value is
then = -rr, i^ vrs;. 'i^ above. It is the first of a system of four forms, the others
of which are
-TW^r%^s-q^i)-pn n(i+9. i.-9+i, i^+^.-Q.
nsn /'^7i -^y~^u(l3 all l + o^-a-^-]
-r(is-5+i)r(i + 5)/c'+=A^ W *H* 9 + i. i + ?a 9.^;-
t + f" — K^
And hence, writing as before x = — "^^ , &c., the four values are
z
= r (I + g[r'(^L g + 1/'" ^-' -^')'"^ /l^ ^"^ (^ +/-«=)-* (<+/0-*- ci^,
where we may in the integrals wi-ite t + k- —f in place of t, making the limits oo , 0 ;
but the actual form is preferable.
91. In the third form, for / write mf, at the same time changing t into mi;
the new value of the disk-integral is
Writing here mf=f+Bf, that is, ni = l + -4, m' = l + — /, and observing that, if
— 5 + i be positive, the factor jm' (<+/')— «*}"'+* vanishes for the value t= — —f^ at
the lower limit, we see that on this supposition, —q + i positive, the value is
viz. the term in S/" is =Bf multiplied by the expression
607] A MEMOIR ON PREPOTENTIALS. 379
that is, multiplied by
which is in fact = hf multiplied by the value of the ring-integral.
92. Comparing for the cases of an interior point k <f and an exterior point
K >f, the four expressions for the disk-integral, it will be noticed that only the third
expressions correspond precisely to each other ; viz. these are : interior point, k </; the
value is
ra.^yAj-,) f"['-^ " +/' - '■>""' <' ■^/■>"'"'"' * - S ^^) •
where, if ^— q be positive (which is, in fact, a necessary condition in order to the
applicability of the formula), the term in e vanishes, and may therefore be omitted :
and exterior point, k >/; the value is
differing only from the preceding one in the inferior limit K^—f^ in place of 0 of
the integral. We have ^ — q positive, and also i^s+q positive ; viz. q may have any
value diminishing from ^ to — ^», the extreme values not admissible.
Annex IV. Examples of Theorem A. Art. Nos. 93 to 112.
93. It is remarked in the text that, in the examples which relate to the s-coordinal
sphere and ellipsoid respectively, we have a quantity 6, a function of the coordinates
(a, . . , c, e) of the attracted point ; viz. in the case of the sphere, writing o" -I- . . . -H c^ = /e^
we have
p + e^e
in the case of the ellipsoid, we have
the equations having in each case a positive root which is called 6. The properties
of the equation are the same in each case ; but for the sphere, the equation being
a quadric one, can be solved. The equation in fact is
^ - ^ (e= + «-"-/»)- e»/» = 0,
and the positive root is therefore
e = ^{^ + K''-f' + V(e= + k" -f^y + 4,^/%
Suppose e to diminish gradually and become = 0 ; for an exterior point, k >f, the
value of the radical is =K^—f^, and we have 6 = K'-—f-\ for an interior point, K<f,
48—2
VIZ.
380 A MEMOIR ON PREPOTENTIALS. [607
the value of the radical, supposing e only indefinitely small, is =/' — «- ■'r-fi \ ^, and
^ + /•j-k'J' '^/•a,^' *''■' ^'^'^^^ ^'^ ^^^ ^*"® thing, ^ = f 1 - ^J;
the positive root of the equation continually diminishes with e, and becomes ultimately
= 0.
If « or e be indefinitely large, then the radical may be taken = e° + «", and we
have 6 indefinitely large, =d' + k^.
94. The result is similar for the general equation
"' ^ a. f - 1 .
the left-hand side is = 0 for 0 = oo , and (as 0 decreases) continually increases, becoming
infinite for ^ = 0 ; there is consequently a single positive value of 0 for which the
value is = 1 ; viz. the equation has a single positive root, and 0 is taken to denote
this root.
In the last-mentioned equation, let e gradually diminish and become = 0 ; then
for an exterior point, viz. if
^, + - + Ja>l. theequation^-h... + ^^=l
has (as is at once seen) a single positive root, and 0 becomes equal to the positive
root of this equation; but for an interior point, or 75+-" + t5<1. the equation just
written down has no positive root, and 0 becomes = 0, that is, the positive root of
the original equation continually diminishes with e, and for e = 0 becomes ultimately
= 0; its value for e small is, in fact, given by ^ = ( 1 — ^^^ — ... — r^J. Also a,..,c, e
(or any of them) indefinitely large, 0 is indefinitely large, = a' -f . . . + c^ 4- e^.
95. We have an interesting geometrical illustration in the case s 4- 1 = 2 ; 0 is
here determined by the equation
a'' 6= gj
f'' + 0^ f + 0^ 0
viz. 0 is the squared a-semiaxis of the ellipsoid, confocal with the conic 2=^ + ^ = 1.
•/ if
which passes through the point (a, b, e). Taking e = 0, the point in question, if
j^+-^>l, is a point in the plane of xy, outside the ellipse, and we have through
the point a proper confocal ellipsoid, whose squared ^-semiaxis does not vanish ; but
if 75 + -?<Ii then the point is within the ellipse, and the only confocal ellipsoid
•/ if
through the point is the indefinitely thin ellipsoid, squared semiaxes (/^ g'', 0), which
in fact coincides with the ellipse.
607] A MEMOIR ON PREPOTENTIALS. 381
96. The positive root 6 of the equation
•^' -^ f^ + d - h? + e 6'
has certain properties which connect themselves with the function
©, == ^9-1 {(5 +/») ... (^ + A«)}-i.
We have, the accents denoting differentiations in regard to 6,
r,de 2a ^ de 1 2a
da e+f^~ ' da~J'e+f''
where
and we have the like formulae for . . , t- , t- .
dc de
We deduce
and to this we may join, rj being arbitrary,
a dd c de e dd 2 ,
0 + r]+/'- da 0 + v + h' dc 0 + v de J' {0+p .0 + v+f
+ ?! , '' ]
0 + h\0 + r) + h'^ 0.0+7,y
Again, defining Vj^ and D^ as immediately appears, we have
and passing to the second differential coefficients, we have
d'0 2 8a- 4aU"
da» ~ J' {0 +p) J''' {0 +pf J'» {0 +py '
where
and the like formulae for .., t-, j-^. Joining to these ^ -5- = j-vn" > we obtain
^. _/d^ d:'0^ d?e 2g+l d^N
_ 2 [ 1 I , 1 ,l + (2g + l))
~ J'\0+f^--'^ 0 + h?^ 0 f
-^,(-K")-^'(n
382 A MEMOIR ON PREPOTENTIALS. [607
where the last two terms destroy each other ; observing that we Lave
e" ^\e+p^'"^e+h*^ e )•
the result is
p.- 2 / 20'\ 40'
97. First example, x' = a' + ... + c^, and ^ the positive root of 7;- — 2 + 3 = l.
F is assumed = | <"«"' (i +f')~^ dt, where 5 + 1 is positive.
J 9
I do not work the example out; it corresponds step by step with, and is hardly
more simple than, the next example, which relates to the ellipsoid. The result is
P-{riyr(q+i)-' [^ /» — j . if ^ + -+^'</';
hence the integral
f 1 '^ ] dx...dz
]{{a -xf + ...+ (c-zy + ^}i'+i '
taken over the sphere a^ + ... + z^ =/^
-^i^v:--'-/-)-*-
98. Second example, d the positive root of -?r — 7; + ... + 5 ^ + -^=1; Q+l
j^ + u n^-\- 0 ff ^
positive.
Consider here the function
V = I r«-i {{t +/») ...(< + h')}-i dt ;
J e
this satisfies the prepotential equation. We have in fact
da ' da' da^ \da/ '
dV
da
d^V d'^V
with the like expressions for ... , -r-j , ~j— ; also
2q + ldV^ Q2q + lde
e de e de'
Hence
DV=-@ad-e'S7,d,
607] A MEMOIR ON PREPOTENTIALS. 383
or, substituting for D^ and Vj0 their values, this is
Moreover V does not become infinite for any values of (a, . . , c, e), e not = 0 ;
and it vanishes for points at x . And not only so, but for indefinitely large values
of any of the coordinates (a, . . , c, e) it reduces itself to a numerical multiple of
(a' + . . . + c" + e^)~i*+' ; in fact, in this case 6 is indefinitely large, =a^ + ... +c^ + e\
Consequently throughout the integral, t is indefinitely large, and we may therefore write
F= /""r^' . t-i' dt, = -T-^ {t-^'-^t, = r-^ ^~^*~*.
that is,
V= -. (a" + . . . + c= + e=)-i*-«.
The conditions of the theorem are thus satisfied, and we have for p either of
the formulae
in the former of them q must be positive ; in the latter it is sufiScient if 3 + 1 be
positiTe.
99. We have W the same function of («,.., z, e) that V is of (a, . . , c, e) ; viz.
writing X for the positive root of
f + X h^ + \ \
the value of W is
= f °°«-«-' {(« +/0 •••(« + 'i')}-* <^<-
Considering the formula which involves e^ W, — first, if t^ + • • • + Fj > 1. then, when
e is =0 the value of \ is not =0; the integral W is therefore finite (not indefinitely
large), and we have e^* IF = 0, consequently p = 0.
3? z^
But if 2^1+ ••• +Ti< 1. then, when e is indefinitely small \ is also indefinitely
^ 3? ^
small; viz. we then have - = 1 ~ Jt~ •••~'u''> ^^^ "^^"^^ of TT is
''- (ri)T3 9UJ ^■^■•■^■' • -(Tiyriq + i)'''^-''^ V p - W*
If
and hence
384
A MEMOIR ON PREPOTENTIALS.
[607
dV ^dd
100. Again, using the formula which involves (e*«+' "j")' ^® ^®^^ ^^''^ 1~~~®~ir '
dd
or substituting for 0 and -r- their values and multiplying by e^+S we find
dV
= 2e'9"" ^'
and therefore
dW
L(/' + e)=
+ ...+
+ ...+
(A'
-?]
2'
<]
(A»
+ x)=
a^
Hence, writing e=0: first, for an exterior point or 7-^+ ... + rj> 1, X is not = 0,
and the expression vanishes in virtue of the factor c^+^ whence also p = 0 ; next,
for an interior point or ^"2+ ••• +^j< 1. X is =0. hence also y^^- \\}- ~ Ji~ • • • ~ Ti]
a?
is infinite ; neglecting in comparison with it the other terms , . , ^^ + . . . , the value is
2gy(/.../o-s=2(i-^-...-i;y(/...An
and we have, as before.
z\<i
P (r^yr (5 + 1)^-^ •••"-* V p "• h\
101. Hence in the formula
„_ /" pdx ... dz
~j{(a-icy+... + (c - z)° + ^\i'+9
J »
p has the value just found, or, what is the same thing, we have
{'^-p---f^'dx...dz
{(a -«)» + ...+ (c - zy + e"}**** '
taken over ellipsoid 7^ + •■• + r^ = l*
= ^^r'as+g) ^^ ^■^-^^ //"'"' '^' +-^') •••('+ '*')'"*
d«.
607] A MEMOIR ON PREP0TENTIAL8. 385
102. We may in this result write e = 0. There are two cases, according as the
(J? (?
attracted point is exterior or interior: if it is exterior, ^+... + ,-^>l, Q will denote
C , -P •. 1 • X ■ « c-^
the positive root of the equation ^^ ^ + ■ ■ • + 7737^ = 1 ; if i* be interior, ^ + ... + ,-<!,
B will be = 0 ; and we thus have
1 — -^„ — . . . — ra ) dx ...dz
\{a-xf+...+{c-zyY"+9
= ^T([/+g)^^ ^-^ • • • ^^ r*''~' [(«+/')•••(«+ ^'))"* <^t' fo'- i'^t^^io'- poi^t ^ + • • • + 1 < 1 ;
but as regards the value for an interior point it is to be observed that, unless q be
negative (between 0 and — 1, since 1 + gj is positive by hypothesis), the two sides of
the equation will be each of them infinite.
103. Third example. We assume here
V^rdtP"!,
J e
1=1-
where
a!
p + t ■■■ h^ + t t'
T=tr^-'{(t+p)...{t+h')\-i;
as before, 6 is the positive root of the equation
r_i_ «! _^L_^ _o
f + e ••• h'+e 6' ~ '
and ^s + q is positive in order that the integral may be finite ; also m is positive.
104. In order to show that V satisfies the prepotential equation [3V=0, I shall,
in the first place, consider the more general expression.
V=r dtI"'T,
J e+r,
where t; is a constant positive quantity which will be ultimately put =0. The
functions previously called J and 0 will be written /„ and 0,,. and J, 0 will now
denote
./, = 1 -
ff + v+P '" S + v + h'' 0 + v'
0, = (^ + v)-''-' {{0 + V +/') ■■.{0+V + h')]-^ ;
C. IX. 49
386 A MEMOIR ON PREPOTENTIALS.
whence also, subtracting from J the evanescent function Jo, we have
and we have thence, by former equations and in the present notation,
[607
say this is
de
0+V+P da
c dd _e^ de_2^p
^0 + V + h!' dc'^e + v de~7„ '
«/o
n^ =
-40.'
J 0^0
In virtue of the equation which determines 0, we have
dv^r
da J t+r,
dtmI"^'j^^T
-J<"B
dd
da'
and thence
d^V
-2
ia"
da^ = t, '' H"- iW^ + - (- - 1) ''"-' (iTpy\ ^
- mJ"'-^
da
dd
da
dd\^
- J*"©
da"
with like expressions for
d'^F d'F
e de J e+r,
' dd'' de» ■
J e
Also
t e de
and hence
d^ e d^N
+ m(m-l)/'»-^4
a de
((+/.).+ •••+(^
+ 4w J'»-' 0
d
S + V+f da
+ ... +
^ + 1? + A» dc "^ 5
e d^\
' + »; de/
(/"•H) n;
vda/
+ ...+
607] A MEMOIR ON PREPOTENTIALS. 387
105. Writing /', T for the first derived coefficients of /, T in regard to t, we have
The integral is therefore
f rf< \2m 7"^i ^,„ 7* + TO (m - 1 ) /"-^ . 47' rl ,
•'«+i (. -' J
= f d< {4m 7'"-! 7" + 4?rt (m - 1) 7"^ FT],
viz. 7*"~' r vanishing for t= x , this is
= - 4to J™-' e.
Hence, writing (/"* 0)' instead of ^ (/"" 0), we have
DF=-4mJ*»-'0
i
. -(j'"e)'v,0
viz. this is
nF=-4»n J'»-'e
•'0
or, writing m /"*-' J'B + /"»«' instead of (J"'"®)', this is
4.1)1 f^—'fit d, f'"
□ F = - ^""'j, (J' -2P + J)- ^,^^ (B'H„ - 0(H>;).
We have here
= »;' Q, suppose.
Also 0'0o - 00o' contains the factor rj, is = i/M suppose.
49—2
388
A MEMOIR ON PREPOTENTIALS.
[607
106. Substituting for /, J'-IP + J, and e'0„-ee„' their values t/P, r)Q, and
ijM, the whole result contains the factor ?;'"■'"', viz. we have
^y_^r^(^^PMy
If here, except in the term ij^+S we \vrite »? = 0, we have
p_ g* c* t - T
a'
the formula becomes
or (instead of J„, 0„) using now J, % in their original significations
J=l-
this is
^Tr^---^"|ii-'^' ^«d © = ^^M(^+/o-(^+An}-*,
or, what is the same thing,
viz. the expression in ( } is
-,+ ...+
+ :*
(^ +/2y^ •••^ {e + h?f^ 6^
1_ 1^_ 2g + 2
We thus see that, rj being infinitesimal, D F is infinitesimal of the order j;"'^' ; and
hence, t} being = 0, we have
nF=0;
viz. the prepotential equation is satisfied by the value
V=rdtI^T,
where »t + 1 is positive.
107. We have consequently a value of p corresponding to the foregoing value
of F; and this value is
p = -
27r4«r((^+l)
e»9+'-^
607] A MEMOIR ON PREP0TENTIAL8. 389
where, writing \ for the positive root of
we have
W
we thence obtain
rfe'
or, multiplying by e^+' and substituting for t~ its value
2e
X
ave
^-f=/>-^"(-,^.--Fi7..-r'«-/-'-<-*')i-*
2g»(+j
where the second term, although containing the evanescent factor
/ _ a? _ _ z'- _ e^y
is for the present retained.
108. I attend to the second term.
x^ z^
1°. Suppose -fi+ ••• +u>^] then, as e diminishes and becomes =0, X does not
become zero, but it becomes the positive root of the equation
X+/' ■■■ X + A''
hence the term, containing as well the evanescent factor e**+' as the other evanescent
factor { 1 - r ?-,— .-.— r — ,.— r) , is =0.
\ X+p X + /i« Xj '
390 A MEMOIR ON PBEPOTKNTIAL8. [607
2*. Suppose i;^ + . . . + TT^ < 1 ; then, as e diminishes to zero, \ tends to become = 0,
but - is finite and = 1 — -^ — . . . — ^j , whence — is indefinitely large ; and since
a? z* a? z^ . .
T\ — 7«v +•••''■ (\A.h^ becomes = ivi + • ■ • + r; , which is finite, the denominator may
g»
be reduced to - , and the term therefore is
-2(1 -^.-...-g(i-^.-...-^,-g" </.../.,-.,
which, the other factor being finite, vanishes in virtue of the evanescent factor
/ «» _ z"- e'V"
Hence the second term always vanishes, and we have (e being =0)
x^ z^
109. Considering first the case ■^+ ... + j->l: then, as e diminishes to zero, \
does not become = 0 ; the integral contains no infinite element, and it consequently
vanishes in virtue of the factor e°*+^.
But if p5+---+p<l. then, introducing instead of t the new variable f, =t. tbat
is, t=T> dt= — ^-^ , and writing for shortness
/' + ! h'+^
the term becomes
=j'df.2m(K-f)'-f»|(/^ + |) ... (^'+1)}"*,
where, as regards the limits, corresponding to i = 00 we have ^ = 0, and corresponding
to < = \ we have f the positive root of R — ^ = 0. But e is indefinitely small ; except
for indefinitely small values of f, we have
iJ = l_^_..._J, and {(/» + |) ...(A' + |)|'* = (/...A)-;
G07] A MEMOIR ON PREPOTENTIALS. 391
and if f be indefinitely small, then, whether we take the accurate or the reduced
expressions, the elements are finite, and the corresponding portion of the integral is
indefinitely small. We may consequently reduce as above; viz. writing now
the formula is
dW
7?-1 ^ •^
^+iyi= d^, 2m (R - ?)"^'^9 (/. . . h)-\
de J jt
= - 2m (/ . . . A)-' ^d^ .^HR- f)'"-' ;
Jo
or writing ^=Ru, the integral becomes =729+"* I dit.M«(l— i«)'"~', which is
Jo
r(l + q)T(m)
~ ra + q+m) ^ '
that is, we have
(k( ^-^ ' r (1 + 5^ + m) '
and consequently
that is,
p u-i^) (ri)«r(i + 9 + m)^ '
viz. p has this value for values of (x,.., z) such that -^ + ... +ri < 1, but is =0 if
3? 2- ^
110. Multiplying by a constant factor so as to reduce p to the value ^«+"', the
final result is that the integral
the limits being given by the equation
is equal to
rc^yrg + ry + m)
l'->-
■■-i) ^
V ... dz
-{a-xf+..
.+{c-zf+..
,.+e2jj.+«'
equation
•2' ,
• + A' = 1'
where d is the positive root of
e e
^ e+f^ ■•■ 6i + /i« 0 ^"
392 A MEMOIR ON PREP0TENTIAL8. [607
In particular, if e = 0, or
|(l-y.---I.J d--
J [{a-xy+... + {c-zy}i
there are two cases:
exterior, -7i + ■•■ +ri>^, ^ is positive root of 1 — 7^ — ... — .^ = 0,
interior, ^ + -" + fj<l, 0 vanishes, viz. the limits in the integral are 00, 0;
q must be negative, 1 +q positive as before, in order that the (-integral may not be
infinite in regard to the element t = 0.
It is assumed in the proof that m and 1 + q are each of them positive ; but,
as appeai-s by the second example, the theorem is true for the extreme value ?» = 0 ;
it does not, however, appear that the proof can be extended to include the extreme
value 5 = — 1. The formula seems, however, to hold good for values of m, q beyond
the foregoing limits ; and it would seem that the only necessary conditions are ^s + q,
1 + m, and l + q+m, each of them positive. The theorem is, in fact, a particular case
of the following one, proved Annex X. No. 162, viz.
F =
{{a -xf + ...+{c- zf + e=)i»+« '
a? z^
taken over the ellipsoid 2^ + • . • + r^ = 1, is equal to
'(-g)^r'^[,+\) \] d* '"'"' {(< +/') -(* + /'=))"* (1 - '^)-'/] ^"'-' <t>{<r+(l-a) x} dx,
e-
where <r denotes ^^ — i+"- + ri — i + T- assuming ^m = (1 — !«)«+"», we have
<^ {<r + (1 - <r) a;} = (1 - o-)«+'» (1 - «)»+'»,
and the theorem is thus proved.
111. Particular cases: ni = 0;
/,
(1 ^. ... i;)dx...dz (ri)«r(i4-g) ,,)rd«r^Ma+/-n (t+h^)]-i
[(a-a;)«+... + (c-2)= + e=']i«+9 r^i^s + q)
Cor. In a somewhat similar manner it may be shown that
hi
ii::fc::g!:;:;,:;=a^i^>(/...^.r^^^'K.v)...(...-)i-.
607] A MEMOIR ON PREPOTENTIALS. 393
Multiplying the first by a and subtracting it from the second, we have
or, writing q+l for g', this is
[^~fi~---~h2) (a-x)dx ...dz (Yl\»T('>A-n\ C n
K»-«)'^..^(c-.)-+^l''-^°r("4^i><-^---*'/.'''?^-'"'"''<'-^^'>---<'-^'''''-'-
and we have similar formulae with .., {c — z), e, instead of {a — x), in the numerator.
112. If m=l, we have
{(a - a;)» + ... + (c - zf + e='|i'+«
which, diflferentiated in respect to a, gives the (a — a;)-formula ; hence conversely,
assuming the a — x, ..,c— z, e-formulae, we obtain by integration the last preceding
formula to a constant prhs, viz. we thereby obtain the multiple integi-al =G + right-
hand function, where C is independent of (a, . . , c, e) ; by taking these all infinite, and
observing that then 0 = oo , the two integrals each vanish, and we obtain C = 0.
In particular, when s = 3, q = — \, then
which, putting therein e = 0, gives the potential of an ellipsoid for the cases of an
exterior point and an interior point respectively.
Annex V. Green's Integration of the Prepotential Equation
113. In the present Annex, I in part reproduce Green's process for the integration
of this equation by means of a series of functions, which are analogous to Laplace's
Functions and may be termed "Greenians" (see his Memoir on the Attraction of
EUip-soids, referred to above, p. 320); each such function gives rise to a Prepotential
Integral,
Green .shows, by a complicated and difficult piece of general reasoning, that there
exist solutions of the form F=0(^ (see post. No. 116), where ^ is a function of the
c. IX. 50
394 A MEMOIR ON PREPOTENTIALS, [607
s new variables a, /3, . . , 7 without 6, such that V ^ = K<f>, k being a function of 6 only ;
these functions ^ of the variables a, /3, . . , 7 are in fact the Greenian Functions in
question. The function of the order 0 is <^ = 1 ; those of the order 1 are <^ = o,
<f> = 0, .. , <l> = y; those of the order 2 are <f> = a/3, &c., and s functions each of the form
^{Aa» + B^+... + Cr]+D.
The existence of the functions just referred to other than the s functions involving
the squares of the variables is obvious enough; the difficulty first arises in regard to
these s functions; and the actual development of them appears to me important by
reason of the light which is thereby thrown upon the general theory. This I accom-
plish in the present Annex ; and I determine by Green's process the corresponding
prepotential integrals. I do not go into the question of the Greenian Functions of
orders superior to the second.
114. I write for greater clearness (a, b,..,c, e) instead of (a, . . , c, e) to denote the
series of (s + 1) variables ; viz. (a, 6, . . , c) will denote a series of s variables ; corre-
sponding to these we have the semiaxes (/, g,- . , h), and the new variables (a, /S, . . , 7) ;
these last, with the before-mentioned function 6, are the s + 1 new variables of the
problem ; and, for convenience, there is introduced also a quantity e ; viz. we have
a = V7M^ a.
b = ^/g' + 0 /S,
c = ^W+'d 7,
e = \fd e,
where \ = a* + ^ + ... +'f + ^.
That is, we have 6 a function of a, h, .., c, e, determined by
and then a, /3, . . , 7 are given as functions of the same quantities a, b, . . , c, e by the
equations
»2 «' 02 ^ " c»
also e, considered as a function of the same quantities, is
-y fi+0 g2^0 ••• h-'+d'
115. Introducing instead of a, b,..,c, e the new variables a, /3, ..,7, 0, the trans-
formed differential equation is
r+^f (. + 2,-H2-^^-...-^,)-HVF=0
607]
where for shortness
A MEMOIR ON PREPOTENTIALS.
395
p^e
h}
— a- —
— ^ fl2 - ~2 + 1 ,
d'V
If/" , h* 1 d'V
^/i» + ^t/» + ^ g' + d'^ ^^'■]dy'
d'V
,2
2g d'V
'p + e.g-'+e dad^
1
-&c.
dF
Also
^j;^e{-'^-'-K7h-e^fhe'--h
dV
dfi
dV
dy '
^ de- ^ ' r r+e ■■• 7= + ^] [r + e^da^-^h'+d^dy ^ de)-
116. To integrate the equation for V, we assume
F=0<^,
where 0 is a function of 6 only, and ^ a function of a, /9, . . , 7 (without ^), such that
« being a function of 6 only. Assuming that this is possible, the remaining equation
to be satisfied is obviously
Solutions of the form in question are
<^ = 1 , « = 0,
1
^ = a , K =
p+e
^* ^ q'+d - h' + e
4> = ^ , K= „ „
■hd'
e
e
h' + 0\
50—2
396 A MEMOIR ON PBEPOTENTIAL8. [607
and it can be shown next that there ia a solution of the form
117. In (act, assuming that this satisfies V ^-K<f> = 0, we must have identically
+ K{^(Aa'' + B^-+...+ Cy') + D j;
so that, from the term in a', we have
■^ f 9^ ^'1 BP Cf*
or, what is the same thing,
with the like equations from /S", . . , 7- ; and from the constant term we have
118. Multiplying this last by /", and adding it to the first, we obtain
viz. putting for shortness n = gf-.. ,+ » . fl + ••• +xrT"g) » ^^i^ i^
il (2o + 2 + n + i/c (Z'' + 6)] + Kf'D = 0:
and fumilarly
5 (2g + 2 + n + i« {g- + 6')} + Kg'D = 0,
C {2g + 2 + n + J/c (^» + ^)) + «A^ Z) = 0.
607] A MEMOIR ON PREPOTENTIALS. 397
To these we join the foregoing equation
f^ + e g^+e h? + e
Eliminating A, B, . . ,C, D, we have an equation which determines k as a function of 6 ;
and the equations then determine the ratios of A, B,..,C, D, so that these quantities
will be given as determinate multiples of an arbitrary quantity M. The equation
for K is in fact
+ 1=0;
^(h'' + e){2q+2 + il + ^K{h''+d)]
and the values of ^, B, . . ,C, D are then
Mp M£ m= _M
2g+2 + fl + i/c(/' + ^)' 2q + 2 + il + ^K{f + ey'2q+2+^ + ^K{h^-Jre)' k'
values which seem to be dependent on 6: if they were so, it would be fatal to
the success of the process; but they are really independent of 6.
119. That they are independent of 6 depends on the theorems; that we have
_ (2g + 2 + n) K,
" 2q + 2-^K,e'
where «<, is a quantity independent of 0 determined by the equation
»
2q+2 + ^K„/'^2q+2 + ^K,g^^-^2q+2+^Koh'
(«o is in fact the value of k on writing ^ = 0) : and that, omitting the arbitrary
multiplier, the values of A, B,..,C, D then are
p g' h' _1
2q+2 + ^K,/" 2q+2 + iK,g 2g + 2 + i«„A»' k,'
or, what is the same thing, the value of <f> is
~2q + 2 + i*„/» ■•" 25- + 2 + ^K^g''^ '" '^ zq + a i-4«„A^ "«„'
120. To explain the ground of the assumption
_i2q + 2 + n.)K,
" 2q + 2-^Koe'
observe that, assuming
2^ H^2 + n + ^/t(/' + g) ^ 2q + 2 + n + ^K(g' + d)
2q + 2 + ^>c,/' 2q + 2+^K,g'
398 A MEMOIR ON PREPOTENTTALS. [607
then multiplying out and reducing, we obtain
viz. the equation divides out by the factor g'—/^, thereby becoming
Ko{2q + 2+n)-{2q+2)a + ^KKod = 0.
that is, it gives for k the foregoing value : hence clearly, k having this value, we
obtain by symmetry
2q + 2+ il + ^K (p + 6), 2q + 2 + n + ^K{g'+ 6),. . ,2q + 2+ n + ^K (h'' + 6),
proportional to
2q + 2 + \Kj\ 2g+2+i«,5f»,.., 25 + 2+^«,A»;
viz. the ratios, not only of .4 . B, but of J. : B : ... : G will be independent of 0.
121. To complete the transformation, starting with the foregoing value of k, we
have
2, + 2 + n + i.(/^ + ^) = (2,+ 2 + n)|±|±|^\ &c.;
so that we have
A{2q+2 + i*Co/'} + Kof'D = 0,
B {2q + 2 + ^K,g'} + Kog'D = 0,
C{2q + 2 + ^Koh'} + K,h?D = 0,
and
_A^,_B_ C (2g + 2 + n)^oZ)
fi + e'^ g^ + e'^---'^h' + 0 2q + 2-iiK,e
Substituting for A, B,.., C their values, this last becomes
_ <o-P I 2g+2 ^1 _ _ /tpZ) I 2q+2 0)
2q + 2 + ^Ko0 \2q+2 + ^Kj' f'' + 0} '" 2q + 2- ^k,0 \2q + 2 + ^K,h' h' + 0\
viz. this is
\ 2q + 2 0 ] ( 2q+2 ^l„„r.«
or, substituting for Xi its value, and dividing out by 2q + 2, we have
1 1 1 , «
2q + 2+y,f 2g + 2 4 i /Co<7= 2^ + 2 + i «„ A^
the equation for the determination of «■„
607] A MEMOIR ON PREPOTENTIALS. 399
122. The equation for «„ is of the order s; there are consequently s functions of
the form in question, and each of the terms a^ /8^, . . , 7- can be expressed as a linear
function of these. It thus appears that any quadric function of a, . yS, . . , 7 can be
expressed as a sum of Greenian functions; viz. the form is
A
+ B(z+ &c.
+ C'a/3 + &c.
jy,( \f-(iC- y^^ ^hy 1 \
\2q+2 + iK;p'^2q+2 + ^K,Y'^-'^2q+2+^Ko'h' kJ
+ D'\ ,. .. „ )
(s lines),
viz. the terms multiplied by U, D", &c. respectively are those answering to the roots
*o'i *o"> • • of tlie equation in a:,.
The general conclusion is that any rational and integral function of a, B,.., 7 can
be expressed as a sum of Greenian functions.
123. We have next to integrate the equation
Suppose « = 0, a particular solution is 0 = 1. Next, suppose
K= ^z — A—2q—2 n— ••• — -n. — -n\\ a particular solution is , ^■^ :
in fact, omitting the constant denominator, or writing 0 = ^/' + 6, and therefore
d% ^ 1 dm ^ 1
d0~2 >/f2~+e ' d^ 4 (/» 4. ^)t '
the equation to be verified is
— 26
Again, suppose k— .„ — ^ — r „ + &c. (value belonging to A = a/8, see No. 116); a
J + " -9 + "
particular solution is ^^ — — Tj : in fact, omitting the constant factor, or writing
+ -7^
400
and therefore
A MEMOIR ON PEEPOTliNTIALS.
[607
d0
_ v/' + g)
the equation to be verified is
/Vg' + g V/' + g\
1 {p + ef ^/p+e^g' + d (g'+d)*)
6 6 6
or putting for shortness O = ^ — >, + - — g + . . . + tt-T/i > t^i^ i^
J ' + 0 g + a li + a
e^g^ + e
26
6'JpV6
{p+ef^ 'Jp+es/g'^ + d (g' + 6)i Wp + d 'Jg^ + 6)^ ^ '
26 ^fTe / g, o , 0 n] + ^Z!±l(.2o-2 + -^-n)-o
'^f^r0^j:re^'JpT'6\ ^ ^^p + e "j + v^^l ^* ^ f+e "j"*^'
which is true. And, generally, the particular solution is deduced from the value
of <^ by writing therein
^p + e
'^g' + 6
^/h'+6
-^P + f + .-. + h?' '^p+sr'+--- + h'"" "^p + g' + .-.+h*
in place of a, /3, . . , y respectively : say the value thus obtained is S = H, where H is
what <f> becomes by the above substitution.
124. Represent for a moment the equation in 0 by
and assume that this is satisfied
by e = HJ
zd6. Then we have
dH rrd2\
de'^^dd)
+ KHlzdd
'I
= 0,
607] A MEMOIR ON PREPOTENTIALS. 401
and therefore
de
80^ + 2PHj z + 4^^^ = 0 ;
jjr
viz. multiplying by -rj,, this is
4^'
or
viz. substituting for P its value, this is
Hence, integi'ating,
ff ^2 = ,- , G an arbitrary constant,
-jp + e.g^ + e.^h^ + e ^
and
CA ' nu i e-i-'de ,.^
W = CU , Y arbitrary,
}xH'-'Jp+e.g^ + e...h^-^e ^ ^
where the constants of integration are C, X; or, what is the same thing, taking I"
the same function of t that H is oi 0 (viz. T is what <f> becomes on writing therein
V/M^^TTTm»' 'Jf'-+g-'+...+h^"" '^p + f+...+h''
in place of a, /8, . . , 7 respectively), then
e=^-CH
P
where x ^^7 ^ taken = « : we thus have
V = m = -CHSr — 1r9-^dt
^J» T'\'f' + t.g'+t...h'' + t
Recollecting that
so that for ^ = x we have a' + 6" + . . . +0^ + 6"=^, the assumption % = 00 comes to
making V vanish for infinite values of (a, b,..,c, e).
125. We have to find the value of p corresponding to the foregoing value of V\
viz. W being the value of V, on writing therein {x, y,. . ,z) in place of (a, 6, . . , c),
then (theorem A)
''" 2(ri)'r((? + i)l^ dW/
c. IX. 51
402 A MEMOIR ON PREPOTENTIALS. [607
Take X the same function of (x, y,..,z, e) that 0 is o{ {a, b, . . , c, e) : viz. take X
the positive root of
1 — ^-4- ..+ — - = 1-
f^ + \ g^ + \ h^ + X X '
and let (f, 17, . . , ?, t) correspond to (a, /S, . . , 7, e), viz.
j,_ _« y_ o^ g = /1 _ ■'^ ^
f-V/T:fx'''~V^»Tx'"'^ VA'Tx' "■ V^ /^ + x ^» + ;
»»
X ■■■ A' + X'
80 that W is the same function of (f , 17,.., X) that V is of (a, yS, . . , ^) : say this is
t-i-^ dt
then we have for p the. value
r {\s^±qi ^,,, ^^^, r 1 - /^^l - - >^
f^ + \ ■■■ A» + x/
V/= + x^ d? ^•••^A=+x^ d? dxr
where e is to be put = 0.
126. Suppose e is =0; then, if 7^ 4-^,+ ... 4- r^ > 1, ^ is not =0 but is the
positive root of ^^^+^,--+...+^^^ = 1: .,=^1 -^^--^^-_^^^- ... -^^^_^^, i8 = 0:
and we have p = 0, viz. p is = 0 for all points outside the ellipsoid 7:^ +,+ ••• + tj = !•
Sj^ -1*2 »2 g3
But if 7-5 + ^+ ••• + r-2< 1, then, on writing e = 0, we have X = 0, t-'=-,
J 9 "" X
rgs + g) e»»+' X/l dTf 1 rfF ^l.lY.o^^'^
2,ri'r(ry+l)- "^ ^ ' 1/=^ d? + ^^^ d^ + - + /.^ ^ df ^dxA=o'
where the term in ( ) is
= -20
Hence
A„/(7.../j-X?+'-
'^ 27rJ"r(5+l)-A„/5r...A W
_ -r(^s + g) 2CVr, / a? f _zy
27ri'r(q+l)-A,fg...hV /» fir' - /iV '
607] A MEMOIR ON PREPOTENTIALS. 403
where i/r^, A,, are what ifr, A become on writing therein X = 0. It will be remembered
that A is what H becomes on changing therein d into X; hence Ao is what H
becomes on writing therein ^ = 0.
Moreover yfr is what (f> becomes on changing therein a, /3,..,y into ^, t],.., ^■.
CC 1J z
writing \ = 0, we have f = -?, V — >•■> ^—tj hence -^o is what <f) becomes on changing
SO IJ z
therein a, ^,..,.7 into -j., -,..,t- And it is proper in (j) to restore the original variables
by writing . , , ,-■, / in place of a, B,..,y.
127. Recapitulating,
y_ r pdx ...dz
~] [{a - (cy + ... + (c-z)- + e=']i»+9 '
where, since for the value of V about to be mentioned p vanishes for points outside
the ellipsoid, the integral is to be taken over the ellipsoid
a^ z- ,
and then, transferring a constant factor, if
<-«-i dt
the corresponding value of p is
where Aq is what H becomes on writing therein ^ = 0, and yjrt, is what yfr becomes
on writing
CC z
f,-,r in place of a, . . , 7.
128. Thus, putting for shortness H = <-»-'{(«+/»)... (<4-A')l-*, we have in the three
several cases d>=l, <6= , -, d>= , — ; — = respectively,
51—2
404 A MEMOIR ON PREPOTENTIALS. [607
For the case last considered
V'+g ^h' + 0 1
S = ±^<f^tf)+,„ + J.^^L+^ _ 1 , r same function with t for ^,
A ^fl + I i^^ 1
'*'°~2<? + 2 + i«./»^""^2? + 2 + i«<A'' «o'
where «» is the root of the equation
^ ,+ - + o, ■ o\ ,...+ 1=0.
2^ + 2 + i/co/' ■■■ ^ 2^ + 2 + i*<A''
Annex VI. Examples of Theorem C. Art. Nos. 129 to 132.
129. First example: relating to the (s + l)-coordinal sphere a^+ ... +z'+ti/'=/'.
Assume
M M
^' = (a'+... + c' + 6n>"-v' ^"=y^. (a constant);
these values each satisfy the potential equation.
V is not infinite for any point outside the surface, and for indefinitely large
distances it is of the proper form.
V" is not infinite for any point inside the surface ; and at the surface V = V".
The conditions of the theorem are therefore satisfied. Writing
we have
where
j {(a - a;)» + . . . + (c - 2)» + (e - J«yji»-i '
^ ~ 4 (ri)«+' I, cJh' ^ ds" ) •
dW^_/xd_ ■^Aj.^A^ ^
d«' ~ [fdx'^ ■■■ '^fdz'^f dw) («»+ ... +z^ + w'')i-i
{s-l)y(x'+... +z' + ii/')M
" {af + ...+z''+iv')i'+i '
I
607] A MEMOIR ON PREPOTENTIALS. 405
which at the surface is
/' ■
Hence
'' " 4 (r^r'/' ' " 2 (r^y+V' ' ' '' '^ constant).
2 (rAy+' /"*
130. Writing for convenience M= p/_f — tt S/ (S/" a constant which may be put
= 1), also a'^+ ... + c- + e^ = «°, we have p = 8/", and consequently
/• BfdS
j {(a-xy+ ...+{c-zy + {e-wf}i'-i
= -ir(^8+^) ii^i exterior point « >/,
= T, , / — . it"^ ?i_, 'or interior point «</
By making a,..,c, e all indefinitely large, we find
viz. the expression on the right-hand side is here the mass of the shell thickness Bf.
Taking s = 3, we have the ordinary formulae for the Potential of a uniform .spherical
shell.
131. Suppose 3 = 3, but let the surface be the infinite cylinder x- + y^=f\ Take
here
F' = i»/ log Va-+>, F" = #log/
each satisfying the potential equation %— + -rij = 0 ; but V, instead of vanishing, is
infinite at infinity, and the conditions of the theorem are not satisfied; the Potential
of the cylinder is in fact infinite. But the failure is a mere consequence of the special
value of 8, viz. this is such that s — 2, instead of being positive, is = 0. Reverting
to the general case of (s + l)-dimensional space, let the surface be the infinite cylinder
as* + . . . + ^- =/- ; and assume
^' = (a'+... + c-)i"-" ■ ^" =/- (^ 'constant).
These satisfy the potential equation ; viz. as regards V, we have
406 A MEMOIR ON PREPOTENTIALS. [607
V is not infinite at any point outside the cylinder; and it vanishes at infinity,
except indeed when only the coordinate e is infinite, and its form at infinity is not
= ilf -=- (a.= + . . . + c* + e'')*"-".
V" is not infinite for any point ^vithin the cylinder; and at the surface we have
V'=V".
We have
where
d^'- =- (^+... + ^)y ■ Ti^r— at the surface; ^^„ =0,
and therefore
P= 4(r^)»+'/" — ■ ^ ^^ constant);
or, what is the same thing, writing M = .- -— | -. p// _ /x . vvhence p=Bf, and writing also
«'+...+(? = «", we have
/
{(a - a;)= + . . . + (c - «)" + (e - ?<;)«}*»-*
^(s-2)r4"7^fc/'^ ^^'^ interior point «</
132. This is right; but we can without difficulty bring it to coincide with the
result obtained for the (s + l)-dimensional sphere with only s — 1 in place of s; we may
in fact, by a single integration, pass from the cylinder of + ... + z^ =/' to the s-dimen-
sional sphere or circle x^ + ... +z-=f^, which is the base of this cylinder. Writing first
dS — dldw, where d2 refers to the s variables {x,..,z) and the sphere a;= + ... +2*=/^;
or using now dS in this sense, then in place of the original dS we have dS dw : and
the limits of w being oo , — x , then in place of e — w we may write simply w. This
being so, and putting for shortness {a — xy+...-'r{c—zf = A'', the integi-al is
r f hfdSdw
and we have without difficulty
/:
dw i^ rir^(«-2)
, {A'' + w«)*"-" ~ A"-^ V^ (s -1) '
\
607] A MEMOIR ON PREPOTENTIALS. * 407
To prove it, write tu = A tan 0, then the integral is in the first place converted into
-jj^ I COS*"' 6 d0, which, putting cos 6=4x and therefore sin 6 = '^\—x, becomes
= -^^ { «i-i (1 - a;)i"-'»-' dx,
which has the value in question.
Hence, replacing A by its value, we have
v\ v\{s-i) r Sfds ^ 47r^'r(i)/'-'g/ f i i i
r^ (s - 1) J {(a-xy+...+(c- zy\i<^'> (s - 2) r^(s - 1) ((a" + . . . + c')h>-^> '^^ f'-^] '
that is,
i{(a-ar)»+... + (c-2)»}i<*-''> (s- 2) ri(s- 2) |(a'' + ... + c")*'*-^' "'"7^4
r^s 1 (a= + . . . + c^)* <»-^i " /"-sj '
viz. this is the formula for the , sphere with s— 1 instead of s.
Annex VII. Example of Theorem D, Art. Nos. 133 and 134.
133. The example relates to the (s + l)-dimensional sphere 00^+ ... + z'^ + w^=f^.
Instead of at once assuming for V a form satisfying the proper conditions as to
continuity, we assume a form with indeterminate coefficients, and make it satisfy the
conditions in question. Write
M
^ = 7-r-r ,^ .,,fa_i for (*'' + ... +c-' + e=>/^;
=^A(a'+... + c' + e')+B for a- + ...+c- + (F </-.
In order that the two values may be equal at the surface, we must have
dV
in order that the derived functions 7-, &c. may be equal, we must have
-(s-l)aM
f>+. = iAa, &c.,
viz. these are all satisfied if "only — ,^^^ — =2 A.
We have thus the values of A and B ; or the exterior potential being as above
M
408 ' A MEMOIR ON PREPOTENTIALS. [607
the value of the interior potential must be
ML. ,,, .. ,,o' + ...+c» + c»l
=yv=i](i« + i)-(i»-4) f, }•
The corresponding values of W are of course
^ J -^ !/i , IX /I ^.a^ + .-. + z' + w^]
(^+... + .'+z^)t'-* ^°^ 7^^ i^** + *^ - ^^' - *) T } '
and we thence find
p = 0, i{ a^ + ... + z'' + w''>p,
^_ r(i.-i) iif _r(^. + f) it/
if ar' + ...+22 + w=</-^
(ri)«+i
Assuming for M the value „ . , sx/''^'. the last value becomes p = 1 ; ^vriting for
shortness a- + . . . + c" + e" = /c% we have
V = / 77 rr- , '" ^^,. r^ojS+i '^'^'^'^ (« + l)-dimensional sphere 0^+ ... +z- + w- =/-,
r(is + |)^-
, for an exterior point k >f,
{(i»+ i)/" — (i*~i) '^}. for an interior point «</.
134. The case of the ellipsoid 77., + ... + .-5= 1 for s + 1-dimensional space may be
worked out by the theorem ; this is, in fact, what is done in tridimensional space
by Lejeune-Dirichlet in his Memoir of 1846 above referi'ed to (p. 321).
Annex VIII. Prepotentials of the Homaloids. Art. Nos. 135 to 137.
135. We have in tridimensional space the series of figures— the plane, the line,
the point; and there is in like manner in (s+l)-dimensional space a corresponding
series of (s+1) terms; the (s + l)-coordinal plane — the line, the point: say these
are the homaloids or homaloidal figures. And, taking the density as uniform, or,
what is the same thing, =1, we may consider the prepotentials of these several
figures in regard to an atti-acted point, which, for gi'eater simplicity, is taken not to
be on the figure.
136. The integral may be written
y_ (' dw ... dt
~ ) {{a-xy+... +(c-zy + {d-wf+ ... +(e-0' + «')**■*•«'
which still relates to a (s+ l)-dimensional space: the (s+1) coordinates of the
attracted point are (a,..,c, d,..,e, «), instead of being (a, . . , c, e) ; viz. we have the
607] A MEMOIR ON PREPOTENTIALS. 409
«' coordinates (a, ..,c), the s—s' coordinates (d,..,e), and the (s + l)th coordinate u:
and the integration is extended over the (s — s')-dimensional figure w = — x to
+ cc ,.. , t = — oc to +00. And it is also assumed that q is positive.
It is at once clear that we may reduce the integral to
dw ...dt
I {(a-xy+...
{{a - x)' + ... + {c - zy + u- + vf + ... + <2}i»+9'
say for shortness
dw ... dt
-I
where A", =(a — x)-+... + {c — z)- + ifi, is a constant as regards the integration, and
where the limits in regard to each of the s—s' variables are — oo , + oo .
We may for these variables write rf, ...,rf, where f'+ ... + 5'-= 1 ; and we then
have w= + . . . + 1^ = ?^, dw ... dt = r*-»'-' dr dS, where dS is the element of surface of
the (s — s')-coordinal unit-sphere f +... + f^=l. We thus obtain
where the integral in regard to ?• is taken from 0 to oo , and the integral IdS
over the surface of the unit-sphere; hence by Annex I. the value of this last factor
is = „/■ ^^ , . The integral represented by the first factor will be finite, provided
l-^ (s — 8 )
only ^8 +q he positive ; which is the case for any value whatever of s', if only q
be positive.
The first factor is an integral such as is considered in Annex II. ; to find its
value we have only to write r = A yx, and we thus find it to be
_ _J: , r a^'-i'-'dx ^ _1_ ^ms-s')r(^s'+q)
~(il»)4«'+?^Jo (H-a;)i'+«' A'"^''! TQs + q)
and we thus have
_j_ (T^y-'^ri^s' + q)
J^<f+iq r{^8 + q)
r{^8 + q) {ia-xy-\-...+{c-zy + u']i''+i'
137. As a verification, observe that the prepotential equation [DV=0, that is,
\dd'^'"^ dd"^ dd^^'"^d^^dv:''^ u du) '
for a function V, which contains only the s' +\ variables (a, . . , c, u), becomes
/d= d?_ d" 2q + l d\„_
Ua»'^'""^dc»'^dM»"^ u du) '
which is satisfied by V, a constant multiple of {(a — «)^+ ... +(c — ^)^-|-M^ji~''~'.
C. IX. 52
410 A MEMOIR ON PREPOTENTIALS. [607
Annex IX. The Gauss-Jacobi Theory of EpispJieric Integrals. Art. No. 138.
138. The formula obtained (Annex IV. No. 110) is proved only for positive values
of m ; but writing therein q = 0, vi = — ^, it becomes
dx ... dz
U^
a?
1 -^,- ... - J, {(a -a:)^ + ... + (c -^)^ + e")*'
=^/....p*.r^(i-^-|^.-...-^,-f)-*((.vr..(^-+m
a formula which is obtainable as a particular case of the more general formula
J {(*$a;, ..,z, wy]i' r (is) j - A V- Disct. f(«$Z, .. ,Z,W,Ty + t{X' + ... + Z'+W' + T^)\ '
(notation to be presently explained), being a result obtained by Jacobi by a process
which is in fact the extension to any number of variables of that used by Gauss*
in his Memoir " Determinatio attractionis quam exerceret planeta, &c." (1818).
I proceed to develop this theory.
139. Jacobi's process has reference to a class of s-tuple integrals (including some
of those here previously considered) which may be termed "epispheric": viz. considering
the (s + 1) variables {x,..,z, w) connected by the equation 0^*+ ... +^'+w'= 1, or say
they are the coordinates of a point on a (s + l)-tuple unit-sphere, then the form is
I UdS, where dS is the element of the surface of the unit-sphere, and U is any function
of the 8 + 1 coordinates; the integral is taken to be of the form I ,. ^ ,.,,..,
j [{*\x,..,z, w, \f\»
and we then obtain the general result above referred to.
Before going further it is convenient to remark that, taking as independent vaiiables
the s coordinates x,..,z, we have d(S= — '^ — , where w stands for ±'J\ — a?— ... —2^;
aw
we must in obtaining the integral take account of the two values of w, and finally
extend the integral to the values of x,..,z which satisfy x^+ ... + z^kI.
If, as is ultimately done, in place of x,..,z we write ->■,•• >r respectively, then
1 dtjc dz I 3? ^
the value of dS is =■ -^ j '— — , where w now stands for + a/ 1 —?;—... —r;;
J ... h w ~ y f 'I
we must, in finding the value of the integral, take account of the two values of w,
and finally extend the integral to the values of a;, ..,^ which satisfy ^-|- ... -|-t^ < 1.
• [Qes. Werke, t. iii, pp. 331—366.]
607]
A MEMOIR ON PKEPOTENTIALS.
411
140. The determination of the integral depends upon formulae for the transforma-
tion of the spherical element dS, and of the quadric function (a;, y,.., z, w, 1)-.
First, as regards the spherical element dS ; let the s + 1 variables x, y,.., z, w
which satisfy x^ ■\- y- + . . . + z- + w"- = \ be regarded as functions of the s independent
variables 6, <f>,.., ■^; then we have
dS =
dy
dx
dd' dd"
dx dy
d(f> ' d^
dx dy
d^' d^''
z , w
dz dw
dO' dd
dz dw
d<f> ' d<p
dz dw
d^' d^
dddA... d^lr, =o& ^'"' ^,' ^"\ded<i>...d^, for shortness.
8(^, <^,.., i|r,»)
Suppose we efifect on the s + 1 variables {x, y,. ., z, w) a transformation
X Y
X, J!,.., z, w=-^, -J,,..
Z W^
thus introducing for the moment s + 2 variables X, Y,..,Z, W, T, which satisfy
identically X'+ Y'+ ...+ Z^+ W'^ — T'' = 0; then, considering these as functions of the
foregoing s independent variables 0, <f>, .. ,y^, we have
dS =
2*+i
X,
Y,..
. z,
W
dX
dY
dZ
dW
dd'
dd'"
• de '
de
dX
d<f>'
dY
d<^''
dZ
• d4>'
dW
d<f>
dX
dY
dZ
dW
d^fr-
d-f' ■
' d^k'
dyfr
ddd4> ... d^=^^ ,^^^——-—Jddd<f> ... d^.
141. Considering next the s + 2 variables X, Y,..,Z, W, T as linear functions
(with constant terms) of the s + 1 new variables f, •»/,.., ?, (o, or say as linear functions
of the « + 2 quantities f, 77, . . , f, w, 1 : which implies between them a linear relation
aX + hY+...+cZ+dW + eT = \:
and assuming that we have identically
x«+ 7"+...+^" + F»-r»=r+i7''+ •■• + ?' +®'-i>
so that, in consequence of the left-hand side being =0, the right-hand side is also
= 0 ; viz. ^, 77, . . , f, w are connected by
f + i7'+... + r' + «'=l:
52—2
412
A MEMOIR ON PREPOTENTIALS.
[607
let d% represent the spherical element belonging to the coordinates f, 17, . . , f, «.
Consideiing these as functions of the foregoing s independent variables 0, <f>,.., ■^,
we have
d2 =
f .
V ••
■. ?>
w
dO'
df)
Td '■
•• de'
ddS
de
d<f>'
dv
d4> "
•' d<i>'
dm
d,f>
d^
dyjr'
drj
df'
•' d^'
d(o
dyjr
d0d<f,...dylr = ^''!'-' ^; '''{dedS...dyb:
142. In this expression we have f, 7;, . . , f, w, each of them a linear function of
the s + 2 quantities X, Y,..,Z, W, T; the determinant is consequently a linear function
of s + 2 like determinants obtained by substituting for the variables any s + 1 out of
the 5 + 2 variables X, Y,..,Z, W, T; but in virtue of the equation
X' + Y'+ ... + Z-' + W' - T' = 0,
these s + 2 determinants are proportional to the quantities X, Y,..,Z, W, T respectively,
and the determinant thus assumes the form
aX + bY+... + cZ + dW+eT
T
A.
where A is the like determinant with {X, Y,. . , Z, W), and where the coefficients
a, 6, . . , c, d, e are precisely those of the linear relation aX + hY+ ... +cZ-\-dW + eT =1;
the last-mentioned expression is thus =mA, or, substituting for A its value, we have
1 3(Z, Y,..,Z, W),^,,
viz. comparing with the foregoing expression for dS we have
which is the requisite formula for the transformation of dS.
143. Consider the integral
dS
/
{{•1x, y,..,z, w, l)»ji«
which, from its containing a single quadric function, may be called " one-quadric'
Then effecting the foregoing transformation,
X, y,..,z, w =
X Y
Z W
rp } rp y • • i rp i my
607] A MEMOIR ON PREP0TENTIAL8. 413
and observing that
y2
{*\x, y,..,z,w, \f = 4 {*\X, Y,..,Z, W, TY;
the integral becomes
_ f rfS
~i{(»$X, Y,..,Z, W, Tr}^
where X, Y, .., Z, W, T denote given linear functions (with constant coefficients) of the
« + 1 variables f , i},.., f, w, or, what is the same thing, given linear functions of the
« + 2 quantities ^, 17, . . , ?, w, 1, such that identically
Z»+ 7-+...+Z«+ W^-T^ = ^ + rf + ... + ^' + oi"--\.
We have then f* + i?^+ ... + ^ + a)-— 1 =0, and d2 as the corresponding spherical element.
144. We may have X, Y,. . , Z, W, T such linear functions of |^, rj,.., f, w, 1 that
not only
Z^+F2+... + Z=+TF'»-r= = f + 7?'+... + ?-^ + «=-l
as above, but also
(*5Z, F,.., Z, W, Ty = A^ + Bv' + ... +C^ + Elo'' - L;
this being so, the integral becomes
dt
( d2
J [AP + Bv' +... + '
where the 8+2 coefficients A, B,.., C, E, L are given by means of the identity
-{d + A)<,0 + B) ... (e + C){d+E){d+ L)
= Di8ct. [{*\X, Y,.., Z, W, T)'' + e{X^+Y-'+ ... + Z^ +W^ - T^)];
viz. equating the discriminant to zero, we have an equation in 6, the roots whereof
axe -A, -B,.., -G, -E, -L.
The integral is
r dS
] {{A- L)^^ + {B - L)rf + ... +(G - L) C+ {E - L)(o']i>'
which is of the form
/• dS
j {af-* + 677» + . . . + cf ^ + ea)»ji» '
where I provisionally assume that a, b,.., c, e are all positive.
14.5. To transform this, in place of the s+1 variables ^, 17, .., f, co connected by
^ + 7;'+ ... + 5^ + a>'=l, we introduce the s+1 variables x, y,.., z, w, such that
fVa 17 V6 tVc G)Vd
a; = -^ , V = - — , . . , z = - — , w = ,
P -^ P P P
414 A MEMOIR ON PREPOTENTIALS. [607
where
and consequently
Hence, writing dS to denote the spherical element corresponding to the point (x, y,,,,z, w),
we have, by a former formula,
'^^-^^ did, <!>,.., f,*) ddd<l>...dyfr
_{ab...ce)i^_
or, what is the same thing,
^^ 1 ..dS.
Hence, integrating each side, and observing that I dS, taken over the whole spherical
surface oc^ + f- + ... + z' + vf = l, is = 2 (r^ )«+'-=- T (^s + ^), we have
{ dl 2 (r^)'+' 1
i {a^ + bTf+... + c^' + eoy'\ii'+»~r{^s + i)(ab... cef
146. For a, b,.., c, e write herein a + 0, b + 0,.. , c + 0, e + 6 respectively, and
multiply each side by 6i~^, where q is any positive integer or fractional number
less than ^s: integrate from ^=0 to d=<x. On the left-hand side, attending to the
relation ^ + ■>]''+... + ^+e>)^=l, the integral in regard to 0 is
f" 09-^ d0
where p^, = a^ + br}^+ ... + c^ + eco^, is independent of ^ as before ; the value of the
definite integral is
^T{^(s+l)-q}r{q) 1
rj (s + 1) p»+i-»9 '
which, replacing p by its value and multiplying by dS, and prefixing the integral sign,
gives the left-hand side; hence, forming the equation and dividing by a numerical
factor, we have
In particular, if q = — ^, then
607]
A MEMOIR ON PREPOTENTIALS.
415
or, if for a,. . , c, e we restore the values A — L,. . , G — L, E — L, then
= ^^^f^dt{(t + A)...(t + 0){t + E)(t + L)}-i;
viz. we thus have
where (t + A) ... (t+ C)(t + E)(t + L) is in fact a given rational and integral function
of t ; viz. it is
= -Disct. K*$X.., Z, W, Tf + tiX^+...+Z'+W'-T^)].
147. Consider, in particular, the integral
dS
j{(a-fxy+..
here
{(a -fxf + ...+{c-hzy + {e- kwf + i-j4« '
^^'{aT -fXf + ... + (cT - hZf + {eT -kWf + PT-^ + tiX-" + ... + Z^ + W^ - T")
= (/' + <) Z» + ... + (A» + 1) Z^ + {h? + t)W^ + {a? + ... + d" + e' + V-t) T^
- 2a/XT - ... - 2chZT - 2ekWT ;
viz. the discriminant taken negatively is
t+/\... , -af
...,< + AS -ch
- a/,...-ch, - (a'+ ... + c'+e^ + f)+t
which is
= (t + A)...{t+C){t + E)(t + L);
and consequently —A, .., —C, — E, —L are the roots of the equation
1-
'' -^1 = 0.
«+/' ■" t + h^ t ■{■!<? t
148. The roots are all real ; moreover there is one and only one positive root.
Hence, taking —L to be the positive root, we have A,.., C, E, —L all positive, and
therefore d fortiori A— L,,., C — L, E — L all positive : which agi-ees with a foregoing
416 A MEMOIR ON PREPOTENTIALS. [607
provisional assumption. Or, Avriting for gi-eater convenience 6 to denote the positive
quantity — Z, that is, taking 8 to be the positive root of the equation
we have
dS
r dS
j [{a -fxf ■ir... + {c-hzf + {e- kwy + i'}**
^^jy
or, what is the same thing, we have
dx ...dz
_1_ f
f...h] ±10 [(a- xy + ... +{c- zf -{■ {e T kvif + i>j*'
la? z^
where on the left-hand side w now denotes a/ 1 — 7; — ■ • • — r-j , and the limiting equation
P
149. Suppose i = 0 : then, if
. a? z" -,
is^+...+p = l
.■ + ?. + ^,>i.
the equation
, a?
c» e"
e+p •■
• ^ + /i^ e + k?
= 0
has a positive root differing from zero, which may be represented by the same letter 0;
but if
then the positive root of the original equation becomes = 0 ; viz. as I gradually
diminishes to zero, the positive root 0 also diminishes and becomes ultimately zero.
Hence, writing 1 = 0, we have
r dS
j {{a -fxf + ... + (c - hzf + {e- kwy]^' '
or, what is the same thing,
1 r d,r ...dz
f...hj ± w {{a - xy+ ... + (c - zy + {e + kwyji* '
607] A MEMOIE ON PREPOTENTIALS. 417
6 now denoting either the positive root of the equation
or else 0, according as
a- c? e- ^
In the case 7; + ... + r;, < 1, the inferior limit being then 0, this is, in fact, Jacobi's
theorem (Crelle, t. xii. p. 69, 1834) ; but Jacobi does not consider the general case
where I is not = 0, nor does he give explicitly the formula in the other case
l = 0,%+... + ^„ + ^>l.
150. Suppose A^ = 0, e being in the first instance not =0: then the former alter-
native holds good; and observing, in regard to the form which contains +w in the
denominator, that we can now take account of the two values by simply multiplying
by 2, we have «
r rfS _ 2 r dx ...dz
j {(a -fxf + ... + (c - hzf + e=ji» ' ~/...hJ w {(a -«)«+... + (c - zf + e=|*« '
/ ^ ?-
(w on the right-hand side denoting A/ 1 — 2=^2 - • • • — 7i . and the limiting equation being
J,+ -+/^»=l)' each
a'
where 0 is here the positive root of the equation 1 — 2, — .. — ... — ^ — r- — ^ = 0, which
u +/ (7 + h u
is the formula referred to at the beginning of the present Annex. We may in the
formula write e = 0, thus obtaining the theorem under two different forms for the cases
a' c'
fi+ ••• +Ti>^ and < 1 respectively.
Annex X. Methods of Lejeune-Dirichlet and Boole. Art. Nos. 151 to 162.
1.51. The notion, that the density p is a discontinuous function vanishing for
points outside the attracting mass, has been made use of in a different manner by
Lejeune-Dirichlet (1839) and Boole (1857): viz. supposing that p has a given value
f(x, .., z) within a given closed surface S and is = 0 outside the surface, these geometers
in the expression of a potential or prepotential integral replace p by a definite integral
which possesses the discontinuity in question, viz. it is —f{x, . . , z) for points inside
c. IX, 53
418 A MEMOIR ON PREP0TENTIAL8. [607
the surface and =0 for points outside the surface; and then in the potential or
prepotential integral they extend the integration over the whole of infinite space, thus
getting rid of the equation of the surface as a limiting equation for the multiple
integral.
152. Lejeune-Dirichlet's paper "Sur une nouvelle m^thode pour la determination
des intdgrales multiples" is published in Gomptes Rendm, t. viii. pp. 155 — 160 (1839),
and Liouville, t. rv. pp. 164 — 168 (same year). The process is applied to the form
1 d C dxdy dz
~ p-\ da J {{a -xy + {b- yf + (c - 2)'j*'J'-')
w^ V* z^
taken over the ellipsoid -7 + o^ + -^ = 1 ; but it would be equally applicable to the
triple integral itself, or say to the s-tuple integral
r dx ...dz
J {(a-xY+...+{c-zy]^+9 '
or, indeed, to
r dx...dz
J {(a~xy+ ... + (c-z)' + c»}*^9
taken over the ellipsoid 7^ + • • • + r^ = 1 ; but it may be as well to attend to the first
form, as more resembling that considered by the author.
153. Since — | , cos X<f> dd> is =1 or 0, according as \ is < 1 or > 1, it
IT Jo <p
follows that the integral is equal to the real part of the following expression,
2 r ^A«^ f/(.^^"+^) dx...dz
irjo "^^ 4> r l{a-xy+...+ic-zy\i'*^'
where the integrations in regard to x,.,,z are now to be extended from — x to + oc
for each variable. A further transformation is necessary : since
1 1 f" .
— = =- e~*^ I d^ . •^•""^ e^*, a- positive, and r positive and < 1,
writing herein {a — xy+...+(c — zy for rr, and ^s + g for r, we have
-I 1 1-"
{(a-xy+... + {c-zy]i'+i r(i*4-g) j^ar-r
and the value is thus
= ^=^7^ . e-<»*^"T rd<t> "^ ^d^lr . -f i«+«-' f c'(7'^ • +S>)* e-*{<«-«»'+-+(-fl»} dx...dz,
vr(i8+q) Jo 4> Jo J
607] A MEMOIR ON PREPOTENTIALS 419
where the integral in regard to the variables («,.., z) is
and here the a;-integral is
and the like for the other integrals up to the ^•-integral. The resulting value is thus
which, putting therein ^ = t , d"^ ~~M^f> ^
t t
r
#- ^{f...h)e-^\ dt , ^ e^V'+r '^A»Wsinrf..<A»-'d<A.
11 r°°
: = ^ e-«^ d<f> . ^«-' e*"* (<7 positive),
Jo
154 But we have to consider only the real part of this expression ; viz. writing
for shortness o' = 75- /+-- + ia — /■ ^® require the real part of
/ *T* t /I *T" C
e-i«" I e'"^ (I)'-' sin </> d<f>.
.'o
Writing here for sin <f> its exponential value a", (e** — e~**), and using the formula
1^_ J^
and the like one
1 1 f
r = =, e*" I d4>. <f>^~^ e"^ (<r negative),
(in which formulae q must be positive and less than 1), we see that the real part in
question is = 0, or is
r3sin(^ + l)7r _ TT 1
2(l-<r)9 ' ~ 2r(l-g) (1 - <r)9'
according as o- > 1 or o- < 1.
a' &
155. If the point is interior, 72+."+rj<l. and consequently also o- < 1, and
the value, writing {V\y instead of tt, is
53—2
420 A MEMOIR ON PREPOTENTIALS. [607
But if the point be exterior, 7^,+ - •• + rj>l, and hence, writing 6 for the positive
a? (?
root of the equation, o-=l; viz. 6 is the positive root of the equation ^^ — o'^'-'^ii — a~^''
then < = 0, <T is greater than 1, and continues so as t increases, until, for t = 6, a
becomes = 1, and for larger values of t we have o- < 1 ; and the expression thus is
viz. the two expressions, in the cases of an interior point and an exterior point
respectively, give the value of the integral
/(
dx ...dz
{(a - a;)' + . . . + (c - ^)«}i»+« "
This is, in fact, the formula of Annex IV. No. 110, writing therein e = 0 and m = — q.
156. Boole's researches are contained in two memoirs dated 1846, " On the
Analysis of Discontinuous Functions," Trans. Royal Irish Academy, vol. xxi. (1848),
pp. 124 — 139, and "On a certain Multiple Definite Integral," do. pp. 140 — 150 (the
particular theorem about to be referred to is stated in the postscript of this memoir),
and in the memoir "On the Comparison of Transcendents, with certain applications
to the theory of Definite Integrals," Phil. Trans, vol. CXLVii. (1857), pp. 745 — 803,
the theorem being the third example, p. 794. The method is similar to, and was in
fact suggested by, that of Lejeune-Dirichlet ; the auxiliary theorem made use of in
the memoir of 1857 for the representation of the discontinuity being
•'-4- = —^ .1 I I da dv ds COS [{a — X - ts) V + ^tVl v's'-^f(a),
V irlij -00 Jo 0
which is a deduction from Fourier's theorem.
Changing the notation (and in particular writing s and ^s + q for his n and i),
the method is here applied to the determination of the s- tuple integral
of z"
where </> is an arbitrary function, taken over the ellipsoid ^-3+ ••• + u — ^-
157. The process is as follows: we have
9 I, , + ... + ,, 1 .. n r'c r»
{(a-a;)'+... + (c-^)» + e»p+9 7rr(i« + g)Jo.'o Jo
COB I^M -/.-••• - |i ~ "^ {(«-'")' +--+{c-zy + e»}) i; + i(i« + q)fr\ ^u;
607] A MEMOIR ON PREPOTENTIALS. 421
viz. the right-hand side is here equal to the left-hand side or is = 0, according as
7^+... + Tj<l or > 1. F is consequently obtained by multiplying the right-hand side
by dx ... dz and integrating from — oo to -(- oo for each variable.
Hence, changing the order of the integration,
F= ^f^-,^^ r r ( ( du dv dr vi'+9 Ti»+«-' (bu . n,
■jrr(^s + q)]oJo Jo ^
where
n = jda: ...dzcos (u-efh--^^- ...-^^+T{{a-xy+ ... -i- (c-zy}\v + ^(^s + q)-7r\.
Now
if
fc- /''^a v_ _ AVc_
158. Substituting, and integi-ating with respect to f, . . , f between the limits — oo ,
-t- » , we have ;
" = f-7^ -?r\ 7T ,„ ,11 ^ cos -^ M — e-T — :; t;- ... — _ -7— D + A OTT K
((1 -I-/V) . . . (1 -f- AV)}i vi* |\ 1 +/V 1 -f AV/ ^ ^ J '
or, what is the same thing, writing - in place of t, this is
that is, writing
we have
^ <-«-' t;9 cos [{u — a)v + ^qir] ^u
or, writing tt*^' = - (F^)*, this is
TT
= ^^y"\^^ fd* . <-'-' K« +/') •••(< + ^')}"* - f 7 <^« ^^ • ''^ cos [(w -a)v + ^qir] <f,u.
I (i8 + q) Jo •"■ JoJ 0
159. Boole writes
- I I dw du »* cos {(a — <r) » -I- J^tt} ^m = f - t- j ^ (o-) ;
viz. starting from Fourier's theorem,
1 n
- I I dudv cos (u — <r) V . <f>u = ip (cr).
422
A MEMOIR ON PREPOTENTIALS.
[607
where ^(o") is regarded as vanishing except when a is between the limits 0, 1,
and the limits of u are taken to be 1, 0 accordingly, then, according to an admissible
theory of general differentiation, we have the result in question. He has in the
formula - instead of my t ; and he proceeds, " Here a increases continually with s.
s
As s varies from 0 to oo , a also varies from 0 to oc . To any positive limits of a
will correspond positive limits of s; and these, as will hereafter appear — this refers to
his note B — , will in certain cases replace the limits 0 and oo in the expression for V."
160. It seems better to deal with the result in the following manner, as in part
shown p. 803 of Boole's memoir. Writing the integral in the form
V^'^^^Xn^^^ P l'<i^dt.t-^-'{(t+p)...{t + h'')}-i<f>{u)j dv.v^cos[(u-a)v + iq-,r},
IT 1 (^5 + q) J 0 J 0 Jo
effect the integration in regard to v ; viz. according as u is greater or less than a,
then
r(£+l)sin(j+l)7r
r
dv . xfl cos ((m — a-)v + i^qir]
IT
r(-g)(M-o-)«+'
alue,
v"* f' + t ••• h' + t
or 0,
or 0;
and consequently, writing for cr its value.
(/)«>, or 0, as above.
161. To further explain this, consider < as an a;-coordinate and it as a y-coordinate ;
then, tracing the curve
^=/'T^+'
+
¥-\-x x'
for positive values of x this is a mere hyperbolic branch, as shown in the figure,
viz. a; = 0, y = oo ; and as x continually increases to x , y continually decreases to zero.
The limits are originally taken to be from « = 0 to m=1 and <=0 to t=x, viz.
over the infinite strip bounded by the lines tO, 01, 11; but within these limits the
607] A MEMOIR ON PREPOTENTIALS, 423
function under the integral sign is to be replaced by zero whenever the values u, t
are such that ii is less than ^^ — - + . . . + -^^ — - + — , viz. when the values belong to a
/ "T" fc «- "f* 6 ti
point in the shaded portion of the strip; the integral is therefore to be extended
only over the unshaded portion of the strip; viz. the value is
V=
V{-q)V{\s
the double integral being taken over the unshaded portion of the strip; or, what is the
same thing, the integral in regard to u is to be taken from ^^=-7^ — i "*"■■• "^ iT" t^~i
(say from « = o-) to u = \, and then the integral in regard to t is to be taken from
t = B to i = 00 , where, as before, 6 is the positive root of the equation o- = 1, that
i«'°f/rr^ + - + A.+^ + ^ = i-
162. Write « = <r + (l — o-)a;, and therefore « — o- = (1 — o-)a;, 1 — ?/ = (1 — o-)(l — «)
and du = {\—<i)dx; then the limits (1, 0) of x correspond to the limits (1, a) of u,
and the formula becomes ,;
where a is retained in place of its value . + . . . + .^37^ + r • This is, in fact, a
form (deduced from Boole's result in the memoir of 1846) given by me, Cambridge
and Dublin Mathematical Journal, vol. 11. (1847), p. 219, [44].
If in particular <f>u = {1 - ti)i+"', then (|) fcr + (1 - o-) «j = (1 - (r)9+™ (1 - «)«+»», and
thence
[ X-9-' l<f><T + (1 - a) x] dx = (1 - o-)"* I a^«-' (1 - x)t+^ dx,
Jo Jo
_T{-q)ril+q + m)
r(l + m) ^ ^ '
and then, restoring for <t its value, we have
as the value of the integral
'k' + t t
{(a-xy+ ... +(c - «)" + e»14'+«
/i
taken over the ellipsoid 7-J+ ■•• + ra = l- This is, in fact, the theorem of Annex IV.
No. 110 in its general form; but the proof assumes that q is positive.
424 [608
608.
[EXTRACT FROM A] REPORT ON MATHEMATICAL TABLES.
[From the Report of the British Association for the Advancement of Science, (1873),
pp. 3, 4.]
It was necessary as a preliminary to form a classification of mathematical (numerical)
tables ; and the following classification was drawn up by Prof. Cayley and adopted by
the Committee.
A. Auxiliary for non-logarithmic computations.
1. Multiplication.
2. Quarter-squares.
3. Squares, cubes, and higher powers, and reciprocals.
B. Logarithmic and circular.
4. Logarithms (Briggian) and antilogarithms (do.) ; addition and subtraction
logarithms, &c.
5. Circular functions (sines, cosines, &c.), natural, and lengths of circular arcs.
6. Circular functions (sines, cosines, &c.), logarithmic.
C. Exponential.
7. Hyperbolic logarithms.
8. Do. antilogarithms (e*) and h . 1 tan (45° + ^ <^), and hyperbolic sines, cosines,
&c., natural and logarithmic.
D. Algebraic constants.
9. Accurate integer or fractional values. Bernoulli's Numbers, A"0'", &c.
Binomial coefficients.
10. Decimal values auxiliary to the calculation of series.
608] EXTRACT FROM A REPORT ON MATHEMATICAL TABLES. 425
E. 11. Transcendental constants, e, ir, y, &c., and their powers and functions.
F. Arithmological.
12. Divisors and prime numbers. Prime roots. The Canon arithmeticns, &c.
13. The Pellian equation.
14. Partitions.
15. Quadratic forms a" + b", &c., and partition of numbers into squares, cubes,
and biquadrates.
16. Binary, ternary, &c. quadratic, and higher forms.
17. Complex theories.
G. Transcendental functions.
18. Elliptic.
19. Gamma.
20. Sine-integral, cosine-integral, and exponential-integraL
21. Bessel's and allied functions.
22. Planetary coefficients for given — , .
23. Logarithmic transcendental,
24. Miscellaneous.
Several of these classes need some little explanation. Thus D 9 and 10 are
intended to include the same class of constants, the only difference being that in 9
accurate values are given, while in 10 they are only approximate ; thus, for example,
the accurate Bernoulli's numbers as vulgar fractions, and the decimal values of the
same to (say) ten places are placed in different classes, as the former are of theoretical
interest, while the latter are only of use in calculation. It is not necessary to enter
into further detail with respect to the classification, as in point of fact it is only very
partially followed in the Report.
C. IX. 54
426 [609
609.
ON THE ANALYTICAL FORMS CALLED FACTIONS.
[From the Report of the British Association for the Advancement of Science, (1875), p. 10.]
A FACTION is a product of differences such that each letter occurs the same
number of times; thus we have a quadrifaction where each letter occurs twice, a
cubifaction where each letter occurs three times, and so on. A broken faction is one
which is a product of factions having no common letter; thus
{a-by{c-d)(d-e){e-c)
is a broken quadrifaction, the product of the quadrifactions
(a-by and (c- d)(d-e) (e-c).
We have, in regard to quadrifactions, the theorem that every quadrifaction is a sum
of broken quadrifactions such that each component quadrifaction contains two or else
three letters. Thus we have the identity
2{a - b){b-c)(c -d) (d- a) = (b -cy. (a -dy-(c- ay. (b-dy + (a- by. (c-d;^,
which verifies the theorem in the case of a quadrifaction of four letters ; but the
verification even in the next following case of a quadrifaction of five letters is a
matter of some difficulty.
The theory is connected with that of the invariants of a system of binary quantics.
610]
427
610.
ON THE ANALYTICAL FORMS CALLED TREES, WITH APPLI-
CATION TO THE THEORY OF CHEMICAL COMBINATIONS.
[From the Report of the British Association for the Advancement of Science, (1875),
pp. 257—305.]
I HAVE in two papers " On the Analytical forms called Trees," Phil. Mag. vol.
xm. (1857), pp. 172—176, [203], and ditto, vol. xx. (1859), pp. 374—378, [247], con-
sidered this theory; and in a paper "On the Mathematical Theory of Isomers," ditto,
vol. XLVii. (1874), p. 444, [586], pointed out its connexion with modern chemical theory.
In particular, as regards the paraffins C„Hj,i+2, we have n atoms of carbon connected
by n — 1 bands, under the restriction that from each carbon-atom there proceed at
most 4 bands (or, in the language of the papers first referred to, we have n knots
connected by n — 1 branches), in the form of a tree ; for instance; n = 5, such forms
(and the only such forms) are
V
<>2
"3
3*
2
2
2
34
3»-
-•3
And if, under the foregoing restriction of only 4 bands from a carbon-atom, we
connect with each carbon-atom the greatest possible number of hydrogen-atoms, as
shown in the diagrams by the affixed numerals, we see that the number of hydrogen-
atoms is 12 (=2.5 + 2); and we have thus the representations of three different
paraffins, CsH,,. It should be observed that the tree-symbol of the paraffin is
54—2
428
ON THE ANALYTICAL FORMS CALLED TREES, WITH
[610
completely determined by means of the tree formed with the carbon-atoms, or say of
the carbon-tree, and that the question of the determination of the theoretic number
of the paraffins CnH^+s is consequently that of the determination of the number of
the carbon-trees of n knots, viz. the number of trees with n knots, subject to the
condition that the number of branches from each knot is at most = 4.
In the paper of 1857, which contains no application to chemical theory, the
number of branches from a knot was unlimited ; and, moreover, the trees were
considered as issuing each from one knot taken as a root, so that, n — 5, the trees
regarded as distinct (instead of being as above only 3) were in all 9, viz. these were
w
v^
V
/;:
V 1
which, regarded as issuing from the bottom knots, are in fact distinct ; while, taking
them as issuing each from a properly selected knot, they resolve themselves into the
above-mentioned 3 forms. The problem considered was in fact that of the "general
root-trees with n knots" — general, inasmuch as the number of branches from a knot
was without limit; root-trees, inasmuch as the enumeration was made on the principle
last referred to. It was found that for
knots 1,
No. of trees was ... 1,
= 1,
2,
3,
4,
5,
6,
7,
8,
1,
2,
4,
9,
20,
48,
115,
A,,
A„
A„
A„
A,,
A,,
A,,
the law being given by the equation
{l-x)-^{\-a?)-^'{\-a?)-^^{l-ai')-'^'... = \ ■{■ A^x^- A^ocF + AsU? + A^ai^ + ...;
but the next following numbei-s A^, A^, A^, the correct values of which are 286,
719, 1842, were given erroneously as 306, 775, 2009. I have since calculated two
more terms, Au, .4i2 = 4766, 12486.
The other questions considered in the paper of 1857 and in that of 1859 have
less immediate connexion with the present paper, but for completeness I reproduce
the results in a Note*.
* In the paper of 1857 I also considered the problem of finding B, the namber with r &ee branches,
with bifurcations at least : this was given by a like formula
(l-x)-'(l-a^-»i(l-x'')-«»(l-x*)-»'... = l+x + 2Bjxa+2JBjar>+2B«a^...,
B,= 1, 2, 6, 12, 38, 90,
r= 2, 3, 4, 5, 6, 7
In the paper of 18S9, the question is to find the number of trees with a given number in of terminal
knots : we have here
^=1 . 2. 3. ..(»»- 1) coefficient of «""' in 5 — — ,
leading to
for
610] APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS. 429
To count the trees on the principle first referred to, we require the notions of
"centre" and "bicentre," due, I believe, to Sylvester; and to establish these we
require the notions of " main branch " and " altitude " : viz. in a tree, selecting any
knot at pleasure as a root, the branches which issue from the root, each with all
the branches that belong to it, are the main branches, and the distance of the furthest
knot, measured by the number of intermediate branches, is the altitude of the main
branch. Thus in the left-hand figure, taking A as the root, there are 3 main branches
of the altitudes 3, 3, 1 respectively: in the right-hand figure, taking A as the root,
there are 4 main branches of the altitudes 2, 2, 1, 3 respectively; and we have
then the theorem that in etery tree there is either one and only one centre, or else
one and only one bicentre ; viz. we have (as in the left-hand figure) a centre A
which is such that there issue from it two or more main branches of altitudes equal
to each other and superior to those of the other main branches (if any) ; or else
(as in the right-hand figure) a bicentre AB, viz. two contiguous knots, such that
issuing from A (but not counting AB), and issuing from B (but not counting BA),
we have two or more main branches, one at least from A and one at least from B,
of altitudes equal to each other and superior to those of the other main branches in
question (if any). The theorem, once understood, is proved without difficulty: we
consider two terminal knots, the distance of which, measured by the number of
intermediate branches, is greater than or equal to that of any other two terminal
knots; if, as in the left-hand figure, the distance is even, then the central knot A
is the centre of the tree; if, aa in the right-hand figure, the distance is odd, then
the two central knots AB form the bicentre of the tree.
In the former case, observe that if 0, H are the two terminal knots, the distance
of which is = 2\, then the distance of each from A is = \, and there cannot be
giving the valnea
0m= 1, 1, 3, 13, 75, 541, 4683, 47293, ...
tat
m= 1, 2, 3, 4, 5, 6, 7, 8, ...
Bat if from each non-tenninal knot there ascend two and only two branches, then in this case ^=ooefSoient
l--Jl-ix
Tiz. we nave tne very su
1. 3.5...2m-3.
of z""' in - -g — , viz. we have the very simple form
givmg
far
<fym:
X . a . u ... 4
■ 1.2.3.
.TO
-2»
'-',
^=
1,
1, 2,
5.
14.
42, ...
m=
1,
2. 3,
4,
5.
7,...
430 ON THE ANALYTICAL FORMS CALLED TREES, WITH [610
any other terminal knot /, the distance of which from A is greater than \ (for, if
there were, then the distance of / from Q or else from H would be greater than
2\); there cannot be any two terminal knots I, J, the distance of which is greater
than 2X ; and if there are any two knots /, J, the distance of which is = 2\, then
these belong to different main branches, the distance of each of them from A being
= X; whence, starting with /, J (instead of Q, H), we obtain the same point A as
centre. Similarly, in the latter case, there is a single bicentre AB,
Hence, since in any tree there is a unique centre or bicentre, the question of
finding the number of distinct trees with n knots is in fact that of finding the
number of centre- and bicentre-trees with n knots ; or say it is the problem of the
"general centre- and bicentre-trees with n knots:" general, inasmuch as the number
of branches from a knot is as yet taken to be without limit; or since (as will
appear) the number of the bicentre-trees can be obtained without diflSculty when the
problem of the root-trees is solved, the problem is that of the "general centre-trees
with n knots." It will appear that the solution depends upon and is very readily
derived from that of the foregoing problem of general root-trees, so that this last has
to be considered, not only for its own sake, but with a view to that of the centre-
trees. And in each of the two problems we doubly divide the whole system of trees
according to the number of the main branches, issuing from the root or centre as
the case may be, and according to the altitude of the longest main branch or
branches, or say the altitude of the tree; so that the problem really is, for a given
number of knots, a given number of main branches, and a given altitude, to find the
number of root-trees, or (as the case may be) centre-trees.
We next introduce the restriction that the number of branches from any knot
is equal to a given number at most ; viz. according as this number is = 2, 3 or 4,
we have, say oxygen-trees, boron-trees*, and carbon-trees respectively; and these are,
as before, root-trees or centre- or bicentre-trees, as the case may be. The case where
the number is 2 presents no diflSculty: in fact, if the number of knots be =n, then
the number of root-trees is either ^(ri-f 1) or ^n; viz. n = 3 and w = 4, the root-
trees are
Iv
/
and the number of centre- or bicentre-trees is always = 1 : viz. n odd, there is one
centre-tree ; and n even, one bicentre-tree ; it is only considered as a particular case
of the general theorem. The case where the number is = 3 is analytically interesting :
although there may not exist, for any 3-valent element, a series of hydrogen compounds
* I should have said nitrogen-trees; bnt it appears to me that nitrogen is of necessity S-valent, as
shown by the compound, Ammonium-Chloride, =NH4C1. Of course, the word boron is used simply to stand
for a 3-valent element.
610] APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS. 431
B„H„+3 corresponding to the paraffins. The case, where the number is =4 or say
the carbon-trees, is that which presents the chief chemical interest, as giving the
paraffins C„H^tn+i'j and I call to mind here that the theory of the carbon-root trees
is established as an analytical i-esult for its own sake and as the foundation for the
other case, but that it is the number of the carbon centre- and bicentre-trees which
is the number of the paraffins.
The theory extends to the case where the number of branches from a knot is
at most =5, or = any larger number ; but I have not developed the formula.
I pass now to the analytical theory: considering first the case of general root-
trees, we endeavour to find for a given altitude iV the number of trees of a given
number of knots n and main branches a, or say the generating function
where the coefficient fl gives the number of the trees in question. And we assume that
the problem is solved for the cases of the several inferior altitudes 0, 1, 2, 3, ...iV— 1.
This being so, observe that a tree of altitude N can be built up as shown in
the figure, which I call the edification diagram, by combining one or more trees of
altitude N — 1 with a single tree of altitude not exceeding .fl^ — 1 ; viz. in the figure,
iV=3, we have the two trees a, b, each of altitude 2, combined, as shown by the
V I
dotted lines, with the tree c of altitude 1 : the whole number of knots in the
resulting tree is the sum of the number of knots on the three trees a, b, c: the
number of main branches is equal to the number of the trees a, b, plus the number
of main branches of the tree c. It is to be observed that the tree c may reduce
itself to the tree (•) of one knot and of altitude zero; but each of the trees a, b,
as being of the altitude N—1, must contain at least N knots.
Taking N = 2 or any larger number, it is hence easy to see that the required
generating function Sn<"a;" is
= (1 - te^)-' (1 - faJ'+i)-'. (1 - te^+2)-'' ... [(i""] (first factor),
x + {t)a^+ (t, P) x' + (t, t\ <») ar* -f- . . . (second factor).
As regards the first factor, the exponents taken with reversed sign, that is, as
positive, are 1 = no. of trees, altitude iV— 1, of iV knots; Z, = ditto, same altitude, of
(J\r-(-l) knots; i,= ditto, same altitude, of N+2 knots, and so on; and where the
432 ON THB ANALYTICAL FORMS CALLED TREES, WITH [610
symbol [<'•••*] denotes that, in the function or product of factors which precedes it,
the terms to be taken account of are those in (', f, <*,...; viz. it denotes that the
term in «°, or constant term (= 1 in fact), is to be rejected.
In the second factor, the expressions x, {t)x', (t, P)a?,... represent, for given
exponents of t, x, denoting the number of main branches and the number of knots
respectively, the number of trees of altitude not exceeding N —\: thus x, =\ fa?
represents the number of such trees, 1 knot, 0 main branch, = 1 ; and so, if the
value of («, <', f, t^)af be {at + ^P + y1? + Ztf)oi?, then for trees of an altitude not
exceeding N —\, and of 5 knots, a represents the number of trees of 1 main branch,
;9 that of trees ,of 2 main branches, 7 that of trees of 3 main branches, Z that of
trees of 4 main branches. It is clear that the number of trees satisfpng the given
conditions and of an altitude not exceeding JV^ — 1 is at once obtained by addition
of the numbers of the trees satisfying the given conditions, and of the altitudes
0, 1, 2, .,,iV— 1; all which numbers are taken to be known.
It is to be remarked that the first factor,
(1 - tx^y (1 - <a^+>)-'' (1 - te^+»)-^ ... [«'•••"],
shows by its development the number of combinations of trees a, b,. . of the altitude
N— 1 ; one such tree at least must be taken, and the symbol [<'■• °°] gives effect to
this condition : the second factor x + {t)x' + (t, t^)a?+ ... shows the number of the
trees c of altitude not exceeding N —\. And this being so, there is no difficulty in
seeing how the product of the two factors is the generating function for the trees of
altitude N.
In the case iV = 0, the generating function, or GF, is = a; ; viz. altitude 0, there
is only the tree (•), 1 knot, 0 main branch.
When N=\, the GF is = (1 - te)"' [«'-'^] «, = to? + f^a? + ^a^ . . .,
viz. altitude 1, there is 1 tree to?, 2 knots, 1 main branch ; 1 tree iW, 3 knots, 2 main
branches; and so on.
Hence N =2, we obtain
GF = {\-ta?)-^(\-taf)-^{l-toi*)-^ ... [«'-"]. (a; + te= + <W + «»a:« + ...);
viz. as regards the second factor, altitude not exceeding 1, that is, =0 or 1, there
is altitude 0, 1 tree x, and altitude 1, 1 tree te", 1 tree Vaf', and so on. And we
hence derive the &!"& for the higher values iV = 3, 4, &c. : the details of the process
will be afterwards more fully explained.
So far, we have considered root-trees; but referring to the last diagram, it is at
once seen that the assumed root will be a centre, provided only that (instead of, it
may be, only a single tree a of the altitude N —\), we take always two or more trees
of the altitude N —\ to form the new tree of the altitude N. And we give effect
610] APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS. 433
to this condition by simply writing in place of [f-'^] the new symbol [i^-"], which
denotes that only the terms t', i^, t*, ... are to be taken account of; viz. that the
terms in f and f^ are to be rejected. The component trees of the altitude iV — 1
are, it is to be observed, as before, root-trees; hence the second factor of the generating
function is unaltered : the theorem is that for the centre-trees of altitude N we have
the same generating function as for the root-trees, writing only p^-''] in place of
[^'•••"J. Or, what is the same thing, supposing that the first factor, unafifected by either
symbol, is
= 1 +x^ (at + 01'+. ..) + a^+^(^a't + ^f + ...) + ...,
then, affecting it with [<*••«'], the value for the root-trees is
= a^{ca + fit' +...) + x^+'(a't + ^r +...) + ...,
and, affecting it with [<^-«=], the value for the centre-trees is
= x^ifit'+...)+x^+'{fi't'+ ...) + ...
It thus appears how the fundamental problem is that of the root-trees, its solution
giving at once that of the ceiitre-trees ; whereas we cannot conversely solve the problem
of the root-trees by means of that of the centre-trees.
As regards the bicentre-trees, it is to be remarked that, starting from a centre-tree
of altitude N+1 with two main branches, then by simply striking out the centre, so
as to convert into a single branch the two branches which issue from it, we obtain
a bicentre-tree of altitude K. Observe that the altitude of a bicentre-tree is measured
by that of the longest main branch from A or B, not reckoning AB or BA as a
main branch. Hence the number of bicentre-trees, altitude N, is = number of centre-
trees of two main branches, altitude N+l.
This is, in fact, the convenient formula, provided only the number of centre-trees
of two main branches has been calculated up to the altitude N+1. But we can find
independently the number of bicentre-trees of a given altitude N: the bicentre-tree
is, in fact, formed by taking the two connected points A, B each as the root of a
root-tree altitude N (the number of knots of the bicentre-tree being thus, it is clear,
equal to the sum of the numbers of knots of the two root-trees respectively) ; and
it is thus an easy problem of combinations to find the number of bicentre-trees of
a given altitude N. Write
iK^+' (1 + fix + ^0^+ Sai> + ...)
as the generating function of the root-trees of altitude N ; viz. for such trees, 1 = no.
of trees with N + I knots, fi = no. with iV -t- 2 knots, and so on ; then the generating
function of the bicentre-trees of the same altitude N is
= af^->-Hl+fi,x + yy+B,a^ +...),
c. IX. 55
434 ON THE ANALYTICAL FORMS CALLED TREES, WITH [610
where
% = 7+i/8(/3+l),
6, =e+/38 + i7(7 + l),
C = r + y86 + yB,
and so on ; or, what is the same thing, calling the first generating function <f>x, then
the second generating function is = ^ K*^)" + <f> (*0!-
It will be noticed that the bicentre-trees are not, as were the centre-trees, divided
according to the number of their main branches; they might be thus divided according
to the sum of the number of the main branches issuing from the two points of the
bicentre respectively ; a more complete division would be according to the number of
main branches issuing from the two points respectively ; thus we might consider the
bicentre-trees {2, 3), with 2 main branches from one point, and 3 main branches from
the other point of the bicentre ; but the whole theory of the bicentre-trees is com-
paratively easy, and I do not go into it further.
We have yet to consider the case of the limited trees where the number of
branches from a knot is equal to a given number at most: to fix the ideas, say the
carbon-trees, where this number is = 4. The distinction as to root-trees and centre-
and bicentre-trees is as before ; and the like theory applies to the two cases respectively.
Considering first the case of the root-trees, and referring to the former figure for
obtaining the trees of altitude N from those of inferior altitudes, then the trees
a, b, ... of altitude N—l must be each of them a carbon-tree of not more than
(4 — 1 =) 3 main branches : this restriction is necessary, inasmuch as, if for any such
tree the number of main branches was = 4, then there would be from the root of
such tree 4 branches phis the new branch shown by the dotted line, in all 5 branches;
and similarly, inasmuch as there is at least one component tree a contributing one
main branch, the number of main branches of the tree c must be (4 — 1 =) 3 at most :
the mode of introducing these conditions will appear in the explanation of the actual
formation of the generating functions (see explanation preceding Tables III., IV., &c.).
The number of main branches is = 4 at most, and the generating functions have only
to be taken up to the terms in P; the first factor is consequently in each case affected
with a symbol [t^-*], denoting that the only terms to be taken account of are those
in t, t', t', t*; hence as there is a factor t at least, and the whole is required only
up to t*, the second factor is in each case required only up to t\
As regards the centre-trees, the generating functions have here the same expressions
as for the root-trees, except that, instead of the symbol [<'••*], we have the symbol
[I?"*], denoting that in the first factor the only terms to be taken account of are
those in t*, <•, t*; hence as there is a factor f at least, and the whole is required
only up to t^, the second factor is in each case required up to C; and we then com-
plete the theory by obtaining the bicentre-trees. The like remarks apply of course to
610] APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS. 435
the boron-trees, number of branches = 3 at most, and to the oxygen-trees, number = 2
at most ; but, as already remarked, this last case is so simple, that the general method
is applied to it only for the sake of seeing what the general method becomes in such
an extreme case.
We thus form the Tables, which I proceed to explain.
Table I. of general root-trees is in fact a Table of triple entry, viz. it gives for
any given number of knots from 1 to 13 the number of root-trees corresponding to
any given number of main branches and to any given altitude. In each compartment,
that is, for any given number of knots, the totals of the columns give the number
of the trees for each given altitude, and the totals of the lines give the number of
the trees for each given number of main branches : the corner grand totals of these
totals respectively show for each given number of knots the whole number of root-
trees : —
viz. knots ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
numbers are ... 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486,
as already mentioned : these numbers were calculated by an independent method.
Table II. of general centre- and bicentre-trees consists of a centre part and a
bicentre part: the centre part is arranged precisely in the same manner as the root-
table. As to the bicentre part, where it will be observed there is no division for
number of main branches, the calculation of the several columns is effected by the
before-mentioned formula,
thus, column 2, we have by Table I. (totals of column 2)
^x=x> + 2a^ + 4ixf + 6af+ lOa^-t- 14a^-t- 21a^+ 29a;"-(- ...,
and thence
<l>,a; = of + 2ai' + 7 of + Uaf + 22x"' + 58x" + llOx'' + 187 X" + ...
As already mentioned, each column of Table I. is calculated by means of a generating
function given as a product of two factors, each of which is obtained from the
columns which precede the column in question ; and Table II., the centre part of it,
is calculated by means of the same generating functions slightly modified : these
generating functions sei-ving for the calculation of the two Tables are given in the
table entitled "Subsidiary Table for the calculation of the GFa of Tables I. and II.,"
which immediately follows these two Tables, and will be further explained.
55—2
436
ON THE ANALYTICAL FORMS CALLED TREES, WITH
Table I. — General Root-trees.
[610
H'S»
s-sl
Index
3r nam'
of kno<
Index I
number
main bran
Altitade or namber of column.
i
0
2
3
4
0
6
7
8
9
10
11
12
18
1
0
1
1
1
^
Total
1
2
1
Total
1
3
1
1
2
1
2
Total
1
2
4
1
1
1
2
1
1
3
1
4
Total
2
1
4
5
1
1
2
1
2
2
1
3
3
1
1
4
1
9
Total
4
3
1
9
6
1
1
4
3
1
2
2
3
1
6
3
2
1
8
4
1
1
5
1
20
Total
6
8
4
1
20
7
1
1
6
8
4
1
2
8
8
4
1
16
3
3
3
1
7
4
2
1
3
5
1
1
6
1
Total
10
18
13
5
1
48
8
1
1
10
18
13
5
1
48
2
3
15
18
5
1
87
3
4
9
4
1
18
4
3
3
1
7
5
2
1
8
6
1
1
7
1
116
Total
14
38
86
19
6
1
115
9
1
1
14
38
36
19
6
1
2
4
30
36
19
6
1
96
3
5
19
14
5
1
44
4
5
9
4
1
19
6
3
3
1
7
6
2
1
3
7
1
1
8
1
1
Total
1
21
76
93
61
26
7
1
286
610]
APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
437
Table I. (continued).
4
^4i
X t, or
ber of
ranche
Altitude or
number of column.
Ind<
or ni
ofk
Inde
num
ain b
— %
a
0
1
2
3
4
5
6
7
8
9
10
11
12
1
13
10
1
1
21
76
93
61
26
7
1
286
2
i
51
89
61
26
7
1
239
8
7
42
41
20
6
1
117
4
6
20
14
5
1
46
6
5
9
4
1
19
6
3
3
1
7
7
2
1
3
8
1
1
9
1
1
719
Total
1
29
147
225
180
94
34
8
1
719
11
1
1
29
147
225
180
94
34
8
1
a
6
90
210
180
94
34
8
1
622
8
8
79
110
67
27
7
1
299
4
9
46
42
20
6
1
124
5
7
u20
^ 9
14
6
1
47
6
5
4
1
19
7
8
3
1
7
8
2
1
3
9
1
1
10
1
1
1842
Total
1
41
277
528
498
308
136
43
9
1
1842
12
1
1
41
277
528
498
308
136
43
9
1
2
5
14S
467
493
308
136
43
9
1
1607
8
10
152
278
208
101
35
8
1
793
i
11
91
lis
68
27
7
1
320
'
B
6
7
8
9
10
11
1
10
7
6
8
2
1
47
20
9
3
1
42
14
4
1
20
5
1
6
1
1
126
47
19
7
3
1
1
4760
Total
1
55
509
1198
1323
941
487
188
53
10
1
4766
13
1
1
55
509
1198
1323
941
487
188
53
10
1
2
6
238
1012
1524
941
487
188
53
10
1
4460
8
12
272
669
376
344
144
44
9
1
1871
4
IS
184
299
213
102
36
8
1
857
6
13
96
116
68
27
7
1
327
6
11
47
42
20
6
1
127
7
7
20
14
5
1
47
8
5
9
4
1
19
9
3
3
1
7
10
2
1
3
11
1
1
12
1
1
1
Total
1 j 76
924
2666
3405
2744
1615
728
261
64
11
1
12486 j
438
ON THE ANALYTICAL FORMS CALLED TREES, WITH
[610
Table II. — General Centre- and Bicentre-Trees.
Index f,
or nnmber
of knots.
Index t, or
number of
ain branchea.
Centre-Trees.
Altitude or number of column.
1
1
1
n
Bicentre-Trees.
Altitude.
.
'
,
B
0
2
3
4
5
6
0
1
2
8
4
5
1
0
1
1
0
1
1
0
1
1
Total
1
2
8
2
1
1
1
0
Total
1
4
2
3
1
1
1
2
1
1
Total
1
.6
2
1
3
4
1
2
3
1
1
Total
1
2
6
2
1
1
3
1
1-
4
5
1
3
3
6
3
2
1
Total
2
3
7
2
2
1
3
2
2
4
1
S
6
7
11
4
2
2
Total
5
1
8
2
2
2
4
3
3
1
4
2
5
1
6
7
11
12
28
11
3
7
1
Total
8
3
12
9
2
8
7
1
8
4
3
7
4
4
1
6
6
2
2
6
1
1
7
8
1
27
47
20
S
14
8
Total
14
11
1
27
610]
APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
439
Table II. (continued).
1
to
-S ^
Sol
Centre-Trees.
3
Bicentre-Trees.
;l2
Altitude or number of column.
1
o
i
£3
Altitude.
Ind(
or nu
111
-"1
1
t3
a
2
S
0
1
2
3
4
5
6
0
1
2
3
4
5
10
2
3
14
3
20
3
6
11
1
18
4
5
3
8
5
4
1
5
6
2
2
7
1
1
8
9
1
1
51
65
106
51
4
32
14
1
Total
1
21
29
4
55
11
2
4
32
14
1
8
7
26
4
87
i
8
12
1
21
S
6
3
9
6
4
1
5
7
8
2
1
H
2
1
9
10
1
1
127
235
108
4
58
42
4
Total
1
32
74
19
1
127
12
2
4
58
42
4
108
8
9
63
19
1
92
4
10
30
4
44
5
9
12
1
22
6
6
3
9
7
4
1
5
8
2
2
9
1
1
10
11
1
1
267
284
551
267
5
110
128
23
1
Total
1
4S
167
66
5
284
13
2
5
110
128
23
1
3
11
132
66
5
214
4
14
78
20
1
113
5
12
31
4
47
6
10
12
1
23
7
6
3
9
8
4
1
5
9
2
2
10
1
1
11
12
1
1
682
1801
619
5
187
834
88
5
Total 1
1
65
867
219
29
1
682
440
ON THE ANALYTICAL FORMS CALLED TREES, WITH
Subsidiary Table for GFs of Tables I. and II.
[610
1
Index of x.
0
1
2
8
4
5
0
7
8
9
10
11
12
18
0
1
GF, colnmu 0.
«
•
-1
GF, column 1.
0
1
2
S
4
6
6
7
8
9
10
11
12
(1)
1
1
1
1
1
1
1
1
1
1
1
1
First factor.
Second factor.
0
1
j
*
-^
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
GF, column 2.
First fictor.
Second factor.
0
1
2
3
4
5
6
(1)
1
1
1
1
1
1
1
2
1
1
2
1
1
3
2
1
1
3
3
1
1
4
4
2
1
1
4
6
3
1
1
5
7
5
2
1
0
1
2
3
4
6
6
7
8
9
10
11
12
1
1
1
1
1
1
1
1
1
1
1
1
1
«
-1 -2
-4
-6
-10
-14 -21
-29
-41
-55
-76
GF, column 3.
First factor.
Second factor.
0
1
2
3
4
(1)
•
•
1
2
4
6
1
10
2
14 ! 21
7 1 14
' 1
29
32
2
41
58
7
55
110
18
1
0
1
2
8
4
5
6
7
8
9
10
11
12
1
1
1
1
1
1
1
1
2
1
1
1
2
2
1
1
1
3
3
2
1
1
I
4
3
2
1
1
1
4
5
5
8
2
1
1
1
4
7
6
6
1
1
1
1
6
8
9
7
6
8
2
1
1
1
6
10
11
10
7
6
3
2
1
1
1
6
12
15
13
11
7
5
8
2
1
1
610]
APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
441
Subsidiary Table for GFs of Tables I. and II. (continued).
H4
Index of x.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
*
-1
-3
-8
-18
-38
-76
-147
-277
-509
-924
GF, coluitin 4.
First factor.
Seoond factor.
0
1
2
3
(1)
•
1
3
8
18
38
1
76
3
147
14
277
42
509
128
1
0
1
2
8
4
6
6
7
8
9
10
11
12
1
1
1
1
2
I
1
3
8
1
1
6
5
3
1
1
7
11
6
3
1
1
11
18
13
6
3
1
1
15
34
24
14
6
3
1
1
22
55
49
26
14
6
3
1
1
30
95
87
.55
27
14
6
3
1
1
42
150
162
102
57
27
14
6
3
1
1
56
244
284
199
108
58
27
14
6
3
1
1
*
•
•
-1
-4 1-13
-36
-98
-225
-528
-1198
-2666
GF, column 5.
First factor.
Seoond fiaotor.
0
1
2
(1)
1
4
13
36
93
225
1
528
4
1198
23
0
1
2
8
4
S
6
7
8
9
10
n
12
1
1
1
1
2
1
1
4
\
1
8
6
8
1
1
15
15
7
\
1
29
31
17
7
3
1
1
53
70
88
18
7
3
1
1
98
144
90
40
18
7
3
1
1
177
306
197
97
41
18
7
3
1
1
319
617
440
217
99
41
18
7
3
1
1
565
1256
953
498
224
100
41
18
7
3
1
1
*
•
•
. ' . .
-1
-5
-19
-61
-180
-498
-1323
-3405
GF, column 6.
First factor.
Second factor.
0
1
3
(1)
1
5
19
61
180
498
1323
1
0
1
2
3
4
6
6
7
8
9
10
11
12
1
1
1
1
2
1
1
4
3
1
1
9
6
8
1
1
19
16
7
3
1
1
42
36
18
7
3
1
1
89
89
4.S
19
7
3
1
1
litl
205
110
4.5
19
7
8
1
1
402
485
264
117
46
19
7
3
1
1
847
1110
648
285
119
46
19
7
3
1
1
1763
2780
1,S29
711
292
120
46
19
7
3
1
1
a IX.
56
442
ON THE ANALYTICAL FORMS CALLED TREES.
[610
Subsidiary Table for GFa of Tables I. and II. (continued).
M
.3
Index of x.
0
1
2
3
4
5
6 1 7 1 8 9 10 ] 11 i 12 1 IS
•
-1
-6
-26
-94 -308
-941
-2744
GF, colamn 7.
First factor.
Second factor.
0
1
(1)
1
6
26
94 i 808
941
0
1
2
3
4
5
6
7
8
9
10
11
12
1
1
1
1
2
1
1
4
8
1
1
9
6
3
1
1
20
16
7
8
1
1
47
37
18
7
3
1
1
108
95
44
19
7
3
1
1
252
231
116
46
19
7
8
1
1
582
679
291
123
47
19
7
3
1
1
1345
1418
749
312
125
47
19
7
8
1
1
8086
3721
1673
813
319
126
47
19
7
8
1
1
*
-1
-7
-34
-136
-487
-1615
OF, column 8.
First factor.
Second factor.
0
1
(1)
1
7
34
136 j 487
1615
0
1
2
3
4
6
6
7
8
9
10
11
12
1
1
1
1
2
1
1
4
3
1
1
9
6
3
1
1
20
16
7
3
1
1
48
37
18
7
8
1
I
114
96
44
19
7
3
1
1
278
238
117
46
19
7
3
1
1
676
613
298
124
47
19
7
8
1
1
1653
1554
784
319
126
47
19
7
3
1
1
4027
4208
1817
848
326
127
47
19
7
3
1
1
*
-1
-8
-43
-188
-728
OF, colamn 9.
First factor.
Second factor.
0
1
(1)
1
8
43
188
728
0
1
2
3
i
5
6
7
8
9
10
11
12
1
1
1
1
2
1
1
4
3
1
1
9
6
3
1
1
20
16
7
8
1
1
48
37
18
7
8
1
1
115
96
44
19
7
3
1
1
285
239
117
46
19
7
3
1
1
710
621
299
124
47
19
7
8
1
1
1789
1597
792
3-20
126
47
19
7
3
1
1
4514
4396
1861
856
327
127
47
19
7
3
1
1
Subsidiary Table for GF's of Tables I. and II. (continued).
M
Index of x.
0 ! 1
2
3
4
5 6
7
8
9
' 10
11
12
13
*
1
-1
-9
-63
-251
GF, column 10.
First factor.
Second factor.
0
1
(1)
1
9
53
251
0
1
2
3
4
0
6
7
8
9
10
11
13
1
1
1
1
2
1
1
4
3
1
1
9
6
3
1
1
20
16
7
3
1
1
48
37
18
7
3
1
1
115
96
44
19
7
3
1
1
286
239
117
46
19
7
3
1
1
718
622
299
124
47
19
7
3
1
1
1832
1606
793
320
126
47
19
7
3
1
1
4702
4449
1870
857
327
127
47
19
7
3
1
1
•
1
- 1 • - 10
-64
GF, column 11.
First factor.
Second factor.
0
1
(1)
1
10
64
0
1
2
3
4
5
6
7
8
9
10
11
12
1
1
1
1
2
1
1
4
3
1
1
9
6
3
1
1
20
16
7
3
1
1
48
37
18
7
3
1
1
115
96
44
19
7
3
1
1
286
239
117
46
19
7
3
1
1
719
622
299
124
47
19
7
3
1
1
1841
1607
793
320
126
47
19
7
3
1
1
4755
4459
1871
857
327
127
47
19
7
3
1
1
«
;
-1
-11
GF, column 12.
First factor.
Second factor.
0
1
(1)
1
11
0
1
2
3
4
5
6
7
8
9
10
11
12
1
1
1
1
2
1
1
4
3
1
1
9
6
3
1
1
20
16
7
3
1
1
48
37
18
7
3
1
1
115
9B
44
19
7
3
1
1
286
239
117
46
19
7
3
1
1
719
622
299
124
47
19
7
3
1
1
1842
1607
793
320
126
47
19
7
3
1
1
4765
4460
1871
857
327
127
47
19
7
3
1
1
«
1 -1
GF, column 13.
First factor.
Second factor.
0
1
(1)
1
0
1
2
3
4
5
6
7
8
9
10
11
12
1
1
1
1
2
1
1
4
3
1
1
9
6
3
1
1
20
16
7
3
1
1
48
37
18
7
3
1
1
116
96
44
19
7
8
1
1
286
239
117
46
19
7
3
1
1
719
622
299
124
47
19
7
3
1
1
1
1842
1607
793
320
126
47
19
7
3
1
1
4766
4460
1871
857
327
127
47
19
7
3
1
1
56—2
444 ON THE ANALYTICAL FORMS CALLED TREES, WITH [610
I proceed to explain the Subsidiary Table, first in its application to Table I.
The Subsidiary Table is divided into sections, giving the OP'S of the successive
columns of Table I., each section being given by means of the preceding columns
of Table I. ; for instance, that for column 3 by means of columns 0, 1, 2 of Table I.
As regards column 0, the Table shows that the GF is =x.
As regards column 1, it shows that the GF has a first factor,
(1-te)"', ={l) + tx + t'aJ' + faf+...,
which is operated on by the symbol [<'•"], viz. the constant term (1) is to be rejected;
and that it has a second factor, —x: the product of these, viz. (tx + fx' + faf + . . .) xx, is
the required GF, the coefficients of which are accordingly given in column 1 of
Table I.
As regards column 2, it shows that the GF has a first factor,
(l-te»)->(l -te»)-(l-te')-' ...,
where the indices —1, —1, —1,.. are the sums of the numbers in column 1, Table I.,
{with their signs changed) : which first factor is
1 +te» + te'+ / t \x*+ ...,
\ + tV
and it is as before to be operated on with [f- "], viz. the constant term is to be
rejected; and further, that there is a second factor =a; + te' + iV+ ..., the coefficients
of which are obtained by summation of the numbers in the several lines of columns
0, 1 of Table I. We have thence column 2 of Table I.
As regards column 3, it shows that the GF has a first factor,
(1 - te')-' (1 - tx*)-^ (1 - te»)-^ . . . ,
where the indices — 1, — 2, — 4, . . are the sums of the numbers in column 2 of
Table I., (with their signs changed) : which first factor is
= l+fcB» + 2te* + 4te»+ / 6t\af+...,
and it is as before to be operated on with [<'"*], viz. the constant term is to be
rejected ; and that there is a second factor
= x + ta?+ I t \a^+ / t \ar*+...,
the coefficients of which are obtained by summation of the numbers in the several
lines of columns 0, 1, 2 of Table I. : we have thence column 3 of Table I.
And similarly, by means of columns 0, 1, 2, 3 of Table I., we form the GF of
column 4; that is, we obtain column 4 of Table I., and so on indefinitely.
610] APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS. 445
To apply the Subsidiary Table to the calculation of the GF's of Table II., the
only diflference is that the first factors are to be taken without the terms in t^:
thus for Table II. column 3, the first factor of the GF
= «V + 2t'af + It^af + /14<n of + &c.,
the second factor being as for Table I,
= a; + te* + / t \ic' + &[c.
The remaining Tables are Tables III. and IV., oxygen root-trees and centre- and
bicentre-trees, followed by a Subsidiary Table for the calculation of the GF's:
Tables V. and VI., boron root-trees and centre- and bicentre-trees, followed by a
Subsidiary Table ; and Tables VII. and VIII., carbon root-trees and centre- and
bicentre-trees, followed by a Subsidiary Table. The explanations given as to Tables I.,
II. and the Subsidiary Table apply mutatis mutandis to these ; and but little further
explanation is required : that given in regard to the Subsidiary Table of Tables III.
and IV. shows how this limiting ease comes under the general method. As to the
Subsidiary of Tables V. and VI., it is to be observed that each * line of the Table
is calculated from a column , of Table V., rejecting the numbers which belong to t^ ;
thus Table V., column 4, the numbers are
13 5 7 8 9...
1 4 10 21 36 ...
«» 1 4 11 26...;
and taking the sums for the first and second lines only, these are
1, 4, 9, 17, 29, 45,..,
which, taken with a negative sign, are the numbers of the line *0F, column 5.
And so as to the Subsidiary of Tables VII. and VIII., each * line of the Table
is calculated from a column of Table VII., rejecting the numbers which belong to t*;
thus Table VII., column 4, the numbers are
t'
1 3 8 15
27
43...
f
1 4 13
33
74...
«•
1 4
14
38...
f
1
4
14...
and taking the sums for the first, second, and third lines only, these are
1, 4, 13, 32, 74, 155,..,
which, taken with a negative sign, are the numbers of the line *GF, column 5.
Referring to the foregoing " Edification Diagram," the effect is that we thus
introduce the conditions that in a boron-tree the number of component trees a, b, ...
is at most (3 — 1 =) 2 and that in a carbon-tree the number of component trees
a, b, ... is at most (4 — 1 = ) 3.
I
446
ON THE ANALYTICAL FORMS CALLED TREES, WITH
[610
Table III. — Oxygen Root-Trees.
Index X,
or number
of knots.
Index t, or
nnmber of
main branches.
Altitude or number of column.
0
1
3
3
i
5
6
7 1 8
9
10
11
12
1
0
1
1
2
1
1
3
1
2
1
1
4
1
2
1
1
5
1
2
1
1
6
1
2
1
7
1
2
1
8
1
2
9
1
2
10
1
2
1
11
1
2
1
1
12
1
2
1
1
1
13
1
2
1
1
1
1
610]
APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
447
Table IV. — Oxygen Centre- and Bicentre-Trees.
Index *,
or number
of knots.
Index t, or
number of
main branches.
Centre-Trees.
Altitude or number of colnmn.
i
s
1
■3
t
%
«
0
Bicentre-Trees.
Altitude.
0
1
2
3
4
5
6
0
1
2
3
4
5
1
0
X
2
i i
0
1
0
1
1
1
1
1
1
1
3
2
1
k
1
1
0
1
0
1
4
5
2
1
1
6
0
7
2
X
1
0
8
1
1
1
0
1
9
2
ii
1
0
10
1
0
1
i 1
11 2
1
1
1
0
12 i
0
1
1
18 1 2
1
1
0
448
ON THE ANALYTICAL FORMS CALLED TREES, WITH
[610
Subsidiary Table for QFs of Tables III. and IV.
M
Index of x.
0
1
2
3
*
6
6
7
8
9
10
11
12
13
0
GF, column 0.
*
-1
GF, column 1.
First factor.
Second factor.
0
1
2
1
1
1
0
1
*
-1
GF, column 2,
First factor.
Second factor.
0
1
2
1
1
1
0
1
1
1
«
-1
GF, column 3.
First factor.
Second factor.
0
1
2
1
1
•
•
1
0
1
I
1
1
*
-1
GF, column 4.
First factor.
Second factor.
0
1
2
1
1
,
,
1
0
1
1
1
1
1
*
-1
GF, column 5.
First factor.
Second factor.
0
1
2
1
1
1
0
1
1
1
1
1
1
610]
APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
449
and so on indefinitely ; viz. observing that the first factors, as shown by the Table,
are (1 -te)-^ [<'-^], (1 -<a^)-' [«»•=], &c., the Table in fact shows that as regards Table III.
the GF's are for
column 0 : cc,
„ 1 : tx + fa^ . X,
2 : tx^ + tV . X + taf,
S: ta? + Pa^ .x + t{x' + x'),
„ i: tx* + tV .x + tiaf + a^ + x*),
„ 5: tx^ + t^x^" .x + t{x' + x^ + x* + x^);
viz. developing as far as t-, that the successive GF's are
column 0 : x,
1 : iar" + t'x>,
2: tx' + t''(x* + x'),
„ '3: tx* + f (x" + af -^ x'),
„ 4 : te» + <" (a;« + a;' + a;» + a?),
5: te» + «»(«' + a^ + «' + *'» + a!>') ;
&c., agreeing with Table III.
And so also it shows that, as regards Table IV. (centre part), the GF's of the
successive columns are for
column 0 : x,
1 : «V . X,
2 : <V . .r,
3 : foi^. X,
4 : fa? . X,
5 : fx'" .x;
viz. that the successive GF'a are x, Va?, tV, fa?, fa?, <W, . . , agreeing in fact with
Table IV.
c. IX.
57
Table V. — Boron Root-trees.
Index X,
or namber
of knots.
Index t, or
namber of
main branches.
Altitude or
namber of oolnnm.
0
a
3
4
6
6
7
8
9
10
11
12
1
0
1
1
1
1
2
1
1
3
3
1
\
3
46
52
29
98
109
68
207
244
147
451
532
887
983
1196
757
Total
1
2
1
Total
1
8
1
2
1
Total
1
2
4
1
2
8
1
1
1
Total
2
1
4
0
1
2
3
2
1
2
1
1
Total
3
«
1
7
6
1
2
3
1
2
2
3
1
3
1
1
Total
3
6
4
1
14
1
7
1
2
3
2
1
6
3
5
4
1
4
1
1
11
12
6
23
23
14
Total
3
10
10
6
1
29
8
1
2
3
2
1
7
7
7
10
4
9
5
1
5
1
1
Total
2
15
21
15
6
1
60
9
1
2
3
1
9
11
8
21
11
17
15
5
14
6
1
6
1
1
Total
1
20
40
37
21
7
1
127
10
1
2
3
1
7
18
9
36
26
29
37
16
32
21
6
20
7
1
7
1
1
Total
1
25
71
82
59
28
8
1
275
H
1
2
3
7
21
7
59
63
45
' 82
43
66
59
22
63
28
7
27
8
1
8
1
1
Total
28
119
170
147
88
36
9
1
598
12
1
2
3
4
26
7
82
102
6G
165
105
127
147
66
125
88
29
81
36
8
36
9
1
9
1
1
Total
80
191
i 336
340
242
125
46
10
1
1320
18
1
2
3
3
26
4
114
175
89
316
236
a31
340
177
274
242
96
213
125
37
117
45
9
44
10
1
10
1
1
Total
29
293
641
748
612
875
171
55
11
1
2936
610] ■ ON THE ANALYTICAL FORMS CALLED TREES.
Table VI. — Boron Centre- and Bicentre-Trees.
451
Index X,
or number
of knots.
Index t, or
number of
main branches.
Centre-Trees.
Altitude or number of column.
2
1
■3
o
■a
s
2
2
1
Bicentre-Trees.
Altitude.
0
1
2
S
4 1 5
6
0
1
1
1
1
1
2
1
2
5
5
6
4
3
1
3
1
3
11
22
44
68
4
1
4
19
53
5
1
5
1
0
1
1
1
18
18
67
71 1
1
0
1
1
1
2
4
6
10
19
36
68
138
1
1
1
2
2
4
6
11
18
37
66
135
265
0
1
0
1
1
2
2
6
18
30
67
127
Total
1
1
2
3
2
1
Total
1
4
2
1
Total
1 1
1
5
2
i
1
Total
K
2
2
6
2
3
1
1
Total
2
2
7
2
3
1
2
1
Total
3
1 4
8
. 2
3
2
2
1
2
3
Total
2
3
5
9
2
3
1
5
3
1
6
4
Total
1
8
1 10
10
2
8
1
5
9
3
1
8
11
Total
1
14 i 4
19
11
2
8
6
14
11
4
1
Total
20
16
1
36
12
2
3
4
21
22
16
4
1
30
38
Total
23
38
5
68
13
2
3
3
24
44
42
19
5
1
Total
27
86
24
1
138
57—2
452
ON THE ANALYTICAL FORMS CALLED TREES, WITH
Subsidiary Table for GFs of Tables V. and VI.
[610
^
Index of x.
•S
•s
0
1
2
8
4
S
6
7
8
9
10
11
12
18
0
1
GF, column 0.
*
-1
GF, column 1.
First factor.
0
1
1
1
2
1
3
1
Second factor.
0
1
•
-1
-1
GF, column 2.
First factor.
0
1
1
1
1
2
1
1
1
3
1
1
1
1
Second factor.
0
1
1
1
2
1
*
-1
-2
-2
-1
-1
GF, column 3.
First factor.
0
1
1
1
2
2
1
1
2
1
2
5
5
6
4
3
3
1
2
5
9
Second factor.
0
1
1
1
1
1
2
1
1
2
1
1
•
-1
-3
-5
-7
-8
-9
-7
-7
-4
-3
GF, column 4.
First factor.
0
1
1
1
3
o
7
8
9
7
7
4
2
1
3
11
22
44
3
1
Second factor.
0
1
1
1
1
2
2
2
1
1
2
1
1
3
4
7
7
7
7
7
4
3
«
-1
-4
-9
-17
-29
-45
-66
-89
-118
GF, column 6.
First factor.
0
1
1
I
4
9
17
29
45
66
89
2
1
4
19
Second factor.
0
1
1
1
1
2
8
5
6
8
8
9
7
7
4
2
1
1
8
5
11
17
80
43
66
86
117
«
-1
-5
-14
-32
-66
-127
-231
-405
GF, column 6.
First factor.
0
1
1
1
5
14
82
66
127
231
2
1
Second factor.
0
1
1
1
1
2
8
6
10
17
25
88
62
78
93
3
1
1
8
6
12
22
45
80
148
251
438
610]
APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
453
Subsidiary Table for GF's of Tables V. and VI. {continued).
4>*
Index of x.
0
2
3
4
6
6
7
8
9
10
11 12
18
*
-1
-6
-20
-53
-125
-274
-571
OF, column 7.
First factor.
Second factor.
0
1
1
1
6
20
53
125
274
0
1
2
1
1
1
1
2 3
1 1 3
6
5
11
12
22
23
39
51
70
101
118
207
200
898
824
773
*
i
-1
-7
-27
-81
-218
-516
GF, column 8.
First factor.
Second factor.
0
1
1
1
i
1
7
27 i 81
213
0
1
2
1
1
1
2
1
3 •
3
6
5
11
12
28
23
49
52
90 171
108 i 235
325
486
598
1015
•
-1
-8
-35
- 117 - 838
GF, column 9.
First factor.
Second factor.
0
1
1
«
»
1
8
35
117
0
1
2
1
1
1
2
:
3
3
6
5
11
12
23
23
46
62
97
109
198
248
406
522
811
1140
1
• i
-1
-9
-44
-162
GF, column 10.
First factor.
Second factor.
0
1
1
1
9
44
0
1
2
1
1
1
2
1
3
3
6
5
11
12
23
23
46
52
98
109
206
244
441
531
928
1185
*
i -'
-10
-54
GF, column 11.
First factor.
Second factor.
0
1
1
1
10
0
1
2
1
1
1
2
3
3
6
5
11
12
23
23
46
52
98 207
109 ; 244
450
532
972
1195
*
i
1
-1
-11
GF, column 12.
0
1
1
1
First factor.
Second factor.
0
1
2
1
1
1
2
1
3
3
6
5
11
12
23
23
46
62
98 207
109 244
451
632
982
1196
*
-12
GF, column 18.
First factor.
Second factor.
0 1
1
i
0
1
2
1
1
1
1
2 1 S
1 i 3
6
5
11
12
23
23
46
52
98
109
207
244
451
532
983
1196
454
ON THE ANALYTICAL FORMS CALLED TREES, WITH
[610
Table VII. — Carbon Root-trees.
Index X,
or number
of knots.
Index t, or
nnmber of
main branches.
Altitude or number of column.
0
1
2
3
4
5
6
7
8
9
10 ! 11
12
1
0
1
1
1
1
Total
1
1
2
1
1
Total
1
1
3
1
1
2 1
1
1
Total
1
1
2
4
1
2
3
1
1
1
1
2
1
1
4
Total
1
2
1
4
5
1
1
2
1
2
2
1
3
3
1
1
4
1
1
8
Total
1
4
3
1
9
6
1
*
3
1
2
2
3
1
6
3
2
1
3
4
1
1
17
Total
5
8
4
1
18
7
1
4
8
4
1
2
2
8
4
1
15
3
3
8
1
7
4
2
1
3
Total
7
16
13
5
1
42
610]
APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
455
Table VII. {continued).
Indez,'x,
or number
of knots.
SO
Index t,
number
main bram
Altitude or number of column.
12
0
1
2
3
4
5
6 ' 7
1
8
9
10
11
8
1
5
15
13
5j 1
39
2
1
13
13
5
1
33
3
3
9
4
1
17
4
3
3
1
7
89
Total
7
30
33
19
6
1
96
9
1
4
27
32
19 1 6
1
2
1
22
33
19
6
1
82
3
3
17
14
5
1
40
4
4
*
4
1
18
211
Total
8
52
78
57
26
7
1
229
10
1
4
43
74
56
26
7
1
2
29
74
57
26
7
1
194
3
3
34
38
20
6
1
102
4
4
18
14
5
1
42
507
Total
7
85
169
156
89
34
8
1
549
11
■ 1
3
67
155
151
88
34
8
1
2
40
154
156
89
34
8
1
482
3
2
64
95
63
27
7
1
249
4
5
38
39
20
5
1
108
1238
Total
7
135
355
394
272
130
43
9
1
1346
12
1
2
97
316
374
267
129
43
9
1
2
46
297
889
273
130
43
9
1
1188
3
1
88
218
184
96
35
8
1
631
4
4
66
100 64
27
7
1
269
3056
Total
5
202
712 t 953
1
770
439
181
53
10
1
3326
18
1
1
136
612
889
743
432
180
52
10
1
2
55
550
929
770
439
181
53
10
1
2988
8
1
127
474 491
309
138
44
9
1
1594
4
4
117
239
190
97
35
8
1
691
Total
5
300
1399
2222
2065
1356
665
243
63
11
1
8329
456
ON THE ANALYTICAL FORMS CALLED TKEE8, WITH [610
Table VIII. — Carbon Centre- and Bicentre-Trees.
Index or,
or number
of knots.
Index t, or
number of
main brancbes.
Centre-Trees.
Altitude or number of oolamn.
1
3
1
Bioentre.
Bicentre-Trees.
Altitude.
0
1
2
8
4
5
6
0
1
2
8
4
6
1
0
1
1
2
3
2
1
9
6
5
38
80
18
174
167
88
1
0
1
1
2
2
6
9
20
37
86
188
419
1
1
1
2
3
5
9
18
35
75
159
857
799
0
1
0
1
1
a
8
9
15
88
73
174
880
1
1
1
2
1
1
1
2
7
12
28
30
42
47
1
3
14
39
108
244
1
4
23
84
1
9
Total
1
1
2
S
2
1
Total
1
i
2
1
Total
1
5
2
1
Total
1
2
6
2
3
1
1
Total
2
2
7
2
8
4
2
2
1
1
Total
' 5
1
6
8
2
3
4
1
3
2
2
1
3
4
2
Total
6
3
9
9
2
8
4
1
3
4
7
3
1
1
Total
8
11
1
20
10
2
3
4
3
4
12
11
3
3
1
15
IS
7
Total
7
26
4
37
11
2
3
4
2
5
28
24
12
14
4
1
1
Total
7
69
19
1
86
12
2
3
4
1
4
80
54
27
39
19
4
4
1
78
75
36
Total
5
111
62
5
183
18
2
8
4
1
4
42
88
63
108
63
20
23
5
1
1
Totel
6
198
191
29
1
419
610] APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
Subsidiary Table for GF's of Tables VII. and VIII.
457
H
Index of x.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
0
1
OF, column 0.
*
-1
GF, column 1.
First factor.
0
1
1
1
2
1
3
1
4
1
Second factor.
0
1
•
-1
-1
-1
GF, column 2.
First factor.
0
(1)
1
1
1
1
8
1
1
2
1
1
3
1
1
2
2
2
1
1
4
1
1
2
2
3
Second factor.
0
1
1
1
2
1
3
1
*
-1
-2
'■-4
-4
-6
-4
-4
-3
-2
-1
-1
GF, column 3.
First factor.
0
(1)
1
1
2
4
4
5
4
4
3
2
1
2
1
2
4
12
23
30
42
8
1
2
7
16
4
1
Second factor.
0
1
1
1
1
1
1
2
1
1
2
2
2
1
1
3
1
1
2
3
3
3
3
2
1
1
♦
-1
-3
-8
-15
-27
-43
-67
-97
-136
-183
GF, column 4.
First factor.
0
<1)
1
1
3
8
15
27
43
67
97
136
2
1
3
14
39
108
3
1
Second factor.
0
1
1
1
1
2
3
4
4
5
4
4
3
2
1
2
1
1
3
5
10
14
23
29
40
46
55
3
1
3
6
12
20
37
56
89
128
•
-1
-4
-13
-32
-74
-155
-316
-612
-1160
GF, column 5.
First factor.
0
(1)
1
4
13
32
74
155
316
612
2
1
4
23
Second factor.
0
1
1
1
1
2
7
12
20
31
47
70
99
137
2
1
1
6
14
27
56
103
194
343
605
3
1
3
7
16
34
76
151
307
602
*
-1
-6
-19
-56
-151
-374
-889
-2032
OF, column 6.
First factor.
0
1
1
1
5
19
56
151
374
889
2
1
Second factor.
0
1
1
1
1
2
4
8
16
33
63
121
225
415
749
2
1
1
3
6
15
32
75
160
350
732
1534
8
1
1
3
7
17
39
95
214
491
1093
C. IX,
58
458 ON THE ANALYTICAL FORMS CALLED TREES, WITH
Subsidiary Table for OPs of Tables VII. and VIII. (contimied).
[610
i
a
Index of x.
0
2
3
4 1 6
6
7
8
9
10
11
12
18
•
-
-1
-6
-26
-88
-267
-743
-1968
OF, column 7.
First factor.
Second factor.
0
1
(1)
1
6
26
88
267
748
0
1
2
8
1
1
1
2
1
1
4
3
1
8
6
3
17
15
7
38
33
17
82
81
40
177
186
101
376
439
241
789
1005
587
1688
2804
1402
*
-1
-7
-34
-129
-432
-1320
GF, column 8.
First factor.
Second factor.
0
1
(1)
1
7
34
129
432
0
1
2
3
1
1
1
2
1
1
4
3
1
8
6
3
17
15
7
39
33
17
88
82
40
203
193
102
464
473
248
1056
622
2381
2743
1540
*
-1
-8
-43
-180
-657
GF, column 9.
First factor.
Second factor.
0
1
(1)
1
8
43
180
0
1
2
3
. 1
1
1
2
1
1
4
3
1
8
6
3
17
15
7
39
33
17
89
82
40
210
194
102
498
481
249
1185
1178
630
2813
2924
1584
•
-1
-9
-53
-242
GF, column 10.
First factor.
Second factor.
0
1
(1)
1
9
53
0
1
2
3
1
1
1
2
1
1
4
3
1
8
6
3
17
15
7
39
33
17
89
82
40
211
194
102
506
482
249
1228
1187
631
2993
2977
1593
•
i
-1
-10
-63
GF, column 11.
First factor.
Second factor.
0
1
(1)
1
10
63
0
1
2
3
1
1
1
2
1
1
4
3
1
8
6
3
17
15
7
39
33
17
89
82
40
211
194
102
507
482
249
1237
1188
031
3048
2987
1594
*
-1
-11
OF, column 12.
First factor.
Second factor.
0
1
(1)
1
11
0
1
2
8
1
1
1
2
1
1
4
3
1
8
6
3
17
15
7
39
33
17
89
82
40
211
194
102
507
482
249
1238
1188
631
3055
2988
1594
«
-1
OF, column 13.
First factor.
Second factor.
0
1
(1)
1
0
1
2
3
1
1
1
2
1
1
4
8
1
8
6
8
17
16
7
39 89
83 82
17 40
211
194
102
507
482
249
1238
1188
634
3056
2988
1594
610]
APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
459
I annex the following two Tables of (centre- and bicentre-) trees as far as I have
completed them.
Table A.
i
Valency
not greater than
Gen.
0
1
2
3
4
5
6
7
8
Oxygen.
Boron.
Carbon.
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
3
1
1
1
1
1
1
1
4
2
2
2
2
2
2
2
5
2
3
3
3
3
3
3
6
7
8
1
4
6
11
5
9
18
6
6
6
11
6
11
23
6
11
23
10
21
11
22
23 1
9
18
35
42
45
46
47 1
47
10
37
75
106
11
66
159
235
12
\
135
367
551
13
I
266
799
1801
Table B.
1
Actual Valency.
0
1
2
3
4
5
6
7
8
1
1
2
1
8
4
1
6
1
1
6
3
1
1
7
5
3
1
1
8
10
7
3
1
1
9
17
17
7
3
1
1
10
36
88
11
1
65
98
12
134
222
18
264
534
In A, the columns 2, 3, 4, and the last column are the totals given by the
Tables IV., VI., VIII., and II., and the remaining numbers of columns .5, 6, 7, 8
have been found by trial; and, in B, the several columns are the differences of the
58—2
460 ON THE ANALYTICAL FORMS CALLED TREES. [610
columns of A. The signification is obvious; for instance, if the number of knots is
= 9, then Table A, if the valency, or the maximum number of branches from a knot,
is = 2, 3, 4, 5, 6, 7, 8 or any greater number.
No. of trees = 1, 18, 35, 42, 45, 46, 47 :
viz. with 9 knots the tree can have at most 8 branches from a knot, so that the
number of trees having at most 8 branches from a knot is = 47, the whole number
of trees with 9 knots ; and so the number of knots being as before = 9, Table B
shows that the number of 47 is made up of the numbers
1, 17, 17, 7, 3. 1, 1;
viz. 1 is the No. of trees, at most 2 branches from a knot,
17 „ „ 3 „ „ at least one 3-branch knot.
17 4 4
■ » » " » » » ^ o
*^ » » " » >i w V n
17 7
■'■»»>" M » >i O »
I annex also a plate showing the figures of the 1 + 1+2 + 3+6 + 11+23 + 47
trees of 1, 2, 3, . . , 9 knots, classified according to their altitudes and number of main
branches; and as to the bicentre-trees, according to the number of main branches
from each point of the bicentre. The afiSxed numbers show in each case the greatest
number of branches from a knot; so that when this is (2), the knots may be oxygen-,
boron-, carbon-, &c., atoms; when (3), boron-, carbon-, &c., atoms; when (4), carbon-,
&c., atoms; and so on.
611] 461
611.
KEPORT OF THE COMMITTEE ON MATHEMATICAL TABLES:
CONSISTING OF PROFESSOR CAYLEY, F.R.S., PROFESSOR STOKES, F.R.S.,
PROFESSOR SIR W. THOMSON, F.R.S., PROFESSOR H. J. S. SMITH, F.R.S.,
AND J. W. L. GLAISHER, F.R.S.
[From the Report of the British Association for the Advancement of Science (1875),
pp. 305—336.]
The present Report (say Report 1875) is in continuation of that by Mr Glaisher,
published in the volume for 1873, and here cited as Report 1873.
Report 1873 extends to all those tables which are at p. 3 (I.e.) included under the
headings : —
A, auxiliary for non-logarithmic calculation, 1, 2, 3;
B, logarithmic and circular, 4, 5, 6 ;
C, exponential, 7, 8 (but only partially to C. 8), other than those tables of C
referred to as "h . 1 tan(45'' + J </>)"; and also partially (see Art. 24, pp. 81—83) to
the tables included under the heading " E. 11, transcendental constants e, it, y, &c.,
and their powers and functions."
A future Report will comprise the tables, or further tables, included under the
headings : —
C. 8. Hyperbolic antilogarithms (e*) and h . 1 tan(45°-|- i^), and hyperbolic sines,
cosines, &c.
D. Algebraic constants.
9. Accurate integer or fractional values. Bernoulli's Numbers, A"©"", &c.
Binomial coefficients.
10. Decimal values auxiliary to the calculation of series.
K 11. Transcendental constants e, ir, y, &c., and their powers and functions.
462
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
[611
The present Report (1875) comprises the tables included under the headings: —
F. Arithmological.
12. Divisors and prime numbere. Prime roots. The Canon arithmeticus, &c.
13. The Pellian equation.
14. Partitions.
15. Quadratic forms a' + b", &c., and partitions of numbera into squares, cubes,
and biquadrates.
16. Binary, ternary, &c., quadratic and higher forms.
17. Complex theories :
which divisions are herein referred to, for instance, as [F. 12. Divisors, &e.].
Report 1873 consists of six sections (§) divided into articles, which are separately
numbered (see contents, p. 174); the present Report 1875 forms a single section
(§ 7), divided in like manner into articles, which are separately numbered ; but
besides this the paragraphs are numbered, and that continuously, through the present
Report 1875, so that any paragraph may be cited as Report 1875, No. — , as the
case may be.
[F. 12. Divisors, <&c.] Divisors and Prime Numbers. Art. I.
1. As to divisors and prime numbers see Report 1873, Art. 8 (Tables of
Divisors — factor tables — and Tables of Primes), pp. 34 — 40. The tables there refen-ed
to, such as Chernac, Burckhardt, Dase, Dase and Rosenberg, ai-e chiefly tables running
up to very high numbers (the last of them the ninth million) : wherein, to save
space, multiples of 2, 3, 5 are frequently omitted, and in some of them only the
least divisor is given. It would be for many purposes convenient to have a small
table, going up say to 10,000, showing in every case all the prime factors of the
number. Such a table might be arranged, 500 numbers in a page, in some such
form as the following: —
Factor Table
1 to 500
2.3.5.13
17.23
2». 7^
3. 131
2.197
5.79
2>.3M1
397*
2.199
3.7. 19
39
where the top line shows the units, and the left-hand column the remaining figures,
viz. the specimen exhibits the composition of the several numbers from 390 to 399 :
a prime number, e.g. 397, would be sufficiently indicated by the absence of any
decomposition, or it may be further indicated by an asterisk.
It may be noticed that, in the theory of numbere, the decomposition is specially
required when the next following number is a prime, viz. that of p — 1, p being a
611]
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
46a
prime: also, that this is given incidentally, for prime numbers p up to 1000, in Jacobi's
Canon Arithmeticus, post, No. 20, and up to 15,000 in Reuschle's Tables, V. (a, b, c)
post, No. 22.
2. It may be proper to remark here that any table of a binary form is really
a factor-table in the complex theory connected with such binary form. Thus in
a table of the form a' + fr", a number of this form has a factor a+ bi (i = V— 1 as
usual); and the table, in fact, shows the complex factor a+bi of the number in
question : a well arranged table would give all the prime complex factors a + bi of the
number. But as to this more hereafter; at present, we are concerned with the real
theory only, not with any complex theory.
3. Connected with a factor-table, we have (i) a Table of the number of less relative
primes ; viz. such a table would show, for every number, the number of inferior integers
having no common factor with the number itself The formula is a well-known one:
for a number N = a'l/cy ..., (a, b, ... the distinct prime factors of N), the number of
less relative primes is
■tir(iV), =a«-i6^-^..(a-l)(6-l)...,
or, what is the same thing, =iV(l j (l - t) ... A small table {N = 1 to 100),
occupying half a page, is given by
Euler, Op. Arith. Coll. t. ii. p. 128; viz. this is 7rl=0, 7r2 = l,..., 7rl00 = 40.
4. But it would be interesting to have such a table of the same extent with
the proposed factor-table. The table might be of like form; for instance.
Number of less relative Primes Table
1 to 500
29
112
192
144
292
84
232
144
198
148
264
It would be of still greater interest to have an inverse table showing the values of iV
which belong to a given value of w (N) ; for instance,
w =
N=
40
41,
55, 75, 82, 88,
100,
110,
42
43,
49, 86, 98,
44
69,
92,
46
47,
94,
48
65,
104, 105, 112,
where, observe, that «r is of necessity even.
464
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
[611
5. Again, connected with a factor-table, we have (ii) a Table of the Sum of the
divisors vi a Number. The formula is also a well-known one ; for a number
N = o'6^ . . . , (a, b,... the distinct prime factors of N), the required sum
I'
iJVis ={l + a+... + a')(l+b + . ..+¥)...,
or, what is the same thing,
a-L ■ 6-1 ••■'
where, observe, that the number itself is reckoned as a divisor.
6. Such a table was required by Euler in his researches on Amicable Numbers
(see post, No. 10), and he accordingly gives one of a considerable extent, viz.
Euler, Op. Arith. Coll. t. i. pp. 104—109.
It ijs to be remarked that, inasmuch as I iV is obviously ^la"^ \h^ ..., the function
need only be tabulated for the dififerent integer powers a" of each prime number a.
The range of Euler's table is as follows: —
a —
a =
2
to 36,
3
„ 15,
5
„ 9,
7
„ 10,
11
„ 9,
13
„ 7,
17
„ 5,
19
„ 5,
23
„ 4.
29 to 997
„ 3,
viz. for the several prime numbers from 29 to 997 the table gives \ <^> \ a^ and I a'.
And it is to be noticed that the values of the sum are exhibited, decomposed into
their prime factors: thus a specimen of the table is
Nam.
Summs Divisorum.
139
2'. 5. 7
139"
3.13.499
139»
2».5.7.9661
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 465
7. The form of the above table is adapted to its particular purpose (the theory
of amicable numbers); but Euler gives also,
Euler, Op. Arith. Coll. t. i. p. 147 — in the paper "Observatio de Summis
Divisorum," (1752), pp. 146 — 154, — a short table of about half a page, N=\ to 100.
of the form (1 = 1. I 2 = 3, .. , I 100 = 217. The paper contains interesting analytical
researches on the subject of I N which connect themselves with the theory of the
Partition of Numbers.
8. It would be interesting to carry the last-mentioned table to the same extent
as the proposed factor-table ; and to add to it an inveree table, as suggested in regard
to the number of less relative primes table.
9. Perfect Number.^. — A perfect number is a number which is equal to the sum
of its divisors, the number itself not being reckoned as a divisor ; e.g.
6 = 1 -I- 2 + 3, and 28 = l-H 2 -I- 4 + 7 -I- 14.
Snob numbers are indicated l^y a table of the sums of divisors |6 = 12, 128=56,
these two being, as appears by the table. Art. 7, the only perfect numbers less
than 100.
10. Amicable Numbers. — These are pairs of numbers such that each is equal to
the sum of the divisors of the other, not reckoning the other number as a divisor ;
that is, each has the same sum of divisors, the number being here reckoned as a
divisor ; say I A=B, I B = A; or, what is the same thing, \ A = j B(=A +B). Thus
for the numbers 220, 284,
j 220 = (H- 2 + 4) (1 -h 5) (1 + 11) - 220, = 284,
f 284 = (1 + 2 -t- 4) (1 + 71) - 284, =220;
or, what is the same thing,
[ 220 = (H- 2 -H 4) (1 -I- 5) (1 -I- 11) = 504 = (IH- 2 -I- 4) (1 -|- 71) =[284.
11. A catalogue of 61 pairs of numbers is given by
Euler, Op. Arith. Coll. t. i. pp. 144 — 145 ; it occupies about one page. The paper,
" De Numeris Amicabilibus," pp. 102 — 145, contains an elaborate investigation of
the theory, by means whereof all but two of the pairs of numbers are obtained.
The first pair is the above-mentioned one, 2^.5.11 and 2=.7l (=220 and 284); and
the fifty-ninth pair is the high numbers
3». 7' . 13 . 19 . 53 . 6959 and V . V .13.19. 179 , 2087.
C. IX. 59
466
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
[611
The last two pairs are refened to as "forraaj diversse a precedentibus ; " viz. these are
f2» . 19 . 41
and
[2».41.467
2» . 19 . 233,
12M99
12. A Table of the Frequency of Primes is given by
OaUM, Tafel der Frequenz der Piimzahlen, Werke, t. II. pp. 4.36 — 443 ; viz. this
extends to 3,000,000.
The first part, extending to 1,000,000, = 1000 thousand, shows how many primes
there are in each thousand : a specimen is
1, 168:
2, 135 :
3, 127
4, 120:
5, 119;
&c. ;
viz. in the first thousand there are 168 primes, in the second thousand 13.5 primes,
and so on.
For the second and third millions the frequency is given for each ten thousand :
s\ specimen is
1,000,000 to 1,100,000.
1
4
21
54
114
171
217
164
126
71
39
12
6
0
1
2
3
4
5
6
7
8
9
1
1
2
1
1
1
1
3
4
2
2
3
1
2
3
3
1
4
2
8
5
4
3
6
9
4
5
8
5
11
10
8
18
12
10
10
12
15
8
6
14
14
18
21
16
22
19
15
17
15
7
26
17
23
23
24
24
17
22
20
21
8
19
19
21
7
14
\f>
20
17
15
17
9
11
13
9
13
14
14
12
13
11
16
10
8
6
8
0
9
5
it
9
7
9
11
6
6
4
6
3
1
3
1
4
5
12
1
1
2
1
1
1
2
2
1
13
1
1
1
1
1
1
14
15
16
752
719
732
700
731
698
713
722
706
737
/
7210,
dx
\ogx
= 7212-99;
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 467
viz. in the interval 1,000,000 to 1,010,000, 100 hundreds, there is 1 hundred containing
1 prime, there are 2 hundreds each containing 4 primes, 11 hundreds each containing
5 primes, . . , 1 hundred containing 13 primes, so that, as
Ix 1= 1,
4x 2= 8,
5 X 11 = 55,
13 X 1 = 13,
100 752,
the whole 10,000 contains 752 primes; the next 10,000 contains 719 primes, and so
on; the whole 100,000 thus containing 752 + 719 + &c. ... =7210 primes, which number
is at the foot compared with the theoretic approximate value
/,
dx
,=^ (limits 1,000,000 to 1,010,000) = 7212-99.
log a;
The integral in question is represented b}' the notation Li. x or li. x.
p. 443. We have the like tables 1,000,000 to 2,000,000 and 2,000,000 to 3,000,000,
showing for each 100,000 how many hundreds there are containing 0 prime, 1 prime,
2 primes, up to (the largest number) 17 primea
13. It is noticed that
the 26,379th hundred contains no prime,
the 27,050th hundred contains 17 primes.
It may be observed that, if iV^ = 2 . 3 . 5 . . . p, the product of all the primes up
to p, then each of the numbers N +\ and N + q (if q be the prime next succeeding
p) is or is not a prime; but the intermediate numbers N+2, iV+3, .., N + q—\
are certainly composite ; viz. we thus have at least q — 2 consecutive composites. To
obtain in this manner 99 consecutive composites, the value of N would be =2. 3. 5... 97,
viz. this is a number far exceeding 2,637,900; but, in fact, the hundred numbers
2,637,901 to 2,638,000 are all composite.
Legendre, in his Eami sur la Tli^orie des Nonibres (Ist edit., 1798 ; 2nd edit.,
1808, supplement, 1816: references to this edition), gives for the number of primes
inferior to a given limit x the approximate formula
log a; -108366'
and p. 394, and supplement, p. 62, he compares for each 10,000- up to 100,000, and for
each 100,000 up to 1,000,000, the values as computed by this formula with the
actual numbers of primes exhibited by the tables of Wega and Chemac. Thu.><
for a; = 1,000,000, the computed value is 78,543, the actual value 78,493.
59—2
468 REPOBT OF THE COMMITTEE ON MATHEMATICAL TABLES. [611
He shows, p. 414, that the number of integera, which are less than n and are not
divisible by any of the numbers 6, X, /*,..., is approximately
=»(-^)(-a(>-D-
and taking d, X, fi,... the successive primes 3, 5, 7,... he gives the values of the
function in question, or, say, the function
2 4 6 10 m-1
3"5711" a> '
» a prime, for the several prime values w = 3 to 1229 in the Table IX. (one page)
at the end of the work.
14. A table of frequency is given by
Olaisher, J. W. L., Bi-itish Association RepoH for 1872, p. 20. This gives for the
second and the ninth millions, respectively divided into intervals of .50,000, the actual
number of primes in each interval, as compared with the theoretic value lia;' — Ha;;
and also deduced therefrom, by the formula log ^ (a;' + x), a table of the average
interval between two consecutive primes; this average interval increases very slowly:
at the beginning and the end of the second million the values are 13'76 and 14".58
{theoretic values 13"84 and 1450); at the beginning and the end of the ninth million
16-02 and 15-95 (theoretic values 1590 and 1601).
15. Coming under the head of Divisor Tables, some tables by Reuschle and
Oauss may be here referred to. These are: —
Reuschle, Mathematische Abhandlung, zahlentheoretische Tabellen sammt einer
dieselien ti-effenden Coi-respondenz mit der uereudgten C. G. J. Jacobi, 4°, pp. 1 — 61*
(1856). The tables belonging to the present subject are
A. Tafeln zur Zerlegung von a" — 1 (pp. 18 — 22).
1. Table of the prime factors of 10" — 1, viz.
(a. pp. 18—19.) Complete decomposition of 10"-1, n=l to 42: and 10"+1, n = l to 21.
Some values of n are omitted.
A specimen is
10" - 1 = 3'' . 53 . 79 . 265371653,
10'» + 1 = 11 . 189 . 10.5831.3049.
(b. p. 19.) List of the specific prime factoi-s / of 10"— 1, or the prime factore
of the residue after separation of the analytical factors, for those values of n for which
the complete decomposition is unknown, and omitting those values for which no factor
is known, m = 25 to 243.
* Titlepage misBing in my copy; but I find from Prof. Kummer's notice of the work, Crelle, t. liii.
(18.57), p. 379, that it appeared as a ProKramm of the Stuttgart Gymnasium, Michaelmas, 1856, and was
separately printed by Liesohing and Co., Stuttgart.
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
A specimen is
469
11
25
/
21401.
The meaning seems to be, residue of 10^-1 is 1 +10»+ 10"+ 10"+ 10* and
this contains the prime factor 21401 ; but it is not clear why this is the " specific
prime factor."
II. Prime factors of a"— 1 for different values of a and tu
(a. p. 20) gives for 41 values of a (2, 3, &c. at intervals to 100) and for the
following values of n the decompositions of the residues or specific factors of a" — 1 ;
viz. these are
n= 1, a — 1 :
2, a + 1 :
3, a- + a+l:
6, a- — a + 1 :
4, a^ + 1 :
5, o* + a' + ft* + a + 1 :
10, a* — a^-\- a- - a + 1 :
8, a* + 1 :
12, a'-vC'-ir 1.
A specimen is
a-\ 1 a»-l
10
3»
11
a?-\
3'. 37
«•-!
7.13
101 ! 41 . 271
9091
a»-l
73.137 I 9901
(b. p. 21.) Specific prime factore for the numbers 2, 3, 5, 6, 7, 10, (the powers
4, 8, 9 being omitted as coming under 2 and 3), for the exponents 1 to 42.
A specimen is
19
2«-l
524287
3»-l
5»-I
1597.363889 191. a:
6»-l
191. a;
7»-l
419. a;
10»-1
where the x denotes that the other factor is not known to be prime. And so, where no
number is given, as in 10"— 1, it is not known whether the number (=1 + 10' +10' +...+10")
is or is not prime.
Addition, p. 22. For a = 2, the complete decomposition of the prime factor of
2"— 1 is given for values of n, =44, 45,... at intervals to 156.
A specimen is
/
44 397 . 2113,
VIZ.
218 + 2" - ... - 2- + 1, = 838861 = 397 . 2113.
m=31, Fermat's prime. ?? = 37, the first case for which the decomposition is not
^ven completely. n = 41, the first case for which no factor is known.
470
REPORT OF THE COMMrTTEE ON MATHEMATICAL TABLES.
[611
16. GauM, Tafel zur Cyclotechnie, Werke, t. IL pp. 478 — 495, shows, for 2452
numbers of the several forms a' +1, a* + 4, a'+9, .,., a' + 81, the values of a such that
the number in question is a product of prime factors no one of which exceeds 200,
and exhibits all the odd prime factors of each such number. The table is in nine
parts, zerlegbare o' + l, zerlegbare a* + 4, &c., with to each part a subsidiary table, as
presently mentioned. Thus a specimen is
zerlegbare o" + 9
1
5
2
13
4
5 . 5
5
17
7
29
8
73
ATIZ.
1411168679 5 . 5 . 13 . 17 . 17 . 89 . 113 . 157 . 173 . 197 . 197 ;
1- + 9, odd prime factor is 5,
2^ + 9, „ „ 13,
4*" + 9, „ factors are 5, 5,
and so on.
And the subsidiary table is
5
13
17
1, 4, 79
2, 11, 41
5, 29, 46, 379, 1042
showing that the numbers a for which the largest factor is 5 are 1, 4, 79 ; those
for which it is 13 are 2, 11, 41 ; and so on.
The object of the table is explained in the Bemerkuiigen, (I. c, p. 523), by Schering,
the editor of the volume, viz. it is to facilitate the calculation of the circular arcs
the cotangents of which are lational numbers. To take a simple example, it appears
to be by means of it that Gauss obtained, among other formulae, the following:
7r
and
7r
J = 12 arctan T*g + 8 arctan ^ — 5 arctan ^^,
= 12 arctan 515 + 20 arctan -^ + 1 arctan ^+24 arctan ^.
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 471
[F. 12. Divisors, etc.] continued. Prime Roots. The Canon Arithmeticus, Quadratic
residues. Art. II.
17. Prime Roots. — Let p be a prime number; then there exist «r(p— 1) inferior
integers g, such that all the numbers 1, 2, ... , p — 1 are, to the modulus j),
= 1, g, g\ ..., g^^ig^^ is of course = 1).
This being so, g is said to be a prime root of p ; and moreover the several numbers
g*, where a is any number whatever less than and prime to p — 1, constitute the series
of the vi{p—\) prime roots of p. It may be added that, if /8 be an integer number
less than p—\, and having with it a greatest common measure = k, so that
(«j^) * = g'' " , = 1, (since t is an integer, and ^tf*"' - 1) »
then g^ has the indicatrix ^-^ : the prime roots are those numbers which have the
indicatrix p — I.
The like theory exists as to any number N of the form p'" or 2p"K There are
here «j(JV), — N (l 1 or ^N'l j, in the two cases respectively, numbers less
than N and pi-ime to it ; and we have then w (w (N)\ numbers g such that, to the
modulus iV, all these numbers are =1, g, (^ ... g'"'^^^^ (g^^^ is of course =1). This
being so, g may be regarded as a prime root of N (=p'^ or 2p"', as the case may
be); and moreover the several numbers g', where a is any number whatever less
than and prime to ta(If), constitute the series of the tT(t!T(N)\ prime roots of N.
Thus iV"=3» = 9, xj(iV0 = O; we have
1, 2\ 2\ 2», 2^ 2',
= 1, 2, 4, 8, 7, 5, mod. 9;
or the prime roots of 9 are 2' and 2', = 2 and 5.
So also iV=2.3'^=18, isr(iV) = 6; we have
1 5* 5- 5' 5** o"
= 1, o, 7, 17, 1.3, 11, mod. 18;
and .5' and o', =o and 11 are the prime roots of 18.
18. A small table of prime roots, p=S to 37, is given by
Euler, Op. Arith. Coll. t. i. pp. .525 — 526. The Memoir is entitled "Demonstra-
tiones circa residua e divisione potestatum per numeros primos resultantia," pp. 516 —
.537 (1772).
472
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
[611
19. A table, p and />'", 3 to 97, is given by
OauBS, " Disquisitiones Arithmeticae," 1801, {Werke, t. I. p. 468). This gives in
each case a prime root, and it shows the exponents in regard thereto of the several
prime numbers less than p or p"*. Thus a specimen is
2
3
5
7
11
13 17
19
23
29 &c.
27
2
1
«
5
16
13
8 15
12
11
29
10
11
27
18
20
23
2 7
15
24
viz. for 27 we have 2 a prime root, and 2 = 2', 5 = 2», 7 = 2", 11=2", &c.; and so
also for 29 we have 10 a prime root, and 2 = 10", 3 = 10", 5 = 10", &c.
20. Small tables are probably to be found in many other places; but the most
extensive and convenient table is Jacobi's Caiwn Anthiieticus, the complete title of
which is
Canon Arithmeticus sive tahida quibiis exhibentur pro singulis numeris primis vel
primorum potestatibus infra 1000 nvmeri ad datos indices et indices ad datos numeros
pertineates. Edidit C. G. J. Jacobi. Berolini, 1839. 4°.
The contents are as follows: —
Introductio
Tabulae numerorum ad indices datos pertinentium et indicum
numero dato correspondentium pro modulis primis minoribus
quam 1000
Tabulae residuorum et indicum sibi mutuo respondentium pro
modulis minoribus quam 1000 qui sunt numerorum primorum
potestates . .
Hujus tabula ea pars quae pertinet ad modulos formse 2", invenitur
The following is a specimen of the principal tables: —
^=19, /)-l = 2.3^
Numeri.
Pages
i to xl
1—221
222—238
239—240.
/
0
1
2
3 1 4
5
6
7
8
9
10
5
12
6
3
11
15
17
18
9
14
7
13
16
8
4
2
1
Indices.
N
0
1
2
3
4
6
6
7
8
9
10
18
17
5
16
2
4
12
15
1
1
6
3
13 11
7
14
8
9
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 473
where the first table gives the values of the powers of the prime root 10 (that 10
is the root appears by its index being given as =1) to the modulus 19, viz.
10* = 10, 10- =5, 10^=12, &c. ; and the second table gives the index of the power to
which the same prime root must be raised in order that it may be, to the modulus
19, congruent with a given number: thus 10"= 1, 10" s 2, 10' = 3, &c. The units of
the index or number, as the case may be, are contained in the top line of the table,
and the tens or hundreds and tens in the left-hand column.
21. There is given by
Jacobi, Crelle, t. xxx. (1846), pp. 181, 182, a table of m for the argument m,
such that
1 + g'" = (/"'■ (mod. p), p = 7 to 103, and m = 0 to 102.
A specimen is
to 103
p
7
11
13
17
19
23
29
31
37 .
9
3
2
6
10
10
10
10
17
5
m
11
•
•
6
4
7
*
27
21
34
for instance, p=\9, 1 + 10" = 10' (mod. 19).
Jacobi remarks that this table was calculated for him by his class during the
winter course of 1836 — 37; and that, by means of the Canon Arithmeticus since published
(in 1839), the same might easily be extended to all primes under 1000. In fact, for
any such number p, putting any number of the table " Indices " = m, the next following
number of the table gives the value of m.
22. We have next, in Reuschle's Memoir {ante. No. 15), the following relating
to prime roots : —
C. Tafeln fiir primitive Wurzeln und Hauptexponenten, oder V. erweiterte und
bereicherte Burkhardtsche Tafel, pp. 41 — 61, being divided into three parts; viz.
these are
a. Table of the Hauptexponenten of the six roots 10, 5, 2, 6, 3, 7 for all prime
numbers of the first 1000, together with the least primitive root of each of these
numbers (pp. 42 — 46).
A specimen is as follows: —
10 5 2 6 3 7 w
p p — \ e n e n e n e n e n e n
53 2^.13 13 4 52 1 52 1 26 2 52 1 26 2 2
where c is the Hauptexponent or indicatrix of the root (10, 5, 2, 6, 3, 7, as the case
p — \
may be), n = , w the least primitive root ; thus
p = 53, 10" = 1, 5»-=l, 2*- = l,
C. IX. 60
474 REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. [611
(2 being accordingly the least prime root),
6« = 1, S^hI, T'-sl.
The number w of the last column is the least primitive root. It is, of course, not
always (as in the present case) one of the numbers 10, 5, 2, 6, 3, 7 to which the table
relates: the first exception is p = l91, w = 19: the highest value of w is m> = 21, corre-
sponding to p = 409.
b. The like table for the roots 10 and 2 for all prime numbers from 1000 to
5000, together >vith as convenient as possible a prime root (and in some cases two
prime roots) for each such number (pp. 47 — 53).
A specimen is: —
10 2
p p — 1 en en to
1289 2>.7.23 92 14 161 8 6, 11
viz. here, mod. 1289, 10"* = 1, 2™ = 1 ; and two prime roots are 6, 11. We have thus
by the present tables a prime root for every prime number not exceeding 5000.
c. The like table for the root 10 for all prime numbers between 5000 and
15000, (no column for w, nor any prime root given), pp. 53 — 61.
A specimen is
p p—1 en
9859 2.3.31.53 3286 3:
viz., mod. 9859, we have 10^^=1. But in a large number of cases we have tt = l,
and therefore 10 a prime root. For example,
9887 2.4983 9886 1.
23. For a composite number n, if N = ia (n) be the number of integers less than
n and prime to it, then if x be any number less than n and prime to it, we have
x^ =1 (mod. n). But we have in this case no analogue of a prime root — there i.s
no number x, such that its several powers «', a?,..., x^~^ (mod. ;i.) are all different
from unity ; or, what is the same thing, there is for each value of x some submultiple
of N, say N', such that x^' ~ 1 (mod. n). And these several numbers N' have a least
common multiple /, which is not = JV^, but is a submultiple of ^V; and this being
so, then for all the several values of x, I is said to be the maximum indicator. For
instance, n=12, N=w(n); the numbers less than 12 and prime to it are 1, 5, 7, 11.
We have, (mod. 12), 1' = 1, 5»=1, 7^=1, 1P = 1, or the values of N' are 1, 2, 2, 2;
their least common multiple is 2, and we have accordingly 7 = 2: viz. a^=l (mod. 12)
has the w(12) roots 1, 5, 7, 11. So « = 24, w(») = 8; the maximum indicator / is in
this case also = 2.
611]
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
475
A table of the maximum indicator n=l to 1000 is given by
Cauchy, Exet: d' Analyse &c., t. il. (1841), pp. 36 — 40, contained in the "Memoire
sur la resolution des Equations ind^terminees du premier degr^ en nombres entiers,"
pp. 1—40.
24. It thus appears that for a composite number n, the cr(w) numbers less than n
and prime to it cannot be expressed as = (mod. n) to the power of a single root ;
but for the expression of them it is necessary to employ two or more roots. A small
table, n = 1 to 50, is given by
Cayley, Specimen Table M = a'^b^ (mod. N) for any prime or composite modulus ;.
QunH. Math. Journ. vol. ix. (1868), pp. 95, 96, and folding sheet, [397].
A specimen is
Nos.
roots
Ind.
M.I.
12
5, 7
2, 2
2
4
1
0, 0
2
3
4
5
1, 0
6
7
0, 1
8
9
10
11
1, 1
viz. for the modulus 12 the roots are 5, 7, having respectively the indicators 2, 2,
viz. b' = \ (mod. 12), 7-=l (mod. 12). Hence also the maximum indicator is =2.
^(=«j(»)) = 4 is the number of integers less than 12 and prime to it, viz. these are
1, 5, 7, 11, which in terms of the roots 5, 7 and to mod. 12 are respectively
= 5« . 7», 5' . 7», 5" . 7', and 5' . 1\
25. Quadratic Residues. — In regard to a given prime number p, a number N is
or is not a quadratic residue according as the index of N is even or odd, viz. g
being a prime root and N = ff', then iV is or is not a quadratic residue according as
a is even or odd. But the quadratic residues can, of course, be obtained directly without
the consideration of prime roots.
A small table, p = S to 97 and iV= — 1 and (prime values) 3 to 97, is given by
Gauss, "Di.squi8itiones Arithmeticae," 1801; Table II. (Werke, t. i. p. 469): I notice
here a misprint in the top line of the original; it should be —1, +2, +3, &c., instead of
60—2
476
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
[611
1, + 2, + 3, &c. ; the — 1 is printed correctly on p. 499 of the French translation Reclierches
Arithmdtiques, Paris, 1807 and on p. 469 of vol. I, of Werke, (Gottingen, 1870).
A specimen is
19
- 1
+ 2
+ 3
+ 5
+ 7
+ 11
+ 13 +17
+ 19
+ 23
-
-
_ 1
-
-
-
&c.
viz. — 1, 2, 3, 13 are not, 5, 7, 11, 17 &c. are, quadratic residues of 19. The residues
taken positively and less than 19 are, in fact, 1, 4, 5, 6, 7, 11, 16, 17.
The same table earned from ja = 3 to .503, and prime values ^V = 3 to 997, is given by
GauBB, Werke, t. II. pp. 400 — 409. A specimen is
19
2
3
5
7
11
13 1 17 19
23
-
-
-
-
-
&e. ;
viz. the arrangement is the same, except only that the — 1 column is omitted.
26. We have also by GS-auss
" Disquisitiones Ai-ithmeticae " Table III. ( Werke, t i. p. 470), for the conversion into
decimals of a vulgar fraction, denominator p or p^, not exceeding 100. The explanation
is given in Art. 314 et seq. of the same work.
But this table, carried to a greater extent, is given by Gauss, Werke, t. ll.
pp. 412 — 434, " Tafel zur Verwandlung gemeiner Briiche mit Nennern aus dem ersten
Tausend in Decimalbruche ;" viz. the denominators are here primes or powers of primes,
p^ up to 997.
To explain the table, consider a modulus ^ (where fi may be =1); if 10 is not
a prime root of p^, consider a prime root r, which is such that r* = 10 (mod. p^),
e being a submultiple of p'^~^(p — l); say we have ef=p^~^{p — l): then 10^=1
N
(mod. p^). Consider any fraction — ; then we may write N = 7^+' (mod. p^), k from
N 10*>-'
0 to /— 1 and I from 0 to e — 1, = 10*^, and consequently — and have the
same mantissa (decimal part regarded as an integer) ; hence, in order to know the
N . . rl
mantissa of every fraction whatever of — , it is sufficient to know the mantissa of — ,
J pi^' p^'
that is, the mantissas of — , — ,-,..., — , or, what is the same thing, the mantissas
p^ p^ p^ p^ °
. 10 lOr 10r*-»
of — , — , .... .
pf- p" ' ' p"-
For instance, ^j" = 11, 10' = 1 (mod. 11), whence /= 2, e = .5; and taking r=2, we
have 10 = r» (mod. 11).
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 477
The required mantissae, denoted in the table by
(0), (1), (2), (3), (4),
are those of
10 10.2 10.2° 10.2' 10.2^
11' 11 ' 11 ' 11 ' 11 '
viz. these fractions are respectively =
(0). (1), (2), (3), (4),
•9090..., 1-8181..., 3-6363..., 7-2727..., 14-.54.54 . . . ;
or their mantissse are 90, 81, 63, 27, 54.
And we accordingly have as a specimen
11
(1)...81, (2)... 6.3, (3)... 27, (4)... 54, (0)...90.
Or again, as another specimen, r = 2 :
27
(1)...740, (2)... 481, (3)... 962, (4)... 925, (5)... 851, (0)...370.
The table in this form extends to ^ = 463 ; the values of ?• (not given in the
body of the table) are annexed, p. 420.
In the latter part of the table p^ = 467 to 997, we have only the • mantissse of
100 .
-— . A specimen is
1828153564 8994515539 305.3016453 3820840950
547 6398537477 1480804.387 5685557586 8372943327
2394881170 0182815356,
viz. the fraction J^ = -182815 ... has a period of 91, =^546, figures.
[F. 13. The Pellian Equation.'] Art. III.
27. The Pellian equation is y^=aa? + \, a being a given integer number, which
is not a square (or rather, if it be, the only solution is y=l, « = 0), and w, y being
numbers to be determined : what is required is the least values of x, y, since these,
being known, all other values can be found. A small table a = 2 to 68 is given by
Euler, Op. Arith. Coll. t. I. p. 8. The Memoir is "Solutio problematuni
Diophanteorum per numeros integros," pp. 4 — 10, 1732 — 33. The form of the table is
a
x{=p)
2/(=?)
2
2
3
3
1
2
5
4
9
68 4 33.
478 REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. [611
Even here, for some of the values of a, the values of x, y are extremely large ;
thus o = 61, a; = 226,153.980, y = 1,766,399,049.
And probably tables of a like extent may be found elsewhere ; in particular, a
table of the solution of y- = aa? ± 1 (— when the value of a is such that there is
a solution of y = aa?—\, and + for other values of a), a = 2 to 135, is given by
Legendre, Thdorie des Nombres, 2nd ed. 1808, in the Table X. (one page) at the
end of the work. For the before-mentioned number 61, the equation is y- = Q\a? —\,
and the values are a; = 3805, 3/ = 29718; much smaller than Euler's values for the
equation y-=&l a? + \.
28. The most extensive table, however, is given by
Degen, Canon Pellianus, sive Taimla simplidsmnam equationis celebratissimw :
y' = aa:^+l, sohitionein, pro singulis numei'i dati valorihus ab 1 usque ad 1000 in
numeris rationalibus, iisdemque integris exhibens. Auctore Carolo Ferdinando Degen.
Hafii (Copenhagen) apud Gerhardum Bonnarum, 1817. 8vo. pp. iv to xxiv and 1
to 112.
The first table (pp. 3 — 106) is entitled as "Tabula I. Solutionem Equationis
y — aa^ — 1 = 0 exhibens." It, in fact, also gives the expression of Va as a continued
fraction ; thus a specimen is
209
14
2 5 3 (2)
1
13 5 8 11
3220
46551
Here the first line gives the continued fraction, viz.
V269 = 14 + ^ 11111111
^ 2 + 5 + 3 + 2 + 3 + 5 + 2 + 28 + 2 + &c.,
the period being (2, 5, 3, 2, 3, 5, 2) indicated by 2, 5, 3 (2). [The number of terms
in the period is here odd, but it may be even ; for instance, the period (1, 1, 5, 5, 1, 1)
is indicated by 1, 1 (5, 5).]
The second line contains auxiliary numbei-s presenting themselves in the process;
thus, if jR- = 239, we have i? = 14+ -,
n
_ Jl 1(^-1- 14) ^^+i4^ 1
"~jB-14 209 -14= 13 /3'
13 13(i?-H2)^ie-H2^ 1
^"JB-12~ 209-12' 5 ''"7'
5 _5(i? + 13) _:»+i3^q , 1
'y-jR_18 209-13-^ 8 S'
&c..
611]
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
479
where the second line 1, 13, 5,... shows the numerical factors of the third column.
The value of this second line as a result is not very obvious.
The third line gives x, and the fourth line y,
29. The second table, pp. 109—112, is entitled "Tabula II. Solutionem sequationis
if — aa?+\=0, quotiescunque valor ipsius a talem admiserat, exhibena " ; viz. it is
remarked that this is only possible (but see infra) for those values of a which in
Table I. correspond to a period of an even number of terms, as shown by two
equal numbers in brackets ; thus a = l.S, the period of Vl3 given in Table I. is
(1, 1, 1, 1) as shown by the top line 3, 1 (1, 1), and accordingly 13 is one of
the numbers in Table II. ; and we have there 13
18.
457422.5
Or take another specimen, 241 _ viz. the first line gives the value of
^ 71011068; ^
X, and the second line the value of y (least values), for which y^ — iia? = — 1.
It is to be noticed that a = 2 and a = 5, for which we have obviously the
solutions (ir=l, y = l) and {x = \, y — 2) respectively, are exceptional numbers not
satisfying the test above referred to ; and (apparently for this reason) the values in
question, 2 and .5, are omitted from the table.
•SO. Cayley, "Table des plus petites solutions impaires de I'dquation a;^— 2)y=±4,
i)=5 (mod. 8)." Crelle, t. Lili. (1857), page 371 (one page), [231].
As regards the theory of quadratic forms, it is important to know whether for
A given value of D {= 5, mod. 8) there does or does not exist a solution, in odd
numbers, of the equation a? — Dy^ = 4. As remarked in the paper, " Note sur TAjuation
a?-Df=±^, D=5 (mod. 8)," pp. 369—371, [231], this can be determined for values
of D of the form in question up to I)= 997 by means of Degen's Table ; and the
solutions, when they exist, of the equation x' — By- = 4, as also of the equation
x' — Dy'= — 'i, can be obtained up to the same value of D. Observe that when the
equation a? — Di/' = — 4 is possible, the equation x^ — Dy^ = 4 is also possible, and that
its least solution is obtained very readily from that of the other equation ; it is therefore
sufficient to tabulate the solution of ar* — Dy^ = ± 4, the sign being — when the
coiTesponding equation is possible, and being in other cases +. Hence the form of
the Table: viz. as a specimen we have
D
+
X
y
757
imposs.
765
+
83
3
773
-
139
5
781.
imposs.
480
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
[611
that is, if Z) = 757 or 781, there is no sohition of either a' — Z)y- = + 4 or = — 4; if
D = 765, there is a solution a; = 83, y = 3 of a? — Dy^ = + 4, but none of a;* — Z)y= = — 4 ;
if D = 77S, there is a solution a; = 139, y = 5 of x- — Dy^ = — ^, and therefore also a
Bolution of a;* — Dy- = + 4 ; and so in other cases.
[F. 14. Partitions.] Art. IV.
31. The problem of Partitions is closely connected with that of Derivations.
Thus if it be asked in how many ways can the number n be expressed as a sum
of three parts, the parts being 0, 1, 2, 3, and each part being repeatable an indefinite
number of times, it is clear that n is at most = 9, and that for the values of
m, =0, 1,.., 9 shown by the top line of the annexed table, the number of partitions
has the values shown by the bottom line thereof: —
0123456789
a»
a^b
a\
aH
abd
acd
atP
bdr-
cd^
d?
ab'
abc
6'
b'c
bH
be*
bed
c"
c'd
But taking a, h, c, d to stand for 0, 1, 2, 3 respectively, the actual partitions of
the required form are exhibited by the literal terms of the table (these being obtained,
each column from the preceding one, by the method of derivations, or say by the
rule of the last and last but one), and the numbers of the bottom line are simply
the number of terms in the several columns respectively.
((the h\^
«','«' . for different
= 0, 1, 2,..,mJ
values of n and in (where the number of letters is =m4-l), would be extremely
interesting and valuable. The tables for a given value of m and for different values
of n are, it is clear, the proper foundation of the theory of the binary quantic
(a, b, c, . . , k'^x, 1 )'", which corresponds to such value of m. Prof. Cayley regi'ets that
he has not in his covariant tjibles given in every case the complete series of literal
terms; viz. the literal terms which have zero coefficients are, for the most part,
though not always, omitted in the expressions of the several covariants.
33. But the question at present is as to the numhe)- of terms in a column,
that is, as to the number of the partitions of a given form : the analytical theory
has been investigated by Euler and others. The expression for the number of
partitions is usually obtained as equal to the coefficient of a;" in the development, in
ascending powers of x, of a given rational function of x: for instance, if there is no
limitation as to the number of the parts, but if the parts are 1, 2, 3, in (viz. a
part may have any value not exceeding in), each part being repeatable an indefinite
number of times, then
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
1
481
Number of partitions of m = coefficient of a" in
(l -x)(l - x'){l - ai') ... (1 -of")'
and we can, by actual development, obtain for any given values of m, n the number
of partitions.
These have been tabulated m=l, 2, ...,20, and 111=00 (viz. there is in this case
no limit as to the largest part), and w = 1 to 59, by
Euler, Op. Arith. Coll. t. I. pp. 97 — 101, given in the paper "De Partitione
Numerorum," pp. 73 — 101, (1750); the heading is "Tabula indicans quot variis modis
numerus n e numeris 1, 2, 3, 4, . . , m, per additionem exhibi potest, seu exhibens
valores formulae n*""." The successive lines are, in fact, the coefficients of the several
powers «", a^, . . , a^ in the expansions of the functions
1 1 1
\-x' \-x.\-a?"" \-x.l-!ii?...\-af^'
34. The generating function for any given value of m is, it is clear, =- —
multiplied by that for the next preceding value of m, and it thus appears how each line
of the table is calculated from that which precedes it. The auxiliary numbers are
printed; thus a specimen is
Valores nunieri n.
m 0 ! 1
1
2
3
4
5 6
7
8
9
10
4
1
1
2
3
1
5
1
6
2
9
3
11
5
15
6
18
9
23
5
1
1 1
2
3
■
5
1
7
1
10
2 3
13 18
5
23
7
30
viz. suppose the numbers in the second 4-line known : then simply moving these each
five steps onward we have the (auxiliary) numbers of the first 5-line ; and thence
by a mere addition the required series of numbers shown by the second 5-line. And
similarly from this is obtained the second 6-line, and so on.
35. More extensive tables are contained in the memoir by
Manano, Hulle leggi delle derivate generali delle fumioni di funzioni et sulla
teoria delle forme di partiziove dei mimen ivtieri, (4°. Genova, 1870), pp. 1 — 281 ;
and three tables paged separately, described merely as "Tavole dei uumeri Cg^r, Sq^e, 'S",,,
citate nel testo colle indicazioni di Tavole I., II., III., ai n' 77, 79, 81 " ; viz. the
reafler is referred to these articles for the explanations of what the tabulated functions
are ; and there is not even then any explicit statement, but the investigation itself
has to be studied to make out what the tables are. It is, in fact, easier to make
this out from the tables themselves ; the explanation is as follows : —
C. IX. 61
482
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
[611
Table I. (16 pages) is, in fact, Euler's table, showing in how many ways the
number n can be made up with the parts 1, 2, 3, .., m; but the extent is greater,
viz. n is from 1 to 103, and m from 1 to 102. The auxiliary numbers given in
Euler's table are omitted, as also certain numbers which occur in each successive
line; thus a specimen is
n =
10 &c.
c,..
1
0
0
0
0
0
0
0
0
0
c,,.
1
1
1
1
1
1
1
1
1
G,..
2
2
3
3
4
4
5
5
<?3..
3
4
5
6
7
8
10
12
c,,.
_
5
9
11
15
18
where the line C;,,,, (ways of making up w — 1 with the parts 1, 2, 3, 4) is 1, 1, 2, 3,
5, 6, 9, 11, 15, 18, &c., viz. we read from the corner diagonally downwards as far
as the 6, and then horizontally along the line : this saves a large number of figures.
The table is printed in ordinary quarto pages, which are taken to come in in tiers
of seven, five, and three pages one under the other, as shown by a prefixed diagram ;
and the necessity of a large folding plate is thus avoided.
The successive lines give, in fact, the coefficients in the expansions of
1 1 1 1
1-a;' l-x.l-a?' 1 - x .l-x- .1 -a?'" ' 1 -x.l -x" ...1 - sc""''
each expanded as far as x^'^.
Table II. (6 pages). The successive lines give the coefficients in the expansions of
8 S 8
8.
where
l-a;' l-x.l-x"''
1
8 =
1-x.l-ar' ...l-it^'
... ad inf.,
{l-x){l-af){l-x')
each expanded as fai' as x^, and further continued as regards the first ten lines,
that is, the expansions of
„ ^ 8 8
' l-;r' l-x.l-x''" l-x.l-a^...l-af'
each as far as x^'".
Table III. (2 pages). The successive lines give the coefficients in the expansions of
8*,
8'
' 1-x' l-x.l-a?'"' l-a;.l-a^...l
each expanded as far as *•".
611]
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
483
36. As regards Tables II. and III., the analytical explanations have been given
in the first instance ; but it is easy to see that the tables give numbers of partitions.
Thus, in Table II., the second line gives the coefficients in the development of
1 .
(l-a;>'(l-a;^)(l-a^)...'
viz. these are 1, 2, 4, 7 12, 19, 30 being the number of ways in which the
numbers 0, 1, 2, 3, 4, &c. respectively can be made up with the parts 1, 1', 2, 3,
4, &c. ; thus
Partitions. No. =
11 •'>
v
2
1,
1
1.
1'
1'.
1'
3
2,
1
2,
r
1,
1,
1
1,
1,
r
1,
V,
r
1',
1',
r
&c. dec.
Similarly, the third line shows the number of ways in which these numbers respectively
can be made up with the parts 1, 1', 2, 2', 3, 4, 5, &c. ; the fourth line with the
parts 1, 1', 2, 2', 3, 3', 4, 5, &c.; and so on.
And in like manner in Table III., the first line shows the number of ways when
the parts are 1, 1', 2, 2', 3, 3',...; the second line when they are 1, 1', 1", 2, 2',
3. 3', &c. ; the third when they are 1, 1', 1", 2, 2', 2", 3, 3', &c. ; and so on.
It is clear that the series of tables might be continued indefinitely, viz. there
might be a Table IV. giving the developments of
s\
• and so on.
\-x' l-a-.l-ar"
An interesting table would be one composed of the first lines of the above
series, viz. a table giving in its successive lines the developments of 8, 8', S", S*, &c.
There are throughout the work a large number of numerical results given in a
quasi-tabular form ; but the collection of these, with independent explanations of the
significations of the tabulated numbers, would be a task of considerable labour.
61—2
484 REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. [611
[F. 15. Quadratic farms a^ + b', tC'C, and Partitions of Numbers into squares, cubes,
and biquadrates.] Art. V.
37. The forms here referred to present themselves in the various complex
theories. Thus N=a^ + b^, =(a + bi){a — bi); this means that, in the theory of the
complex numbers a + bi (a and b integers), N is not a prime but a composite
number. It is well known that an ordinary prime number = 3, mod. 4, is not
expressible as a sum a' + b^, being, in fact, a prime in the complex theory as well
as in the ordinary one: but that an ordinary prime number =1, mod. 4, is (in one
way only) = a' + 6* ; so that it is in the complex theory a composite number. A
number whose prime factors are each of them = 1, mod. 4, or which contains, if at
all, an even number of times any prime factor = 3, mod. 4, can be expressed in a
variety of ways in the form a^ + ¥ ; but these are all easily deducible from the
expressions in the form in qiiestion of its several factors = 1, mod. 4, so that the
required table is a table of the form p = a^ + b', p an ordinary prime number = 1 ,
mod. 4: a and 6 are one of them odd, the other even; and to reader the decom-
position definite a is taken to be odd.
p — a^+b-; viz. decomposition of the primes of the form 4n + 1 into the sum
of two squares: a table extending from p = b to 11981 (calculated by Zornow) is given by
Jacob!, Crelle, t. xxx. (1846), pp. 174—176.
This is carried by Reuschle, as presently mentioned, up to jo = 24917. Reuschle
notices that 2713 = 3''+ 52'' is omitted, also 6997 = Sg'' + 74^ and that 8609 should be
= 47' + 80».
38. Similarly, primes of the form 6n + 1 are expressible in the form p = a' + 36^.
Observe that, w being an imaginary cube i-oot of unity, this is connected witii
p' —{a + bm) (a + bio^), =a'' — ab + b\ viz. we have 4p' = (2a — hf + 36- ; or the form
a^ + Sft" is connected with the theory of the complex numbers composed of the cube
roots of unity.
p = a^ + 36'^ ; viz. decomposition of the primes of the form 6w + 1 into the form
a" + 36": a table extending from p=7 to 12007 (calculated also by Zornow) is given by
Jacobl, Grelle, t. xxx. (1846), itt supra, pp. 177 — 179.
This is carried by Reuschle up to p= 13369, and for certain higher numbers up to
49999, aa presently mentioned. Reuschle observes that 6427 = 80^ + 3.3* is by accident
omitted, and that 6481 should be =41= + 3. 40'.
39. Again, primes of the form Sre + 1 are expressible in the form jo = a" + 26»
(or say = c" + 2d'), the theory being connected with that of the complex numbers
composed with the 8th roots of unity (fourth root of — 1, = /=-)•
p — c* + 2d* ; viz. decomposition of primes of the form 8n + 1 into the form c* + 2rf' :
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 485
a table extending from p=l6 to 5943 (extracted from a MS. table calculated by
Struve) is given by
Jacobi, Crelle, t. xxx. (1846), ut supra, p. 180.
This is carried by Reuschle up to p= 12377, and for certain higher numbers up
to 24889, as presently mentioned.
40. Reuschle's tables of the forms in question are contained in the work: —
Reuschle, Mathematische Ahhandlung, &c. (see ante No. 15), under the heading
" B. Tafeln zur Zerlegung der Primzahlen in Quadrate" (pp. 22 — 41). They are as
follows : —
Table III. for the primes 6w + 1,
The first part gives p = ^' + 3fi= and 4p = i= + 27i^/^ from p = 7 to 5743. The table
gives A, B, L, M; those numbers which have 10 for a cubic residue are distinguished
by an asterisk. A specimen is
p A B L M
37^^ 5 2 11 T'
viz. 37 = .52 + 3.2', 148=1P+27.P; the asterisk shows that o-'^s + lO (mod. 37) is
possible : in fact 34" = 10 (mod. 37).
The second part gives p = A- + S^ only, from p = 5749 to 13669. The table gives
A, B; and the asterisk implies the same property as before.
The third part gives p = A-+SB', but only for those values of p which have 10
for a cubic residue, viz. for which a;' = 10 (mod. p) is possible, from p = 13689 to
49999. The table gives A, B; the asterisk, as being unnecessary, is not inserted.
Table IV. for the primes 4w + 1 in the form A' + B', and for those which are
also 8n + 1 in the form C- + 2I>.
The first part gives p = ^» + 5^, =C'' + 2Ifi, from p = 5 to 12377. The table gives
A, B, C, D; those numbers which have 10 for a biquadratic residue, viz. for which
ic* = 10 (mod. p) is possible, are distinguished by an asterisk ; those which have also 10
for an octic residue, viz. for which a^ = 10 (mod. p) is possible, by a double asterisk.
A specimen is
p A B C D
229
15
2
—
—
233
13
8
15
2
241**
15
4
13
6
The second part gives /) = .4' + fi^, from p= 12401 to 24917 for all those values of
p which have 10 for a biquadratic residue (ar* = 10 (mod. p) possible). The table gives
A, B ; those values of p which have 10 for an octic residue, viz. for which a^ = 10
(mod. p) is possible, are distinguished by an asterisk.
486 REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. [611
The third part gives p = C' + 2D', from jj = 12641 to 24889 for all those values of
p which have 10 for a biquadratic residue. The table gives C, D; those values of p
which have 10 as an octic residue are distinguished by an asterisk.
41. A table by Zornow, Crelle, t. xiv. (1835), pp. 279, 280 (belonging to the
Memoir " De Compositione numerorum e Cubis integris positivis," pp. 276 — 280), shows
for the numbers 1 to 3000 the least number of cubes into which each of these numbers
can be decomposed. Waring gave, without demonstration, the theorem that every
number can be expressed as the sum of at most 9 cubes. The present table
seems to show that 23 is the only number for which the number of cubes is
= 9(= 2.2' + 7.1'); that there are only fourteen numbers for which the number of
cubes is = 8, the largest of these being 454 ; and hence that every number greater
than 454 can be expressed as a sum of at most 7 cubes; and further, that every
number greater than 2183 can be expressed as a sum of at most 6 cubes. A small
subsidiary table (p. 276) shows that the number of numbere requiring 6 cubes gradually
diminishes — e.g. between 12" and 13' there are seventy-five such numbers, but between
13' and 14" only sixty-four such numbers ; and the author conjectures " that for
numbers beyond a certain limit every number can be expressed as a sum of at most
5 cubes."
42. For the decomposition of a number into biquadrates we have
Bretschneider, "Tafeln fiir die Zerlegung der Zahlen bis 4100 in Biquadrate,"
Crelle, t. xlvi. (1853), pp. 3—23.
Table I. gives the decompositions, thus: —
N
1*. 2S 3*,
4^ 5«,
696
6 1 2
2
3 2 5
1
0 3 8
viz. 696 = 6 . 1* 4- 1 . 2* + 2 . 3« + 2 . 4<, &c.
And Table II. enumerates the numbers which are sums of at least 2, 3, 4, . . ,
19 biquadrates. There is at the end a summary showing for the first 4100 numbei-s
how many numbers there are of these several forms respectively: 28 numbers are each
of them a sum of 2 biquadrates, 75 a sum of 3,..., 7 a sum of 19 biquadrates.
The seven numbers, each of them a sum of 19 biquadrates, are 79, 159, 239, 319, 399,
479, 559.
[F. 16. Binary, Tm-nary, etc. quadratic and higher /orwis.] Art. VI.
43. Euler worked with the quadratic forms oaf ± cy" (p and q integers), particularly
in regard to the forms of the divisors of such numbers. It will be sufficient to refer
to his memoir: —
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 487
Euler, " Theoremata circa divisores numerorum in hac forma pa? ± qh^ contentorum,"
{Op. Aiith. Coll. pp. 35 — 61, 1744), containing fifty-nine theorems, exhibiting in a
quasi-tabular form the linear foi-ms of the divisors of such numbers. As a specimen : —
"Theorema 13. Numerorum in hac forma a^ + 766^ contentorum divisores primi
omnes sunt vel 2, vel 7, vel in una sex formularum
28WI-I-1, 28ni-l-ll,
28ot + 9, 28m + 15,
28m -I- 25, 28m -t- 23,
sen in una harum trium
14m + 1,
14hi + 9,
14»n,-(- 11,
sunt contenti"; viz. the forms are the three I4m + 1, 14m + 9, 14m -I- 11.
But Euler did not consider, or if at all very slightly, the trinomial forms
ax' + bicy + cy^, nor attempt the theory of the reduction of such forms. This was first
done by Lagrange in the memoir
Lagrange, Mem. de Berlin, 1773. And the theory is reproduced by
Legendre, Theorie des Novibres, Paris, 1st ed. 1798; 2nd ed. 1808, § 8, "Reduction
de la formule Ly' + Myz + Nz' a I'expression la plus simple," (2nd ed. pp. 61 — 67).
44. But the classification of quadratic forms, as established by Legendre, is
defective as not taking account of the distinction between proper and improper
equivalence ; and the ulterior theory as to orders and genera, and the composition
of forms (although in the meantime established by Gauss), are not therein taken
into account; for this reason the Legendre'a Tables I. to VIIL relating to quadratic
forms, given after p. 480 (thii-ty-two pages not numbered), are of comparatively little
value, and it is not necessary to refer to them in detail.
The complete theory was established by
Gauss, Disquisitioiies Arithmetiae, 1801.
It is convenient to refer also to the following memoir :
Lejeune Dirichlet, " Recherches sur diverses applications de I'Analyse a la theorie
des Nombres," Crelle, t. XIX. (1839), p. 338, [Ges. Werke, t. I. p. 427], as giving a
succinct statement of the principle of classification, and in particular a table of the
characters of the genera of the properly primitive order, according to the four forms
D=PS\ P=l or 3 (mod. 4), and D=2PS', P=l or 3 (mod. 4), of the determinant.
45. Tables of quadratic forms arranged on the Gaussian principle are given by
Cayley, Crelle, t. lx. (1862), pp. 357—372, [335]; viz, the tables are-
Table I. des formes quadratiques binaires ayant pour d^tei-minants les nombres
n^gatifs depuis D = -\ jusqu'a Z) = - 100. (Pp. 360—363 : {Coll. Math. Papers, t. v,
pp. 144—147].)
488
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
[611
A specimen is
D
Glasses
a /S
i e 1 <<
Cp
= 26
1, 0, 26
+
+
1
3,-1, 9
+ *
+
9"
3, 1, 9
+
+
9*
5, 2, 6
9
2, 0, 13
-
-
g'
5,-2, 6
-
-
^
where a, /8 denote, as there explained, the characters in regard to the odd prime
factors oi D; B, e, Se those in regard to the numbers 4 and 8. The last column
shows that the forms in the two genera respectively are 1, g", g* and g, g", g',
where 5r* = l, viz. the form g, six times compounded, gives the principal form (1, 0, 26).
Table II. des formes quadratiques binaires ayant pour determinants les nombres
positifs non-carr^s depuis i) = 2 jusqu'a Z) = 99. (Pp. 364f— 369: [I.e., pp. 148—153].)
The arrangement is the same, except that there is a column " P^riodes " showing,
in an easily understood abbreviated form, the period of each form. Thus D = 7,
the period of the principal form (1, 0, —7), is given as 1, 7, —3, i, 2, i, —3, 7, 1,
which represents the series of forms (1, 2, -3), (-3, 1, 2) (2, 1, -3), (-3, 2, 1).
Table III. des formes quadratiques binaires pour les treize determinants n^gatifs
irreguliers du premier millier. (Pp. 370 — 372 : [I.e., pp. 154 — 156].)
The arrangement is the same as in Table I. It may be mentioned that the thirteen
numbers, and the forms for the principal genus for these numbers, respectively are : —
576, 580, 820, 900
884
243, 307, 339, 459, 675, 891
755, 974
Principal genus
(1, e»)(l, e.O
(1, e=) (1, i\ i\ i")
(1, d, d')(l, d„ d,')
(1, d, d')(l, d„di>)(l,e').
where cP' = d,' = l, e* = ei*=l, r* = l, viz. (1, e'-)(l, e/) denotes four forms, 1, e^, e^^ e^gj^;
and 80 in the other cases.
46. Gauss must have computed quadratic forms to an enormous extent ; but,
for the reasons (rather amusing ones) mentioned in a letter of May 17, 1841, to
Schumacher (quoted in Prof. Smith's Report on "The Theoiy of Numbei-s," Brit.
Assoc. Report for 1862, p. 526, [and Smith's Coll. Math. Papers, t. I. p. 261]), he did
not preserve his results in detail, but only in the form appearing in the
"Tafel der Anzahl der Classen binarer quadratischer Formen," Werke, t. II. pp.
449 — 476 ; see editor's remarks, pp. 521 — 523.
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 489
This relates almost entirely to negative determinants, only three quarters of
p. 475 and p. 476 to positive ones; for negative determinants, it gives the number
of genera and classes, as also the index of irregularity for the determinants of the
hundreds 1 to 30, 43, 51, 61, 62, 63, 91 to 100, 117 to 120; then, in a different
arrangement, for the thousands 1, 3 and 10, for the first 800 numbers of the forms
— (15n + 7) and — (15n+13); also for some very large numbers, and for positive
determinants of the hundreds 1, 2, 3, 9, 10, and for some others.
A specimen is
Centas I.
G II. (.58) ... (280)
1. 5, 6, 8.
9, 10, 12,
13, 15, 16,
18, 22, 25,
28, 37, 58,
2. 14, 17, 20,
HSumma 233 477
Irreg. 0 Impr. 74 ;
viz. this shows, as regards the negative determinants 1 to 100, that the determinants
belonging to G II. 1, viz. those which have two genera each of one class, are 5, 6,
8, 9, &c., in all fifteen determinants; those belonging to G II. 2, viz. those which
have two genera each of two classes, are 14, 17, 20, &c.; and so on. The head
numbers (58) . . . (280) show the number of determinants, each having two genera, and
the number of classes; thus,
G II. 1 X 15 = 15
2 X 17 = 34
3 X 17 = 51
4 X 6 = 24
5 X 2 = 10
6x1= 6
58 140
X 2
= 280;
and the bottom numbers show the total number of genera and of classes, thus
G I. 17 X 1 = 17 61
II. 58 X 2 = 116 280
IV. 25 X 4 = 100 136
100 233 477 ;
c. IX. 62
490
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. [611
viz. seventeen determinants, each of one genus, and together of sixty-one classes;
fifty-eight determinants, each of two genera, and together of 280 classes: and twenty-
five determinants, each of four genera, and together 136 classes, give in all 233
genera and 477 classes. These are exclusive of 74 classes belonging to the improperly
primitive order ; and the number of irregular determinants (in the first hundred) is = 0.
The irregular determinants are indicated thus:
243(*3*),
307(*3*), 339(»3*),
459C),
576(*2*), 580(*2*),
675(*3*),
755(*3*),
891(*3*), 820(*2*), 900(*2*), 884(*2*), 974<*3*),
*3* 243, 307, .339, 459?, 675, 755, 891,
*2» 576, 589, 820, 884, 900, 974,
•which is a notation not easily understood.
As regards the positive determinants, a specimen is
Centas I.
Excedunt determinantis
quadrati 10.
G I.
...(12),
1. 2,
5, 13,
17,
29, 41,
53.
61, 73,
89,
97,
3. 37 ;
viz. in the first hundred, the positive determinants having one genus of one class
are 2, 5, 13, &c. ... (eleven in number); that having one genus of three classes
is 37, (one in number); 11 + 1 = 12. The irregular determinants, if any, are not
distinguished.
47. Binary cubic forms. — The earliest table is given by
Arndt, " Tabelle der reducirten binaren kubischen Formen und Klassen fiir alle
negativen Determinanten -D von Z)=3 bis Z) = 2000," Ch-utiert's A7-chiv, t. XXXL
1858, pp. 369—448.
The memoir is a sequel to one in t. xvil. (1851). The binary cubic form
(o, b, c, d), of determinant - Z) (= (6c - a<i)= - 4 (6' - ac) ((^ - 6d)), is said to be reduced
when its characteristic <fi, =(A, B, C), =(2(6»-ac), be -ad, 2(cf'-bd)), is a reduced
quadratic form, that is, when in regard to absolute values B is not > ^A, G not < A.
611] REPORT OF THE COMMITTEE OX MATHEMATICAL TABLES. 491
A specimen is
X> Itedaced forms, with cliaraoters Classes
44
(0, 1,
(2,
0,
0,
-11)
22
(1.
-1, -
6,
2
0)
8)
(0,
-1,
0,
")
(0,
-2,
±1.
1)
Two subsidiary tables are given, pp. 351, 352, and 353 — 368.
48. It appeared suitable to remodel a part of this table in the manner made
use of for quadratic forms in my tables above referred to ; and it is accordingly
divided into the three tables given by
Cayley, Quart. Math. Jonrii. t. xi. (1871), where the notation &c. is explained,
pp. 251—261. [496]; viz. these are:—
Table I. of the binary cubic forms, the determinants of which are the negative
numbers = 0 (mod. 4) from - 4 to - 400 (pp. 251 — 258 ; [Coll. Math. Papers, t. viii.,
pp. 55—61]).
A specimen is
Det. 4 X
Classes.
Order.
Charact.
Comp.
11.
0, - 1, 0,
"I
1
on
1, 0, 11
1
0,.r2, -1,
PP PP
3, 1, 4
d
0, - 2, 1,
3,-1, 4
d\
Table II. of the binary cubic forms the determinants of which (taken positively)
are = 1 (mod. 4) from — 3 to — 99, the original heading is here corrected, [tc, pp.
61, 62] ; and
Table III. of the binary cubic forms the determinants of which are the negative
numbers - 972, - 1228, - 1336, - 1836, and - 2700, [Lc, pp. 63, 64] ; viz. - 972 = 4
x-243, .., — 2700 = 4 X — G75, where —243, .., — 675 are the first six irregular numbers
for quadric forms.
4 X — 675, = — 2700 is beyond the limits of Arndt's tables, and for this number
the calculation had to be made anew; the table gives nine classes (1, d, d^) (1, d,, di'-)
of the order ip on pp, but it is remarked that there rnay possibly be other cubic-
classes based on a non-primitive characteristic ; the point was left unascertained.
49. The theory of ternary quadratic forms was discussed and partially established
by Gauss in the Disquisitimies Arithmetiae. It is proper to recall that a ternary
quadratic form is either determinate, viz. always positive, such as x^-^y^ + ^, or always
negative, such as — a? — y- — z'^ ; or else it Ls indeterminate, such as af + y^ — z^
But as regards determinate forms, the negative ones are derived from the positive
ones by simply reversing the signs of all the coefficients, so that it is sufficient to
attend to the positive forms; and practically the two cases are positive forms (meaning
thereby positive determinate forms) and indeterminate forms; but the theory for
positive forms was first established completely, and so as to enable the formation of
tables, in the work
Seeber, Uebei- die Eigenachaften der positiven terndren quadratischen Formmi,
(4to. Freiburg, 1831),
62—2
492 REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. [611
which is reviewed by Gauss in the CoM. Oelehrte Anzeigen, 1831, July 9 (see Gauss,
Werke, t ii. pp. 188—193). The author gives (pp. 220—243) tables "of the classes
of positive ternary forms represented by means of the corresponding reduced forms"
for the determinants 1 to 100. A specimen is
-'•^ a;J:J).(-;;-;;S.
/8, 8, 3\ / 7, 7, 4\
lo, 0, %)' \ 4, 4, 27 •
Zugeordnete (S, 8, 3^ ( 7, 7, 4^
Formen
where it is to be observed that Seeber admits odd coefficients for the terms in
yz, zx, xy; viz. his symbol ( ' ' j denotes
(jud' + hy^ + cz- + fyz + gzx + lucy,
and his determinant is
4a6c - ap -bf- ch? +fgh.
Also his adjoint form is
/ ^c-f^, 4ca-o», 4a6-A» \ ,,, .,, „ „ . , ^x
W-df, 2hf-ihg, 2fg-,ch)' =(^-r)^ +-n2gh-iaf)yz ^ ...
In the notation of the DisquisHimies Arithineticce, followed by Eisenstein and
others, the symbol ( ' ' j denotes
««;» + 6y* + cz^ + %fyz + 2gzx + ihxy ;
the determinant is
= - {ahc - a/" - hg^ - ch^ + 2fgh),
a positive form having thus always a negative determinant. And the adjoint form is
-{gh-i}. hf-bg.fg-on)' = -i^o-^)'^- ...-2igh-af)yz- ...
Hence Seeber's determinant is = — 4 multiplied by that of Gauss, and his tables really
extend between the values — 1 and — 2.5 of the Gaussian determinant.
50. Tables of greater extent, and in the better form just referred to, are given by
Eisenatein, Crelle, t. xll (1851), pp. 169 — 190; viz. these are
I. "Tabelle der eigentlich primitiven positiven temaren Formen fiir alle negativea
Determinanten von -1 bis -100," (pp. 169—185).
A specimen is
z>
Anzahl
Bfldncirte Formen fflr -D
10
3
n, 1, 10\ /I, 2, 5\ /2, 2, 3\
Vo, 0, o; ' Vo, 0, o; ' Vo, -\, o)'
8=8 8 = 4 8^.4
611]
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES.
493
II. " Tabelle der uneigentlich primitiven positiven ternaren Formen fur alle
negativen Determinanten von — 2 bis — 100," (pp. 186 — 189).
A specimen is
D
Anzahl
Bedacirte Formen filr - D
10
1
1% % 4\
VI, 1, ir
8 = 6.
And there is given (p. 190) a table of the reduced forms for the determinant
— 385 (= — 5 .7 . 11), selected merely as a largish number with three factors; viz.
there are in all fifty- nine forms, corresponding to values 1, 2, 4, 6, 8 of h.
It may be remarked that S denotes, for any given form, the number of ways in
which this is linearly transformable into itself, this number being always 1, 2, 4, 6,
8, 12, or 24. The theory as to this and other points is explained in the memoir
(pp. 141 — 168), and various subsidiary tables are contained therein and in the Ankang
(pp. 227 — 242); and there is given a small table relating to indeterminate forms, viz.
this is ^
"C. Versuch einer Tabelle der nicht aquivalenten unbestimmten (indiflferenten)
ternaren quadratischen Formen fiir die Determinanten ohne quadratischen Theiler
unter 20," (pp. 239, 240).
A specimen is
10
Indifferente ternare quaJratiscbe Formen
/O, 1, 10\ /I, 2, - 5\
[O, 0, \) ' [o, 0, 0;* '
/O, 0, 10\
Vo, 0, i) '
where, when the determinant is even, the forms in the second line are always improperly
primitive forms.
[F. 17. Complex Tlteories.] Art. VII.
.51. The theory of binary quadratic forms (a, b, c), with complex coefBcients of
the form a + /9t, (i=V— 1 as usual, a and /3 integers), has been studied by Lejeune
Dirichlet, Prof H. J. S. Smith, and possibly others; but no tables have, it is believed,
been calculated. The calculations would be laborious; but tables of a small extent
only would be a sufficient illustration of the theory, and would, it is thought, be of
great interest.
494 REPORT OF THK COMMITTEE ON MATHEMATICAL TABLES. [611
The theory of complex numbers of the last-mentioned form a + /Si, or say of the
numbers formed with the fourth root of unity, had previously been studied by Gauss ;
and the theory of the numbers formed with the cube roots of unity (a + /3«i), ay' + a + I = 0,
a and /8 integers) was studied by Eisenstein ; but the general theory of the numbers
involving the nth roots of unity (k an odd prime) was first studied by Kummer. It
will be sufficient to refer to his memoir,
Kummer, " Zur Theorie der complexen Zahleii," Bei-l. Monatsb., March, 1845 ; and
Vrelle, t xxxv. (1847), pp. 319 — 326; also "Ueber die Zerlegung der aus Wurzeln der
Einheit gebildeten complexen Zahlen in ihre Primfactoren," same volume, pp. 327 — 367,
where the astonishing theory of "Ideal Complex Numbers" is established.
52. It may be recalled that, p being an odd prime, and p denoting a root of
the equation pP~^ + p^""^ + . . . + p + 1 = 0, then the uunibera in question are those of
the form a + bp+ ... + kp^*~'\ where (a, b,..,k) are integers; or (what is in one point
of view more, and in another less, general) if rj, r]iy>Ve-i are "periods" composed with
the powers of p (e any factor of p—l), then the form considered is aj? + 6171 + ... ■¥hr)f_^.
For any value of y or e there is a corresponding complex theory. A number (real or
complex) is in the complex theory prime or composite, according as it does not, or does,
break up into factors of the form under consideration. For p a prime number under 23,
if in the complex theory iV is a prime, then any power of N (to fix the ideas say iV')
has no other factore than N or N- ; but if jj = 23 (and similarly for higher values of p),
then N may be such that, for instance, N^ has complex factore other than N or N'^ (for
jp = 23, N = 4,7 is the first value of iV, viz. 47' has factors other than 47 and 47-) ;
say iV has a complex prime factor A, or we have vA as an ideal complex factor
of N. Observe that by hypothesis A^ is not a perfect cube, viz. there is no complex
number whose cube is = J.. In the foregoing general statement, made by way of
illustration only, all reference to the complex factors of unity is purposely omitted, and
the statement must be understood as being subject to coirection on this account.
What precedes is by way of introduction to the account of Reuschle's Tables
{£erline7' Monatsberichte, 1859 — 60), which give in the different complex theories p = 5,
7, 11, 13, 17, 19, 23, 29 the complex factors of the decomposable real primes up to in
some cases lOOU.
It should be remaiked that the form of a prime factor is to a certain extent
indeterminate, as the factor can without injury be modified by affecting it with a
complex factor of unity ; but in the tables the choice of the representative form is
made according to definite rules, which are fully explained, and which need not be
here referred to.
611]
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLKS.
495
53. The following synopsis is convenient :-
The foregoing synopsis of Reuschle's tables in the Berliner MonaUherichte was
written previous to the publication of Reuschle's far more extensive work. It is
496 REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. [611
allowed to remain, but some explanations which were given have been struck out, and
were instead given in reference to the larger work, which is
Reuschle, Tafeln complexer Primzahlen, welche aus Wurzeln der Einheit gebildet
mid. Berlin, 4° (1875), pp. iii — vi and 1—671.
This work (the mass of calculation is perfectly wonderful) relates to the roots of
unity, the degree being any prime or composite number, as presently mentioned, having
all the values up to and a few exceeding 100 ; viz. the work is in five divisions,
relating to the cases:
I. (pp. 1 — 171), degree any odd prime of the first 100, viz. 3, 5, 7, 11, 13, 17,
19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97;
II. (pp. 173—192), degree the power of an odd prime 9, 25, 27, 49, 81 ;
III. (pp. 193 — 440), degree a product of two or more odd primes or their powers,
viz. 15, 21, 33, 35, 39, 45, 51, 55, 57, 63, 65. 69, 75, 77, 85, 87, 91, 93, 95, 99, 105;
IV. (pp. 441 — 466), degree an even power of 2, viz. 4, 8, 16, 32, 64, 128;
V. (pp. 467—671), degree divisible by 4, viz. 12, 20, 24, 28, 36, 40, 44, 48, 52,
56, 60, 68, 72, 76, 80. 84, 88, 92, 96, 100, 120 ;
the only excluded degrees being those which are the double of an odd prime, these,
in fact, coming under the case where the degree is the odd prime itself.
It would be somewhat long to explain the specialities which belong to degrees
of the forms II., III., IV., V. ; and what follows refers only to Division I., degree an
odd prime.
For instance, if X = 7, X— 1=2.3; the factors of 6 being 6, 3, 2, 1, there are
accordingly four divisions, viz.
II. ij„ = o + a-', 17, = a« + or-, % = a' + or^, or ?? a root of
ij» + »?= - 27? - 1 = 0,
I. a a prime seventh root, that is, a root of 0° + o" + a' + a^ + a + 1 = 0 ;
Vi = 2 + »?i, %' = 2 + 17,, &c.
■noVi=Vi-^Viy &C.;
III. 170 = o + a* + 0*, 17, = a» + a" + a", or 1; a root of 1;= + t; + 2 = 0 ;
IV. Real numbers.
I. ^ = 7m + 1. First, it gives for the several prime numbers of this form 29,
43, . . , 967 the congruence roots, mod. p ; for instance.
p
a
a»
a»
0*
a»
«•
29
- 5
-4
-9
-13
+ 7
-6
43
+ 11
-8
- 2
+ 21
+ 16
+ 4.
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 497
This means that, if a = -b (mod. 29), then o? = 25, s. - 4, a' =20, s - 9, &c., values
which satisfy the congruence a«+ o" + a* + a'+ 0^ + 0+ 1 = 0 (mod. 29).
Secondfy, it gives, under the simple and the primary forms, the prime factors /"(a)
of these same numbers 29, 43, . . , 967 ; for instance,
p f{cL) simple. /{"■) primary.
29 a + o''' - a» 2 f 3a - a^ + .5a' - la* + 4a=
43 a» + 2a* 2o - 2a- + 4a^ - a'* - 5a«.
The definition of a primary form is a form for which f{a.)f{a-^) = f{l)- mod. \,
and y(o) =f{X) mod. (1 — a)-. The simple forms are also chosen so as to satisfy this
last condition ; thus /(a) = a + a- - a', then /(I) -/(a) = 1 - a - a= + a» = (1 - a)" (1 + a), =0
mod. (1 — a)".
II. p=7m— 1. First, it gives for the several prime numbers of this form 13,
41,.., 937 the congruence roots, mod. p\ for instance,
P Vt Vi Vi
13 -3 - 6 - 5
^1 - 4 +14 - 11 ;
and secondly, it gives, under the simple and the primary forms, the prime factors /{r]) of
these same numbers 13, 41, . . , 937 ; for instance,
p fiv) simple. f{r)) primary.
13 17. + 2% 3 + 77,1
41 4+ 7;„ - 11 +777,-77;.,.
Thus 13 =(770 + 2775) (7/, + 2770) (7/2 + 2771), a.s is easily verified; the product of first and
second factors is =4+ 377„ + 877, + 0774, and then multiplying by the third factor, the
result is 42 + 29 (77, + 17,), = 13.
III. p = 7m + 2 or Tm + 4. First, it gives for the several prime numbers of this
form 2, 11,.., 991 the congruence roots, mod. p: for instance,
p Vo Vi
2 0-1
11 4 -.5;
and secondly, it gives the primary prime factors /(77) of these same numbers; for instance,
p fin)
2 77„
11 1 - 2771.
c. IX. 63
498 REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. [611
IV. p = 7m + 3 or 7m + 5. The prime numbers of these forms, viz. 3, 5, 17,
19,.., 997, are primes in the complex theory, and are therefore simply enumerated.
The arrangement is the same for the higher prime numbei-s \ = 23, &c., for which
ideal factors make their appearance ; but it presents itself under a more complicated
form. Thus X = 23, \- 1=2.11, and the fivctors of 22 are 22, 11, 2, 1. There are
thus four sections.
I. o a prime root, or o** + a^ + . . . + a- + a + I = 0 :
II. T}o = a + a-\..., 7j,o = a" + a~", or tj a root of »;" + i?'" - lO?/' +. . .+ 1 017= - 617 - 1 = 0 ;
III. r}„ = a + 0-, 7)i = «-• + «"■-'. or 7} a root of »;- + 17 + 6 = 0 ;
IV. Real numbers.
I. p=2Sm + l. First, it gives for the prime numbers of this form 47, 139,.., 967
congruence roots, mod. p, and also congruence roots, mod. p^ * ; these last in the fonn
a + bp + cp^, where a is given in the former table ; thus first table : —
p a or* o' . . . a-
47 6 - 11 - 19... + 8;
and second table —
p a a- /x* ... oc-'
47 -irp--2p' + 13jo - 23jj- + U)p - Hp-. .. +22p + 22^.
The meaning is that, ^ = 47, the roots of the congruence
o»» + oP + ... + a= + a+l = 0 (mod. 47»)
are
a = 6 + p - 2pS a» = - 11 + 13/> - 23^^ &c.
Secondly, it then gives /(a), the actual ideal prime factor of these same primes
47, 139,.. , 967 ; viz. the whole of this portion of the table X = 28, I. (2) is,
having actual prime factors,
P /(«)
.599 o + a'« - a"
691 a» 4- a-' + a-
829 a» + a^^ + a« ;
having ideal factors, their third powers actual,
P ./''(a)
47 a'' + a-^ + a" -I- i'» + a'" - a» + a*"
139 1 -a»-a' + a'' + a" + a'Ma'^+a" + a"' + a«'+o"
277 a" - a* - a« + o' - a'" - a" - a" + a-' + «==
461 a - a» + a» - a" + a" - 2a"
967 a» - a» - a» + a"» + a"> - 2a'<' + a" + a".
I repeat the explanation that, for the number 47, this means /(a)/(a-) .•./(a=')= 47'.
* Where, as presently appearing, 3 is the index of ideality or power to which the ideal {actors have to
be isised in order to become actual.
611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 49&
And the like further complication presents itself in the part III. of the same
table, \ = 23 (not, as it happens, in part II., nor of course in the concluding part IV.,
which is a mere enumeration of real primes). Thus III. (1), we have congruences,
(mod. p'),
p = 2, rj = -% p = S, 7/„ = +12, &c.;
and having actual prime factors,
P
/(v)
59
0 - 2i/,
101
I -Hi
and having ideal prime factors, their third powers actual,
P f'M
2 1-7,,
3 1-2,,;
i
as regards these last the signification being
2* = (1 - 7;„)(1 - ■>;,), '7o + '7] = — li ^^ = 6 (as is at once verified),
3' = (l-2,„)(l-2,,,);
but the simple numbers 2, 3 are neither of them of the form (a + 6%) (« + bvi).
Contents of Report 1875 on Mathematical Tables.
§ 7. Tables F. Arithmological.
Page
Art. I. Divisors and Prime Numbers 462
II. Prime Roots. The Canon Arithmeticus, Quadratic residues 471
III. The Pellian Equation 477
IV. Partitions 480
V. Quadratic forms a- + b" &c., and Partitions of Numbers into
8(|uares, cubes, and biquadrates ..... 484
VI. Binary, Ternary, &c. c]uadratic and higher forms . . 486
VII. Complex Theories 498
63—2
500 [612
612.
NOTE SUE UNE FORMULE D'INTEGRATION INDEFINIE.
[From the Comptes Rendua de VAcademie des Sciences de Paris, torn. Lxxviii. (Janvier—
Juin, 1874), pp. 1624—1629.]
En ^tudiant les M^moires de M. Serret (JourtMl de Liouville, t. x., 1845) par
rapport k la repr&entation geomdtrique des fonctions elliptiques, avec les remarques
de M. Liouville sur ce sujet, je suis parvenu a une formule d'int^gration indefinie
qui me parait assez remarquable, savoir: en prenant d entier positif quelconque, je
dis q»ie I'int^grale
r(a; + jj)"'+"-* (x + qf dx
J " af+^{x+p + qY+^
a une valeur algebrique
(x + p)'»+''-»+i (x+p + fy)-» «-'» (A+Bx + Caf + ... + Kx'-^),
pourvu qu'une seule condition soit satisfaite par les quantity m, n, p, q. Cette
condition s'dcrit sous la forme symbolique
([m]p'' + [n]qr = 0,
en d^notant ainsi I'^uation
[m]»p^ + y [m]»-' [nfp^"- q- + ... + [«]» q^ = 0,
oil, comme a I'ordinaire, [»»]* signifie m(w— 1) ... (m — ^+ 1).
Je rappelle que les formules de M. Serret ne contiennent que des exposants entiere,
et cellos de M. Liouville qu'un seul exposant quelconque : la nouvelle formule contient deux
exposants quelconques, m, n. Je remarque aussi I'analogie de la condition ([»0j9^+[«] ^•)*=0
avec celle-ci
(ot etant un entier positit), qui figure dans les M^moires cit^
612] NOTE SUR UNE FORMULE d'iNTEGRATION IND^FINIE.
Pour d^montrer la formule, j'ecris
it = .r-'" (A+Bx+Caf'+ ....+ Ka^-^),
et aussi pour abreger
X = (x + p)»»+»-»+i {x + p + (/)-»,
ce qui donne
501
X
{x+p)Ue+p + q)
L'equation a verifier est done
= < *• + ;>)•»+«-» (x+p + q)-"-\
X (x + q)* dx
„ j A. {x + qy ax
Xu= \ ——7-, — ^^ — r7-
?)'
ou, en differentiant et divisant par X,
X a!'»+'(A-+j9)(a; + p + 5')'
ou enfin
vtil
ou m' denote ^, 11 ne s'agit done que d'exprimer que cette Equation ait une integrate
En Kupposant que cela soit ainsi, et en effectuant la substitution, les termes en
ar**** se detruisent, et Ton obtient une equation qui contient des termes en a;~"'~\
«"*•,..., x'"'"^*"', savoir (^+1) termes. On a ainsi, entre les 6 coefficients A, B, C, ...,
K an systfeme de (^+1) equations lineaires, ce qui implique une condition entre les
constantes in, n, p, q ; mais, cette condition satisfaite, les equations se rMuisent a 0
^nations ind^pendantes, et les coefficients seront ainsi d^termin^s.
Par exemple, soit 0 = 2; l'equation diflferentielle est
[m — 1 p + m ■^- n — 1 q + m — I x^u + [p- + pq + X (2p + q)+ a?] u' = *•-'"-' (q + xf,
laquelle doit etre satisfaite par u = Ax~"^ -^ Bx~^'^'^^. Cela donne
/p~"*~l X~^*^ x~^^'^^ «.— m+a
-m{p'+pq)A
-f
(m — lp + m + n—lq)A, (m—lp + m + n—lq)B,
(m -1)A
-{m-lJipf+pqyB,
— m (2p + q)A
it
-H
-(m-l)(2^ + g)fi,
— niA
-1
(w-l)5
-{m-\)B
= 0,
k
502
NOTE 8UR UNE FORMULE d'iNT^GRATION TND^FINIE.
[612
UimI
+
+
+
CI
SI,
r— \
+
"a.
J.
n
o
"o
..
3^
'S
"o^
'S
Cm
0
«
1
1
^
1
1—1
1
a,
1
a
2
T
a
T
a
f— 1
1
f— 1
T
a
2
1
a.
9)
C
&>
a,
r 1
a,
5?
r— t
eo
ST
rH
■S
V ■•
s
1
1
1
1
l:§
1— J
1
1
r—i
1
1
g
1
CM
1
1
'g'
^
T
S^
Cm
1 1
(—1
1 — 1
^
V.
® 3
1 — 1
^
Oh
!>•
^
e.
,^^
,>v
?-.
^
^^
II
II
II
1 S
« o
+
!>
+
'5^
+
Cm
+
a.
Cm
+
a.
CM
+
CM
+
a
CM
+
a
c«
+
a
CM
+
a
a
o
u
a
3
c .^
^*^
^ — ■
+
I— 1
CM
r-t
1
+
"a,
1
a.
a.
"a.
a,
a
a
a
s
a
1
T
a.
c8
c
8-1
II
^-1
1
II
f^^■
1
I-H
+
II
1
+
l-H
1
II
1
«5
+
1
+
a,
+
o
&i
—
r-(
4S
CO ^
TO
+
<u
S c
1
1:
II
i
.2 »
1
t
5n
+
+
u >
O 01
•
•
TO
+
g
+
a,
*
T
a.
s
CT"
1
Cm
n
1
a
I— t
r-<
a
3
8
s
+
1
1
©
J
^ W
1
1
I-H
1
g
.
+
+
«M
o
o
a>
3
3
.2 II
,
(N
a
+
+
(N
a
+
g
rH
01
"oS
P-^
■
"
"
*
""
*
-
••
-
a
>
n
&<
CM
9
-« b
l-H
_„^^
rH
.-^
C
g
'3
s
3
§1
3
i-(
1
T
a
+
s
+
1— (
<>
1
1
+
•
I-H
1
s
1
a
I— 1
1
g
+
a
1
g
•
•
m
o
O
u
^
-h3
+
r
1
+
1
+
+
1
1
a.
la,
f-1
'
1
a.
5,
•
•
1
a
5,
•
•
1
ii.
1
1
1
1
eo
CO
&
'
1
1
1
1
1
r
¥
S
g
S
§
g
g
f— 1
o
-"
Oi
"&<
l-H
i
^
1;.
i-(
i-
:k'
■b.
T
II
II
II
o
o
o
0
612] NOTE SUE UNE FORMULE d'iNTEGRATION INDEFINIE. 503
et ainsi de suite. Les notations ([to]jo — [«] ?)', {[m] p — [n] q)', . . . ont des significations
semblables a celles de {[m]p-+ [)}]<i^y, ([in] p- + [n] q-)-, . . . , auparavant expliquees. On a,
par exeniple,
([m]/) — [«] (/)- = [«i]'p- — 2 [m]' [nYpq + [nf q".
Considerons, pai- example, le deuxi&me ddtermiuant: ceci contient trois termes en
1, 2q, (f respectivement ; le premier terme est
1 . {iu - 1) {pp- + p(() . m {p- + pq),
c'est-i,-dire
[mY}fip + qy;
le deuxieme terme est
2q. — m{p^+ pq) [(m — 1 ) jo - nq],
c'est-k-dire
- 2 [m]'p (p + q)q([m- 1] p - [n] q)' ;
le tnjisifeme terme est
<f [(m -lp-nq){m+lp-n-lq)- {m - l)(p^ +i«Z)].
c'est-^dire
<f [(m* -m)jii- 2mnpq + (n- - n) q-] = q- ([m] p - [n] qf.
Et de meme le troisifeme determinant est compose de quatre termes en 1, Sq, 3g', 5^
respectivement, lesquels sont les quatre termes de la premiere expression transformee ;
et ainsi jjour le quatrifeme determinant, etc. Au moyen de ces premieres transformees,
on obtient sans peine les expressions finales ([ni]p- + [n.]q^y, {[m]p' + [n]q'y, ....
En dcrivant z — ^(p + q) au lieu de or, et puis ^{p + q) = a, ^(p — q) = a, la formule
devient
) {z ^F+* {z + a)"+' '
et la valeur alg^brique
= (2 + «)'»+''-•-> {z - a)-^» {z + a)-" {A' + Ifz+ ... + K'z»-'),
pourvu qu'on ait entre les quantites m, n, a, a la relation
{[m]ia + ay + [n]{a-ay\» = 0.
Eu tfcrivant 0 = in, on a la formule de MM. Serret et Liouville, laquelle, en y
ecrivaiit %— '- =? et ^"^"=f-l, peut s'ecrire sous la forme {[w]5' + [n] (?-!))»= 0.
Je remarijue fjue I'dquation en f ne donne pas toujours pour f des valeui-s reelles,
positives et plus grandes que I'unite: par exemple, pour ^=1, on a f= — , valeur
m + n
qui ue peut pas satisfaire a ces conditions. Je n'ai pas cherche dans quel cas ces
conditions (qui ont rapport k I'application des fonnulcs k la representation des fonctions
elliptiques) subsistent.
504 [613
613.
ON THE GROUP OF POINTS G^ ON A SEXTIC CURVE WITH
FIVE DOUBLE POINTS.
[From the Mathematische Annalen, vol. viii. (1875), pp. 359 — 362.]
The present note relates to a special group of points considered incidentally by
MM. Brill and Nother in their paper "Ueber die algebraischen Functionen und ihre
Auwendung in der Geometrie," Math. Annalen, t. Vll. pp. 268 — 310 (1874).
I recall some of the fundamental notions. We have a basis-curve which to fix
the ideas may be taken to be of the order n, =j9+l, with ^p(p— 3)dps, and
therefore of the " Geschlecht " or deficiency p ; any curve of the order n — 3, =p — 2
passing through the Jp (/) — 3) dps is said to be an adjoint curve. We may have, on
the basis-curve, a special gi-oup Gg of Q points {Ql^lp — 2); viz. this is the case
when the Q points are such that every adjoint curve through Q — q of them — that
is, every curve of the order p — 2 through i^pip — 3) dps and the Q—q points — passes
through the remaining q points of the group: the number q may be termed the
" speciality " of the group : if ^ = 0, the group is an ordinary one.
It may be observed that a special gi-oup Gq is chiefly noteworthy in the case
where Q — q is so small that the adjoint curve is not completely determined : thus
if p = h, viz. if the basis-curve be a sextic with 5 dps, then we may have a special
group Oi, but there is nothing remarkable in this ; the 6 points are intersections
with the sextic of an arbitrary cubic through the 5 dps — the cubic of course intersects
the sextic in the 5 dps counting as 10 points, and in 8 other points — and such cubic
is completely determined by means of the 5 dps and any 4 of the 6 points. But
contrariwise, there is something remarkable in the group G^ about to be considered:
viz. we have here on the sextic 4 points, such that every cubic through the 5 dps
and through 3 of the 4 points (through 8 points in all) passes through the remaining
one of the 4 points.
The whole number of intersections of the basis-curve mth an adjoint, exclusive
of the dps counting as p{p—'A) points, is of course = 2p — 2 : hence an adjoint
through the Q points of a group G\ meets the- basis-curve besides in R, —2p—2 — Q,
613]
ON THE GROUP OF POINTS 6r ' ON A SEXTIC CURVE.
50&
points ; we have then the " Riemann-Roch " theorem that these R points form a
special group Gk, where
Q + R=2p-2,
as just mentioned, and
Q-R = 2q-2r;
viz. dividing in any manner the 2p — 2 intersections of the basis-curve by an adjoint
into groups of Q and R points i-espectively, these will be special groups, or at least
one of them will be a special group, Gq, G'r, such that their specialities q, r are
connected by the foregoing relation Q — R=2q — 2r.
The Authors give (I.e., p. 293) a Table showing for a given basis-curve, or given
value of p, and for a given value of r, the least value of R and the corresponding
values oi q, Q : this table is conveniently expressed in the following form.
The least value of
P
R=P-,+ l + r'
and then
P
-1,
^ r+1
, Q-P+^,-r-2,
where -£-r- denotes the integer equal to or next less than the fraction.
It is, I think, worth while to present the table in the more developed form ;
n
P
Dps
r=
12 3 4 5 6
4
3
0
G,' g:- .
G," g: .
.
.5
4
2
<?,' G,' G,"
6V G," g;
•
.
6
0
5
g: g,» <?,» G,*
g; G° G^" (?,»
.
7
6
9
(?,' G," G," G,* G,,"
G,' <?,' G," G," G,o
8
7
14
G,' 67 G,' G,o* e„» G,,'
6-V' &V G,o G," G^" G,"
:
where the table shows the values of ^ for any given values of p, r.
C. IX.
64
506 ON THE GBOUP OF POINTS (r/ ON A [613
I recur to the case p=h and the gi'oup 0^, which is the subject of the present
note : viz. we have here a sextic curve with 5 dps, and on it a gioup of 4 points
G^, such that every cubic through the 5 dps and through 3 points of the gi-oup,
« points in all, passes through the remaining 1 point.
MM. Brill and Nother show (by con8iderati(jn of a rational transformation of the
whole figure) that, given 2 points of the group, it is po.ssible, and possible in .5
different ways, to determine the remaining 2 points of the group.
I remark that the 5 dps and the 4 points of the group form " an ennead " or
system of the nine intersections of two cubic curves: and that the nuestion is, given
the 5 dps and 2 points on the sextic, to show how to determine on the sextic a
pair of points forming with the 7 points an ennead : and to show that the number
of solutions is = 5.
We have the following " Geiser-Cotterill " theorem:
If seven of the points of an ennead aie fixed, and the eighth point describes a
curve of the order n passing a,, o.^,.., iv, times through the seven points respectively,
then will the ninth point describe a curve of the order v pas.sing a,, o.,.., a- times
through the seven points respectively : where
V —%n — 32a,
a, = 3n — tti — la,
a- = 3?! — a- — la,
and conversely
n =8i/-3Sa,
O] = Sv — 3] — Sa,
Oy = Sv — a, — la.
(Geiser, Crelle-Borchardt, t. Lxvii. (1867), pp. 78 — 90; the complete form, aa just
stated, and which was obtained by Mr Cotterill, has not I believe been published) :
and also Geiser's theorem " the locus of the coincident eighth and ninth points is a
sextic passing twice through each of the seven points."
The sextic and the curve n intersect in 6?i points, among which are included the
seven points counting as 22a points: the number of the remaining points is
= 6n— 22a. Similarly, the sextic and the curve v intersect in 6v points, among which
are included the seven points counting as 22a points : the number of the remaining
points is Qv — 22a (= 6n — 22a). The points in question are, it is clear, common
intersections of the sextic, and the curves n, v. viz. of the intersections of the
curves n, v, a number 6n — 22a, = 61* - 22a, = 3n + 3j/ — 2a - 2a lie on the sextic.
The curves n, v intersect in nv points, among which are included the seven
points counting 2aa times: the number of the remaining intersections is therefore
613] SEXTIC CURVE WITH FIVE DOUBLE POINTS. 507
nv — lcM, but among these are included the 3n+3j'— 2a— Sa points on the sextic;
omitting these, there remain nK- 3 (« + v)— 2aa + 2a + 2a points, or, what is the same
thing, (»i — 3)(i/— 3)— 2 (a— l)(a— 1) — 2 points: it is clear that these must form pairs
such that, the eighth point being either point of a pair, the ninth point will be the
remaining point of the pair : the number of pairs is of course
i [(„ _ 3)(,. - 3) - 2 (« - l)(a - 1) - 2],
and we have thus the solution of the question, given the seven points to determine
the number of pairs of points on the curve n (or on the curve v) such that each pair
may form with the seven points an ennead.
In paiticular, if «=6; O], a,, a^, a^, Oc, a,, 07 = 2, 2, 2, 2, 2, 1, 1 respectively, viz.
if the curve be a sextic having 5 of the points for dps, and the remaining two for
simple points, then we find i' = 12; a,, ctj, O3, «,, O5, a,, 07=4, 4, 4, 4, 4, 5, 5
respectively, and the number of pairs is
= i[3.9-5(2-l)(4-l)-2], =H27-15-2), =5,
viz. starting with the 5 dps and any 2 points of the group Ot we can, in 5 different
ways, determine the remaining (2 points of the group.
In reference to the number Sp ~ 3 of parameters in the curves belonging to a
given value of p, it may be remarked as follows. Such a curve is rationally trans-
formable into a curve of the order p + 1 with \p (p — 3) dps, and therefore containing
^{p+ l){j) + i), — ^/»(p — 3), =4/j-f-2 parameters. Employing an arbitrary homographic
transformation to establish any assumed relations between the parameters, the number
is diminished to 4p + 2 — 8, = 4j> — 6 ; and again employing a rational transformation
by means of adjoint curves of the order p — 2 drawn through the dps and p — S
points of the curve — thereby transforming the curve into one of the same order
p + \ and deficiency p — then, assuming that the p ~ 2 parameters (or constants on
which depend the positions of the p — H points) can be disposed of so as to establish
/) — 3 relations between the parameters and so further diminish the number b}^ p — S,
the required number of parameters will finally be 4p — 6 — (p — 3) = 3p — 3.
Cambridge, 26th October, 1874.
64—2
o08
[614
614
ON A PROBLEM OF PROJECTION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiil. (1875),
pp. 19—29.]
I MEASURE off on three rectangular axes the distances fiX=f2F=nZ, =0; and
then, in a plane through fl drawing in arbitrary directions the three lines CIA, ilB, CIC,
= a, b, c respectively, I assume that A, B, C (fig. 1) ai-e the parallel projections of
X, Y, Z respectively; viz. taking flO as the direction of the projecting lines, then
a A, ilB, no being given in position and magnitude, we have to find 6, and the
position of the line CIO.
Fig. 1.
This is in fact a case of a more general problem solved by Prof. Pohlke in 1853,
(see the paper by Schwarz, " Elementarer Beweis des Pohlke'schen Fundamentalsatzes der
Axonometrie," Crelle, t. lxiii. (1864), pp. 309—314), viz. the three lines CIX, ClY, CIZ
may be any three axes given in magnitude and direction, and their parallel projection
614] ON A PROBLEM OF PROJECTION. 509
is to be similar to the three lines ilA, D,B, ilC. Schwarz obtains a very elegant
construction, which I will first reproduce. We may imagine through D, a plane
cutting at right angles the projecting lines, say in the points X', Y', Z' ; we have
then in piano a triad of lines D,X', D.Y', D.Z' which are an orthogonal projection of
nX, HY, ilZ; and are also an orthogonal projection of a plane triad similar to
D,A, ilB, ilC; quk such last-mentioned projection, the triangles nY'Z', D,Z'X', ilX'Y',
must be proportional to the triangles ilBC, QCA, ilAB; that is, we have to find
an orthogonal projection of nX, flY, D.Z, such that the triangles ilY'Z', D,Z'X\
fiX'F', which are the projections of VlYZ, ilZX, ClXY respectively, shall be in given
i-atios. There is no difficulty in the solution of this problem; referring everything to
a sphere centre fl, let the normals to the planes ilYZ, ilZX, ilXY, meet the sphere
in the points X", Y", Z" respectively, and the projecting line through H meet the
sphere in the point 0, then the projection of H YZ is to il YZ as cos OX" : 1 ; and
the like as to the projections of ilZX and Q.XY: that is, in the given spherical
triangle X"Y"Z", we have to find a point 0, such that the cosines of the distances
OX", OY", OZ" are in given ratios: we have at once, through X", Y", Z" respectively,
three arcs meeting in the required point 0.
The projecting lines being thus obtained, say these are the three parallel lines
X', Y', Z', we have next to draw through D a plane meeting these in the points
A', B', C such that the triangle A' EC is similar to the given triangle ABG; for
this being so, the triangles ilB'G', ilC'A', ilA'B' being the projections of, and therefore
proportional to ClY'Z', nZ'X', nX'Y', that is, proportional to nBC, nCA, D.AB, will,
it is clear, be similar to these triangles respectively; that is, we have the triad
CIA', CIF, nC", a projection of fiX, ilY, nZ, and similar to the triad D,A, nB, flC,
which is what was required.
It remains only to show how the given three parallel lines X', Y', Z', not in
the same plane, can be cut by a plane in a triangle similar to a given triangle ABC.
Fig. 2.
I
Imagine the three lines at right angles to the plane of the paper, meeting the
plane of the paper in the given points X, Y, Z (fig. 2) respectively. On the base
510 ON A PROBLEM OF PROJECTION. [614
YZ describe a triangle A"YZ similar to the given triangle ABC; and through A", X
with centre on the line YZ, describe a circle meeting this line in the points D
and E. Then in the plane, through YZ at right angles to the plane of the paper,
we may draw a line meeting the lines Y, Z in the points B', G" respectively, such
that joining Zfi", XC" we obtain a triangle XB"C" similar to A"YZ, that is, to
the given triangle ABC.
Taking K the centre of the circle, suppose that its radius is =1, and that we
have KY=fi, KZ=y; also FZ = <r, ZX = t; YA" = a", ZA" = t". If for a moment
X, y denote the coordinates of A', then
■T^ = (x-y)- +y', =l' + 'f -2yx,
and thence
7<7= - /9t=^ = 7 (Z^ + /?■)- /3 (^^ + 7^,
that is,
7<7-'-;8T= = (7-^)(i'-/37);
viz. this is the equation of the circle in terms of the vectors a, t ; we have therefore
in like manner
7tr"=-/3T"» = (7-^)(/-'-/37).
We may determine ^ so as to satisfy the two equations
or"' = <r^ cos= e + {l-\- /9)' sin^ 6,
t"2 = t'^ cos= eJr{l + 7)- sin^ 6 ;
in fact, these equations give
r/a"^ - $t"' = (70--' - ^r) cos'' ^ + {7 {I' + ^') - ^ {l- + 7=)} sin= 8,
which, the left-hand side and the coefficients of cos^6^, and sin'^ on the right-hand
side being each = (7 — ;8) (I- — ^y), is, in fact, an identity.
But in the figure, if 0, determined as above, denote the angle at B, then
{XB"f = XF» -t- YB"'' =a-' + {l + fff tan= e,
(ZCy = XZ' + ZC"' = r" + (i + 7)8 tan" 0,
that is,
XB" = a" sec 0, ZG" = t" sec 0,
or, since B"G" = YZsqc0 [={y — ^)^c0\, the triangle XB"C" is, as mentioned, similar
to the triangle A"YZ.
I was not acquainted with the foregoing construction when my paper was
written ; but the analytical investigation of the particular case is nevertheless
interesting, and I proceed to consider it.
Taking (fig. 1) ft as the centre of a sphere and projecting on this sphere, we
have A, B, C given points on a great circle; and we have to find the point 0, such
\
614] ON A PROBLEM OF PROJEC!TION. 511
that there may be a trirectaiigular triangle XYZ, the vertices of which lie in OA,
OB, OC respectively, and for which
sin OX _a sin OF _ 6 sin OZ _ c
amOA'O' sin~OjS~^' siiTOC'"^'
I take the arcs BC, CA, AB = a, 0, y respectively, a + /S+7=2Tr; and the required
arcs OA, OB, 00 are uken to be ^, tj, f respectively; these are connected by the
relation
sin a cos f + sin yS cos •// + sin y cos f = 0,
to obtain which, observe that from the triangles OAB, OAC, we have
. cos 7) — cos ^ cos 7 _ cos f — cos f cos yS
cos A. = ; — z — : -; iT ~ a >
sm f sin y am f sm p
that is,
sin /3 (cos 7} — cos f cos 7) + sin 7 (cos f — cos ^ cos /8) = 0,
which, with sin a = — sin (yS + 7), gives the required relation. We have
sin OX = ^sin P, sin 0Y=^ sin 7), sin OZ = ^ sin f ;
p p p
and then from the triangles OBC, OCA, OAB, and the quadrantal triangles OYZ,
OZX, OXY, we have
y A/fl-^sin^'/lA/fl-Ssin"?)
o^„ COS a - cos 77 COS f V V ^ / V V ^ / 0
cos BOO = r- ^-~ — -* = - -5^— ^ j ^ , &c. ;
sm 17 sin c be . . ^
0i sm V sin f
that is,
be (cos a - cos 17 COS i;) = — n/(&' — b^ sin- 17) V(^ - c^ sin^ 0>
ca (cos /3 - cos f cos f ) = - V(^ - c'' sin- f) ^(^ - a-" sin' ^),
oi (cos 7 — cos f cos 57) = — \/(^ — a" sin- ^) \/(^' — ^'^ sin- ■»;),
which, when rationalized, are quadric equations in cos ^, cos 17, cos f. The first
equation, in fact, gives
6»c» (cos a - cos 77 cos f)' = (^ - 6= + 6^ co8» (^- - c= 4- c^ cos" ^,
that is,
(^- 6')(^ - c=)- 6V cos" a + (^ - 6'')c= cos^ f + (^ - 0^)6= cos' 97 + 2b'<f cos aco8i7 cos f = 0,
or, what Is the same thing,
- (,1 - 6. - ^7c~::^j + c^:r^ ''•'" ^ + i^r^ '=°"' " " (6r_-^y(^rr^) ^'^^ « '^"^ " «°« ?= <^-
Completing the system, we have
-(,i-cr_-^:^?:r^J+^rr^cos'| + ^,--^cos'r-(^,_g,)^^,_g,^cos^co8rcosg=o,
/, a2fr'cos'7 \ 6» a» ^ 2a'6'
" l^ " a^^r^TF^^j + If^re^ "°" " + S^^^ ''•^"^ ^ - (tt^-g^x^^-ga) ^os 7 cos ^ cos ,7 = 0,
512 ON A PROBLEM OF PROJECTION. [614
and, as above,
sia a cos f + sin /8 cos 17 + sin 7 cos ^= 0.
It seems difficult from these equations to eliminate f, ij, f, so as to obtain an equation
in 6; but I employ some geometrical considerations.
Taking II as the pole of the circle ABC, and drawing TLX, YIY, HZ to meet
the circle in p, q, r respectively, then, if a", /S", 7" are the cosine-inclinations of 0
to X, Y, Z respectively, we have
sin Xp, sin Yq, sin Zr = a", 0", 7".
From the right parallel triangles BYq and CZr, we have
sin Yq = sin fi F sin B,
sin Z»' = sin CZ sin C,
and, thence,
or, since
and thence
we obtain
sin Yq _ sin BY sin OG
sInZr ~ sin CZ " sinO B '
BY=OB-0 Y, CZ =0C- OZ,
sin J9F = - ^ { V(^' — h- sin- ■>j) — b cos 7;},
sin CZ = „ WiS'' — c- sin* ?) ~ c cos f },
/3" \/(^ - h- sin= r))-b cos »;
7" x/(^- c'-sin- f ) — ccos f
We have thence
j8" V(^ - C sin^ ?) - 7" V(^' - 1" 8in» t)) = yS"© cos f - y"b cos 1?,
or, squaring and reducing
/3"= (^^ - c") + y"^ (^ - b-) + 2j8"7'' [- V(^ - c= sin" ?) V(^' - 6' sin"^ 17) + 6c cos 17 cos f) = 0.
that is,
ff'"- {6^ - c") + 7"- (^- - 6=) + 2/3"7" . 6c cos a =0 ;
and, similarly,
7"2 (^ - a") + a"^ (^ - C-) + 27"a" . ca cos /3 = 0,
a"s (^ _ 6=) + ^"2 (^ _ (,.) ^ 2a"/3" . a6 cos 7 = 0,
or, what is the same thing,
r' , y"' 26c cos g o^/ » _ 0
7"' g"' 2cacos^
a"» , /9"» 2a6cos7 , ,
a'-ff'^b^ -0"- a^-d'.b^-e'
614] ON A PROBLEM OF PROJECTION. 513
writing
a", ff', 7" = X >J{a' - &% Y V(6- - ff'}, Z V(c-" - ^),
and
be cos a ca cos /3 a6 cos 7 _ . ,
the equations are
F^ + Z'= _ 2/y'Z' = 0,
Writing the last two under the form
Z" - 2gZ'X' + ^'^ = 0,
X'^-2hY'X'JrY" = 0,
and eliminating Z', we have
- 4 (1 - i^=) (1 - A") F'=^'^ + ( F'^ + Z'^ - 2gh Y'ZJ = 0,
which, in virtue of the first eqpation, is
- 4 (1 - (^») (1 - AO F'»Z'' + 4 {gh -ff Y'^Z" = 0,
that is,
(l-g^)(l-h^)-(gh-fy = 0;
or, what is the same thing,
l-f'-f--h^+2/gh = 0.
I remark that we may write
gh-f=^{l-f)^{l-h'),
V-5r=Va-A')V(l-/=).
y5r-A = v(i-/')V(i-n
the signs on the right-hand side being either all +, or else one + and two — , so that
the product is +. In fact, multiplying the assumed equations, we have
/yh' -fgh (/' +5r» + h?) + flC + A'/« +/y -fgh = l-p- f -h? + fh' + h\p +fy -/yL\
that is,
l-f'-f-h? +fgh (1 +/= + ^= + P) - 2/yA= = 0,
or,
(1 -f^-g^-k-+2fgh){\ -fgh) = 0,
which is right; but with a different combination of signs the result would not have
been obtained.
Substituting for /, g, h their values, we have
(a' -e^){¥- e--) (c' - &') - 6V (a* - ^) cos^ a - cW (6= - ^) cos> /S
- «?¥ {f - ff') cos' 7 + 2a'6V cos a cos y9 cos 7 = 0,
C. IX. 65
514 ON A PROBLEM OF PROJECTION. [614
where the term independent of 0 is
a^b^c? ( 1 - cos" a — cos* 0 — cos' 7 + 2 cos a cos /3 cos 7),
which is = 0 in virtue of o + y8 + 7 = Stt. We have, therefore, for ff' the quadric
equation
fc»c» sin' a + c-a'' sin' /S + a=6» sin" 7 - (a' + 6« + c*) ^ + ^ = 0,
giving for ^ the two real positive values
^ = i{a» + 6> + c>+V(n)},
where
I write now
and also
n= = (a' + fr" + (fy - 4 (6*0= sin' a + Cf'a^ sin' 0 + ci'b- sin' 7)
= a* + b' + (^+ 26'c' cos 2a + 2c'a' cos 2/3 + 2a=6' cos 27
= {a? + ¥ cos 27 + c' cos 2/8)' + (6' sin 27 - c' sin 2/3)'.
a cos ^ h cos 17 c cos K _ v v 7
^J{^W)' V(6^ - ^) ' V(c-^)~ ' ' '
sin a, r suiyW, — sin 7 = 4, B, 0.
a b
The equations for cos f , cos ?;, cos f become
F' + .^' - 2/F.? - (1 -/') = 0,
Z^ + X^- 2gZX - (1 - g') = 0,
Z'+ F' - 2AZ F- (1 - A') = 0,
and
AX + BY-\-GZ=0,
in virtue of the relation between /, g, h. The first three equations are satisfied by a
two-fold relation between X, Y, Z; viz. treating these as coordinates, the equations
represent three quadric cylinders having a common conic.
To prove this, I write
1-/', l-5r', 1-/6', gh-f, hf-g,fg-h = a, b, c, f, g, h.
We have, as usual,
be — f , ca — g", ab — h', gh — af, hf - bg, fg — ch, each = 0 :
the equations
aZ + hF+g-^=0, hZ + bF+fZ=0, gX + fF + cZ = 0,
represent each of them one and the same plane, which I say is that of the conic in
question.
614] ON A PROBLEM OF PROJECTION. 515
The three given equations are
Y' + Z'-2fYZ -a = 0,
Z^ + X'- 2gZX - b = 0,
X^+ Y'>-2hXY-c = 0,
say these are U=0, V = 0, W=0; it is to be shown that cF-bTT, aW-cU, bCT-aF,
each contain the linear factor in question. We have
cV-hW = (c-h)X'-hY'' + cZ^-2cgZX + 2hhXY;
or, what is the same thing,
a (cF - b TT) = a (c - b) Z^ - h»P + g'Z' - 2gg^ZX + 2hh?XY.
Assuming this
we have
that is,
= (aX + hF + gZ) (\Z - hF+ g^,
a\ = a (c — b),
g{ a+\) = -2<,
h (- a + \) = 2/ih^
\ = c— b, a + \ = — 2gg, — a + X = 2Ah ;
but X = c — b, = — h'4-g°, and the other two equations are a + c— b + 2^g = 0, a+b— c+2Ah=0,
which are identically true.
The values of X, Y, Z are thus determined as the coordinates of the intersection
of the conic with the plane AX+BY+CZ = 0; or, what is the same thing, of the line
AX+BY+CZ=0,
aZ + hF + g^ = 0.
with any one of the three cylinders.
We may, however, complete the analytical solution in a different manner as follows :
Assuming as above v'(bc) = f, V(ca)=g, \/(ab) = h, and thence h V(c) — g V(b) = 0,
we obtain from the second and the third equations
Y = hX + ^(c) V(l - X'), Z=gX- V(b) V(l - -X"-),
(the signs are one + the other — , in order that this may consist with the equation
a.X+hY+gZ = 0). Substituting in AX + BY+CZ = 0, we have
iA+Bh + Cg) X + [B V(c) - C V(b)} V(l - X^) = 0,
that is,
{A + Bh + Cgf X' - (^^c + C'h - 2BCi) ( 1 - X^ = 0,
65—2
516 ON A PBOBLBM OF PROJECTION. [614
or say
iA+Bh + CgyX'+{B'{l-l>P)+C'(l-g')-2BC(gh-f)}(X'-l) = 0.
that is,
(^» + £» + C» + 2BC/+ 2CAg + 2ABh) Z» = [5» + G^ + 250/- (Bh + Cgf},
or writing
A^ + B^ + C^+2BG/+2CAg + 2ABh = A,
say we have
AX' = B' + G' + 2BCf - {Bh + Cgf,
AF' = C= + 4» + 20 Ag - (Cf +Ahy,
AZ'==A'' + B' + 2ABh-{Ag + Bfy.
Now attending to the vahies of ^, B, C, f, g, h, we have
BGf, CAg, j45/t=sin/3sin7Cosa, sin 7 sin a cos /3, sin o sin /3 cos 7,
and thence
A=8in^a(l-^^+sin-^y3(l-^;) + sin'7(l-^)
+ 2 (sin /9 sin 7 cos a + sin 7 sin a cos y3 + sin o sin /3 cos 7) ;
in virtue of a + /8 + 7 = 27r, the last term is
= 2 (cos a cos /3 cos 7 — 1),
whence
. ^/sin'a , sin^/S , sin'-7\ ^.- ■ />,*
A = - ^- --- -\ i— 4- — -^ , say this is = - ff^A.
Moreover Bh + Cg= -. ^ ^. , whence the value of AX^ is
1 — -p] + sin" 7 (1 - -2) + 2 sin y8 sin 7 cos a — (1 A sin- a.
Here the constant term is
= sin" /8 + sin' 7 + 2 sin /3 sin 7 cos a,
that is,
= 1 — (1 — sin" /8) (1 — sin* 7) + sin- /9 sin" 7 + 2 sin /S sin 7 cos a
= 1 — cos" yS cos" 7 — cos" a + (cos a + sin /8 sin 7)"
= 1 — cos" a, = sin" a,
or the whole is
''-('-,7=^)-K'^^'^1'
614] ON A PROBLEM OF PROJECTION. 517
which is
so that we have
Similarly,
-^ / sin'' a sin" /S sin" 7
_ / sin" g sin" ff sin" 7\
W-^- 6' c" /•
\ y _ Z' s"t" a sin" ^ sin" y\
~\ a" "^6"-0"'^ c= /'
_3_ /sin^ sin"ff sin" y\ .
~l a" "^ 6" ^c"-6*»/'
and hence also
^^^ "^^^ o"(a"-^)' ^^^ ^^ 6"(6'-^)' ^^^ '''' c"(c"-^)'
where
sin" a sin";3 sin" 7
The equation in X is
that is,
or
/ a"cos"f\ _ - g" sin" a
A(a"sin"f-^) = -^^''~,
« • o t. /v. /, sin" aN
a"sxn"?=^(l-^,
, , ,., „ ^ ,„ •,- ,. sin' a sin" 18 sin" 7
and the like for rj, q. Wnting for greater convenience — — , — 1^ — , — ^ = p, q, r,
then A =^ + 7 + r, and we have
. ,^ ^ 7 + r . „ $' r+p ■ .y. ^ P + q
sin" f = -, — , sin" v = TT. —. — , . '51^1 f = -.; ~ — ^ ,
* a^p + q + r ' b^p + q+r' * c^p + y+r'
(whence also a"8in"f + 6"sin"7; + c"sin" f=2^: as a simple verification, observe that, if
the projection is rectangular, the axes being all equally inclined to the plane of pro-
jection, then f = «7 = ?=90", a = 6=c=^sins, and the equation is 3sin"s = 2; s, s are
here the sides of an isosceles quadrantal triangle, the included angle being 120°, that
is, we have cos 120° (= -J) = — cot's, that is, cot"s = ^, or sin"s = §, which is right).
I remark, that a geometrical solution may be obtained upon very different principles.
We have on a sphere the trirectangular triangle XYZ, which by parallel lines is projected
into ABC. Every great circle of the sphere is projected into an ellipse having double
518 ON A PROBLEM OF PROJECTION. [614
contact at the extremities of a diameter with the ellipse which is the apparent con-
tour of the sphere. Moreover, if the arc of great circle XY is a quadrant, then the
radius through X and the tangent at Y are parallel to each other, whence, if fi be
the projection of the centre, and AB the projection of the arc XY, then in the pro-
jection the line ilA and the tangent at B are parallel to each other. It is now easy
to derive a construction : with centre il, and conjugate semi-axes (Q,B, f2(7), (0,0, HA),
(ilA, CIB) respectively, describe three ellipses ; and find a concentric ellipse having
double contact with each of these (there are in fact two such ellipses, one touching
the three ellipses internally, and giving an imaginary solution ; the other touching
them externally, which is the ellipse intended). Drawing then through the ellipse a
right cylinder (there are two such cylindei-s, but only one of them is real), and
inscribing in it a sphere, and projecting on to the surface of the sphere by lines
parallel to the axis of the cylinder, the three ellipses are projected into three great
circles cutting at right angles, or, say, the elliptic arcs BC, CA, AB are projected into
the trirectangulai" triangle XYZ.
615] 519
615.
ON THE CONIC TORUS.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. Xlir. (1875),
pp. 127—129.]
The equation p + >i/{qr) + i/(st) = 0, where p, q, r, s, t are linear functions of the
coordinates {x, y, z, w), and as such are connected by a linear relation, belongs to a
tjuartic surface having a nodal conic (p = 0, qr — st=0); and four nodes (conical points),
viz. these are the intersections of the line q = 0, »• = 0 with the quadric surface
p' — qr — st = 0, and of the line r = 0, s = 0 with the same surface. The quartic surface
has also four tropes (planes which touch the surface along a conic); viz. these are
the planes 5 = 0, r = 0, s = 0, t=0, the conic of contact or tropal conic in each plane
being the intersection of the plane with the before-mentioned quadric surface
p- — qr — st = 0. The planes q = 0, r = 0, and also the conies in these planes pass
through two of the nodes, say A, C; and the planes s = 0, ^ = 0, and also the conies
in these planes pass through the remaining two nodes, say B, D; so that the relations
of the surface aie as is shown in fig. 1. It is to be added that AB, BC, CD, DA
(but not AC or BD) are lines on the surface.
The planes q = 0, )• = 0, which contain the tropal conies through A, C, are in
general distinct from the planes ABC, ADC which contain the line-pairs BA, BC and
DA, DC respectively: and .so also the planes s = 0, t = 0, which contain the tropal
conies through B, D, are in general distinct from the planes ABD, CBD which contain
the line-pairs AB, AD and CB, CD respectively.
If, however, the identical linear relation contain only p, s, t, then the planes g = 0,
r = 0 will be the planes ABC, ADC respectively: and the tropal conies in these planes
will consequently be the line-pairs BA, BC, and DA, DC respectively. But the planes
8 = 0, t = 0 will continue to be distinct from the planes ABD, CBD: and the tropal
conies in the planes f = 0, t = 0 will remain pioper conies.
520
ON THE CONIC TORUS.
[615
A surface of the last-mentioned form is
VIZ + V(a^) + V(w' - Z-) = 0,
viz. this has the nodal conic z = 0, a^ — w= = 0, the nodes {x = 0, y=0,{m" +l)z^ — V)' = 0],
and (^ = 0, w = 0, x= 0), (^^ = 0, ?« = 0, y — 0), and the tropes « = 0, y = 0, z + w = 0,
^ — «; = 0 ; but the planes z + w = 0 and ^ — w = 0 are ordinary tropal planes each
touching the surface in a proper conic ; the planes x = 0, y = 0 special planes each
touching along a line-pair.
Fig. :.
D
The equation in question, writing therein w = 1 and x + iy, x — iy in place of {x, y)
respectively, is
y{a? + f) + vizY = l-z\
which is derived from
{x + mzy =\-z-,
by the change of x into \l{a? + y^) ; and the surface is consequentlj' the torus generated
by the rotation of the conic {x + mzf =\— z" about its diameter. Or, what is the
same thing, the surface
mz + V(«y) + V(m^ - z') = 0,
regarding therein {x, y) as circular coordinates and ?y as being = 1, is a torus. The
rational equation is CT = 0, where we have
U = {(to^ + 1) ^» _ w2 + xyY - ^i'z'^xy
= {ajy + (1 - m') z^ - v)'}^ + hn'z^ (z^ - vf)
= a?y- + {m^+\yz*jfW* + (2- 2m')z^xy - (2 + 2m") z'w- - 2oeyw-.
I find that the Hessian H of this function U contains the factor xy + (l- m=) «» - v/*,
viz. that we have
H={xy+(l- m') 2» - w»} H',
615] ON THE CONIC TORUS. 521
where
H' = a?f {\- ni')
+ a?y^ {(3 + 8m= + m^ ^'^ + (- 3 + m») it}"]
+ xy {(3 + 11m' + 9m* + m«) 2^ + (- 6 - 12m'' + 6m«) z^^ + (3 + m=) w*}
+ (1 + m') {(1 - m') ^= - M)'} {(1 + w?) z" - vf]-,
giving without much difficulty
H'= 2? (l+m»)Hl-m»)
+ 22* [(1 + 4?tt= + m*) an/ - (1 - m*) w^] (1 + m»)
+ 2= («2/ - Mr") [(1+ 12m' - m*) a^ - (1 - m*) w/^]
+ [(1 - m') «?/ - (1 + m=) i^r*] U;
say this is
where
= z^H" + [(1 - m») a;y - (1 + m») w=] IT,
H"= ^(\+m?y (l-m=)
+ 2^' (1 '+ m») [(1 + 4m2 + m*) a^ - (1 - m*) vf]
+ (a;y-M;»)[(l + 12m=-m«)a;y-(l-m*)w=],
or, what is the same thing,
H" = ay (1 + 12m» - m*)
+ 2a^ [(1 + 4m» + m«) (1 + m=) a= + (- 1 + 6m') «;=]
+ (1 - m*) {(1 + m») z" - w;>)^
It consequently appears that the complete spinode curve or intersection of the
quartic surface and its Hessian, being of order 4x8,= 32, breaks up into
£/"= 0, an/ + (1 - m") 2' - w' = 0,
conic 2 = 0, xy — vP = 0 twice, order 4
conic 2 + w = 0, xy — mhv" = 0, „ 2
conic 2 — w = 0, ocy — m V = 0, „ 2
Cr=0, 2»F" = 0,
U=0, 2- = 0, conic 2 = 0, xy— 10^ = 0 four times, „ 8
proper spinode curve {7 = 0, Zf " = 0, „ 16
32;
viz. the intersection is made up of the conic z = 0, xy — w" = 0 six times, the conies
2±w = 0, a;y — mV = 0 each twice, and the proper spinode curve of the order 16.
C. IX. 66
that is,
and
that is,
522 [616
616.
A GEOMETRICAL ILLUSTRATION OF THE CUBIC TRANSFORMA-
TION IN ELLIPTIC FUNCTIONS.
[From the Quarterly Journal of Pure and Applied Mathematics, vol xin. (1875),'
pp. 211—216.]
Consider the cubic curve
a?-\-y>-^z^ + &lxyz=Q.
If through one of the inflexions z = 0, x + y—O, we draw an arbitrary line
z — tt,{x+ y), we have at the other intersections of this line with the curve
u {«' (x + yY + 6lxy\ +a^ — xy + y''=0;
that is,
(u? + l)(x' + /) + 2*1/ (u' + 3lu - J) = 0 ;
and from this equation it appeai-s that the ratio x : y is given as a function involving
the square root of
(u'+'siu-^y-iu' + iy,
which, rejecting a factor 3, is
= (2u' + Slu + ^){lu-^).
It maybe noticed that lu — ^ = 0 gives the value of u, which in the equation z = u{x+y)
belongs to the tangent at the inflexion ; and 2m' + Slu 4-^ = 0 gives the values which
belong to the three tangents from the inflexion.
It thus appeai-s that the cooi-dinates x, y, z of any point of the curve can be
expressed as proportional to functions of n involving the radical
VKi«-i)(2?*= + 3iu + J)},
and the theory of the curve is connected with that of a quasi-elliptic integral
depending on this radical.
61 6 j A GEOMETRICAL ILLUSTRATION OF THE CUBIC TRANSFORMATION. 523
Taking m an imaginary cube root of unity, write
ma- + m-y — 2lz = a:',
orx + (01/ — 2lz = y',
x+ y — 2lz = z' ;
then we have
x'y'z' = oo'+y^- Sl'z' + eix-yz = ar" + 1/' + ^^ + Glxyz - (1 + 8/») z\
Also
- 6lz = x' + y' + z', z>= 2jg~ (a;' + y' + zj,
whence
216?' 216P
¥ + y' + zJ - j-^23 afy'z' = j—g^3 (a^ + y. + ^3 ^. g;^^^ .
so that, putting
or, what is the same thing,
t^ 8?»m» + Z» + »n ' = 0,
the equation of the curve is
(a:' + y' + z*)' + 216»>iVyV = 0 ;
and if we write
a! -.y' : z! = X^ : Y' : Z»,
then the original curve is transformed into
(Z» + F' + ^r')» + 21Gm'>X'Y'Z' = 0,
a curve of the ninth order breaking up into three cubic curves, one of which is
Z' + F' + Z» + 6mZF^ = 0,
and for the other two we write herein mw and mco- respectively in place of m.
Attending only to the first curve, we have
a;3 + y' + z^ + Glwyz = 0,
X'+Y' + Z' + drnXYZ^O,
as corresponding cur\'e8, the corresponding points being connected by the relation
ax + w^y - 2lz : m'x + my - 2lz : x + y-2lz = X' : Y' : Z\
or, for convenience, we may write
a>x + la'y - 2lz = X'-\ giving Zx = o)=Z' + wF' + Z\
«'ar + wy- 2lz = Y\ Sy = a>X' + w'F' + Z',
x+ y-2lz = Z\ -Glz= X'+ Y'^ + Z\
66—2
524 A GEOMETRICAL ILLUSTRATION OF THE [616
This is a (1, 3) correspondence; viz. to a given point on the curve (»i), there
corresponds one point on (0; but to a given point on (/), three points on (m).
As to the first case, this is obvious. As to the second case, if the point (x, y, z) is
given, then the corresponding point {X, Y, Z) on the other curve will lie on one of
the three lines
F= {(OX + (ohj- 2lz) - X' (<o^x + 0)1/- 2lz) = 0 ;
each of these intersects the curve (m) in three points: but of the points in the
same line it is only one which is a corresponding point of (x, y, z), and the number
of the corresponding points is consequently the same as the number of lines, viz. it
is =3.
We infer that the above equations lead to a cubic transformatiou of the quasi-
elliptic integral
into one of the like form
jdv-i- s/{(mv - ^) {2if + 2mv + ^)\ ;
and this is now to be verified.
We have, as before, the line z = ii(x + y) meeting the curve (l) in the points
(u'> + l)(a?+y^) + 2xy(u' + 3lu-^) = Q;
and if similarly through an inflexion of the curve (m) we take the line Z=v{X+Y),
this meets the curve in the points
{i!^ + l)(X'+Y')+2XY(v' + 3mv-i) = 0.
Then if (x, y, z), (X, Y, Z) are taken to be the corresjwnding points as above, we
can obtain t) as a function of w. We, in fact, have
_ _ - 2f^ ^ X^+Y' + Z^ _ X'>+Y^ + v'{X+Yf
a;-|-y -Z'-7= + 2Z» - (Z» + F») + S^H^ + F)»
X^-XY+Y^ + ^iX+Y^P
~ -X^ + XY- Y"- + 2^{X+Yy'
_ (v'-\-l){X^+Y^) + {2v'-l)XY
~ {2^ - 1) {X' + Y-) + (4t;» + 1) XY'
or, since we have
that is,
the equation becomes
{}fi + 1) (Z^ + Y-) + 2Z F(v' + 3wiM - i) = 0,
A'-'+P : XY = -2v'-Qmv + l : ifi+l,
-2lu= -6»M;(t;'-|-l)
(2t;» - 1) (- 2«» - 6mv + 1) + (4«;» + 1) (tH + 1)
- Qmv (if + l)
-Sv{'imv>-Sif-2m)'
616] CUBIC TRANSFORMATION IN ELLIPTIC FUNCTIONS. 525
or say,
— lu = m (tfi + 1) (-=-), where the denominator = imir' — 3v^ — 2m.
This may also be written
— {lu — i) = Sv' (mv — i) -^ .
Proceeding to calculate 2jt^ + Slu + ^, omitting the denominator (imv^ — Hv^ — 2/)*)',
this is
-^ (if+iy- 3m (ir" + 1) {4mif -3v^- 2m)= + ^ (imv" -S^f- 2ni)' ;
or, observing that
,_ -I'
*" ~l + 8l''
that is,
the numerator is
= 2 (1 + 87?i') {^ + iy-3vi(i^ + 1) (4mw» -Sii'- 2m)= + ^ (4m«* - St^ - 2m>',
which is found to be identically
viz. we have
2m» + Sill + h = (2v'+ Smv + J) (v" + Gmi) - 2)= -;- (•imiP - Sv- - 2m)\
and hence
(lu - ^) (2m' + Slu + ^) = -3 (mv - i) (2?;=' + 3m?; + i){^ + Qmv - 2)- if -f- (4:iHif -St^- 2my.
Moreover, we find
Idu = Smdv .v(if + 6mv - 2) -r (4mt;' - 3«^ - 2my',
and we thence have
Idti . nidv
= V(- «5)
VK'«-i)(2«='+3i?t + i)} ^^ V{(TOt>-^)(2t;' + 3mD + i)}'
viz. this differential equation corresponds to the integral equation
~lu = m(v' + l)-i- ('imv'-3if^ - 2m),
where 8Pm' + f' + m' = 0, which corresponds to the modular equation.
It may be remaiked that, if v is the same function of u', I, in that u is of
V, m, I; viz. if
-mv = l (u'^ + 1) -^ {Uu'" - Sti'" - 2m'),
then
mdv ,, „. — Ww'
= V(- o)
Vl('»«-i)(2o' + 3mi; + i)} ^^ "VK^w'-i)(2M'» + 3k' + i)}
and consequently
dw — 3di^
>J\(lu, - \) {2.U? + 3/« + \)\ V{(iw' - \) {2u'^ + 3;m' + i)} '
which accords with the general theory of the cubic transformation.
526 A GEOMETRICAL ILLUSTRATION OF THE CUBIC TRANSFORMATION. [616
We may inquire into the relation between the absolute invariants of the two
curves. Taking the absolute invariant to be
_ 64/8"- T'
64S» '
where S and T bear the usual significations, we have for the one curve
n= (1+8?/
64Z»(1-P)»'
and for the other curve
(1 + Hm*y
n' =
647ft»(l-m»)>'
and, as above, Slbn' + 1' + m' = 0: writing herein
' 8a" "" 8^'
the relation between a', ^ is simply a' + /8' = 1 ; and the values of fl, fl' are found
to be
64a (1-a')' 64/^(1 -^)».
(1 + %oiy • (1 + 8,8')' '
viz. the required relation is given by the elimination of a', ^ from these three
equations. Or, what is the same thing, \vriting a' = | + ^, and therefore /S' = ^ — ^, we
have
(5 + 80)' fi = 4 (1 + 2^) (1 - 2^)^
(5 - %ey fi' = 4 (1 + ^ey (i - 26),
and the elimination of 0 from these equations gives the required relation between
n and n'.
It of course follows that, if we have a cubic transformation
dx _ Cdx
V{(a, 6, c, d, e\x, l)p " VK«'. i'. C, d', ^\x, l)Y '
then the absolute invariants fl, fi' of the two quartic functions are connected by
the above relation. I have obtained this result, by reducing the radicals to the
standard forms
V(l - *•» . 1 - k'a?), ^(1 -x'K\- \'x'%
from the known modular equation as represented by the equations
, _aM2+a) _a(2 + a)^
l + 2a ' ~(1 + 2a)''
viz. the values of the absolute invariants
21J' , 27J''»
/_ 27 J- 27J^'»\
are
()fc* + 14fc» + 1)» ' (V+14\=+l)''
but the method of effecting this is by no means obvious.
617] 527
617.
ON THE SCALENE TRANSFORMATION OF A PLANE CURVE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. Xlll. (1875),
j^ pp. 321—328.]
The transformation by reciprocal radius vectora can be effected mechanically by
Sylvester's Peaucellier-cell. But, employing a more general cell (considered incidentally
by him) which may be called the scalene-cell, we have the scalene transformation in
question*; viz. if, in two curves, r, r are radius vectoi-s belonging to the same angle
(or say opposite angles) 6, then the relation between r, r is
rr' (r + /) + (m' - f-) r + (m- - n^) r = 0 ;
or, as this may also be written,
r^+ (/ ; — jr+7n--ji-=0.
The transformation is, it will be seen, an interesting one for its own sake, independently
of the remarkably simple mechanical construction, viz. the scalene cell is simply a
system of 3 pairs of equal rods PA, QA; PB, QB; PC, QC (fig. 1, p. .528), jointed
together at and capable of rotating about the points P, Q, A, B, G; the three lengths
PA, PB, PC (say these are =1, m, n) are all of them unequal: in the case of any
two of them equal, we have Peaucellier or isosceles cell. The effect of the arrangement
is that the points A, B, C are retained in a right line, the distances BA, =r', and
BC, =r, being connected by the above-mentioned equation; so that taking jB as a fixed
point, if the point A describe any given curve, the point G will describe the corre-
sponding or transformed curve.
In the case where the given curve is a right line or a circle, we may through
B draw at right angles to the curve the axis x'Bx: viz. in the case of the circle,
* The transformation itself, and doubtless many of the results obtained by means of it, are familiar to
Prof. Sylvester; and I abandon all claim to priority.
528
ON THE SCALENE TRANSFORMATION OF A PLANE CURVE.
[617
the axis x'Bx passes through its centre ; and we measure the angle 6 from this line,
viz. we write Z xBG = Z x'BA = 6.
Suppose, first, that the locus of ^ is a right line, or a circle passing through B.
Its equation is r' = - — ^ or =ccos^; and we accordingly have for the transformed curve
a -TV r + m^-n^ = 0,
cos P CCOS0J
or else
r° + ( c cos d — -I r + m- — n° = 0:
\ c cos 6 J
viz. multiplying in each case by r cos 6, and then writing r cos 6 = x, r^ = .t^ + y", the
equations become
x(x- + y^) + cia^ + y-) a
and
x(se' + y^) + CO?
c
+ {m- — n-) X = 0,
{a? + y") + (??i^ — ?»'-) A- = 0 ;
viz. in each case the curve is a circular cubic passing through the origin B and
having an asymptote parallel to the axis of y. The curve is nodal, if m = n, viz. in
this case the origin is a node : or if c = >J{1" — n") + V(»i° — «')•
Suppose next that the locus of .4 is a circle, centre at a distance = y along Bx'
and radius =h: we have
r'- - 27?-' cos d + y--h' = 0,
viz. if
rf — h- = — (l- — m'),
or, what is the same thing,
h" + 7n- = rf + l\
then we have
7- ;r = "y COS a,
617] ON THE SCALENE TRANSFORMATION OF A PLANE CURVE. 529
and the transformed curve is
7^ + 27?- cos e + nv"- n^ = 0,
or, as this may be written,
r= + 27rcos ^ +7^ -/" = 0,
where 7* —f^ = m^ - n-, that is, f^ + m- = 7- + n- ; viz. this is a concentric circle radius f.
The theorem may be presented as follows. Consider two concentric circles, centre 0
and radii h, f respectively; take an arbitrary point B, distance OB = y; and taking m
arbitrary, determine I, n by the equations
then drawing through B an arbitrary line to meet the circles in -4, C respectively;
also describing a circle, centre B and radius =m; and through 0 drawing a line
perpendicular to ABC to meet the last-mentioned circle in two points P, Q : for these
points, the distances from the points A, B, C are = I, m, n respectively.
To verify this, take 0 as the origin, OB for the axis of x, 6 the inclination of
ABC to this axis, BA = ?•', BC = ?■ ; the coordinates of C, B, A are
7 + »• cos d, r sin 0,
7 , 0 ,
7 + r' cos d, — ?•' sin 6,
whence, taking (w, y) for the coordinates of P (or Q), the equations to be verified are
(x — 'f — r cos 6y + {y — r sin 6f = n-,
(a; -7)- +2/'^ =7w',
(a; - 7 + r' cos df + (y + / sin dy = l\
By means of the second equation, the other two become
— 2 (a? — 7) r cos 6 — 2yr sin ^ + r* = «' — m?,
2 (a; - 7) / cos ^ + 2yr' sin 6 + r'''=P -Tn?;
or, substituting for v? — m?, P — m° the values f- — 'f and h" — 7°, the equations are
— 2xr cos 0 — 2yr sin 0 + 1^ + 2'yr cos 0 + '^ —f"- = 0,
%cr' cos 0 + lyr^ sin 0 + r'-- 2yr' cos ^ + 7= - A'' = 0,
viz. in virtue of the equations of the two circles, these reduce themselves each of
them to
X cos 0 +y8iQ0 = O,
which equation, together with the second equation
(x-yy + y' = m\
determine (x, y) as above.
C. IX. 67
530 ON THE SCALENE TRANSFORMATION OF A PLANE CURVE. [617
Reverting to the case where the locus of A is the circle
r'-' - 2yr' cos ^ + 7" - /i- = 0,
this gives
r' = 7 cos ^ + V(/t' - y sin-" 6),
1 7 cos g - \/{h' - 7° sin- 6)
r'~ rf-h' '
so that for the transformed curve we have
r= + r (1 - J^') 7 cos <> + »• (1 + ^'7^) V(/*' - 7" si"' 6) + »i» - n» = 0.
Putting for .shortness ^—n = \ and for r, rcos^, rsin^, writing ^{a? + 'f), x, y respect-
ively, this is
a? + f + {l-\)riX + (l + \)^{h-{od'-<ry-)-ry^\+m'-n- = Q,
or, what is the same thing,
{a? + y= + (1 - \) 70,- + m' - )i^Y = {I + ^f {h? (of + y») - 7^},
a bicircular quartic. In the case \ = — 1 , it reduces itself to the circle
ir" + y + 27a; + m^ - «'' = 0
twice, which is the case considered above; and in the case \=1, or i* + A' = »t' + 7^,
the equation is
(a^ + 2/= + m^ - ii^y = 4 {h- (x' + rf) - y-tf],
so that the curve is symmetrical in regard to each axis. In the case 7 = 0, the locus
is a pair of concentric circles, centre B.
The equation
\a? + y' + (l-\)yx + m^- nf = (l + \y {h^af + yf) - yY},
which contains the four constants \, 7, k and W4'^ — n', may be written in the form
(a^ + y- + Ax + By = oar' -I- ey\
(where the constants A, B, a, e are also arbitrary). This is, in fact, the equation of
the general symmetrical bicircular quartic, referred to a properly-selected point on the
axis as origin, viz. the origin is the centre of any one of the three involutions formed
by the vertices (or points on the axis); say it is any one of the three involution-
centres of the curve.
To show this, assume
(a; — a) (a; — /3) (a; — 7) (a; — S) = a;* —pa? + qa? — rx + s:
617] ON THE SCALENE TBANSFORMATION OF A PLANE CUBVE. 531
then, taking B arbitrary, the equation of the symmetrical bicircular quartie having for
vertices the points x = a, x — ^, a; = 7, x = h, is
(a? + y"-- ^px + By = (25 +\f-q)ofi + {r- pE) x+{-s + B^)\
in fact, this is the form of the general equation, and writing therein y = 0, it becomes
a^ — pa? + qa? — ra; + s = 0, that is, (x — a){x — /3) {x - 7) (« — S) = 0. Hence, writing for
convenience
A = -^p,
a = 2B+y--q,
b = r — pB,
c =-s+R-,
the equation is
(ai' + y^+Ax + By = aaf + bx + c.
This may be written
(a? +f + Ax+ B + dy = (a + 2e)x' + 2ey- + (b + 2eA)x + c + 2B9 + ff',
viz. assuming &= — „. in order on the right-hand side to destroy the term in x, the
equation is
^ai' + y'+Ax + B-^J = (a-^)x'-^f + ^^^ (¥ - iABb + iA'-c),
which is of the form
(«= + y- + Ax + By = ax' + ey''+/;
and if/=0, that is, if ¥ — 'iABb + iA'c = 0, then it is of the required form
{id' + f + Ax + By = aa? -^ ey".
We have
b'-^ABb + 4:A^c = ir-pBy + 22)B(r-pB) ^p^-s + B*)
= r'' — p's,
or the required condition is r- — p-s = 0. But we have
j»»s - r= = (aS - y37) (/3S - 72) (7S - a/3),
as is easily verified by writing
p = 8+p„, q = Spa + q„, r = Sg'„ + r„, s = Sro,
where p,, q„, r„ stand for
a + /3 + 7, /37 + 7a + a/3, CL0y,
67—2
532 ON THE SCALENE TRANSFORMATION OF A PLANE CURVE. [617
respectively. The required condition thus is
{aS - /37) (08 - yx) {yS - a/3) = 0,
viz. the origin (that is, the fixed point B of the cell) must be at one of the three
involution-centres.
Comparing the equation
[a? + y^ + {l -\)yx + m" - n^\- = (1 + \)'= [h^ (a,-= + y^) - y'y^}
with the equation
{a? + f-\-Ax + BY = aaf + ey',
we have
^=(1-X)7.
5 = m» - n\
a = (1 + xy A?,
e=(l+xy(h'-'f),
and thence a — e = (1 + \)' y. Consequently . ^ = ( :. — - ) , which gives \ : and then
A^ = p: — — - , y' = -p: — -^j , m' — n^ = B; viz. we thus have the values of X, h, y and
m" — n' for the description of a given curve (a:^ + y- + Ax + Ef = aa?-\- ey-. In order that
the description may be possible, a and a- e must be each of them positive.
For the Cartesian a is = e, whence 1 + \ = 0, and the equation becomes
(a!» + y»+ 1yx->r'ne-n^y = 0,
which is a twice repeated circle; hence the Cartesian cannot be constructed by means
of a cell as above.
To obtain a construction of the Cartesian, it may be remarked that, if a symmetrical
bicircular quartic be inverted in regai-d to an axial focus, viz. if the focus be taken
as the centre of invemon, we obtain a Cartesian. The axial foci of the curve
(a? + y» + ila; + 5)" = oar" + cy»
are points on the axis, the abscissa x^B being determined by the equation
e ( ^ + 4 (? + £)» - a ( ^^ - 5)» - ae^ = 0.
The equation referred to a focus as origin is therefore
{a? + y + (il + 2^) a; + £ + ^}» = oar' + ey» + 2a^a; + ^ ;
then mverting, viz. for x, y writing -j > -^ (^ arbitrary), we have, as may be verified
the equation of a Cartesian.
617]
ox THE SCALENE TRANSFORMATION OF A PLANE CURVE.
533
The inversion can be performed mechanically by an ordinary Peaucellier-cell ; the
complete apparatus for the construction of a Cartesian is therefore as in fig. 2, viz.
we have a cell BAG as before, B a fixed point, locus of A a circle (for convenience
of drawing, the aiTangement has been made BAC instead of ABC), and we connect
with C a Peaucellier-cell CA'B', arms n, n', m', the fixed point B' being on the axis,
which is the line joining B with the centre of the circle described by A. This being
so, then A describing a circle, C will describe a symmetrical bicircular quartic, and
A' will describe the inverse of this, being in general a like curve ; but if the position
of fi* be properly determined, viz. if B' be at a focus of the first- mentioned quartic.
Fig. 2.
then A' will describe a Cartesian. A further investigation would be necessary in order
to determine how to adapt the apparatus to the description of a given Cartesian.
A more convenient mechanical description of a Cartesian is, however, that given
in the paper which follows the pi*esent one [618].
The equation
{af + y' + il- X) yx + m' - «=]' = (1 + X)'' {h' {a? + rf) - yhf]
may also be written
{a?-\-^+{\ - \)7a;- ^ (H- Xf^h'-'f) + m? - ii'Y
= {l+\y {'fai' - (1 - \) (h^ - 7=) 7fl; -«- i (1 + ^T (A' - i'Y - (m? - n') {h' - 7=)1.
Tiz. the equation is now brought into the form
(ai' + If' + Aa; + By = oaf + bx + c.
Expressing the coefficients A, B, a, b, c in terms of \, 7, h, m'—n-, it appears by
what precedes, that we should have identically ¥ — iABb + 4.4 ^c = 0, viz. this is the
■equation which expresses that the origin is an involution-centre.
If, instead of the original cell, we consider a new cell obtained by substituting
for the arms PB, BQ, the arms pb, bq, jointed on to the points p, q on the arms
CP, CQ respectively, and instead of B, making b the fixed point; then writing GTp = kn,
pb = km, so that the parameters of the cell are I, m, n, k, and taking Ch = s, hA = s',
534 ON THE SCALENE TRANSFORMATION OF A PLANE CURVE. [617
we have 8 = kr, s + s' = i-+ r', that is, »" = r . '"' = , s + a'. Substituting in the equation
between r, r\ written for greater convenience in the form
(r + r') {rr' + ni« - V) + {I* - n«) r' = 0,
the relation between s, s' is found to be
(»+of-^^ + '^+™'-^^) + (^=-«')(";-»+«')=o.
On account of the term in s", this equation in its general form does not, it would
appear, give rise to transformations of much elegance. If, however, l = n, then the relation
becomes
(fc - 1) s= + /bs' + ^= (m- - Z=) = 0 ;
and in particular, if k = 2, then
s= + 2ss' = 4 (i" - m=), or say (s + s')° - «'' = 4 (i' - m-),
viz. taking A instead of b as the fixed point, the relation between the radii AG, Ah
is p' — p'- = 4 (i- — wi^) ; the cell is in this case Sylvester's "quadratic-binomial extractor."
618]
535
618.
ON THE MECHANICAL DESCRIPTION OF A CARTESIAN.
[From the QuaHerhj Journal of Pare aiid Applied Matliematics, vol. xiil. (1875),
pp. 328-330.]
H
Suppose that in two different curves the radius vectors r, r, which belong to
the -same angle 6, are connected by the equation
^ + (Mr' + iV + ^r + fi=0;
then, taking one of the curves to be the circle
P
Mr' + ^, = A cos e,
r
the other curve is
?•"+ (il cos ^ + JV) r + £ = 0,
viz. this is a Cartesian. It perhaps would not be difficult to contrive a mechanical
arrangement to connect the radius vectoi-s in accordance with the foregoing equation ;
but the required result may be obtained equally well by means of a particular case
of the relation in question ; viz. taking this to be
r' + (-»•' + iV) r + iJ = 0,
then, taking the one curve to be the circle r' = — A cos 6, the other curve is the
Cartesian,
»-» + (2ico8^ + 5)r + i)=0, that is, r- + {Acme + N)r+ B = Q.
The relation between the radius vectors may in this case be written
/ = iV + r + - ,
r
which can be constructed mechanically by a simple addition to the Peaucellier-cell,
viz. if we joint on to C (fig. 1, p. .536) a rod CD A, having a slot, working on a pin
at A, so that the rod is thereby kept always in the line BAC, then, making B the
536 ON THE MECHANICAL DESCRIPTION OF A CARTESIAN. [618
fixed point and taking BA =r, we have AC = — , whence BC = r-\- , or D
being a point at the distance CD, = a, from the point C, and denoting BD by /, we have
?•' = r H h a, which is an equation of the required form ; whence, if the point D
T
describe a circle passing through B, then the point A will describe a Cartesian.
Fig. 1.
The equation of the Cartesian is i^ + {A cos Q + N) r + B = 0, viz. this is
id' + y^ + Ax + B = - N »/(a^ + y'), or writing N^=a, it is
(x' + y" + Ax + B)- = aa!^+ ay\
which is the form considered in the preceding paper. It may be further observed in
regard to it that, starting from the focal equation r = ls + m, where r, s are the
distances of a point {x, y) of the Cartesian from any two of its three foci, this
equation gives i^ — iV + m- = 2mr, or writing r^ = oi? + y', s'' = (« — a)* + y-, the function
on the left-hand is of the form {\ — 1-) {a? + y- -\- Ax + B), whence, assuming ;. — ij = V(o).
the equation becomes as above
{ai' + y"" -^^ Ax-\- By = a{x' + y%
Taking the distance r = \/{x' + y-) to be measured from a given focus, it is easy to
see that, no matter which of the other two foci we associate with it, we obtain the
same equation {a? + y- + Ax + Bf = a {a? + y) ; viz. starting with any one focus, we
connect with it a determinate circle a? + y- + Ax + B = 0, and a determinate coefficient a,
such that taking this focus as the origin, the equation of the curve is
{x' + f + Ax+By = a{x' + y^);
but there are for the given curve three such forms of equation, according as the origin
is taken at one or other of the three foci.
(Addition, Feb. 1875.) It is obvioiisly the same thing, but I find that it is
mechanically more convenient to derive the Cartesian from the Lima9on r'=—N—AcosO,
by the transformation r'=r-\ — : I have on this principle constructed an apparatus
whereby the Cartesian is described on a rotating board by a pencil moving in a
fixed line.
619] 537
619.
ON AN ALGEBRAICAL OPERATION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiii. (1875),
pp. 369—375.]
I CONSIDER
ilF(a, x),
an operation il performed upon F{a, on) a rational function of (a, x); viz. F being
first expanded or regarded as expanded in ascending powers of a, the coefficients of
the several powers are then to be expanded or regarded as expanded in ascending
powers of x, and the operation consists in the rejection of all negative powei-s of x.
In the cases intended to be considered, F contains only positive powers of a : but
this restriction is not necessary to the theory.
The investigation has reference to the functions A (x) of my " Ninth Memoir on
Quantics," Phil. Trans., t. CLXi. (1871), pp. 17 — 50, [462] ; for instance, we there have
a.s regards the covariants of a quadric
^<->4^© = r
l-a;-^
aa^ . 1 — a . 1 — ax~" '
and consequently, in the present notation.
A{x) =n
l-aa?.l -a.l- ax-'' '
by a process of development and summation, the value of this expie.ssion was found
to be
1
l-cw^.l-a*'
C. rx. 68
538 ON AN AL(JEBRAICAL OPERATION. [619
and in the other more complicated cases the value of A (x) was found only by trial
and verification. What I purpose now to show is that the operation fl can be
performed without any development in an infinite series ; or say that it depends on
finite algebraical operations only.
It is clear that if F{a, x), considered to be developed as above contains only
positive powers of x, then
ilF(a, a:) = F{a, x);
And if it contains only negative powers of x, then HE {a, x) = 0.
Consider now il '—^ — , where ^ (a;) is a function containing only positive powers of
os "~ a
x\ we have
x — a x — a x — a'
and thence
n ^^ = n ^ (^) - */* (■"■) + n ±(^
x—a x—a x—a
^<f>{x)-<f)(a)
x—a '
since — ™ is a rational and integral function of a, which when developed
contains only positive powers of x, and ^ when developed contains only negative
powers of x.
Consider next Q ^ , where <j> (x) is a rational and integral function of *• ;
writing this =/(«") + xg {a?), we have
a?— a af — a a? — a
of — a a? — a
As regards the last term, notice that
(c^ — a !t? — a a? — a'
s'hich - '^ ' — ^-^/i is a rational and integial function of (a, x), and therefore
when developed contains only positive j^owera of x, while f when developed
contains only negative powers of x.
We thus have
^ <f>(x) ^f{a?) + xg(a?)-f(a)-xg(a)
of'— a of- (I
_ </) (x) -/(a) - xg (o)
3? — a
m wr
619] ON AN ALGEBRAICAL OPERATION. 539^
Similarly, if if,(x)=fiaf) + a-g{a,^) + x'h(af), then
n ^(^) = ^ (^) -/(") - ^9 (a) - ^h (a) .
a? —a x' — a '
and so on.
Consider now the above-mentioned function
A{x), =n
1 — euf . 1 — a . 1 — cm;"'- '
Writing
we have
!-«-= P Q 5
1— aa:'. 1 — a. 1 — aa;~* 1— cwr' 1 — a 1— cm;"-'
VI - a . 1 - aar-V«.»-.' 1-ar*' l-cv*'
that is,
l-ar-s -a;« 1 1 1 1 1
1— a^.l-a,l— aar» 1 — a:*l— cwr' 1 — a^l— a 1— ar'a^ — a'
and thence
1-a;-- -ar* 1 1 ^ -1
^-i :r^ 5 =j,= , — Ti i ; + ! tt + H
1 — cutf . 1 — a . 1 — oar'-' 1 — a:* 1 — cub^ 1— a^.l — a 1— a^a::= — a"
Here, as regards the last tenn,
<^ (ar') -/(g) _ 1 / -1 1 \ a;*-CT° _ _ a;- + a
ai'-a sc'-a [l - a^ '^ 1 - aV "^ a;»-a. 1 -a;«.l - a=~ l-a^.l-a»'
and we have
^ 1 — ar" - *■< 1 x' + a
I —ax' A - a. 1 — ax-- l-x*.! — aaf 1 —x'.l—a 1— ar*.!— a'*
1 -f-ar'. 1 +a
l-x^.l-a'
1 4- sP 1 + a
The second term is =;j — ^jAj ^: combining this with the third term, the two
together are = ^-— ±^-_^.
68—2
540 ON AN ALGEBRAICA.r. OPERATION.
Hence the value is
_ 1 / -of' l + gaA
~l-«*\l-a«''*' 1-aV'
which is
[619
being, in fact, the expression for this function when decomposed into partial fraction
of the denominators 1 — aa? and 1 — a" respectively. Hence finally
^'^'^ \-aa?.\-a.\-a(ir^^\-ao?.\-a^'
as it should be.
For the cubic function, we have
A{x) = 0.
\-x-'
1 — am? .\—ax.\— ax~^ . 1 — ax~' '
the function operated upon, when decomposed into partial fractions, is
a;" 1 x^ 1_^
l-x-^.l-afl-aa^ 1 —of .l-a^l -ax
X 1
— X
\—3f.\—x*x — a \ - af .\ —afi a? — a'
Hence we require
The first of these is
n
+ ft
— X
1— «*.!— ar'a;— a \—a^.\—oifa? — a'
which is
_ 1 J X a \
~ x-a\\ -a?.\-a* 1 - aM - a^ '
l-a?.\-af.\-a?.\-a*
( 1
+ x (a + a' - a")
+ X' (a' - a*)
+ .<-•»(« -a')
-x*a^
- x^a
As regards the second, the function operated on may be expressed in the form
— af — x — afi 1
1-x'^.l-af af-a'
/
619]
ON AN ALGEBRAICAL OPERATION.
541
whence /(a), g (a), A (a), and therefore / (<t) + a;^ (a) + x'^h (a), respectively, are = — a',
— 1, —a, —a? — x — a3?, each divided by 1 — a-.l-a^; or the term is
which is
_ 1 j —X a' + if + aar* )
~ «'-a (1 -a:*. 1 - a* "^ 1 - aM - a*J '
l-^'.l-a^'.l-aM -«/
ft
— a-
+ *■(-((. - f»' + aO
+ ar' . - 1
■V a? . — a
+ *•*.(«■' -1)
+ «".«-
+ a' . (»'
+ «».l
+ «» . a
To combine the two terms, we multiply the numerator and denominator of the
first by 1 + a^ + ar*, thereby reducing its denominator to 1 — a;* . 1 — a^ . 1 — a* . 1 — a**, the
denominator of the second term ; then the sum of the numerators is found to be
= \~a? viz. this is = (1 — a-)
' 1
+ x (a* — a*)
+ a^af'
+ ai'(a -a»)
+ af>(a -a") "^
+ (a + a^) x'
+ (a + a") a.'"
+ af(a'- a*)
+ aV
+ af{l -a»).
+ o,f.
Hence we have
1 — a;"'-'
^ ^•^^ " " 1 - fw^ . 1 - oa; . 1 - cta;-M - a«-»
.«"» 1
l-ar'.l-ai'l-aa?
^l-a;»,l-a:* l-oa;
l-^af+{a^ + a^)a + {af + af) a' + (a^ + .-/.») a' 1
■^ l-ar'.l-a;' l-a«
which is. in {&
, ^, • f l-aa; + aV
ct. the exnresHioii for ^ — = :.
decnmnospr
into partial
542 ON AN ALGEBRAICAL OPERATION. [619
fractions with the denominators 1 — aa?, \—aic, \—a* respectively. This is most easily
seen by completing the decomposition, viz. we have
4 {1 +a!*+(ar' + a.'°)a + (a? + a:«)a= + (a^ + a.'»)a»!
= (1 +a:>)»(l +a;»)(l + a)(l +«») +(1 -a,')Ml + a.^)(l -a)(l +«') + 2(l -a^)(l -a^)(l -a=),
and thence the expression is
a;'" 1
~ \—a^.\—ofi\—aa?
-x* 1
l—a:^.l—x*l—ax
^*l-a?.l-a,'» l-o*l-a:>.l+ar'l+a'l+a;'H-o»
1 — aa; + aV
1 — ««' .l—ax. I —a*'
as above. Hence finally
1 — aofi .l—aw.l- «^"' . 1 —
ax~
1 —ax+ a'af
l — ax'.l—ax.l—a*
1 + a V
l-uic>.l-a*a^.l-a*
1 - aV
l-f/.r».l-a=ir'.l-aV.l-
-«*■
620] 543
620.
CORRECTION OF TWO NUMERICAL ERRORS IN SOHNKE'S
PAPER RESPECTING MODULAR EQUATIONS.
[From the Journal fiir die reine mid angeiuandte Matheumtik (Crelle), t. Lxxxi. (1876),
p. 229.]
In Sohnke's paper " Aeijuationes modulares pro ti-ansformatione functionum ollipti-
carum," Crelle, t. xvi. (1837), there is, on p. 118, an obvious error in the expre.ssion
of u", viz. the temi g" is given with the same numerical coefficient as it had in «":
this remark was made to me by Mr W. Barrett Davis, who finds that the term of
n* should be
+ 13.569463 7'".
In the expression of «'* (I.e., p. 115), I had remarked that, in the coefficient of g'", a
figure must have dropped out. Mr Davis has verified this, and finds that the figui-e
omitted is a 1 in the unit's place, and thus that the correct value* is
+ 801770337817".
Camhidge, 26 October, 187.5.
* [The former of these numbers should replace the nmnber 15063859 in the Table, p. 128 of this volume;
the Utter has been iotrodaced in the Table, p. 129.]
544
[621
621.
ON THE NUMBER OF THE UNIVALENT RADICALS C„H.
[From the Philosophical Magazine, series 5, vol. III. (1877), pp. 34, 35.]
I HAVE just remarked that the determination is contained in my paper "On the
Analytical Forms called Trees, &c.," British Association Report, 1875, [610]; in fact, in
the form C„H»„+j, there is one carbon atom distinguished from the othei-s by its being
combined with (instead of 4, only) 3 other atoms ; viz. these ai"e 3 carbon atoms, 2
carbon atoms and 1 hydrogen atom, or else 1 carbon atom and 2 hydrogen atoms
(CH;,, methyl, is an exception; but here the number is =1). The number of carbon
atoms thus combined with the first-mentioned atom is the number of main branches,
which is thus = 3, 2, or 1 ; hence we have, number of radicals OnHj^i+i is =
No. of carbon root-trees €« with one main branch,
4- No. of „ „ with two main branches,
4- No. of „ „ with three main branches ;
and the three terms for the values n = \ to 13 are given in Table VII. (pp. 454, 455
of this volume) of the paper referred to.
Thus, if n = 5, an extract from the Table (p. 454 of this volume), is
Index X, or
number of
knots
Index t, or num-
ber of main
branches
Altitude
0
1
3
3
4
5
1
1
2
1
4
2
2
1
3
3
1
1
4
1
1
1 1 Total ...
1
4
3
1
9
621]
ox THK NUMBER OF THE UNIVALENT RADICALS C„H.,„^.i .
545
and the number of the radicals C5H1, (isomeric amyls) is 4 + 3+1=8: or, what is the
same thing, it is 9 — 1, the corner-total less the number immediately above it. The
tree-forms con-esponding to the numbers 1, 2, 1 ; 2, 1 ; 1 in the body of the Table
are the trees 2 to 9 in the figure, p. 428 of this volume.
The numbers of the radicals CnRm+u as obtained from the Table in the manner
just explained, are : —
n=
1
Number of radicals C„H„^4.,.
1
1
Methyl.
2
1
1
Ethyl.
3
1
1
Propyl.
4
4
4
Butyls.
5
9
- 1
8
Aniyls.
6
18
- 1
17
Hexyls.
7
'42
- 3
39
Heptyls.
8
96
- 7
89
Octyls.
9
229
- 18
211
Nonyls.
10
549
- 42
507
Decyls.
11
1346
- 108
1238
Undecyls.
12
3326
- 269
3057
Dodecyls.
13
8329
- 691
7638
Tridecyls.
The question next in order, that of the determination of the number of the
bivalent radicals C„H2„, might be solved without much diflBculty.
Cambridge, November 20, 1876.
*
C. IX.
69
546 [622
622.
ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTFS
PROBLEM.
[From the Proceedings of the London Mathematical Society, vol. vii. (1875 — 1876),
pp. 38—42. Reafl December 9, 1875.]
I CONSIDER the equations
X, = by' + cz' - 2fyz - a (be -f% = 0,
y, = cz^ + ax- - 2yzx -h (ca— g% = 0,
Z, =aaf + by-- 2hxy - c (aft - h% = 0,
where the constants (a, 6, c, /, g, h) are such that
K, =abc- ap - bg' - cA» + Ifgh, = 0.
Hence, writing as usual (.4, B, C, F, 0, H) to denote the inverse coefficients
{be -f\ ca - g-, ab - /t^ gh - of, hf- bg, fg - ch),
we have (A, B, G, F, G, H^x, y, z)- = the square of a linear function, = (aa; + /Sy + 72)-'
suppo.se ; that is,
(A, B, C, F, G, H) = {(e, ^. rf. ^y, ya, a^).
It is to be shown that the three quadric surfaces Z = 0, F=0, Z = 0 intersect in a
conic 0 lying in the plane aaa; + b^y + cyz = 0, and in two points /, J ; or more
completely, that
the surfaces Y, Z meet in the conic Q and a conic P,
» ^t -^ i> >i )> Wi
» '^t ^ ») 11 » ."»
622] ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTl's PROBLEM. 547
where the conies P, Q, R each pass through the two points /, J, and meet the conic
0 in two points, viz.,
the conies P, 0 meet in two points Pj, P„,
Q, ® .. „ Qu Q„
„ R, © „ „ Pi, Pj.
For this purpose, writing
V = Aa-Ff, =Bb-Gg, =Gc- Hh, = ahc -fgh,
n = ^ (X + F+ Z), = aa? + h\f + cz- —fyz — gzx — hxy — V ,
0 = aax + b^y + crfz,
^ Aa Ff Ff
Og Bb Gg
^ Hh Hh Cc
then we have identically
tPfl- VF = %
cC£l-yZ =^f.
In fiict, the first of these equations, written at full length, is
cCA{aa^ + by'' +cz^—fyz — gza: — hxy— V)— V {by" + cz- — 2/yz — ciA)
= (acue + b0y + cyz)(~^a;+-^y + ^z],
where on the left-hand side the constant term is = 0. Comparing, first, the coefficients
of x', y', 2', on the two sides respectively, these are Aa", {Aa—V)b, (J.a— V)c, and
Aa?., Ffb, Ffc, which are equal. Comparing the coefficients of yz, zx, xy, the equations
which remain to be verified are
-(a4-2V)/=P/(c|+6|).
— aAg = Faf- + Aac - ,
-aAh =Faf^ + Aab^;
or, as these may be written,
- {aA - 2 V) ySy = F{cr + b^),
- Ag-ja = Ffa? Jr Acrf,
- Aha0 =Ffa" + Ab^';
69—2
548 ON A SYSTEM OF EQUATIONS [622
and, substituting for a*, )8*, 7*, (Sy, 7a, a^ their values, these may be verified wthout
diflSculty.
It thus appears that the equations of the three (]uadric surfaces may be written
in the form
and we thus obtain the conies Q, P, Q, ^ as the intersections of the surface fl = 0
by the four planes
fl-n ^_X-ft ^-J^-n ^ _ '' -n
• Bb Cc ' Cc Aa~ ' Aa Bb~ '
respectively. There is no difficulty in verifying that the conies intersect as mentioned
above, and that the coordinates of their points of intersection are
p,P^:Nbc, ^, ^\ i-^bc, - ^, -^L\-
\ ' "Jbc' Vbc' V ' Vic' Vic'
Q,Q. :(;i. VoS. ;a (-^. -Vca, -/);
^•■^■Uab' 'Jab' "*> \ ^ab' ^ab' "'V'
I, J : (f,g, h), (-/, -g, -h).
In a paper "On a system of Equations connected with Malfatti's Equation and on another
Algebraical System," Camb. and Dublin Math. Journal, vol. iv. (1849), pp. 270 — 275, [79],
I considered a system of equations which, writing therein ^ = 1, and changing the signs
of (f, g, h), are the equations here considered, X = 0, F=0, Z=0: only the constants
(a, b, c, f, g, h) are not connected by the equation /f = 0, but are perfectly arbitrary.
The three quadric surfaces intersect therefore in 8 points, the coordinates of which
are obtained in the paper just referred to, viz. making the above changes of notation,
the values are
a? = ^{aic +fgh -f^BC + g -JCA + h ^/AB),
f=~(abc +fgh +/ ^BC - g ^GA + A -^ AB),
z"-=^^(abc +fgh +f\/BC + g -JCA - h \'AB),
yz = h igh + af + ^BG),
^a; = i {hf +bg +VCA),
«y = i (f9 +ch + 'JAB) ;
where the radicals are such that ^/BCWCA.s/AB = ABC, so that the system (a^, if, z-,
yz, zx, xy) has four values only, and consequently {x, y, z) has eight values.
622] CONNECTED WITH MALFATTl's PROBLEM. 549
It is very remarkable that, introducing the foregoing relation K=0, there is not
in the solution any indication that the intersection has become a conic and two points,
but the solution gives eight determinate points, viz. the before-mentioned points
P, Pu Q, Qu R, R^, and /, /.
To develope the solution, remark that, in virtue of the relation in question, we have
y/BC=±F, ^CA = ±G, \/AB = ±H,
where the signs must be such that the product is = FGH (viz. they must be all
positive, or else one positive and the other two negative); for, taking the product to
be +FGH, the equations give
Q^ABC-FGH^
that is,
0 = Al,BG - F')- F(GH - AF), =K(Aa-Ff), = K^ ,
which is true in virtue of the relation K=0. Taking the signs all positive, we have
for a?, y, 2^, yz, zx, xy, the values /^ ^r', h\ gh, hf, fg, viz. we have thus the points
tf> a, fi\ (-/. -g, -'<■),
which are the points /, J. Taking the signs one positive and the other two negative,
say ^IBC=F, 'JCA = — G, 'J AB = —H, we find for a?, y^, z^, yz, zx, xy the values
be, J-, — , gh, bg, ch, viz. we have thus the points
o c
^ ' \/bc' *Jbc'' \ ' Vbc' 'Jbc'
which are the points P, P, ; and the other two combinations of sign give of course
the points Q, Qj and R, R^ respectively.
If the coefficients (a, b, c, f, g, h), instead of the foregoing relation K=0, satisfy
the relation
abc-af'--bg'--ch'-2fgh = 0, .say K' = 0,
the quadric surfaces intersect in 8 points, the coordinates of which are given by the
general formulae : but the expressions assume a very simple form. Writing for shortness
F' = gh + af, G' = hf+bg, H'=fg + ch,
then, in virtue of the assumed relation,
s/BG =±F', 'Jga = ±G', 'Jab = ± H',
where the signs are such that the product of the three terms is positive, viz. they
must be all positive, or else one positive and the other two negative. For, assuming
it to be so, we have
0==ABG-F'G'H',
550 ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTl'S PROBLEM. [622
that is,
0 = A (BC - F'-^ - F' (G'H' - AF'),
= K'{Aa + F'/), =K'{aJbc+fgh);
which is right, in virtue of the relation K' = 0. Taking the signs all positive, we find for
(a;*, y', z^, yz, zx, xy) the values {A, B, C, F', 0', H'), giving two points of intersection
Taking the signs one positive and the other two negative, say
^BC = F', 'JCA = - G', '^AB = - H',
we find for (or, y", z", yz, zx, xy) the values
(«, '^l f , r. 0, 0),
viz. we have thus two intersections
(»V?.^VS> (».-«■^/?.-^VS)^
and the other combinations of signs give the remaining two pairs of intersections
(«Vi;. »• yt> (-«Vi. »; V¥).
and
(^/f."VI.»)■ (Vf.-*V|.»)-
But the most convenient statement of the result is that the values of {jui?, by", cz^, yz, zx, xy),
for the four pairs of points respectively, are
(aA, bB, cC, F', 0', H'),
(0, cC, bB, r, 0, 0),
(cC , 0 , aA, 0 , G', 0 ),
(bB, aA, 0 , 0, 0, H');
there is no difficulty in substituting these values in the original equations, and in
verifying that the equations are in each case satisfied.
623]
551
623.
ON THKEE-BAR MOTION.
[From the Proceedings of the London Mathematical Society, vol. vii. (187.5 — 1876),
pp. 186—166. Read March 10, 1876.]
The discovery by Mr Roberts of the triple generation of a Three-Bar Curve,
throws a new light on the whole theory, and is a copious source of further develop-
ments*. The present paper gives in its most simple form the theorem of the triple
generation ; it also establishes the relation between the nodes and foci ; and it con-
tains other reseai'ches. I have made on the subject a further investigation, which I
give in a separate paper, "On the Bicursal Sextic," [624]; but the two papers are
intimately related and should be read in connection.
The Three-Bar Curve is derived from the motion of a system of three bars of
given lengths pivoted to each other, and to two fixed points, so as to form the three
sides of a quadrilateral, the fourth side of which is the line joining the two fixed
points ; the curve is described by a point rigidly connected with the middle bar ; or,
what is more convenient, we take the middle bar to be a triangle pivoted at the
extremities of the ba.se to the other two bars (say, the radial bars), and having its
vertex for the describing point.
Including the constants of position and magnitude, the Three-Bar Curve thus
depends on nine parameters ; viz. these are the coordinates of the two fixed points,
the lengths of the connecting bars, and the three sides of the triangle. It is known
that the curve is a tricircular trinodal sextic, and the equation of such a curve contains
27—6 — 6—3, =12 constants. Imposing on the curve the condition that the three
nodes lie upon a given curve, the number of constants is reduced to 12—3, =9: and
it is in this way that the Three-Bar Curve is distinguished from the general tricircular
* See his paper "On Three-Bar Motion in Plane Space," I.e., vol. vii., pp. 15 — 23, which contains more
than I had supposed of the resnlts here arrived at. There is no question as to Mr Roberts' priority in all his
results.
552 ON THREE-BAR MOTION. [623
trinodal sextic ; viz. in the Three-Bar Curve the two fixed points are foci, and they
determine a third focus * ; and the condition is that the nodes are situate on the circle
thiough the 3 foci.
The nodes are two of them arbitrary points on the circle ; and |the third of them
is a point such that, measuring the distances along the circle from any fixed point
of the circumference, the sum of the distances of the nodes is equal to the sum of
the distances of the foci. Considering the two fixed points as given, the curve
depends upon five parameters, viz. the lengths of the connecting bars and the sides of
the triangle. Taking the fortn of the triangle as given, there are then only three
parameters, say the lengths of the connecting bars and the base of the triangle; in
this case the third focus is determined, and therefore the circle through the three
foci ; we may then take two of the nodes as given points on this circle, and thereby
establish two relations between the three parameters, in fact, we thereby determine
the differences of the squares of the lengths in question : but the third node is then
an absolutely determined point on the circle, and we cannot make use of it for com-
pleting the determination of the parameters ; viz. one parameter remains arbitrary. Or,
what is the same thing, given the three foci and also the three nodes, consistently
with the foregoing conditions, viz. the nodes lie in the centre through the three foci,
the sum of the distances of the nodes being equal to the sum of the distances of
the foci : we have a singly infinite series of three-bar curves.
In reference to the notation proper for the theorem of the triple generation, I
shall, when only a single node of generation is attended to, take the curve to be
generated as shown in the annexed Figure 1 ; viz. 0 is the generating point, 0(7i5,
the triangle, (7, B the fixed points, CC^ and BB^^ the radial bars. The sides of the
Fig. 1.
triangle are a,, 6,, Ci ; its angles are 0, = A, B^, = B, C, , =C: the bars CC, and BBi
are =0^ and Oj respectively, and the distance CB is = a. The sides «i, 6,, Ci may be
put =Ai(sin.d, sin 2?, sin C), and the Hues a,, Oj, 03= (A-,, k.j, A:s)sin4, viz. the original
data Ui, 6,, c,, Oj, a.,, a^, may be replaced by the angles A, B, C {A+B + C = tr) and
the lines A,, kj, k,. And it is convenient to mention at once that the third focus
A is then a point such that ABC is a triangle similar and congruent to OB^Ci.
* A focus is a point, given as the intersection of a tangent to the curve from one circular point at infinity
with a tangent from the other circular point at infinity ; if the circular points are simple or multiple points on
the curve, then the tangent or tangents at a circular point alvould be excluded from the tangents from the
point ; and the intersection of two such tangents at the two circular points respectively is not an ordinary
focus; but, as the points in question are the only kind of foci occurring in the present paper, I have in the
text called them foci
*
623] ON THBEE-BAB, MOTION. 553
It may be remarked that, producing 00^ and BBi to meet in a point a, this is
the centre of instantaneous rotation of the triangle, and therefore aO is the normal to
the curve at 0,
I proceed to show that the three nodes F, G, H are in the circle circumscribed
about ABC, and that their positions are such that (the distances being measured along
the circle as before) we have the property, Sum of the distances of F, 6, H is equal
to the Sum of the distances of J., B, C.
Supposing 0 to be at a node F, we have then the two equal triangles FB^G^,
FBiCi, such that (7,, C/ are equidistant from G, and B^, B{ equidistant from B. Hence
the angles B^FB^, G^FCi are equal; consequently the halves of these angles GFC^ and
Fig. 2.
BFBi are equal; whence the angle GFB is equal to the angle G/FB^', that is, to the
angle A ; ot F lies on a circle through B, C such that the segment upon BG contains
the angle A, that is, upon the circle through A, B, G. To complete the investigation
of the nodes, suppose GF = t, BF = a : then the condition Z GFC^ = Z BFBi gives
fci' + T'-c'^Ci' + g'-a,'
26,T 2c,a
that is,
c,<r (6,» + T» - a,») - 6,T (c,= + o-' -«,») = 0 ;
and the condition that F is on the circle gives
cr' + t' — 2o-T cos A = a".
These equations give six values of (a, t) corresponding in pairs to each other; viz. if
(o-,, T,) is a solution, then (— cr,, — t,) is also a solution; and to each pair of solutions
corresponds a single point on the circle, viz. we have thus the three nodes F, G, H.
Writing the foregoing equation in the form
{c, (6,' - a,") o- - 6, (ci" - a,') t} (o-» + t' - 2<rT cos A) + a" (c,o-t» - 6,aV) = 0,
and putting the left-hand side = M (a- — Pit) (a- — p.iT) {a — p^r) ; then, if a, /3, 7 denote
008.4 +1 sin .4, cos J5 + 1 sin 5, cosG+iainG respectively, putting first o- = ot and next
<r = - , and dividing one of these results by the other, we find
C — b,a_ g -/),. a-pi. a-p,
Cia — bi~l-pi'x.l-pia.l-p3a'
c. IX. 70
554 ON THREE-BAR MOTION. [623
The left-hand side is here
_ sin (7 — a sin 5 _ sin 0— sin 5 (cos A +i sin A)
~ a sin 0— sin B sin (7 (cos A +i sin 4) — sin B
_ sin A (cos B — i sin B) _ y
~ —sinA (cos C — i8mC)~ /8 '
or the equation is
a-p^.a-p^.a-p, ^ y
l-pia.l-p^a.l-psa /3'
Also, \vriting / for the angle FOB, we have cr = . , a f\ t, viz. the values of
sin (A. +j )
sin/ sin g sin A „, , , „ ,
Pi, Pt, P> are . . /, -. , . , .\ — r , . ,.,,.. We thence find
■^ ^ -^ sm (^ -t-/) sin (A+g) sm (il -I- h)
a -jpi _ sin (A +f) (cos .4 + z sin A) — sin/_ cos (A +f) + i sin (A +/)
1 — op, sin (A +f ) — (cos ^ + 1 sin j4 ) sin/~ cos/— i sin/
= cos (4 -t- 2/" ) + 1 sin (^ + 2/) ;
with the like values for the other two values. Hence, writing also
- I = - cos((7-£)- isin(C- £) = cos (tt + C-B) + isin (tt + C-B),
the equation becomes
cos(3il + 2/+ 25r-H2A)-|-isin(3-4 -I- 2/+ 2g + 2h) = cos(w + C- B) + i ain {-n- + C - B),
that is,
SA + 2/+2g + 2h = -TT + C-B,
or, what is the same thing,
2f+2g + 2h = '7r + C-B-SA.
Fig. 3.
73
Reckoning the angles round the centre from a point 0 on the circumference, if
A', B', C, F, G', H' are the angles belonging to the points A, B, G, F, G, H
respectively, then
A' = \ + 2C, F' = X+2/,
B' = \, G' = \ + 2g,
C' = X + 2C + 2B, H' = \ + 2h;
and therefore
A' + B' + C' = SX + 4,C + 2B, F'+G' + H' = S\ + 2f+2g-^2h, =3\ + 'ir + C-B -SA;
623] ON THKEE-BAR MOTION. 555
that is, A' + B'+C'-F' -G'-H' = --ir + 3 (A +B+G); or, omitting an angle 2-n; this
is A' + B" + C' = F' + G' + H', the equation which determines the relation between the
three nodes on the circle ABC
Reverting to the equation CiO- (6,^ + t^ — Oa") — iir (cr + a-^ — a,^) = 0, which belongs to
a node : if we consider the form of the triangle as given, and write bi, c, = ki sin B,
ki sin G, this becomes
a sin C (bi' - a^) - t sin 5 (Ci' - af) + ar (t smC-a sin 5) = 0 ;
viz. considering the node as given, then the values of tr, t are given, and the equation
establishes a relation between the values of bi — a^ and Ci — a^*. If a second node
be given, we have a second relation between these same quantities, and the two
equations give the values of the two quantities, viz. the values of k^&hx^B — k^sm^ A,
k' k' k^ k^
ki^sin^C — kj'airi'A, or, what is the same thing, the value of -r-^— i . f „, .-l~. — • ! n-
° am'A Bin'B smM sm^B
It thus appears that, if Z,, L, I, are any values of k^, k^, k, belonging to a given system
of three nodes, the general values of ki, k,, ks belonging to the same system of three
nodes are
ki'^li' + u sin- A, Atj" = ij» + M sin= £, iV = 's" + m sin" (7,
where u is an arbitrary constant.
It may be added that there will be a node at B, if the equation is satisfied by
k k«
T = 0, 0- = a, for the condition is b^ —a^=0: that is, if -^-^. = . '„ : similarly, there will
smil sm.S •'
k k,
be a node at C, if <>.^—a^ = 0, that is, if ^— i-j = ^ — =; and a node at A, if
sm ^ sm jB
-r-*7i = -.— Si . If two of these equations are satisfied, the third equation is also
Bin C smB ^ ^
k k k
satisfied, viz. we then have ^-^.^ = . - „ = • ^^ ; and the three nodes coincide with the
svaA smij sinO
three foci respectively.
If, in Figure 2 (p. 553), the points C,, Gi coincide on the line GF, and therefore
also the points fi,, 5,' coincide on the line BF, then, instead of a node at F, we have
a cusp. We have in this case a triangle the sides of which are a^+b^, Os + Ci, a, and
the included angle between the first two sides is = J. : we have, therefore, the relation
a'' = (oj + bif + (tts + Ci)' - 2 (a, + b^) (a, + <h) cos A.
Substituting herein for a, Oj, bi, &c., the values A; sin .4, kiSmA, kiSinB, &c., the
equation is
l(?Bixi^A = (Jcie,inB->t-hsmAf + {kiSia G+kiSiuAf
— 2 (i, sin B + kisinA) {ki sin C + Ajj sin .4) cos A.
* Considering, in the equation, Oj and a^ aa the distances of a variable point P from the points G and B
respectively, the equation represents a circle having its centre on the line GB. Similarly, when a second
node is given, the corresponding equation represents another circle, having its centre on the line GB, and
the intersections of the two circles determine a, and a,, the lengths of the radial bars, in order that the
curve may have the given nodes.
70—2
556 ON THREE-BAR MOTION. [623
Expanding the right-hand side and reducing by means of A + B + C=ir, the whole
becomes divisible by sin' J, and we have
jt» = jfc,» + jfcj» -f- ^' — 2^^, cos A + 2ktk, cosB + 2kiki cos C ;
viz. considering A, B, C, k^, k^, k^ as given, this equation determines k so that the
curve may have a cusp. The equation is one of the system of four equations
A« = jfc,» -f Ar," -I- ki^ — 2kiki cos A + ^k^k^ cos £ + 2^^, cos G,
l(?z=k^ + kt' + kt* + 2kikt cos A - 2k,ki cos B + 2kiki cos G,
k^=zki^ + k^ + k^ + 2i2^'8 cos A + 2^,^;, cos B — 2kiki cos C,
k' = ki* + A,' + k,^ — 2^2 ij cos A — 2kiki cos JS — 2A, A-, cos G,
which belong to the different arrangements CG^F or GFCi, BB^F or BFB^, of the three
points on the lines BF and GF ; if k has any of these four values, the curve will
have a cusp. If two of the equations subsist together, we have a curve with two
cusps. Taking ki, k^, k^, and also cos 4, coaB, cos C, as positive, viz. assuming that the
triangle is acute-angled, the fourth equation cannot subsist with any one of the others:
but two of the others may subsist together, for instance, the first and second will do so,
k ki
if k,ktCosA=k,kiCoaB, that is, if — ^-= — =, and then }<? = ki' + k^^ + k,^ + 2kiki cos G :
coaA coaB
the curve has then two cusps. Similarly, the three equations may subsist together, viz.
we must then have
n?! iCj K^
i» = ii' + k^ + k^ + 2ktkt cos A ;
cos A cos B cos G '
writing herein ki, k^, Ai, = \cos4, \co8 5, XcosO, we find
A» = X." (cos' A + cos' B + cos' G -|- 2 cos J. cos 5 cos (7) = X» ;
viz. if ki,k.,,k, are respectively = A; cos J., kcoaB, kcosG, the curve has then three cusps.
It will be recollected that, if
ki : ki : k3 = sin A : sin B : sin G,
the nodes coincide with the foci ; the two sets of conditions subsist together, if
A— B = G = 60°; ki = kt = k, = ik, viz. we have then a curve with three cusps coinciding
with the three foci respectively.
Before going further, I will establish the theorem for the triple generation of
the curve.
The theorem which gives the triple generation may be stated as follows. See
Figures 4, 5, 6*.
Imagine a triangle ABG and a point 0, through which point are drawn lines
parallel to the sides dividing the triangle into three triangles OBfi^, OGtA^, OAiB,,
* Figure 6 (substantially the same as Fig. 5) belongs to the same curve as Figures 1 and 2, and it
exhibits the triple generation of this carve : the generating point 0 being taken at a node (the same node
as in Figure 2), and the two positions OBjCj and OBi'Ci of one of the triangles being shown in the figure.
623]
ON THREE-BAR MOTION.
557
similar inter se and to the original triangle, and into three parallelograms OA^AA,,
OBjBB^, OCiCCj. Then, considering the three triangles as pivoted together at the
point 0, and replacing the exterior sides of the parallelograms by pairs of bars
AtAAt, B^BBi, GfiCi pivoted together at A, B, C, and to the triangles at A^, A^, B^, B^,
C„ C„ the figure thus consisting of the three triangles and the six bars; let the
Fig. 4. Fig. 5. Fig. 6.
4'
three triangles be turned at pleasure about the point 0, so as to displace in any
maimer the points A, B, C: we have the theorem that the triangle ABG will remain
always similar to the original triangle ABC, that is, to each of the three triangles
OBiCi, OG^Ai, OAiBt'. and further, that, starting from any given positions of the
three triangles, we may so move them as not to alter the triangle ABG in magnitude :
whence, conversely, fixing the three points A, B, G, the point 0 will be moveable in
a curve.
Assuming this, it is clear that the locus of the point 0 is simultaneously the
locus given by
The triangle OBiGi , connected by bars BiB and (7,(7 to fixed points B, G,
„ OGfAi, „ G^G „ A2A „ G, A,
OA,B„ „ A,A „ B,B „ A,B;
or, that we have a triple generation of the same three-bar curve. It may be
remarked that the intersection of the lines BB^ and CCi is the axis of the instantaneous
rotation of the triangle OB^Gi, so that, joining this intersection with the point 0,
we have the normal at 0 to the locus; and similarly for the other two triangles.
It of course follows that the intersections of BB^ and (7(7,, of (7Cj and AA^, and of
AAi and BB^, lie on a line through 0, viz. this line is the normal at 0.
The result depends on the following theorem : viz. starting with the similar
triangles OB^Gu AjOG^, A^BtO, say, the angles of these are A, B, G, so that the
sides are
fc,(sin.4, sin 5, sin C), kj{sinA, sin 5, sin (7), kt(eixiA, sin 5, sin (7);
then it follows that the sides of the triangle ABG are
k (sin A, sin B, sin G),
the value of k being given by the equation
ifc« = jfc,« + A^» + A;,' -f- 2kjc, cos{X-A) + 2kA coa (Y-B)+ 2kA cos {Z- G),
558 ON THREE-BAR MOTION. [623
where X, F, Z denote the angles A^OA,, BfiBi, CiOC^ respectively: whence, since
A +B + C=7r, we have also X+Y + Z=7r. If therefore the angles X, Y, Z vary
in any manner subject to this last relation and to the equation 4" = const., the triangle
ABC will be constant in magnitude.
There is no difficulty in proving the theorem. Writing 00 =j, and OB = a, also
/L COCi = -^, and Z BOB^ = 4>, we have
. , , , , , o, „ sin i/r sin Z , 6, + a, cos Z
T* = Oi* + a,* + zoiaa cos Z, = , cos -Jr = — = ,
. , , , , - „ sin <f> sin F , Cj + a, cos F
<r» = Ci» + o,* + 2cia3 cos F, -— , cosd»=-= = ;
and then
a^=r''-\-(j^- 2x0- cos {A -V -^ ■\- (f)
= t" + o"' — 2to- cos A cos 4> cos •</r + 2to- cos A sin <^ sin i/r
+ 2to- sin A sin t^ cos <^ + 2t<7 sin A cos ■^ sin ^
= 6i» + Ci= + ttj' + a^ + 26iaj cos Z + 2cia, cos F
— 2 cos ^ (6j + a^ cos ^) (ci + a^ cos F)
+ 2 cos ^ . ajOj sin F sin Z
+ 2 sin ^ sin Z . a^ (ci + a, cos F)
+ 2 sin ^ sin F. 03(61 + 03 cos Z)
= bi" + Ci" - 26,c, cos A + a.j» + O3'
+ 2ojja3 [— cos A (— sin Fsin Z+ cos Fcos Z)
+ sin A (sin Fcos Z + cos Fsin Z)]
+ 203 [(C] — 6, cos A) cos F + 61 sin J. sin F]
+ 2aj [(6j — Ci cos J.) cos ^ + Ci sin A sin Z].
We have here 61° + Cj' — 2i»iCi cos ^4 = Oi" : the second line is = - 2aj05cos(ul + F+Z)
which, by virtue of F+^=7r — X, is = 20303 cos (X — ^): and in the third and fourth
lines
Ci — 61 cos il = Oi cos 5, 61 sin ^ = o, sin B,
bi — Ci cos A = ai cos C, Ci sin 4 = a, sin C ;
whence these lines are 20301 cos ( F — J5), 2ai02 cos (Z — 0) : the equation therefore is
a» = Oi' + a,' + o,» + 20,03 cos (X -A) + 2a^ cos (F- 5) + 2o,as cos {Z - 0),
which, putting therein for a,, Oj, a, the values A;isin J, k^sin A, Ar,8in4, and assuming
OR above
A:* = Ai" + Jks» + ifc,» + 2A;A cos (Z - .4 ) + 2^-3^ cos ( F - J?) + 2A;iA;, cos (Z - (7),
becomes a'' = jfc'sin'il, or say a = fc8in^; and similarly h = k8mB, c = ksaxC, that is,
{a, h, c) = A(8in^, sinJB, sinC), the required theorem.
623] ON THREE-BAR MOTION. 559
Before proceeding to find the equation of the curve, I insert, by way of lemma,
the following investigation: —
Three triads (A, B, G), {F, G, H), {I, J, K) of points in a line, or of lines
through a point, may be in cubic involution ; viz. representing A, B, &c. by the
equations x — ay = 0, x — hy = 0, &c., then this is the case when the cubic functions
{x-ay){x-hy){x-cy), (x -fy) (x - gy) (x - hy), (x-iy){x-jy)(x-ky),
are connected by a linear equation. Regarding I, J, K as given, the condition
establishes between {A, B, C) and (F, G, H) two relations: viz. these are
(i_a)(i-6)(i-c) : (j-a)(j-b){j-c) : (k- a)ik-b)(k-c)
= (i -/) (^■ - 9) (i -h):U -f) (j - g) (j - h) : (k -f) {k -g){k- h).
But, if if be regarded as indeterminate, then the condition establishes only the
single relation
(i - a) (i - b) (i - c) : ( j - a) (j - b) (j - c)
= (»• -/) (i -g){i-h): {j -f) (j - g) U - h),
which relation, if i = 0, j = <x> , takes the form abc=fgh. When K is thus indeter-
minate, we may say that the triads {A, B, 0), {F, G, H) are in cubic involution
with the duad /, J.
li A, B, &c. are points on a conic, then, considering the pencils obtained by
joining these points with a point 0 on the conic, if the cubic involution exists for
any particular position of 0, it will exist for every position whatever of 0 ; hence,
considering triads of points on a conic, we may have a cubic involution between
three triads, or between two triads and a duad, as above.
Taking x = 0, y=0 for the equations of the tangents at the points I, J respectively,
and z = 0 for the equation of the line joining these two points, the equation of the
conic may be taken to be xy—!? = 0, and consequently the coordinates of any point
A on the conic may be taken to be x : y : z = a : - : \. It is then readily shown
that a, y9, 7, f, g, h referring to the points A, B, G, F, G, H respectively, the condition
for the cubic involution of (^1, B, C), {F, G, H) with the duad (/, J) is a0'y=fgh.
And we thence at once prove the theorem, that there exists a cubic curve
J^IgI(;FGH, viz. a cubic curve passing through /, and having there the tangent JA,
having at / a node with the tangents IB, IG to the two branches respectively, and
pa.ssing through the points F, G, H ; viz. that the triads {A, B, G), (F, G, H) being in
cubic involution with (/, J) as above, there exists a cubic curve satisfying these
2 + 5+3, =10 conditions. In fact, the equation of the cubic curve is
JJJcFGH; [y-^^{x-^z){x-'iz)
9
+ — {(«- oir) {x - ^z) {x - riz) - {x -fz) {x - gz) {x - hz)] = 0,
560 ON THREE-BAR MOTION. [623
where observe that second term is an integral function - 2I' (— Mx + Nz), if, for shortness,
d
M=a + ^ + y-f-g-h,
N = ^y + ya + a^ - gh- h/-/g.
In fact, the equations of the lines J A, IB, IC are y — = 0, x — ^z = 0, x — yz = 0,
respectively, and we at once see that these lines are tangents at the points /, J
respectively; moreover, at the point F, we have x, y, z—f, 51 !• Substituting these
values, the equation becomes
(j-^)(/-)8)(/-7) + |.(/-a)(/-y3)(/-7) = 0,
viz. the equation is satisfied identically, or the curve passes through F; and similarly
the curve passes through 0 and H.
In precisely the same manner there exists a cubic curve Ij^J^JcFGH; viz. this is
IJ^JoFQH; {X -az)(y- 1) (y - J)
.f|(.-9(.-|)(.-5)-(,-|)(.-i)(.4))=».
where the second term is an integral function, az' (— M'y + N'z) ; if, for shortness,
a P y f g h a^y '
PI 7a «/3 gh hf fg a^y
in virtue of the relation a^y= fgk; so that the second term is in fact =-^(— Gx+ Bz).
py
Writing for shortness J^, I^ to denote these two cubics respectively, we have
four other like cubics, Jb{=J^IcIaFGH), Ib{= IbJcJaFGH), Jc{=JcIa^bJ^GH), and
!(,{= IcJaJbFGH); the equations being
Ja; (y-^)(^-yS^)(^-7^) + f {-Mx+Nz) = o,
Jb\ (y-|)(«-7^)(*--a2)+| {-Mx+Nz) = 0,
Jc\ {y--Mx-az){x-fiz)-\-- {-Mx + Nz) = 0,
\ y/ 7
/.; ix-az)(y-'^{y-'^ + ^^{-Ny + Mz)^0,
h\ (^-7^)(y-~)(y-|)+^(-iVy + ii/^) = o.
623] ON THREE-BAR MOTION.
We require the differences of the products I^Jaj IrJb, IqJc- We find
IsJB = {»=-oiz)(.<e- ^z)(j^-l^)[y-^-^[y - -^{y -^^ + ^;^i-Mx + Nz){- Ny + Mz)
■^^<r-Ny + M^) {y- 1) (« - 7^) (« - o^)
561
7«
z'
, J(- i»f^ + Nz) {x - ^z) (y _ J) (y - i) ;
let fl denote the sum of the two expressions in the first line. Similarly, we have
7e Jc = n + ^ (- % + -A^^) (y - -) (a; - 0^) (a; - /3z)
+ ^(^-Mx^Nz){x-rfz)(y-'-){^y-'-^).
We have thence
IsJb - IcJc = Z-' j^ (»,- o^) (- Ny + ilfz) - (y - ^) (- Mx + iV^)J
the factors in { } are respectively
fl 1
BO that we have
= (i - i) {,xy - z^) and {m - ^) {xy - z%
IsJh - IcJc = [-^-^^i^-M)z^xy- z^f.
The constant factor
M
IB
U 7] (a
= (S-f)-(|-f)'----
if P,, P,, P, denote respectively the functions
N_ M N M N M
/S*/ a ' 7a /9 ' a/8 7
Attending to the equation 0/87 ^fgh, it appears that we have
D 1/ a ^ LX /I 1 1 1 1 1\
with like values for P, and P,.
C. IX.
71
562 ON THREE-BAR MOTION. [623
We have thus
I^J„ - I,.Ja = - (A - Ps) z' {xy - z^r.
and similarly
I„J„ -I^J^=- (P, - A) z^ {xy - z%
IaJa - IbJh = - (-P. - P.) ^ (^ - zj.
Any function IaJ a + ^' (xy — z^f, where \ is arbitrary, can of course be expressed
in the form IaJa + (^+ Pi)z^ix!/ — z")-, where 6 is arbitrary, and therefore in the
three equivalent forms
lAJA+(0 + Pi)z^(xy-2j,
lBJB+(S + Pd^(xy-z'y,
IaJc+(0 + Ps)z^xy-zJ.
We have z = 0, the line IJ : and xi/ — z^ = 0, the conic IJABCFOH. The equation
I^J^+X2i'{an/ — z''y=0 may thus be written in the more complete form
I A J„JcFOH . J A InlcFOH + \ (IJy {IJABCFOH f = 0,
and we hence see that it is the equation of a sextic curve, having a triple point
at /, the tangents there being I A, IB, IC; having a triple point at J, the tangents
there being JA, JB, JC ; and having a node (double point) at each of the points
F, G, H. There are thus in all (6 + 3)4-(6 + 3) + 3 + 3 + 3, =27 conditions, and these
would in general be sufficient to determine the sextic. The data are, however,
related in a special manner; viz. regarding the points /, J, F, 0, H as arbitrary,
the lines I A, IB, IC, J A, JB, JC are not arbitrary, but satisfy the conditions that
A, B are arbitrary points, and C a determinate point, on the conic IJABC. And
the foregoing result shows that, this being so, there exists a sextic satisfying the
foregoing conditions, but containing in its equation an arbitrary constant \ or 0, and
that the equation may be presented under the three forms
IaJbJcPOH . J A IbIcFGH + {6 + P,) {IJf {IJABCFGHy = 0, &c.,
corresponding to the partitions A, BC; B, GA; C, AB o{ the three points A, B, C.
In the case where /, J are the circular points at infinity, the conic IJABCFGH
is a circle passing through the six points A, B, C, F, G, H; and the condition
of the cubic involution of the triads {A, B, C) and {F, G, H) with the points (/, /)
is easily seen to be equivalent to the following relation, viz. the sum of the
distances (measured along the circle from any fixed point of the circumference) of
the three points A, B, C is equal to the sum of the distances of the three points
F, G, H.
The sextic is a tricircular sextic having the three points A, B, C for foci, and
having three nodes F, G, H, on the circle ABC, two of them being arbitrary points,
and the third of them a determinate point on this circle. And it appears that there
exists a sextic satisfying the foregoing conditions, and containing in its equation an
arbitrary parameter.
623] ON THREE-BAR MOTION. 563
I proceed to find the equation of the curve.
Consider the curve (see Fig. 1, p. 552) as generated by the point 0, the vertex of
the triangle OC\Bi, connected by the bars Gfi and BiB with the fixed points C and B
respectively; and suppose, as before, CB = a, 0^0 = a^, BiB = a3, BiCi = ai, OCi = hi,
OBi = Ci ; and draw as in the figure the parallelograms 0^00^0 and BiBBfi ; then 0
may be considered as the intersection of a circle, centre G^ and radius C.2O, with a
circle, centre B^ and radius B^O. Take zGfiB=d, /.B,BG = <^: the lines GG,_, BB^
are parallel to 00,, OjB, respectively, and consequently 0 + <f>=-7r — A, a relation between
the two variable angles 6, <f>.
Taking the origin at G and the axis of x along the line GB, that of y being
at right angles to it : the cooi-dinates of G« are (61 cos d, 61 sin 6), and those of iJ,
are (a - Ci cos if>, c, sin (f>) ; the equations of the circles thus are
(x — 6] cos 6y +(y — bi sin tif = a^,
(x — a + Ci cos ^y + (2/ — Ci sin <^)^ = a^ ;
whence
+ 26ja- COS 9 + 2b^ sin 6= a? + y"-->r b,' - Oa',
— 2ci {x — a) cos <^ + 2ciy sin tf>=(x — ay + y' + c{- — a^,
which equations, writing therein for 6 its value ='ir — A — <f> and eliminating the
single parameter <f>, give the equation of the curve.
We in fact have
- 26,a; cos (A +<f>) + 2% sin (-4 + <^) = a;' + y' + 61' - (h\
— 2ci{x — a)cos<f) +2cjysm<p ={x-af + y*-\-c^-a^;
or say these are
P cos <^ + Q sin </) = R,
P' cos </) + Q' sin ^ = R,
where
P = - 26,a; cos A + 26;jr sin A, P' = - 2ci (a; - a),
Q = 26,a; sin 4 + 26iy cos j1, Q' = 2ciy,
R= a? + ^^ + 6,^ - ((2-, i2' = (a:-a)» + y= + c,^-a,l
The equations give therefore
cos(^ : sine/) : -l = QR-Q'R : RF - R'P : FQ' - P'Q,
whence
(Qii' - e'ii)= + (RF - R'Py = (PQ' - FQy ;
and it hence follows that the nodes are the common intersections of the three curves
QR' - Q'R = 0, RF- RP = 0, PQ' - FQ = 0.
71—2
564 ON THREE-BAR MOTION. [623
We have, retaining R and R' to denote their values,
QR -Q'R^-i [(Re, - R\ cosA)y- R% sin A . x],
RP' -R'P=-2 [(Rci - R\ cos A) {x-a)-\- R% sin ^ (y - a cot ^1)],
PQ' - P'Q = - 4 6,Ci [x{x - a) -iry {y- a cot A)].
Observing that iZ = 0, iJ' = 0 are circles ; the equation QR' — QR = 0 is a circular
cubic through the point x = Q, y = 0 ; the equation RP" — R'P = 0, a circular cubic
through the point x=a, y = a cot A ; and the equation PQ' — PQ = 0, a circle through
these two points (and also the points a; = 0, y = a cot A; x = a, y = 0). Hence the
first and third curves intersect in the point (« = 0, y = 0), in the circular points at
infinity, and in three other points which are the nodes; viz. the curve has three nodes,
say these are F, 0, H. The second and third curves intersect in the point (x = 0,
y = acot A), in the circular points at infinity, and in the three nodes. As regards the
first and second curves, it is readily shown that these touch at the circular points at
infinity ; viz. they intersect in these points each twice, in the two finite intersections of
the circles jR = 0, R' = 0, and in the three nodes.
The three nodes F, G, H thus lie in the circle
x(x — a) + y(y — a cot A) = 0,
which passes through the points (x = 0, y = 0) and (x = a, y = 0), that is, the points
G and B. Assuming 6= r „-, the circle also passes through the point x = bcosG,
y = 6 sin C, that is, the point A of the figure. Thus the three nodes F, 0, H lie in
the circle circumscribed about the triangle ABC.
Writing, for greater convenience,
R = a?+y^-e!', R' = a? + y^ -2ax-f\
the nodes F, 0, H lie on the two curves
Ciy(a^+2/''-e=)-6,sin A {x -\-y cot A)(a? + y'' -2ax-p)=0,
af + y^=a(x + y cot A).
The first of these is
[c,3/ - 6, sin A(x + y cot A)]{a^ + y^)
+ [6j sin A(x + y cot A)p - c,e=y]
+ 2a6, s\nA(x + ycotA)x = 0.
We may combine these equations so as to obtain the equation of the triad of
lines OF, CO, CH ; viz. multiplying the second and the third terms of the first equation
(X^ -I. •J/3\* /»>'- 4. 7y2
by „t/^,.,„^t A\a ^^^ ~l — ; . A\ (each = 1 in virtue of the second equation), the
a' (x + y cot Af a{x-\-ycotA) ^ ''
equation becomes divisible by a? + y'': and, throwing this out, the equation is
c^y — 6, sin A{x-\-y cot .4)
+ [6, sin 4 (. + j,cot ^)/' - c,e'<,] ^.^^
+ 26,a; sin A = Q,
623] ON THREE-BAR MOTION. 565
where the first and the third terms together are = (ci — b^ cos A) y + b^x sin A, viz. this
is =aisin B{x + y cot B). Hence, writing also in the second term Oj sin B for &i sin A ,
the equation is
1
Ce^ 1
{x + ycotAy(x + ycotB) + -^(x + ycotA)p-^^^^y'^{aP + f) = 0;
or say this is
(x sinA+y cos Ay (x aia B + y cos B)
sin A sin B
-^^^ i^{x sin A+y cos A)/' -^y'^iaf + f)=0;
viz. there is a term in af + y-, and another term
{x sin A + y cos A)' (x sin B + y cos 5).
Suppose for a moment that the angles FBC, GBC, HBG are called F, G, H ; then
the function on the left hand must be
= ilf (a; sin ^— y cos F) (x sin G — ycos G) (x siuH -y cos H).
Writing in the identity x = iy, we have
(cos A +iBinAy (cos £ + j sin J?) = — ilf (cos F — i sin F) (cos G — i sin G) (cos H — i sin iT) ;
and similarly, writing x = — iy, we have the like equation with — i instead of +i;
whence, dividing the two equations and taking the logarithms,
4.A+2B = 2?«7r -F-G-H,
which leads as before to the relation A' +B' + C = F' + G' + H'.
Ip completion of the investigation, observe that M is determinately +1 or — 1 :
and that
sin AainB \. . , . . -, c, , )
Ux sin A+y cos ^ )/' - ^ ^I/[
is the linear factor of
M {xainF—y cos F) (x sin G—y cos G)(x8inH—y cos H)
— (xsin A +y cos Ay{xsinB + y cos j5),
which remains after throwing out the factor of + y". Calling this linear factor px + qy,
we have
. °'P. p=/'sin^, . ":\ j,=/^cosA-'^e',
Bin Asm B •' siaAmnB •' 6,
or, as this last equation may be written,
a'q ,. . sin (7 ,
. . ■ — D =/ " cos 4 - ^ f^e\
sin il sm ij •' sm B
Hence, writing a = A;8in.4, we have
f^=-=-^n^ e^=-. ^(pcosA—qsinA):
•' sin 5 smC^^ ^ ' ■
566 ON THREE-BAK MOTION. [623
substituting for/* and e* their values, we have
- /t,> sin^C + kt^Bin" A = ^^ + ^sinM,
sin 5
- A,' sin' B + h^ sin' A = -. — ~ ( p cos A— q sin A),
smC ^ ^
or, what is the same thing.
^-,— f + --rn = ■ i A • — p • .nip + S"l' A Sin B)
sin' A sm' C sin' A sin B sm' (7 ^ ^
A-- sin ^sin G sin ii/
sin A sin C . sin A sin ^ sin 0 '
r-T— . + .-vn = ■ > . — . , „ ■ -, (» COS 4 — o sin ^ )
sm' A sin' B sm- A sin- 5 sm G ^^ ^ '
_ ^--^ sin (A - F) sin (^ - G) sin (^ - H)
sin ^ sin if . sin A sin JS sin C
which are the relations connecting k^, k.,, k^, when the foci and nodes are given.
It is to be remarked that if, for instance, F = 0 and G = A, then i, : k. : ks
= sin .4 : sin £ : sin C ; the nodes in this case coincide with the foci. A simple example
is when A=B = G; the three triangles are here equal equilateral triangles. The general
equations show that, if l^, l^, I3 are values of ki, k,, k, belonging to a given set of
nodes and foci, then the values ki' = li- + usiQ^ A, kfl' = 1^- +usm^ B, k3^=ls- + usui-C (where
u is arbitrary) will belong to the same set of nodes and foci.
I write the equation of the curve in the form
{{QR -Q:R) + i {RP' - B'F)} [QE -QR-i {RF - R'F)\ - (PQ' - P'Qf = 0,
where
iQR' - QR) + i (RP' - R'P) = (A;, - Rlh cos A)i{x- a- iy) - R% sin A{x-i(y-a cot A)}.
Calling /, J the circular points (« , a; + ii/ = 0) and (oc , x — iy=0), this is a nodal
circular cubic having / for an ordinary point, but J for a node. Moreover, one of the
tangents at J is the line x — iy = 0, that is, the line JB ; in fact, writing as before
R = x- + y'- — e-, R' = x- +y" — 2ax —/-,
then, when x — iy = 0, we have R = -e-, R' = —2ax—f', and the equation becomes
{-ce^ + b, cos A (2cuc +/')] (- ia) + 6, sin .4 {2ax -f-/') (ia cot -4 ) = 0 ;
viz. the term in x here disappears, or the three intersections are at infinity. The
other tangent at J is the line x — a — iy = 0, that is, the line JG ; in fact, when
a — a-iy = 0, that is, y = — i{x — a), we have R = 2aa; - a' - e', R' = — a^ — /-, and the
equation becomes
{C (2cw; - a« - e') -I- 6, cos .4 (a» +/')} . 0 -f 6, sin .4 (a' +/') . a (1 -f t cot .4 ) = 0,
623] ON THREE-BAR MOTION. 567
viz. the three intersections are here at infinity. The tangent at / is the line
x — b cos C + i(y — bsmC) = 0, that is, the line lA ; in fact, writing this in the form
y = ix — ib (cos C + i sin O) = ix — iby,
(if for a moment cos C + ismC= y, and similarly cos A + isinA = a, cos B + iainB= 0);
then, y having this value, we find
R = 2bxy - by - e-, R'=2(by-a)x- by -f%
and the equation becomes
= _|%_6y_/=;
Ci{2byx-b"-y- -e')^
2c
i (2a; — a — by)
— bicoa A (— r, "" — b-'f — e')
— btsmA (— -^x— by — ^](2x+ ia cot A — by) = 0.
The coefficient of x' is here *
2.-(fc„.^.!).
or, since &iC = &c,, this is
= 2ibc.{y + ^), =0,
in virtue of the relation A + jB + C = tt, giving a^Sy = — 1 : hence there is only one
finite intersection, or the line IA is a tangent.
The cubic in question
QR' -QR + i {RF - R'P) = 0 .
is thus a nodal circular cubic which it is convenient to represent in the form
(IjJ„JrFGH) = 0;
viz. this is a cubic, through / with the tangent I A, having / as a node with the
tangents JB, JC, and through the points F, G, H. Observe that, if F, 0, H were
arbitrary, this would be 2 + 5 + 3, = 10 conditions. The before-mentioned relation is, in
fact, the condition in order to the existence of the cubic.
Similarly the cubic
QR' - Q'R - i (RP' - R'P) = 0
is the cubic
{JJ„I,FGH)^0.
The circle PQ'-P'Q^O is the conic through /, J, A, B, C, F, G, H; or it may
in like manner be written (IJABCFGH) = 0 ; and we may write (/./) = 0, as the
equation of the line infinity. The functions denoted as above contain implicitly
568 ON THREE-BAR MOTION. [623
constant multipliers which give, in the equation of the three-bar curve, one arbitrary
parameter — and the equation thus is
{I^JsJcFQH) (JJJoFOH) - e (Ijy (IJABCFOHy = 0,
a form which puts in evidence that /, / are triple points having the tangents
lA, IB, IC, and JA, JB, JC respectively (whence also A, B, C are foci), and that
F, 0, H are nodes ; viz. the result is as follows : —
Taking A, B, C, F, 0, H points in a circle, such that, Sum of the distances
(being the angular distances from a fixed point in the circumference) oi A, B,G \s equal
to the sum of the distances of F, G, H : then there exist the cubics {IaJb Jc^OH) = 0,
{J^IglcFGH) = 0, and the sextic is as above.
Writing for shortness
il^J^JaFGH) = I A , {JJJcFGH) = J A ,
then the above form is clearly one of three equivalent forms
= la Jo - e,£i\
This implies an identical linear relation between the functions IaJa> ^b/b. IcJc\ whence
also U and Q? are each of them a linear function of any two of these quantities.
I originally obtained the equation of the curve in a form which, though far less
valuable than the preceding one, is nevertheless worth preserving; viz. the equation
{QR - qny + (rp' - Rpy = {pq - rqy
may be written
(if - i^ - (?) (E'' - F' - Q'-') - {RR - PP' - QQJ = 0,
which equation, substituting therein for P, Q, R, P', Q, R their values, gives the form
in question.
Proceeding to the reduction, we have
^ ={!i? + ff-2 (6,' + a,') (a? + y') + (6,^ - a,J
= (x' + y'-bi + <h){a? + y--b,-a^);
R'^-P'»-Q'* = (x-a + y''+ c' - (h'Y - ici'ix-a +f)
= {x-a + y')= -- 2 (c» + a,») (x - a + y*) + (c," - a,>)«
■■{x-a +i/'-Ci + (h){x-a +y^-Ci-at)-
623] ON THREE-BAR MOTION. 569
But the reduction of RR' — PP'—QQ' is somewhat longer. We have
RR' - PF -Q(^ = (*•= + y' + b,^ - ai) (x -a' + y'+ c/-' - a,')
— 46iCi {xx — a + y^) cos A — 46,Ci ay sin A :
and here
26iCi cos A = b^- + c,- — ft]-, 2x (x — a) + 2y'- = afl + y^ + {x — a)- + y- — a-,
also h,CiSm A = OiPi, if jw, be the perpendicular distance of 0 from the base Bfii.
Hence the second line is
— (bi- + c,- — Ui^) {x' + y^ + x — a +y- — a?') — ^aa^piy,
and the whole is
whence, finallj', we have f*
RR' - PP' - QQ' = (jf + y' + a,-- a/ - c,») (x-a +y' + a,- - a^' - 6,-)
+ (a= + a,= - a.f - a^") (6,^ + c{' - Oi^) - 4>aa^p^y.
Hence the equation of the curve is
(ar + y"- -bi + ai)(af + y^-bi-a,)(x- a + y- - Ci + a, ) (x -a + y--Ci- a,)
- {(x' +y'- + a,' - «./ - cr) (x-a +f + a," - a,= - 6,-)
+ (a» + ai' - ai - ct,=) (6,-^ + c,' - a.^) _ ^aa^p.yY = 0,
where pi is given in terms of the constants a,, 6,, Ci by the equation
There are in the equation two terms, (ar' + y')', {x-a +y''y, which destroy each other,
and the remaining terms are of the order 6 at most. Hence the curve is a sextic ;
and it is, moreover, readily seen that the curve is tricircular. Assuming this, it
appears at once that the lines x+iy = 0, x — iy=0 are tangents to the cui-ve at the
two circular points at infinity. In fact, assuming either of these equations, we have
■r= + y= = 0, and the equation becomes
(6,' - a,») (— -Zax + a:' — Cj + Os ) (— 2ax + a' — Ci - Ua )
- {(oi" - a^' - ci') (- 2ax + a= + a,'-' - a,' - bi')
+ (a' + ar - a,' - a-/) {b,' + c,= - 0,'} - '^aa.p.yY = 0,
a quadric equation. Hence there are on each of the two lines only two finite inter-
sections, or the number of intersections at infinity is = 4 ; viz. the line is a tangent
c. IX. 72
570 ON THEEE-BAR MOTION. [623
to one of the branches at the triple point. Similarly, the lines a; — a + ty = 0,
x — a — iy = 0 are tangents. Thus the points C and B are foci. It might with some-
what more difficulty be shown from the equation that the point a; = 6 cos C, y = 6 sin (7
(where, as before, 6= .^p ), viz. the point A of the figure, is a focus; but I have
not verified this directly. It clearly follows, if we generate the curve by means of the
triangle OAfit and the fixed points G, A. Hence A, B, G are a triad of foci, and
the theorem as to the nodes is that these lie on the circle drawn through the three
foci A, B, G.
I prove in a somewhat different manner, for the sake of the further theory which
arises, the theorem of the triple genei-ation ; for this purpose, constructing the foregoing
Figure 2 (p. 553) by means of the three triangles OBfi^, OG^A^, OA^B:,, but without
assuming anything as to the form or position of the triangle ABG, I draw through
0 a line Ox, the position of which is in the first instance arbitrary, say its inclination
to OO, is =v; and drawing Oy at right angles to Ox, I proceed, in regaixl to these
axes, to find the cooixiinates of the points G, B. We have, for G,
X — (u cos v + bj cos ( V + Z), y = a„ sin y + 6, sin {v +Z);
for B,
a; = Ci cos (w + .4 + ^) + a, cos {v + A + Z + Y),
y = Ci&\n{v + A + Z) + a-iSm {v\- A + Z Jf Y);
or, writing for Y+Z the value tt — X, so that
v + A+Z+Y=ir + v + A-X,
the coordinates of B are
x = Ci cos (v + A + Z) — a3Cos(v + A — X),
y = Cisin (v + A + Z) - a-jsin (v + A — X).
Taking the two values of y equal to each other, the equation to determine v is
aj sin 1/ + 6i sin (v + Z) — c, sin (v + A + Z) + Ossin (v + A - X) = 0.
We make the line Ox parallel to BG, so that, writing
X — lu cos V +6, cos {v + Z),
X — a = Ci cos (v + A + Z) — a3 cos (v + A — X),
we have
a - rtj cos u + 6, cos (v + Z) — c,cos{v+A+Z) + a3Cos{v + A - X),
which determines the distance BG, = a. And moreover, writing
y = O2 sin w + 61 sin (v + Z),
= c, sin ( v + yl + Z) — a, sin {v-\- A — X),
623] ON THREE-BAR MOTION. 571
we have y as the perpendicular distance of 0 from BG, and *■ and (a — a;) as the
two parts into which BG is divided by the foot of this perpendicular. In the reduction
of the formulse we assume that the three triangles are similar; viz. we write
(«!, k, Ci), {CU, b^, Co), (tta, 63, c)
= ^-, (sin.4, sin 5, sinC), A;j(sin^, sin 5, sinC), &s(sin^, sin 5, sin (7);
and we use when required the relation A -{■ B + G = ir.
The equation for v becomes
ki sin (u - G + Z) + L sin v + k; sin {v + A - X) = 0,
which may be written
Z sin w — ilf cos u = 0,
where
L =L + i-, cos (Z -G) + ^'3 cos (X — A),
M= -k,8m{Z-G)-\-k,8m(X-A);
hence, putting
k^ = fc,- + /;/ + k,^ + 2kjc^ cos (Z - .4 ) + 2^-^ cos ( F - 5) + 2^1^:3 cos {Z - G),
we have D + M^ = k', that is, •J'D-^M- = k, and therefore
^ sin V = Af, k cos v = Z,
which gives the value of v; and then, after all reductions,
kx = k^%m B cos G + k.^ sm A +k^.O ■\-kjC3»va. A COS {X — A)
+ ^/i [— sin B cos ( 1' f A)]
+ kjci [.sin (5 - A) cos {Z + A) + 2 sin .4 sin 5 sin (.2^ 4- .4)],
k(a — x)= ki' sin C cos B + k^- . 0 + A;/ sin A + Lk^ sin A cos (AT — .4 )
+ k^; [sin (C- ^) cos(r + ^) + 2 sin A sin C sin (F+ A)]
+ ^,io [- sin Gcoa (Z+A)],
and
Ay = A;,' sin B nin C+k^Jc, sin .4 sin (X - .4) + k^ki sin B sin ( 1'+ .4) + k^k^ sin Csin (Z + 4).
The first and second equations give ka^k'sin A, that is, a = A'sin.4; and, similarly,
b = ksm B, c = A; sin (7 ; viz. we have
(a, b, c) = A;(sin .4, sin B, sin 0),
or the triangle ABG is similar to the other three triangles, its magnitude being given
by the foregoing equation for k'. These are the properties which give the triple
generation.
72—2
572 ON THREE-BAR MOTION. [623
Changing the notation of the coordinates, a!id writing (x, y, z) for the perpend-
icular distances from 0 on the sides of the triangle ABC, we have, as above,
kx = ir,' sin BsinC + kjc^ sin A sin {X — A) + kjct sin 5 sin (F+ ^) + A;,AjSin CsiniZ + A),
and therefore
ky = k/ sin 5 sin C + A-^ sin A?.m{X + B} + k^kj sin B sin {Y-B) + A-,A-j sin G sin (Z + B),
kz = k^ sin Csm A + kjc^ sin A sin {X + 0) + kjc^ sin B sin ( F + C) + A:,^;™ sin C sin (^ — (7),
values which give, as they should do,
X sin A-\-ysinB + zsmC=k? sin A sin £ sin C.
Taking {x, y, z) as simply proportional (instead of equal) to the perpendicular
distances, then {x, y, z) will be a system of trilinear coordinates in which the equation
of the line infinity is
a; sin ^4. + y sin 5 + 2: sin (7 = 0 ;
and considering (x, y, z) as proportional to the foregoing values, and in these X, Y, Z
as connected by the equation X +Y+ Z^ir and by the equation which determines Ar",
the coordinates {x, y, z) are given as proportional to functions of a single parameter,
so that the equations in effect determine the curve which is the locus of 0.
But to determine the order, &c., the trigonometrical functions must be expressed
algebraically ; and this is done most readily by introducing instead of X, Y, Z the
functions
cosX + isinZ, cosF + isinF, cos^ + isin^, =f, 7;, ?';
and we may at the same time, in place of ^, B, G, introduce the functions
cos A +ism A, cos B + i sin B, cos (7 + i sin 0, = a, /3, 7.
The relation X +Y + Z =Tr gives fj7^=-l; and similarly A + B+G = tt gives
a/37 = -l.
We have
cos(Z-4) = j(| + |), isin(Z-^)=j(^-|), &c.;
the equation k' = A;i'' + &c. becomes
A:^ = A;,^ + A-,^ + AV + kJc, (| + |) + kk Q + f ) + ^^^' (7 + ?) ■
or, as this may be written,
(_ k^ + k,^ + 1^ + k,') + t,k, (I -«'??)+ kA (I - /Sr?) + A^ k, (^ - y^v) = 0.
Also the value of x is proportional to
623] ON THREE-BAR MOTION, 573
or, what is the same thing, to
with the like expressions as to the values of y and z. Introducing for homogeneity
a quantity as, viz. writing - , - , - in place of f , rj, ^, we have the parameters
{?. '7. ?. «*>) connected by the homogeoeous equations
(- /c' + k,- + L' + k,') «» + k, A.-3 (^^ - a,,?) + tA (^^ - ;Sr|) + k, k, (^- - y^rj^ = 0,
and the ratios of the coordinates are
»^ + hk. (^ - ^) («,a, + f ) + A-,L (7 - ^) (<« + ^^)
+ A..A.. (/3 - J) (r;« + f ) + ^'.A;, (7 - ^) (f^ + 7?,) .
Suppose, for shortness, these are x : y : z = P : Q : R. Observe that the form of
the equations is fj;5'+(a' = 0, 11 = 0, and x : y : z-F : Q : B, where fi and F, Q, R
are each of them a quadric function of the form ((o\ a>^, cori, wif, Tjf, ff, f ??), the terms
in ^, T)^, ^ being wanting.
Treating (^, 77, f, ta) as the coordinates of a point in space, the equation ^rj^+m^^O
is a cubic surface having a binode at each of the points (? = 0, o) = 0), (77 = 0, ft)=0),
(f=0, a) = 0), and the second equation is that of a quadric surface passing through
these three points; hence the two equations together represent a sextic in space, or
say a skew sextic, having a node at each of these three points. The equations
X : y : z = F : Q : R establish a (1, 1) correspondence between the locus of 0 and
this skew sextic. To find the degree of the locus we intersect it by the arbitrary line
ax + by + cz = 0; viz. we intersect the skew sextic by the quadric surface aF +bQ + cR = 0.
This is a surface passing through the three nodes of the skew sextic, and it there-
fore besides iotersects the skew sextic in 12 — 2.3, =6 points. Hence the locus is
(as it should be) a sextic.
574 ON THREK-BAR MOTION. [623
I consider the point 17 = 0, 5' = 0, a> = 0, or say the point (1, 0, 0, 0), of the
skew sextic. This is a node, and for the consecutive point on one branch we have
7] : f : 0) = me : le' : ne, where e is infinitesimal. The equation of the cubic surface
gives Ivi + n" = 0, and the equation of the quadric surface gives kjc, . kjc^fq = 0,
that -is, k3a> = ayki7], which, in fact, determines the ratio I : m; but it will be
convenient to retain the equation in this form. For the corresponding values of
(x, y, z) we have
a=:y:z= ^•,(a-^) 1 + ^,(^,-1)5
' ■ ■.k,(a-^'ia>+h{y-^^7,,
which, writing for k^a its value = ay^ji;, become
a- : y : 2= (a--) y+(y--]-= 07-'^ + -
^ \ aj \ yj a ' a. a. ya.
(a--)a/37+(7-l)^ :_„ + i_ay: + «
(
« - -j ay' + (7 - -] 7 : *V - 7' + 7' - 1.
the last set of values being obtained by aid of the relation 0/87 = — 1 ; viz. we
thus have
that is,
1 ,
^ 7
which are, in fact, the values belonging to one of the circular points at infinity.
For the consecutive point on the other branch we should obtain in like manner
X : y : z = y : —\ : - , which are the values belonging to the other circular point at
infinity; viz. the node (1, 0, 0, 0) of the skew sextic corresponds to the circular
points at infinity. But, in like manner, the other two nodes (0, 1, 0, 0) and (0, 0, 1, 0)
each correspond to the circular points at infinity, or say we have in the skew sextic
the three nodes each corresponding to one circular point at infiuity, and the same
three nodes each coiresponding to the other circular point at infinity; viz. we thus
prove that each of the circular points at infinity is a triple point on the locus of 0.
In order not to interrupt the demonstration, I have assumed the formulae which,
in the system of coordinates defined by taking x, y, z proportional to the perpendiculai-s
on the sides of a triangle ABC, or where the equation of the line infinity is
xsinA+yamB + zsinC=0,
623] ON THREE-BAR MOTION. 575
give the cii-cular points at infinity ; viz. writing
cos A +ismA, cos B + i sin B, cos C + i sin C = a, /3, 7,
the coordinates for the two points respectively are
X : y : z = — I : 7-o ^^^^ x : y : z = — \ : -:/8
= -:— l:a =7:— 1:-
7 a
= y3:^:-l. =!:«:- 1,
a ^
the three vahies for each point being equivalent in virtue of the relation a/S^ = — 1.
This is, in fact, under a different i'orni, the theorem given in my Smith's Prize
paper for 1875; viz. the theorem was: If \, /t, v are the inclinations to a fixed
line of the perpendiculars let fall from an interior point on the sides of the funda-
mental triangle ABC, then, in the system of trilinear coordinates in which the
co<jrdinates of a point P are proportional to the triangles PBC, PGA, PAB (or
where the equation of the line infinity is x + y-\-z = 0), the coordinates of the circular
points at infinity are proportional, those of the one point to e'^ sin {fi — v), e*'* sin {v — \),
e'" sin (X, — /*), and those of the other point to e~'^ sin (/a — i^), e"'" sin (y — X), e"*" sin (\ — /i).
In the plane curve, the lines drawn from A, B, C to the circular points at
infinity are :
To the one point. To the other point.
From A, ay + z = 0, y + az = 0;
„ B. ^z + a;=0, z+^x=0;
„ C, yx + y = 0, a;+'yy = 0.
Each of these lines, quot, tangent at a triple point, meets the curve in the circular
point at infinity counted four times, and in two other points. The corresponding
points on the skew sextic .should be a node counted twice, the two other nodes
counted each once, and two other points. The proof that this is so would show
that the points A, B, C are a triad of foci. There is also the question of the
determination of the values of (^, rj, f, «) which correspond to the nodes of the
plane curve. But I have not further pursued the theory.
Addition. — Since writing the foregoing paper, I have found that the relation
between the nodes and foci (sum of angular distances of the foci = sum of angular
distances of the nodes) may be expressed in a different form ; viz. the triangle of
the foci and the triangle of the nodes are circumscribed to a parabola (having its
focus on the circle); and I have made in relation to the question the following
further investigations: —
Considering a circle : and a parabola having its focus at K, a point of the circle ;
then if, as usual, /, ./ are the circular points at infinity, we have UK a triangle
inscribed in the circle and circumscribed to the parabola ; hence there exists a
576 ON THREE-BAR MOTION. [623
singly-infinite series of in- and circumscribed triangles, so that, drawing from a point
A of the circle tangents to the paiabola again meeting the circle in the points B
and C respectively, BC will be a tangent to the parabola; or, what is the same
thing, starting with the triangle ABC inscribed in the circle, we can, with the
arbitrary point K on the circle as focus, describe a parabola touching the three sides
of the triangle ABC; viz. the parabola described to touch two of the sides of the
triangle will touch the third side.
Taking, then, a circle radius ^k, and upon it the three points A, B, C determined
by the angles 2a, 2/8, 27 respectively (viz. the coordinates of A are x, y = ^k cos 2o,
^ A sin 2a, &c.), and a point K determined by the angle 2ie (suppose for a moment
the origin is at K), the equation of a parabola having K for its focus will be
xr + y" = {x cos 26 + 1/ sin 20 — p)",
or, what is the same thing,
{x sin 2^-7/ cos 2^)- + 2p (x cos 20 + y sin 26) - f = 0,
where 0, p are in the first instance arbitrary ; and the condition in order that
(x + riy + f = 0 may be a tangent is easily found to be
P (?' + ¥) + 2f cos 20 + 27j sin 26 = 0.
It is to be shown that p, 6 can be determined so that the parabola shall touch
each of the lines BC, CA, AB. ,
Taking the origin at the centre, the equation of BC is
X cos (/3 -f 7) + 1/ sin (/3 + 7) — i i' cos (f3—y) = 0,
as is at once verified by showing that this equation is satisfied by the values
x,y — ^k cos 20, JA; sin 2/9, and =^4 cos 27, ^ A sin 27.
Hence, transforming to the point K as origin, the equation is
[x + ^k cos 2k] cos (j8 + 7) -I- [y + ^i sin 2«] sin (/8 -I- 7) - ^k cos (/3 — 7) = 0 ;
viz. this is
a; cos ()9 -(- 7) -H y sin (/8 -f 7) - P [cos (/9 - 7) - cos (/S + 7 - 2k)] = 0 ;
or, finally, it is
a; cos (y9 -I- 7) -f y sin (/8 + 7) — k sin (« — 0) sin (« — 7) = 0.
Hence the condition of contact with the line BC is
p = 2ksm(K — 0) sin (k - 7) cos (20 — y9 — 7) ;
and, similarly, the condition of contact with the line CA is
p = 2k sin (k — 7) sin (k — a) cos (20 — 7 — a) ;
»
623] ON THREE-BAR MOTION. 577
viz. these conditions determine the unknown quantities p, 0. It is at once seen that
we have
2^-/3- 7 = ^77 -(«-«), that is, 26 = ^Tr - k + ci + ^ + y;
and then
p = 2k sin (k — a) sin (k — /3) sin (k — y);
from symmetry, we see that the parabola touches also the side AB.
Suppose, next, F, G are points on the circle determined by the angles 2/", 2g;
retaining p and 0 to denote their values,
2) = 2k sin (k — a) sin (« - yS) sin (« — 7), and 26 = ^tt — «+a+/3 + 7,
the condition, in order that FG may be a tangent, is
p = 2k sin (« -/) sin (« - ^r) cos (2^ -f-g);
viz. determining A by the equation
a + 0 + y=f+ff + h,
this is
p = 2t sin (« — /) sin (« — 51) sin (k — h),
or, what is the same thing,
sin (« — a) sin (/c — yS) sin (« — 7) = sin (« — y ) sin (« — g) sin (« — A) ;
viz, this equation, considering therein h as standing for a + yS + 7 —f— g, is the
relation which must subsist between f and g, in order that the line FG may be a
tangent to the parabola. And then, h being determined as above, and the point H
on the circle being determined by the angle 2h, it is clear that the lines GH, HF
will also be tangents to the parabola; viz. FGH will be an in- and cii-cumscribed
triangle, provided only f, g, h satisfy the above-mentioned two equations. The latter
of these, if / g, h satisfy only the relation a + ^ + y=f+g + h, serves to determine
k; and then, 6 and k denoting as above, the equation of the parabola is
a~' + y^ = {x cos 26 + y sin 26 - pf ;
and it thus appears that the condition in question, a + ^+y=f+g + h, is equivalent
to the condition that the triangles ABC, FGH shall be circumscribed to the same
parabola.
It is to be remarked that the distances KA, KB, &c. are equal to A; sin (k - a),
k sin (« — /3), &c. ; hence the condition
sin (k — a) sin (/c — B) sin (« — 7) = sin (k —f) sin {k — g) sin {k - h)
becomes
KA.KB.KC = KF.KG.KH;
viz. the focus if is a point on the circle such that the product of its (linear)
flistances from the foci A, B, G is equal to the product of its (linear) distances
from the nodes F, G, H.
c. IX, 73
578 ON THREE-BAB MOTION. [623
It is to be remarked that the foregoing equation in k determines a single position
of the point K ; viz. it determines tan k, and therefore sin 2/c and cos 2/c, linejirly.
The equation is, in fiwit, a cubic equation in tan «, satisfied identically by tanK = i
and tan k= —i, and therefore reducible to a linear equation.
Write for a moment tan /c = w, and
(tan AC — tan a) (tan k - tan y9) (tan k — tan 7) = w' — pw* + 50) — r,
(tan K — tan/) (tan « — tan g) (tan k - tan h) = to' —p'a>- + 9'a) — r ;
also
M = cos/cos ^f cos h -T- cos a cos ^ cos 7.
Then we have
«' — pto' + g'ft) — r = ilf (o)' — ^w' + 5'a) - r'),
where
^ = r + 3/(r'-p), g=l + if (g^'-l).
Substituting these values, the equation becomes
o,3 _ ru)'' + a —r = M {(o"— r'co^ + w - r'),
viz. dividing by m^ + 1, this is co — r = M(<o — r'); or substituting for r, r', M their
values,
(cos a cos yS cos 7 — cos/ cos ^r cos h) tan /c = (sin a sin /3 sin 7 - sin/sin ^r sin h),
which is the value of tan*, and then
. „ 2 tan K -. 1 — tan' k
Sm ZK = :;— -— -— , COS '2k = :; ^ ., .
1 + tan- K 1 -f tan'' «
It may be further noticed that, if the parabola intersect the circle in a point L,
and the tangent at L to the parabola again meet the circle in M, then, if 2i, 2m
are the angles for the points L, M, we have I, m, m for values of / g, h, whence
I, m are determined by the equations
I + 2m = a + /8 + 7, sin (« — Z) sin- {k — m) = sin (k — a) sin (k — /3) sin {k — 7) ;
but as the circle intersects the parabola not only in two real points, but in two
other imaginary points, there is no simple formula for the determination of I and m.
To determine the linkage when the nodes are given, suppose that, in the generation
by 0, the vertex of the triangle OBiCi, we have 0 at the node F: then, if t, a
are the distances of G, B from the node in question, we have, as in the memoir,
(6,= -I- T» - Oj'') c,<r = (ci" -t- <T- - 0,=) 6,T,
623] ON THREE-BAR MOTION. 579
that is,
(61" - a,') CO- - (c,^ - a/) 61T + <TT (CjT - 6,0-) = 0,
or, what is the same thing,
(61= — a^-) ca — (Ci" — tta") br + ar (ct — ba) = 0.
Suppose, as in the figure, that F is between B and A; then, if AF=p, we
have CT = 60- + ap, and the equation becomes
(6,« - aa^) CO- - (c,'^ - 03') 6t + a/wrr = 0.
Similarly, if, as in the figure, G is on the other side of A, that is, between A
and C, and if /»', <r', t' be the distances A G, BG, CG, then ba' = ct' + ap', that is,
ct' — ba = — ap', and the corresponding equation is
We hence find
But we have
that is,
ahio,
and
(6,* - a./) ca' - (Ci» - a^) br' - apaW = 0.
{b{- - a,") c (<7t' - (t't) + aW {pa + pV) = 0,
(ci'-' — a,^) 6 (o-t' — ct't) + aaa (pr + /j't') = 0.
BG.CF=BF.GG + BC.FG,
a'r = ar' -f a . FG, or ai' — o't = — a . /^G ;
pa + pV = .4/'. £i' + ^(? . BG, = i^'G . C//, = FG . t",
pr+p'r' =AF.GF+AG.CG, = FG . BH, =FG.a",
as may be shown without difficulty, p", a", t" being the distances AH, BH, GH.
Hence the equations become
c (6,' - a^') - tt't" = 0,
h (c' - a/) - aa'a" = 0,
showing that, the foci being as in the figure, b^^ — ai and Ci° — a^ are each of them
positive; viz. that, in the generation by the triangle OC^Bi, the radial bars a^, a^ are
shorter than the sides 6,, c, respectively. Substituting for 6,, &c. the values k^sinB, &c.;
also, instead of A^, k^, k,, introducing the quantities Xj, X^, \.j, where
ki, k«, ki = \i&\nA, XiSmB, A^sinO,
these equations become
c (\,^ - V) sin' A sin'' i^ = tt't",
b (X,» - \^) sin' ^ sin= C = aa'a" ;
or, as these may be written, putting for shortness Jlf = sin .4 sin B sin G,
M'k' (X,» -W) = c tt't",
M'k'i\^-\') = baa'a".
73—2
580 ON THREE-BAR MOTION. [623
All the quantities have so far been regarded as positive, and the formulae are
applicable to the particular figure ; but, to present them in a form applicable to any
order of the nodes and foci, we have only to write the equations in the forms
M' (X,» - V) = *' sin (o - /9) sin (/- 7) sin (g - 7) sin (h - 7),
M^ (X,» - V) = Ic' sin (a - 7) sin (/- /3) sin (g - p) sin {h - ff) ;
And these may be replaced by the system
M-' (V - V) = ^ sin (/3 - 7) sin (/- o) sin (g - a) siu {h - a),
M' (V - V) = i^ sin (7 - a) sin (/- ff) sin (g - ;8) sin {h - /8),
M' (V - V) = it" sin (a - /3) sin (/ - 7) sin (jr - 7) sin (h - 7),
since the first of these equations is implied in the other two ; and then, reverting
to the original form, we may write
M^l<f(K?-\:?) = BC.FA.OA.HA,
M^t- (V - Xr) ^CA.FB.GB. HE,
M^h? (\,' - V) = AB.FC.GC . HC.
it being understood that the distances BC, FA, &a, which enter into these equations,
are not all positive, but that they stand for isin(/3 — 7), k8m(f—a), &c., and that
their signs are to be taken accordingly. Or, again, these may be written
BC {c^ - b./) = FA.GA.HA,
CA (a,' - cr) =FB.GB. HB,
AB{h('-a.?) = FC .GC .HC,
■where the signs are as just mentioned. We may say that ±(02'- — 63') is the modulus
for the focus A ; and the formula then shows that this modulus, taken positively, is
equal to the product of the distances FA, GA, HA of A from the three nodes
respectively, divided by BC, the distance of the other two foci from each other.
624] 581
624.
ON THE BICURSAL SEXTIC.
[From the Proceedings of the London Mathematical Society, vol. vii. (1875 — 1876),
pp. 166—172. Read March 10, 1876.]
In the paper " On the mechanical description of certain sextic curves," Proceeditigs of
the London Matlienuiticul Society, vol. IV. (1872), pp. 105 — 111, [504], I obtained the bicursal
sextic as a rational transfoi-mation of a binodal quartic. The theory was in effect as
follow.s: taking fi, P, Q, R, each of them a function of \, fi of the form (*5l, X.)"(l, /*)*,
and considering (\, m) -^ connected by the equation fl = 0, (viz. X, /x. being coordinate.s,
this represents a t)inodal quartic), then, if we assume x : y : z = P : Q : R, the locus
of the point {x, y, z) is a curve rationally connected with the binodal quartic, viz.
the points of the two curves have with each other a (1, 1) correspondence; whence
the locus in question, say the curve U=0, is bicursal. The degree is obtained as the
number of the intersections of the curve by an arbitrary line, or, what is the same
thing, the number of the variable intersections of the corresponding \/*-curves
n=0, aP + 0Q + yR=O,
viz. each of these being a quartic curve having the .same two nodes, the nodes each
count as 4 intersections, and the number of the remaining intersections is 4.4 — 2.4, =8,
and thus the curve 6'=0 is in general of the order 8. But if the curves fl = 0,
P = 0, Q = 0, R = 0 have (besides the nodes) k common intersections, then these are
also fixed intersections of the two curves fl = 0, aP + 0Q + yR = 0, and the number of
variable intersections is reduced to H — k: we have thus H — k as the order of the
curve U= 0. In particular, if k = 2, then the curve is a bicursal sextic.
The theory assumes a different and more simple form if, in the several functions
n, P, Q, R, we suppose that the terms in X-, /a* are wanting. The curves H = 0,
P = 0, Q = 0, R = 0 are here cubics having two common points; the curve 17=0, qua
582 ON THE BICURSAL SEXTIC. [624
rational transformation of the cubic ft = 0, is still a bicursal curve; but its order is
given as the number of the variable intersections of the cubics
n = 0, aP + 0Q+'YR = O,
viz. this is =3.3-2, =7. But if the curves n = 0, P = 0, Q = 0, ^ = 0 have (besides
the before-mentioned two common points) k other common points, then the number of
the variable intersections is =7 — k: and this is therefore the order of the curve
U=0. In particular, if k=l, then the curve is a bicursal sextic. And, in the present
paper, I consider the binodal sextic as thus obtained, viz. as given by the equations
(1 = 0, X : y : z = P : Q : R, where fl = 0, P = 0, Q = 0, R = 0 are cubics, having (in all)
three common points.
The bicursal sextic has in general 9 nodes ; but 3 of these may unite together
into a triple point: this will be the case if, in the series of cui-ves aP +^Q + yR = 0,
there are any two curves which have 3 common intersections with the curve n = 0.
(Observe that we throughout disregard the 3 common points of the curves 12 = 0,
P = 0, Q = 0, jR = 0, and attend only to the 6 variable points of intersection of the curves
12 = 0 and aP + ^Q + yR=0, — the meaning is, that there are two curves of the series
such that, attending only to the 6 variable intersections of each of them with the
curve f2 = 0, there are three common intersections.) For, supposing the two curves to
be aP + ^Q + yR = 0 and a'P + ^Q + y'R = 0, then any curve whatever
aP + 0Q + yR+e(a'P + 0Q + y'R) = O
has the same three intersections with the curve 12 = 0, say these are the points
Ai, A3, Af, the coordinates of which are independent of 0. Hence the line
{ax + 0y+yz) + 0 (a'x + 0y + y'z) = 0
intersects the curve U^=0 in six points, three of which, as corresponding to the points
Ai, At, A3, are independent of 6, viz. they are the same three points for any line
whatever of the series; and this means that the curve U = 0 has at the point
(ax + ^y + yz = 0, a'x + ^'y + y'z = 0)
a triple point; and that to this triple point correspond the three points Ai, A^, A3.
We may, in the series of lines ax + ^y + yz+6 {a'x + fi'y + y'z) = 0, rationally determine
0 80 that one of the three variable points of intersection shall correspond to .4,, A.;,,
or A.i ; viz. 0 must be such that the curve aP + ^Q + yR + 0 {a'P + /3'Q + y'R) = 0 shall
touch the curve 12 = 0 at one of the points A^, A.j, A3. The three lines thus determined
are the three tangents to the curve at the triple point : and the three branches may
be considered as corresponding to the three points Ai, A„ A3, respectively.
There is no loss of generality in assuming that the triple point is the point
(x=0, ^ = 0); the condition then simply is that the curves P = 0, R = 0 shall have
three cximmon intersections with the curve f2 = 0 ; and the tangents at the triple point
are x+ 0z = O, 0 being so determined that one of the three variable points of inter-
section shall correspond to one of the three points .4,, A^, A3: in particular, if this
ifl the case for the line x = 0, then this line will be one of the tangents at the
triple point.
624] ON THE BICURSAL SEXTIC. 583
The bicui-sal sextic may have a second triple point, viz. three other nodes may
unite together into a triple point. The theory is precisely the same : we must have
two other curves aP + /SQ + 'yR = 0, a'F + ^'Q + y'E = 0, having with the curve ft = 0
three common interaections Bi, B.,, B^: there is then a second triple point
{ax + fiy + <y2= 0, a'x + ^'y + 7'^ = 0) ;
and, to find the tangents at this point, we must determine 6 so that one of the
variable points of intersection of the line
ax+^y + yz+6 {a'x + fi'y + y'z) = 0
with the sextic shall correspond with £,, B«, or B^; viz. 0 must be such that the curve
al^-i- ^Q + yR+e (a'P + yS'Q + y'R) = 0 shall touch the curve fi = 0 at one of the points
Bi, Bj, B3. In particular, if, as before, the curves P = 0, R = 0 have three common
intersections with the curve 0 = 0, and if, moreover, the curves Q = 0, R = Q have
three common intersections with the curve 12 = 0, then the bicursal sextic will have
the two triple points (a; = 0, ^ = 0) and (y=0, 2 = 0); and it may further happen that
the line x = 0 is a tangent at the first triple point, and the line y =0 a, tangent at
the second triple point. The sextic may in like manner have a third triple point, but
this is a special case which I do not at present consider.
I write for greater convenience - , - in place of \, u., so as to make il, P, Q, R
V V
each of them a homogeneous cubic function of (X,, fi, v); and I give to these functions,
not the most general values belonging to a bicursal sextic with two triple points, but
the values in the form obtained fijr them, as appearing further on, in the problem of
three-bar motion ; viz. the equations £1 = 0, x : y : z = P : Q : R are respectively taken
to be .
V (hv\ +/\») + 11(^1^ + evK + gK?) + /t' (/v + h\) = 0,
X : y : z=\fi {a\ + bfi) : 1^ (c\ + d/x.) : X/j-v.
The four curves n = 0, P = 0, Q = 0, R = 0 have thus the three common intersections
(^4 = 0, i'=0), (i' = 0, X, = ()), (\ = 0, fj, = 0),
represented in the figure by the points A, B, C ; the curve drawn in the figure is the
curve n = 0, and the points F, G, U are the third points of intersection of the cubic
with the lines BG, CA, AB respectively.
The equation P + 6R = 0 is here \/t (a\ + V + ^^) = 0, which intersects £1 = 0 in
the points C, A, G, G, B, F, and the three intersections by the line a\ + bfi + 6i> = 0;
584 ON THE BICUR8AL 8EXTIC. [624
viz. excluding the fixed points A, B, C, the six intersections are C, F, G, and the three
intersections by the line. Hence, of the six intei-sections, we have C, F, G independent
of 0, or we have (a; = 0, z = 0) a triple point, say /, corresponding to the three points
C, F, G, viz. these are the points
(X, /i. v) = {0. 0. 1), (0, g, -/), (h, 0. -/), (C, F, G).
The equation Q + 6R = 0 is i' {cXv + dfiv + 6\fi) = 0 ; viz. the line v = 0 meets fl = 0
in the three points A, B, H, and the conic c\v + dfiv + 6\ix = 0 meets fi=0 in the
points A, B, C, and three other points: hence, rejecting the points A, B, C, the six
points of intersection are the points A, B, H, and the three variable points of inter-
section by the conic ; or we have {y = 0, ^ = 0) a triple point, say J, corresponding to
the three points A, B, H, viz. these are the points
{\, ^, v) = {\, 0, 0), (0, 1, 0), {h, -g, 0), {A,B, H).
To find the tangents at the triple point 7, these are x+ 6z = Q, where ^ is to be
successively determined by the conditions that the line aX + bfi+ 6v = 0 shall pass
through the points 0, F, G* ; viz. we thus have
^ = 0, x = 0, the tangent conesponding to the point C, (0, 0, 1),
0= + ^j, fx+bgz=0, „ „ „ „ F,iO,g,-f),
0^ + j, fx + ahz = 0, „ „ „ „ G,(h,0,-f).
And similarly, at the triple point J, the tangents are y + 0z = O, wheie 0 is to be
successively determined by the conditions that the conic cvX + dv/j. + 0\fjL = 0 shall pass
through the point H, and shall touch the cubic at the points A, B; viz. we thus have
^ = 0, y = 0, the tangent corresponding to the point H, {h, — g, 0),
S = j, fy + cgz = o, „ „ „ „ ^,(1,0,0),
e=j, fy+dhz^a, „ „ „ „ B.{o,i,o).
The two last values of 0 are obtained by the consideration that the equations of
the tangents to fl = 0 at the points A, B respectively, are gfi+/v = 0, h\+fp = 0,
where X, fi, v are current coordinates of a point on the tangent : it may be added
that the equation of the tjingent at the point C is h\ + gfj, = 0.
* Obuerve the somewhat altered form of the condition : 9 is to be determined so that the cubic
\n{a\ + bn + 6ii) = 0 shall touch the cubic fi = 0 at one of the points C, F, G : but, as the first-mentioned
cubic breaks up, and the component curve a\ + bfi + 0y = O does not pass through any one of these points,
this can only mean that 0 shall be so determined as that the line shall pass through one of these points,
viz. that there shall be at the point, not a proper contact, but a double intersection, arising from a node
of the cubic X/t(aX + fc^ + ei') = 0. And the like case happens for the other triple point; viz. there the cubic
y{cy\ + dvfii + 6\ii) = Q is to touch the cubic fi=0 at one of the points ^1, B, H ; the component conic
e»\+diiii, + e\iJ.=Q passes through the points A and B but not through H; hence the conditions for 6 are,
that the conic shall touch the cnbio at A or B, or that it shall pass through H.
624] ON THE BICURSAL SEXTIC. 585
The three-bar curve may be represented by means of a system of equations of
the last-mentioned form, viz. x : y : z=\fi, {aX + bfj.) : v' (c\ -t- dfi) : Xfiv, where \, fi, v are
connected as above ; or, taking X, Y as ordinary rectangular coordinates, x, y, and z
are here the circular coordinates - = X + iY, - = X—iY, and z=l; and the parameters
- , - denote like functions cos 0 + i sin 6, cos <f> + i sin <b of angles which are the
V V r CD
inclinations of two bars to a fixed line. Using, for convenience, Figure 2 of my paper
on Three-bar Motion, (p. 553 of this volume), the curve is considered as the locus of
the vertex 0 of the triangle OC\Bi, connected by the bars 0,(7 and BjB with the fixed
points B and C respectively; and we have CGi = a.2, Of, =6,, OBi=Ci, (7,5, = a,, B^B^a^.
Also, to avoid confusion with the foregoing notation of the present paper, instead of call-
ing it a, I take BC^a^: the angle OGiBi is = C,, and cosC, + isinCi is taken = y.
Hence, taking the origin at G, the axis of X coinciding with GB and that of Y
being at right angles to it : taking also 6, <f), i/r for the inclinations of CO,, C,£,, and
BiB to GB, we have
a., cos d + a, cos <f> — a = — u^ cos ■\}r,
o^ sin ^ + a, sin <f> = Og sin i^ ;
viz. writing cos ^ -)- i sin ^ = \, cos <f> + i Bin <f> = fi, these give
a.j\ + Uifi — tto = — Ha (cos y(r — i sin •^),
"i c + «i «» = — flj (cos y + I sm yjr),
that is,
(«,,\ + ff,/t — a) f ftj - -h a, «o j — a-j- = 0 ;
VIZ.
(a," + a,' -1- a./ - a,=) + «,aj (^ + "j - "oOs i^ + yj- o^o^i (/* + -) = 0,
for the relation between the parameters \, /i. And then
X = 03 cos 0 + bi cos {<f> + Gi),
Y = Hj sin 0 -1- 6, sin (<^ + 0,) ;
viz. if X, y = X +iY, X —lY, then
1 , 1
y = flj - -t- 62 — ,
which equations determine the coordinates (a;, y) in terms of the parameters \, fi con-
nected by the foregoing relation.
C. IX. 74
586
ON THE BICUR8AL 8EXTIC.
[624
Writing for homogeneity - , - in place of \, ft, and - , - in place of x, y, the
equations become
(a,' + a,' + «/ - a^) \fiv + a, cu (V + /t») v - a,a,/i (i/^ + X-') - rt„a,X (/*' + 1^=) = 0,
and
X : y : z = (ai\ + b.,y,ti)XiJi : f ~ \ + a.,/i ) >/- : Xfiv.
Comparing with the foregoing equations
e \fiu +/(X' + ti^)v + g(i^ + X') fi+ h(ijL^ + V-) \ = 0,
and
X : y : z = (a\+ hfi) X/i : (cX -t- dfi) v- : Xfiv,
the equations agree together, and we have
/=
+ Oitt-j,
9 =
-Woaa,
/t =
-ttoCt,,
rt =
fl2,
6 =
ti7..
c =
6.
7i
The tangents at the triple points thus are
x=0,
aiX — aobiyiZ=0,
X — a„z = 0,
a,y z = 0,
7i
?/ - «,0 = 0 ;
viz. restoring the rectangular coordinates, and for y substituting the value cosC+ism 0,
for a« writing a, and taking b= ' , we have
i: + tF=0, X-iF=0,
Z + iF= 6(cos C + isinO), X-iY^ b (cos C-isin C),
X + iF= a„, JV — jF = a„ ;
viz. the first two intersect in the point (0, 0), the second two in the point (6 cos (7,
6 sin (7), the third two in the point (a, 0): the first and third of these are the points
B and C, the second of them i.s the point A of the figure ; viz. the formuhe give
the point A, forming, with B and C, a triad of foci.
625] 587
625.
ON THE CONDITION FOR THE EXISTENCE OF A SURFACE
CUTTING AT RIGHT ANGLES A GIVEN SET OF LINES.
[From the Proceedings of the Loudon Mathematical Society, vol. Vlll. (1876 — 1877),
pp. 53-r57. Read December 14, 1876.]
In a congruency or doubly infinite system of right lines, the direction-cosines
a, /8, 7 of the line through any given point («, y, z), are expressible as functions
of X, y, z; and it was shown by Sir W. R. Hamilton in a very elegant maimer
that, in order to the existence of a surface (or, what is the same thing, a set of
parallel surfaces) cutting the lines at right angles, adx + ^dy + '^dz must be an exact
differential : when this is so, writing V = I {adx + ^dy + '^dz), we have V = c, the
equation of the system of parallel surfaces each cutting the given lines at right angles.
The proof is as follows : — If the surface exists, its differential equation is
adx + ^dy + <^dz = 0, and this equation must therefore be integrable by a factor.
Now the functions a, ^, 7 are such that a' + /3^ + 7''= 1, and they besides satisfy a
system of partial differential equations which Hamilton deduces from the geometrical
notion of a congruency ; viz. passing from the point {x, y, z) to the consecutive
point on the line, that is, to the point whose coordinates are x + pa, y + p/S, z+ py
(p infinitesimal), the line belonging to this point is the original line; and conse-
quently a, )8, 7, considered as functions of x, y, z, must remain unaltered when these
variables are changed into x + pa, y + p^, z + py, respectively. We thus obtain the
equations
da doL da _
dtc dy dz '
dx dy dz '
dx dy dz
74—2
588 ON THE CONDITION FOR THE EXISTENCE OF A [625
Combining herewith the equations obtained by differentiation of a-' + yS"- + 7= = 1 , viz,
— 4-S—+ ^ =0
dx dx dx ~ '
da ndB dy „
<^d,^^d,-'ydl-''
da . ^rf/8 dfy .
and subtracting the corresponding equations, we obtain three equations which may be
written
. o . _ <^/3 dy d^ da da rfy3
dz dy ' dx dz' dy dx'
or, what is the same thing,
d^ _dy dy _da da ^dff _, j.o 7.
dz dy' dx dz' dy dx ' ' ' ''
and, multiplying by a, /S, 7, and adding,
k - (^ - ^^ a (^ - ^"^ (— - ^^\
\dz dy) \dx dz) \dy dx)'
We thus see that, if the function on the right-hand vanish, then k = 0, and conse-
quently also
d^ _dr/ d/y '^^ da d^ 1, _ n .
dz dy' dx dz' dy dx ~ '
viz. if the equation adx + 0dy + ydz = 0 be integrable, then adx + 0dy + ydz is an
exact differential ; which is the theorem in question.
But it is interesting to obtain the first mentioned set of differential equations
from the analytical equations of a congruency, viz. these are x = viz + p, y = nz + q,
where m, n, p, q are functions of two arbitrary parametei-s, or, what is the ssxme
thing, p, q are given functions of vi, n ; and therefore, from the three equations,
m, n are given functions of x, y, z. And it is also interesting to express in terms
of these quantities m, n, considered as functions of x, y, z, the condition for the
existence of the set of surfaces.
We have
MRU
^ m n 1 , n r, i — — :
a, yS, 7= p, p, p, where ii = VI -|- w* + «';
and thence without difficulty
/ d , _ d , d\ 1 f/i ^s f dm dm , dm\ ( dn dn , dn\\
('£-4/^S-i[ - ( . ) - ( . )].
625]
SURFACE CUTTING AT RIGHT ANGLES A GIVEN SET OF LINES.
589
SO that the required equations in a, /8, 7 will be stitisfied if only
dm dm dm
wt j^ + « -7- + -y = 0,
dx ay dz
dn , dn dn .
m-r- +n-j- +-T- =0,
oar dy dz
and it is to be shown that these equations hold good.
Writing for shortness dp = Adm + Bdn, dq = Cdm + Bdn, the equations of the line
give
_ dm . dm „ dn
dx dx dx'
dm , dm „ dn
0 = z J- + A , +B ^ ,
dy dy dy
dm . dm r, dn
— m = z ,~+A J + a -jf ,
dz dz dz
dn ^ dm , ,, d)i
0 = z J +C J- +B -J- ,
(Ix ax ax
, _ dn „ dm „ dn
dy dy dy'
dn „ dm . r. d>i
n = z J- + C 'j~ + D~j-;
dz dz dz
or, writing
so that identically
dm dn dm dn dm dn dm dn dm dn dm dn
' ^' ~ dy dz dz dy ' dz dx dx dz ' dx dy dy dx'
dm dm dm _ -
dx dy dz '
^ dn dn dn „
X , + M J +Vj- =0,
dx dy dz
then in each set, multiplying by X, fi, v and adding, so as to eliminate A, B, G, D,
we find
X — mv = 0, fi — nv = 0.
Substituting these values of X, /a in the last preceding equations, v divides out, and
we have the two equations in question.
The foregoing equations give further
.,,,,,, 1 dn 1 dm
A, B, C, D = -z + - J-, J-,
V dy V dy
1 dm
V dx '
-3 +
1 dm
V dx '
Taking for a, /9, 7 the before-mentioned values, we find
da dfi _1 (dm dn\ m f dm dn\ _]}_ ( dm dn\
dy~ dx~ R\dbj~dx)~W r' rf7 "^ " dy) R" V' dx'^"' dx)
1 (., „, dm ,, , ,.dn , fdm dn\[
and similarly, but using the equations
dm dm dm . dn , dn . dn
dx dy dz dx dy dz
590 ON THE CONDITION FOR THE EXISTENCE OF A [625
to eliminate the coefficients ^ , j- which in the first instance present themselves,
we find
dy _da _ n \ ^, )
di'dz'R'X " " " J-
whence, multiplying by 7, a, /3, and adding,
fd0 dy\ ^/dY_da\ (doc _d^\
''\dz~dy)^^\dx dzJ^'^Kdy da;)
1 {.. , „.dni ,, , .,^dn , /dm dn\
1 + m' + n- (^ ' dy ^ ' dx \dx dy ] '
or we have
„ „, dm ,, .,v dn /dnt d?i\
as the condition for the existence of the set of surfaces.
It is clear that the condition is satisfied when the lines are the normals of a
given surface: seeking the surfaces which cut the lines at right angles, we obtain
the parallel surfaces; and we are led to the theorem that any parallel surface is
the locus of the extremity of a line of constant length measured off from each point
of the surface along the normal — or, what is equivalent thereto, the parallel surface
is the envelope of a sphere of constant radius having its centre on the surface. I
will verify the theorem for the case of the ellipsoid. Taking X, Y, Z as the
X- r» Z^
coordinates of a point on the ellipsoid — ^ +-Tr +^;; = 1. and x, y, z as current co-
ordinates, the equations of the normal are
a' 6° c°
•^ (a; - Z) = ^(2/ - F) = ^(^ - Z), (= A, suppose).
We have therefore
and thence
' ' a'-i-X' 6'-t-\' c= + \'
gV fry c''^° _ 1
an equation which determines X as a function of x, y, z.
X Y Z
The direction-cosines a, fi, y of the normal are proportional to — ,, ji, -^, that
625] SURFACE CUTTING AT RIGHT ANGLES A GIVEN SET OF LINES. 591
is, to , ,—^ — , — — , and the equation a-+/8-+7-=l then determines their
absohite magnitudes: the equation adx + fidy + •ydz = dV thus is
xdx ydy zdz
+ #^ +
^/,
a» + \ i^ + X c» + \ ,„
^^ J^r- = dV,
a? v" z^
..+ ' -"
{a? + Xf (hfi + Xf {(f + \)-
viz. the left-hand side, considering therein \ as a given function of V, is an exact
differential. We verify this by finding the value of V, viz. writing down the two
equations
ar' y- z- V'^
rt=+x "^ PTx "^ c=Tx X ~ '
these are equivalent in virtue of the equation that determines X: and it is to be
shown that, regai-ding F as given by either of them, say by the second equation, we
have for dV its foregoing value. In fact, differentiating the second equation, the
term in rfX disappears by virtue of the first equation, and the result is
xdx ydy zdz V dV
«= + X 6= + X C-' + X X
V
in which substituting for — its value from the first equation, we have for dV the
value in question. Regarding F as a given constant, the two equations give, by
elimination of X, an equation (^ (x, y, z, V) = 0, which is, in fact, the surface parallel
to the ellipsoid an<l at a constant normal distance = F from it.
592
[626
626.
ON THE GENERAL DIFFERENTIAL EQUATION '%+%=0,
VA Vi
WHERE X, Y ARE THE SAME QUARTIC FUNCTIONS OF
X, y RESPECTIVELY.
[From the Proceedings of tlie Londmi Mathematical Society, vol. Vlll. (1876 — 1877),
pp. 184—199. Read February 8, 1877.]
Write (d=a+b6 + c0' + d0' + ed*, the general quartic function of 6 ; and let it be
required to integrate by Abel's theorem the differential equation
We have
a particular integral of
da; ,dy_
X-, X, 1, iJX =0,
y% y, 1. VF
Z-, z, 1, ^'Z
W-, lu, 1, VTF
dx dy_ dz dw _ .
and consequently the above equation, taking therein z, w as constants, is the general
integral of
viz. the two constants z, w must enter in such wise that the equation contains only
a single constant; whence also, attributing to w any special value, we have the general
integral with z as the arbitrary constant.
^ dx dii
626] ON THE GENERAL DIFFERENTIAL EQUATION 71?+ 'JY~
593
Take w = 00 ; the equation becomes
a?, X, 1, tJX
= 0,
y^ 2/, 1. ^Y
Z-, z, 1, v^
1 , 0, 0, Ve
a relation between x, y, z which may be otherwise expressed by means of the identity
eiO' + ^0 + 'if-{ee* + de^ + 00' + he + a) = {2^e-d){d -x)(d - y){d - z),
or, what is the same thing,
e (27 + /3^) - c = - {2Be -d){x + y + z),
e 2/S7 -6= {l^e - d) iyz + zx -\- ooy),
67- —a = — (2/3e — d) xyz,
where /3, 7 are indeterminate coefficients which are to be eliminated.
•I
Write
then we have
giving
/9a; + 7 + P = 0, /32/ + 7 + Q = 0;
/8 : 7 : 1=Q-P : Py-Qx : x-y.
Substituting these values in the first of the preceding three equations, we have
that is.
^2(Py-Q.)(.-y) + (Q-P)»_^^_{2(Q-f)e_^|^^^^^^^_
{x-yf
■y
or, reducing by
j^,-P^)^(Q-P. 2JQ-P) ) ^^^
( x-y {x-yY x-y )
+ ^);
Qy — Px = y' — a? +
xsJX-yJY
this is
Q-P-v'-^^''^. -,.-^+to-.,* if J/ = ^4^^r
gj_v_v ^_v — '_j^2xy+ 1(x + y) -r - iix + y) z + Iz -T
{ »Je{x-y) " e ^ ^' >je ^ ' y> ^ ^^
= c + d(x + y+z) + e(x + yy.
We have Euler's solution in the far more simple form
M''=C + d(x + y) + e(x + yy,
C. IX.
75
I
dx dy ^ r^nr.
594 ON THE GENERAL DIFFERENTIAL EQUATION / y + lY~^- L^Sfi
where G is the arbitrary constant. It is to be observed that, in the particular case
where e = 0, the first equation becomes
M'' = c + d {x •{■ y + z);
and the two results for this case agi-ee on putting G = c-\-dz.
But it is required to identify the two solutions in the general case where e is
not =0. I remark that I have, in my Treatise on Elliptic Fuivctions, Chap, xiv., further
developed the theory of Euler's solution, and have shown that, regarding G as variable,
and writing
g = ad- + h'e - 2bcd + C [- 4ae + 6d + (C - c)=],
then the given equation between the vai-iables x, y, G corresponds to the differential
equation
dx dy dG ^
a result which will be useful for effecting the identification. The Abelian solution
may be written
e^J^^.^0^-a?-f + ~-2{x + y)~\-c-d{x + y) = z\d + 2e(x + y)-2M^e]-,
and substituting for M its value, and multiplying by {x — yY, the equation becomes
2^e{x-y){x^/X-y^/Y)-eia^ + f■)(x-yy + WX-^y)^'
-2(x'-f)WX-^/Y)^e-c(x-yf-d{x+y)(x-yy
= z(x-y)ldix-y) + 2e{af-f)-2WX-^/Y)'Je}.
On the left-hand side, the rational part is
X + Y+c(- af + 2xy - ■f) + d{- X-'' + x^y + xy" - f) + e (- x* + 2a?y - 2a? y'' + 2a;i/' - y*\
which, substituting therein for X, Y their values, becomes
= 2a + 6 (a; + y) + c . 2xy + dxy{x + y) + e. 2xy {a? — xy+y^);
and the irrational part is at once found to be
= 2^/e{x-y){x^/Y-y^/X)-2^TY.
The equation thus is
2a -I- 6 (a; + y) + c . 2xy + dxy(x + y) + e. 2xy (a?-xy + y^M
+ 2^e(x-y)(x^ Y-y^X)-2 VZ F
z =
{x - y) [d (x-y) + 2e (a? - y=) - 2(^X- JY) /e]
which equation is thus a form of the general integral of -j^+tv-^^' ^^^ *'®^ ^
particular integral of "Tr + ^ + 7^= ^-
-, dx dii
626] ON THE GENERAL DIFFERENTIAL EQUATION y= + -^ = 0. 595
Multiplying the numerator and the denominator by
d{x-y)^2e{x^-f)-{-2{^/X-'JY)^e,
the denominator becomes
= {x-yr \\d + 2e{x + y)Y-4.e (^^^Z^Yl ,
\ x — y J
which, introducing herein the C of Euler's equation, is
= (a;-y)»(d=-4eC).
We have therefore
z{x — yf (ci' — 4e(7) = {2a + b(x + y) +c. 2xy + dxy{x + y) + e. 2xy {a? — xy->r- y-)
^■2^e{x-y)(x^JY-y>JX)-2^XY] x {d(x-y) + 2e(ay'-y') + 2^eWX - ^Y)}.
Using S to denote the same value as before, the function on the right-hand is, in
fact,
= (x - yy {2be -cd + dC + 2'^e VS} ;
and, this being so, the required relation between z, C is
z (d» - 4e(7) ={2be-cd + dC+2>^e VS).
To prove this, we have first, from the equation
to express 6 as a function of x, y. This equation, regarding therein (7 as a variable,
gives
and we have therefore
dx dy dC _
.JX'^7fY'^^~ '
dx ay
viz. V-^ J will bs * symmetrical function of x, y. Putting, as before
x-y
we have
C=M^-d{x + y)-e{x-ir yf,
and thence
We have
dM^ 1 X' yz-VF
dx ^ x — y 2»JX i^ — yY '
75—2
(id* cIai
596 ON THE GENERAL DIFFERENTIAL EQUATION -7-v' + -7v^=0' [626
and hence
^/g(a;-y)•: ^X {x-yy\2M~ - d-2e(a; + j/)^
= -(x-y)X'{^/X-^Y) + 2iX+V-2'^TV)^X
+ (d + 2ex + y)(x-yy^/X
= [(x -y)X' + 2X + 2Y + (d+2eiTy) {x - yf] ^X
+ [(x-y)X'-4X]^Y.
We obtain at once the coefficient of \/Y, and with little more difficulty that of
it/X; and the result is
VS (x - y)' = - [4a + Sbx + 2caf + dx' + y(b + 2cx + 3d^ + 4eaf)] -JY
+ [4a + nby + 2cy- -{■ df + x{b + 2cy + My- + 4ev')] i/X.
We have also
C(x-yy = WX-^Yr-d(x + y)(x-yy-e{x + yy(x-yy
= X -irY -d{a? - x-'y -xxf + y') - e{af -23?f-\-y*) -2 -JT?
= 2a + 6 (a; + y) + c(a!= + y») + da;i/ (« + y) + 2ea?y^ - 2 \/XY,
or, say
C(x-yy = 2a{x-y) + b{af'-y-) + c(x'- a?y + xy''-if) ->r d xy {a? - f)
+ 2ear'«/» (x-y)-2(x-y) ^/XY.
We can hence form the expression of
(x - yy {2be -cd + dC+2>Je v'S),
viz. this is
= {ibe -cd)(x-yy + 2ad {x-y)+ bd (of - y^) + cd {a? - a?y -if xy' -f) + d^ xy {x? - y^)
+ 2de xY (x-y)- 2d (x - y) vT?
+ 2 Ve {[- (4a + 36a; + 2c«» + daf) -y(b + 2cx + Sdof + iex")] V Y
+ [(4o + 3% + 2cy' + df) + x{b + 2cy + My- + 4ey^)] ^/X],
and this should be
= {2a + 6 (« + y) + c . 2xy -\-dxy{x + y) + e. 2xy (af — xy-i- if)
-\-2^/eix-y)(x^Y-y^/X)-2s/XY} x {d(x- y) + 2e{af- y^) + 2 y/eWX - >s/Y)\.
The function on the right-hand is, in fact,
= {2a + b(x + y) + c.2xy + dxy{x-\-y) + e. 2xy (of - xy + y^ - 2 -s/XY]
X [d{x-y)+2e(x'-f)\ +ieix-y)WX - s/Y)(x^Y-y ^/X)
+ 2 Ve WX - i^Y) {2a + b {x + y) + c .2xy + d xy{x + y) + e .2xy {x' - xy + y^) - 2 >JXY\
■^2^Je{x-y){x^Y -y ^X) [d{x - y)-^2e(a?-y%
626]
dx dy
ON THE GENERAL DIFFERENTIAL EQUATION Ty"^"/!^"
597
viz. this is
= {2a+h{x + y) + c .ixy + dxy {x + y) + e. 2xy {a? - xy + y^)]
X {d(x-y) + 2e{a^-y')] + 4e(x -y){-xY- yX)
- 2 ^XY [d(x - y) + 2e(afi - y'-)} + 4.eix - y)(x + 7/) \/XY
+ 2>/e( <s/X ['2a + b{x + y) + c.2xy + dxy(x + y) + e.2xy(af-xy + y^y
+ 2Y-(x-y)y[d{x-y) + 2e(a^-f)]}
— <JY {2a + b{x + y) + c. 2xy + dxy{x + y) + e. 2xy {x:- — xy + y-)
+ 2X-ix-y)x[d(x-y) + 2e{af-y')]]
which is, ia fact, equal to the expression on the left-hand side.
To complete the theory, we require to express \/Z as a function of x, y. It
would be impracticable to effect this by direct substitution of the foregoing value
€i 1* fi'ii fi P'
of z; but, observing that the value in question is a solution of -7^ + "7w-+ ~rw = 0.
or, what is the same thing, that —ps- + -j^ j~—^> 'ly + Ty ;/" ~ ^' ^^® "^"^ iro\x\. either
of these equations, considering therein ^ as a given function of x, y, calculate '^Z.
Writing for shortness
J-2^/ey{x-y)l^X + 2^/ex(x-y)^Y-2^JXY
where
R-2'Je{x-y)s/X + 2'Je{x-tj)>JY
R=={x-yy\d + 2e{x-\-y)],
J=2a + b{x + y) + 2cxy + dxy{x + y) + 2exy(a? — xy + y^) ;
N
or, if for a moment ■8 = 7^, then
that is,
dx IJP\ dx dx) s/X'
,_ ^Xf„dD j.dN\ il
dX dR dJ
dx ' dx' dx'
■or, writing for shortness X', R', J to denote the derived functions
dY
i Y' is afterwards written to denote -5— , but as the final formulae contain only
dy ^
X', =-j-, and Y', =-=- , this does not occasion any defect of symmetry), we find
n= N[R'^X-2^JeX-^e{x-y)X' + 2^Je^XY]
-D{J'^/X-2s/eyX-^/e(x-y)yX' + 2^e(2x-y)^XY-X'^Y};
dx cly
598 ON THE GENERAL DIFFERENTIAL EQUATION , t> + -;>, = 0.
and substituting herein for N, D their values, and arranging the terms, we find
where
[626
^ — J\2X + {x-y)X']
-2{x-y)yR'X
^Ry{2X+{x-y)X']
+ 2{x-y)XJ'
+ 2(x-y)X'Y,
6 = — 4ey (x — y) X
-2e(x-y)x[2X + {.T-7/)X'}
-2R'X
+ RX'
+ 2e{x-y)y{2X + ix-y)X']
■{■4ie(x-y)(2x-y)X,
S)= JR'
+ 2e{x-y)y{2X + (.T-y)X'}
+ ^x{x — y) Y
-RJ'
-2e{x-y)y[2X+{x-y)X'\
-4ie(x-y)(2x-y) Y,
2)= 2J
+ 2{x-y)xR'
+ 2{2X+{x-y)X']
-2{2x-y)R
-2(x-y)X'
-2{x-y)J\
where the terms have been written down as they immediately present themselves ; but,
collecting and arranging, we have
21 = 2X {-J + Ry -2Y) + {x- y) [2XJ' + 2X'Y- X'J - 2yR'X + yRX'],
33= JR'-J'R-^e{x-yyY,
6 = - -IXR' + X'R + 4e (*• - yy X-2e{x- yf X',
<!>= 2J+4>X-2Rx + 2{x-y)(xR'-R-J').
To reduce these expressions, writing
M = d + 2e{x + y),
A =c + d(x + y) + e{x' + y"),
we have R = (x — yf M, and therefore R = 2{x — y) M + 2e {x - yf ; also
J = X+Y-{x-yyA;
also, from the original form,
J' = b + 2cy + d {2xy -^y") + e(6x'^y -4!xy'+ 2^).
The final values are
?l = -X^-QXY- P + (x-ijy {A"-+(-b + day)M+xyM%
8= {x-y)M{^Y-¥{,x-y)Y'\+2e{x-yyY',
(^ =.-(x-y)M {iX -(x-y) X'} -2eix -yy X'.
D= ^(X+Y) + 4,e(x-yy,
which, once obtained, may be verified without difficulty.
^ dx chi ^„„
626] ON THE GENERAL DIFFERENTIAL EQUATION np + Jy — ^' ^^^
Verification of 21. — The equation is
- Z^ - 6ZF- Y^ + {x- yy | A-" + (- 6 + dxy) M+xyM''}
= 2X(-J + Ri/-2Y) + (x- y) {2XJ' + 2X'Y- X'J- 2yRX + yRX'} ;
or, putting tor shortness
A- + (-b + (h;y)M + xyM-== V,
this is
(x-yy'7= Z-^ + 6ZF+F=
+ 2X{-X-3Y+{x-yyA+(x-yfyM]
+ (x-y}{ 2XJ' + 2X'Y-X'J- 2yR'X + yRX'} ,
= -X-'+ Y' +2(x- yy XA + 2(x- yy yXM
+ (x-y) \2XJ' + 2Z' F- X'J - 2yR'X + yRX'} ;
we have -X'' ■{■ Y^= -{X - Y){X +Y), where X-Y divides by x-y, =(x-y)il
suppose ; hence, throwing out the factor x — y, the equation becomes
(x-yfS^ =-n{k+Y) + 2(x-y)XA + 2(x-y)yXM
+ 2XJ' + 2Z'F - Z' {Z + F- (« - yy A}
-2yX[2(x-y)M + 2(x- yy e} + {x - yy yMX',
= -n(Z+F) + 2ZJ'-Z'(Z-F)
+ 2{x-y)XA-2(x-y) yXM
+ {x-yy X'A - 4 (a; - yy eyX + {x- yy xjMX'.
We have 2ZJ" = J'(Z+ Y) + J'{X-Y), and hence the first line is
= (-n + J')(^ + F) + J"'(Z-F);
— H + J', as will be shown, divides by x — y, or say it is ={x — y) ^, and, as before,
Z — F is = (a; — y) n ; hence, throwing out the factor x — y, the equation becomes
(a.— y)» V = * (Z + F) + ft ( J' - Z') + 2ZA - 2yXM + {x-y) [X'A - 4eyZ + yMX'}.
We have
il=^h + c{x -^^ y) + d{a? + xy +f)Jt eia? ■\- x^'y + xy" + _»/'),
and thence
- ft + ./' = c (- a; + y) + d (- a.'^ + a^) + e (- j? + oai?y - bxy"" + if) ;
or, dividing this by (x — y), we find
^ = -c — dx — e(x'— ixy + y"),
or, as this may be wiitten,
4> = — A + rfy + ^xy.
600
dx dy ^ r
ON THE GENERAL DIFFERENTIAL EQUATION -y= + -y-^=0. [626
We find, moreover,
J'-X' = 2c(-a;+y) + d(-3ic' + 2xy + f) + e(-^ + Gofiy - ixy* + 2y»),
which divides by {x — y), the quotient being
- 2c - d (3a; + y) - e {^ -2xy+ 2y-),
viz. this is
= -2A-{x-y)(d + 2ex).
Hence the equation now is
(a; - y)= V = (Z + F) [- A + dy + 4>eay\ + 2ZA - 2yXM
+ {x-y) n {- 2A - (a;-y){d+ 2ex)}
+ {x-y) { X'A-'^yX+yMX' ].
The first line is
{X + Y) [-K + yM-\-2{x-y)ye] + 2ZA - 2yXM,
which is
= {A-yM){X-Y) + 2{x-y)ey{X+Y);
hence, throwing out the factor x — y, the equation becomes
{x-y)V ={A-yM)il + 2eyiX + Y)-2An + X'A-4>eyX + yMX' -{x-y)il(d + 2ex)
= iA+yM){- n + X') - 2ey {X - ¥)- (x -y) n(d + 2ex).
We have
- n + X' = c(x- y) + d{2x' - xy - y-) + e (Sx^ -x'y-xy-- f),
which is ={x—y){A-\-xM): also (Z — F) = (« — y) li, as before; whence, throwing out
the factor x — y, the equation is
V =(A + xM){A + yM) - 2eyD. -{d + 2ex) 12,
that is,
V = (A + xM) (A + yM) - Mil ;
viz. substituting for V its value, reducing, and throwing out the factor M, the
equation becomes
— b + dxy = (a; + y) A — 12,
which is right.
Verification of 33. — The equation is
J [2{x-y) M -^-^eix-yYl-J' {x-yf M - ^{x-y)' Y
= 4:{x-y)M Y +ix- yf MY' + 2e (a; - yf Y',
which, throvnng out the factor x-y, is
0=2M(-J+2Y) + {x-y)M(,J'+Y') + 2e{x-y){-J-v2Y) + 2e{x-yf Y'.
dec UAJ
626] ON THE OENERAL DIFFERENTIAL EQUATION -pF? + -p^r=^- 601
Here —J+IY, = — (X—Y) + (x — yy A, is divisible by (x — y): hence, throwing out the
factor X — y, the equation is
0=M{-2b-2c(a; + y)-2d(a^+xy + y^)-2e(ic' + a^ + mf + y>)]
+ M(J'+Y')+2M(x-y)A+2e(-J+2Y) + 2e(x-y)7'.
In the first and second terms, the tiactor which multiplies M is
c(-2a;+2y) + d{-2a^ + 2y') + e (- 2x' + 'iafy - 6xy"- + if),
which is divisible hy x-y; also —J-\-2Y, = - (X — Y) + (x - yf A, is divisible bj' (x—y)z
hence, throwing this factor out, the equation is
0 = M{-2c + d{-2x-2y) + e(-2ofi+2xy-iy^)}+ 2MA
-¥ 2e {- b - c (x + y) - d (a-' + xy + y^) - e(af + af^ + xy^ + f)]
+ 2e{x-y)A + 2er.
Here in the first line the coeflScient of Jf is = e {2xy — 2y^) : hence, throwing out the
constant factor 2e, the equation is
0 = -b-c{x + y)-d(x- + xy-\-y^)-e(af+a^y+ xf + f)+ Y' + {x-y)yM + (x- y) A.
The first five terms are
= c{-x+y) + d{-a?-xy + 2y"-) + e(- x" -x^y-acy- + Sy'),
which is divisible by x—y; throwing out this factor, the equation is
0 = - c - d(x + 2y) - e (x- + 2xy + Sy^) + A + yM,
which is right.
Verification of 6. — We have
-2X\2{x-y)M+2e{x-yf\-\-{x-yyX'M + A>e{x-yyX-2e{x-yyX'
= -{x -y)M [iX -{x- y)X'] -2e{x- yf X' ,
which is, in fact, an identity.
Verifi^atioii of 2). — The equation may be written
4X + 4,Y + 'ieix-yy
= 2X + 2Y-2(x-yyA
+ 4,X-2x{x-yyM
+ 2(x-y){2(x-y)xM+2ex(x-yy-M{x-yy-J'\,
viz. this is
0 = 2X -2Y - ie{x -yy - 2{x-yy A+ 2x{x- yy M
■>r iea:{x- yy -2M (x -yy -2(x -y)J'.
c. IX. 76
dx dy ^ r^^ .
602 ON THE OENBEAL DIFFERENTIAL EQUATION -7^ + -y'v — ^- L^^O
The first terra 2 (X — F) is divisible by 2{x — y); throwing this factor out, the
equation becomes
0 = 6 + c (a; + y) + rf (a^ + a:y + y*) + e (a!* 4 a!»y + «y» + y ) - J'
— 2c (a; — y)' - (a; - y) A + x(x—y) M+2ex(x — yy — M(x — yy.
Substituting for J' its value, the first line becomes
c (x — y) -{■ d (i^ — xy) + e{a^ — 5a^y + 5xy' — y*),
which is divisible by (x—y); hence, throwing out this factor, the equation is
0 = e + dx+e{a^- 4ary + y" ) - A + xM -2e{x — yy + 2ex{x-y) — M{x- y),
where the sum of all the terms but the last is = d (a; — y) + e {2a? — 2xy) : hence, again
throwing out the factor x—y, the equation becomes
0 = d + 2ea; - 2e (a; - y) + 2«a; - ilf,
which is right.
Recapitulating, we have for the general integral of -jy "*■ 7 v ~ ^' ^^ ^^^ *
. . . ^ , „ dx dy , dz ^
particular mtegral 01 -jy + r^ + 77 = ",
^ J - 2 y/e (a; - y) y yZ + 2 Ve (a; - y) a; V F - 2 VZF
'~ (a;-y)=il/-2 Ve(a;-y)VZ + 2Ve(a;-y)VF '
the corresponding value of ^JZ being
-Je [- Z» - 6ZF -Y'-\-{x- yY {A^ + (-b + dxy) M + xyM"]]
+ [{4F+ (a; -y) F') M+ 2e{x-yf F'] (a;- y) V^
- [{4Z - (a; - y) Z'] ilf + 2e (a; - y)^ X'] (a; - y) V F
,„^+i 4(Z+F)+4e(^-yy] VZF
" {{x-yyM-2^e{x-y)^X + 2^e{x-y)s/Y\''
where, as before,
3/ = d + 2e (a; + y),
A = c + d(a;+y) + e(a;» + y»),
.A = 2a + 6 (a; + y) + 2ca;y + dxy {x-\-y) + exy {a? — xy + y') :
also X is the general quartic function o + 6a; + car* + dx* + e«*, and F, ^ are the same
functions of y, z respectively.
In connexion with what precedes, I give some investigations relating to the more
simple form % = a + c6°- ■¥ e6\ or, as it will be convenient to write it, 0 = 1-/^+^.
626]
dx dy
ON THE GENERAL DIFFERENTIAL EQUATION ^^^+-r^,= 0.
603
w
e have
X, s/X
2/, VF
a?, X, >JX
y*. y, s/Y
^, 2, V^
«»,
X,
a;=V^. -JX
y".
y.
f'sjY, VF
z".
z,
z^- V-^, ^/Z
vfi.
w,
vP^jW, >JW
and so on ;
viz
in taking
a?, X, 'JX
f, y, VF
^,
z, >JZ \
= 0 a particular integral
. dx d« .
= 0 the general integral ^
a pai'ticular integraH
\ f dx dy dz ,.
Y^^^M^z^""'
= 0 the genenal integral j
a particular integral '
dw
. dx dy dz — . _
^°f V^ + VF + V^ + VTF=^'
= 0 as the general integral of t^-s^ + jL. = 0,
we consider z as the constant of integration : and so in other cases.
It is to be remarked that it is an essentially different problem to verify a
particular integral and to verify a general integral, and that the former is the more
difficult one. In fact, if U=0 is a particular integral of the differential equation
Mdx-\-Ndy = 0, then we must have N -^ ilf^ =0, not identically but in virtue
of the relation f/ = 0, or we have to consider whether two given relations between
X and y are in fact one and the same relation. In the case of a general solution,
this is theoretically reducible to the form c=V, c being the constant of integration,
and we have then the equation N j ~ ^ j —^' satisfied identically, or, what is the
same thing, U a solution of this partial differential equation.
Hence it is theoretically easier to verify that
ar', X,
f' y.
2^, Z,
•JY
^Z
= 0
is a general solution, than to verify that
w, \/X
y, -JY
= 0
76—2
604
fix Cm It
ON THE GENERAL DIFFERENTIAL EQUATION ,^+ -r4r=0.
J A. J I
dx dy
[626
is a particular solution of the differential equation / y + ^ = t). Moreover, taking the
first equation in the before mentioned form
— z-
x-JY-ys/X'
and writing therein ^ = oo , we see that the second equation
X, sJX
y, VF
= 0
is, in fact, a particular case of the first equation, so that we only require to verify
the first equation ; or, what is the same thing, to verify that
z =
a?-y^
is the general integral of
X ijY — y >JX
dx dy
To verify this, we have to show that dz = £l{ -jy + -p^ , viz. that \JX -=- = n,
dz
a symmetrical function of {x, y); for then \/Y -^ =D., and we have the relation in
question.
We have
{,x^Y-ys/Xr^X^=^X^{a?-f)(^JY-,l^^^)-^{x>JY-y>JX)'
^^X\^{a?-f- 2a?) V F - ^'^~^^^~ + ^y V^}
= - (a.^ + 2/0 VZF + 2xyX - i (^ - f) yX'.
Writing here X = 1- ln^' + x*, then X' = — 2lx + 4^, and we have the last two terms
= 2xy{l - la? + x*) + (a? - f) xy (I - 2x^)
= xy{2-^ax^+2x^ + {a?-f)(l-2a?)}
= xy{2-lix'+y'') + 2xiy}.
Hence the equation is
(xs/Y- y ^/Xy ^/X ^ = -(.'^ + f)^/TY + xy {2 - l{af + y*) + 2aiyl
or we have
" = (^^;^y^^^;^ {- (^ + i/0 ^^^I" + *i/ (2 - Ua^ + y") + 2^y)).
626] ON THE GENERAL DIFFEEENTIAL EQUATION yxr + y.---0.
605
which is symmetrical in {x, y), as it should be. And observe, further, that since
dsc du dz
the equation is a pai-ticular solution of j^ + ~^-\- ,„ = 0, we must have il = — '\/Z;
viz. we have
•JZ(x^Y-y^Xf = -(ar= + t/n VZF + «y {2 - i (a^ + j/^) + tic'y^
Proceeding to the next case, where we have between x, y, z, w & relation which
may be written
{a?, X, ar'VX 'slX) = Q>,
then here a, 6, c, d can be determined so that
(c^ + d)Ml + /3^ + 7^) - (a^ + hdY = <i^{e^- a?) {&' - y') (^ - z-) {&- - vfi),
viz. we have d' = c^ a^y^zHiP, or say d = c\/y xyzw. And, supposing the ratios of a, b, c, d
determined by the three equations which contain (x, y, z) respectively, we have
a :b : c : d={x, af'^X, -JX) : -{af>, ofis/X, ^X) : (oc^, x, ^X) : -(««, x, a?^X),
or in particular
d_- (af , X, a? yZ) _ -^r^^^^^^VZ)
c (a?, X, >JX) ' («*, X, >JX) '
whence we have
w= —
{x', 1, x^X)
(of, X, y/X)
a new form of the integral equation; viz. written at full length, this is
of, 1, x^X
j_
f, I, y^Y
z", 1, z s/Z
a?.
«:,
vx
f,
V'
VF
z".
z,
^JZ
and taking «; = 0 and = oo respectively, we thus see how
a?, 1, X tjX
= 0,
y\ 1, y-JY
z\ 1, z^Z
a?.
X, ^/X
t>
2/. VF
^,
z, sjZ
= 0,
are each of them a particular integral of
^X^ s/Y^ ^Z~^-
Reverting to the general form
^" ~ (X\ X, y/X) '
dx dv
606 ON THE GENERAL DIFFERENTIAL EQUATION -Jy + ^- = ^- [626
this will be a general integral if only
viz. if we have
— i\/X T- p-5 — '■ — .y. = X2, a symmetrical function of {x, y, z).
The expression is
or, writing for shortness
a=x (y- - z\ a = yz {y- - z-),
fi = y{2i'-ofi), b=zx(z^-x"),
y=z{a^-y% c = xy (te' - y-),
we have
and the formula is
(a?, 1, x>/X) = a>^X + ^^Y+y^/Z,
(ijfi, X, s/X)==^as/X+b VT+c .JZ;
= (a VX + /9 ^1'+ 7 V^) {(y'^ - yz") hX' + (- 3a!»2 + z») VZF + (Sary - y) V'Jr^j
-{a>JX + b v/r+ c VZ) {(2/^ - z'') (X + hX'x) - 2xy -JXY- 2xz '^XZ\
= (as/X + fi^Y+y^Z){L+M'^XY + N'^XZ)
- (a ^X+bi^Y+ cs/Z){P+Q V'ZF+ R 'JXZ), suppose.
vz
+ VF
+ '^Z
+ ^XYZ
oZ
+ aMX
+ aNX
+ /3if r
+ /8X
+ 0N
+ rfNZ
+ yL
+ yM
-aP
-aQZ
-aRX
-bQY
-bP
-bR
-cRZ
-cP
-cQ
viz. this is
{aL-aP + Y(^M-bQ) + Z{yN-cR)}^/X
+ {X(aM-aQ) + fiL-bP } VF
+ \X(eLN-aR)+yL-cP ]-jZ
+ i^N + yM-bR-cQ )'JTYZ.
dx dv
626] ON THE GENERAL DIFFERENTIAL EQUATION -jy + iy~
607
The coefficient of wXYZ is here
which is
= y{z^-a?){^y-if)
-\-z{a?- y) (- Sa^z + z')
— zx {z- — nf) (2xz)
-a=y{a?-f){-2a-y)
= f{z'-a?){^a?-f)
■\-z^{x'-y^){-^a? + z^)
- laf'z- (^2 - a?)
+ 2aff(x'-y'),
= Qoe^y-z^ — y-z* — y^z- — z'-x* — z*ii'' — n-^y'^ — afy*.
The coefficient of V^^ is
= [^ (.y* - •z*) (- '^^ + z^) + yz {y-' - z') Ixy] X
+ y(2'- Ofi) iX' (fz -yz>)-zx{l^- of) (f - z') (X + ^X'x)
= - 2xz (a^ - f) (y^ -z')X-z(af- y') (y'' - z'') (z' - «=') J X'
= -(a^-f)(f-z')z{2xX + ^iz--aP)X'],
where the term in j } is
= 2x{l-lx' + x') + (z- -a^}(-la; + 2a^),
or the whole coefficient is
= -{a^-f)(f-ji')zx[2-l (z' + ai") + 2^Vj.
We obtain in like manner the coefficient of tjZ, and with a little more trouble that
of ijX; and the final result is
n {a?, X, y/Xy = -(z'- x') (a^ - f) yz{2-l (/ + z'^) + 2fz^\ s/X
-{of- yO (f - ^') zx{2-l (z^ +«") + 22V} ^/Y
-{f- z') {z* - a^) xy {2 - I {of + y") + 2a!' f] s/Z
+ {Gx'y-z' - y'z^ - y*z^ - z^ar' - z*a? - a?y* - a*y^) ^XYZ.
And inasmuch as the equation is a solution of
dx dy dz dw _ _
it follows that n = — n/W, viz. that 'JW is by the foregoing equation expressed as a
function of x, y, z.
The equation (a;', x, a?\jX, iJX) = 0, that is,
a^, X, a? \JX , -JX
z", z, z' sJZ , sJZ
= 0,
608
gives
dx dy
OS THE GENERAL DIFFERENTIAL EQUATION ^^ + -^ = 0.
(of', 1, XnjX)
[626
«; =
where the numerator and the denominator are deteraiinants formed with the variables
ic, y, z.
Writing - for w, it follows that the equation
«•, X, arn^X, \/X
f, y, y'-JY, VF
^», z, z^sjZ, kIZ
= 0
gives
?<; =
{a?, X, yz)
' {a?, 1, .rVZ)'
which last equation is a transformation of
a^, a?, 1, a;VZ =0.
t, y-, 1, yVF
i:^, Z-, 1, ^ VZ
w*, W-, 1, w\/W
The two equations, involving these determinants of the oixier 4, are consequently
equivalent equations.
627]
609
627.
GEOMETEICAL ILLUSTRATION OF A THEOREM RELATING TO
AN IRRATIONAL FUNCTION OF AN IMAGINARY VARIABLE.
[From the Froceedings of the London Mathematical Society, vol. Vlll. (1876 — 1877),
pp. 212—214. Read May 11, 1876.]
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 ii there belong two or more values, in general
imaginary, x' + iy' 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, of v, u respectively. We must for this
purpose take x, y a& the coordinates of a point P in a plane II, and x', y' as the
cooi-dinates of a corresponding point P" in another plane II'. The series of values
X + iy of u is then represented by means of a curve in the first plane, and the series
of values x' + iy of v by means of a corresponding curve in the second plane. The
correspondence between the two points P and P' is of course established by the two
equations into which the given equation f{x + iy, a/ + iy') = 0 breaks up, on the
assumption that x, y, x', y are all of them real. If we assume that the coefficients
in the equation are real, then the two equations are
f(x + iy, x' + iy) +f{x - iy, x' - iy') = 0,
f(x + iy, x' + ii/) -f{x - iy, x - iy') = 0 ;
viz. if in these equations we regard either set of coordinates, say (x, y), as constants,
then the other set (pcf, i/) are the coordinates of any real point of intersection of the
curves represented by these equations respectively.
I consider the particular case where the equation between u, v is m' + «' = a' : we
have here {x + iy)' + (x' + i^f = a' : so that, to a given point P in the first plane, there
c. IX. 77
€10 GEOMETRICAL ILLUSTRATION OF A THEOREM RELATING TO [627
correspond in general two points P,', P/ in the second plane : but to each of the
points A and B, coordinates (a, 0) and (—a, 0), there corresponds only a single point
in the second plane.
We have here a particular case of a well-known theorem: viz. if from a given
point P we pass by a closed curve, not containing within it either of the points A
or B, back to the initial point P, we pass in the other plane from P,' by a closed
curve back to P,' ; and similarly from P/ by a closed curve back to P/ : but if the
closed curve described by P contain within it A or B, then, in the other plane, we
pass continuously from P,' to P/; and also continuously from P./ to P,'.
The relations between (x, y), («', y') are
x'^-y"- = a?-{a?-'f),
x'y' = - xy,
whence also
(«'» + y'^y- = a*- 2a» (af-y') + {a^ + ff.
And if the point {x, y) describe a curve a? + y- = ^{a? — y'), then will the point (x', y')
describe a curve «'' + y'" = V^ {x'^ — y'% obtained by the elimination ofaf—i/' from the
two equations
x"--y''= a^- {a?-f),
{x^ + y-)" = «* - 2a= {a? -y-) + <f> («" -f);
viz. this is
(x' + yy =-a* + 2a' (a;'» - f-) + </> {a= - («'» - y'%
In particular, if the one curve be {of + y^y = a + /9 (a^ - y") ; then the other curve is
{x' + y'^y =-a* + 2a' {x" -y'') + a + ^{a' - (x'' - y'%
that is,
{x'' + y''y = a' + ^{x''-y'%
where
a' = -a* + fia' + a, /9'=2a'-/9.
Writing for greater simplicity a=l, then a'=-l+a + /9, /3' = 2-/9; in particular, if
0 = 0, then a'=-l +/3, /8' = 2-/3.
Supposing successively /8<1, &=1, and /8>1, then in each case P describes a
closed curve or half figure-of-eight, as shown in the annexed P-figure ; but in the
first case the point A is inside the curve, in the second case on it, and in the third
case outside it, as shown by the letters A, A, A of the figure; and, corresponding
to the three cases respectively, we have the three P'-figures, the curve in the first of
them consisting of two ovals, in the second of them being a figure of eight, and in
the third a twice-indented or pinched oval: the small figures 1, 2, 3, 4 in the P-figure,
and 1, 2, 3, 4 and 1', 2', 3', 4' in the P'-figures serve to show the corresponding
627]
AN IRRATIONAL FUNCTION OF AN IMAGINARY VARIABLE.
611
positions of the points P and P/, P/ respectively ; and the courses are further indi-
cated by the arrows. And we thus see how the two separate closed curves described
P-Figure.
i"-Fig. 1.
P-Fig. 2.
P'-Fig. 3.
by P,' and P/, as in figure 1, change into the single closed curve described one half
of it by Pj' and the other half of it by P^' as in figure 3.
77—2
612 [628
628.
ON THE CIRCULAR RELATION OF MOBIUS.
[From tho ProceedingH of the Londun Afathemutiail Society, vol. viil. (1876 — 1877),
pp. 220—22.'). Hciul April 12, 1877.]
In representing a given imagimiry oi- complex qinintity a, » a; + iy, by nioanH of
the point whoBO coordinateH are to, y, wc iiHRumu in the firnt instance that a, y are
real, — but in the rcHiilts this roHtriction may bo abandoned — for instnnco, if tho imaginary
cjuantitioH u, u', c are connected by the equation «' + «'• — c"; then, writing wx + iy,
u ^ m' + iy', c-tt+W, we have of — y'' + a/' — y'* ^ a* — b^, xy + a/y'miab, equations con-
necting the points U, U', C which serve to represent tho quantities it, u', c, and which
(regarding C as a fixed point) establish a correBpondence between the two variable
points U, U' : any given value m, ^w + iy, is represented by tho point U, and corre-
sponding hereto wc have (in the present case) two points V, viz. those are the real
intersections of the curves of' — y"" ^ a' — b' — (of — y*), xy'^ab — xy, and then the
coordinates x', \J of either of thest) give tho value of -k-iy of «'.
But, the two curves once arrived at, we may for other purposes be concerned
with their intersections as well imaginary as real ; or, still more generally, all tho
({uantities entering into tho two equations may be regarded as imaginary.
Theoretically wo seem to require two imaginary roots of unity, incommensunibic
and convertible, viz. taking these to be i, I, then i'« — 1, /*■■— 1, il^Ii, but without
any relation between i, I: thus, in what precedes, writing / instead of t, viz.
a, //.', cw+Iy, x' + fy', a + bl, here each of the ([uantities x, y, m', y', a, b pxm be
ah initio an imaginary (|uantity of the form \ + fxi. Wnt, conforming to tho onlinary
practice, I use t only, writing for instance w = a; + iy, without any uxpross statement
that a, y are real ; on the understamling that any e(|uation containing such ({uantities,
and therefore ultimately of the form /' + iQ=-0, denotes the two equations /' — O, Q^O
(or, what is tho same thing, thai, wo have not oidy the original eijuation, but, in
tuldition to it, the like equation with each such ()uantity x + iy replaced by tho con-
628]
ON THK OIROULAK HKLATION OK M^BIUH.
013
jugjitf (|unntity ;r — I'y): and tho further luidoi-HtaiMliiiji; that, in the jMvir of tH^uationt*.
oach of tho (]uantitieB a\ y, i&e. ontoring thcntin iiiay it^olf Ih< ooimi(l(>r«Hl iin an
imaginary ipiantity of the fonn \+fii,
Thi' forogoinj{ oxplanation is rc(|nin>il, for othorwiMO it would appoar an if the
circular i-olation of Mol)inN • about to bo oxplainoil was of n»)00HHity a mlation bctwiuin
real pointn: I hold that thin in not the caH«. But in nil that folloWN I do, in fact,
oonaidor priinai'ily tho cjiso of real points ; ami induiHl the oooiwion doim not arino for
any oxplioit conHidomlion of tho ciwo of imajfinary pointH.
The circular relation ia iim followH. If in tho fii-Ht inotniico wo havo four pointn
U, A, B, C on a lino, and u, a, h, o their diNtanoos from any Hxod point on that
lino; and a^iin, U', A', ff, C four other poiiitN on a lino (tho naino or a difforont
line), and «'. a', b', c' their diHtanooH from any fixed {)oint on that lino ; then tho Hiinio
e(piation between u, o, b, c, «', o', b'. c which oxproHMOH tho honioj^raphic relation
between the two rauKtm of pointH U, A, li, 0 and U', A', ff, C, expi-eMMen, when
differently intorpretod. the circular relation betwe<(n the four pointM U, A. H, (■ in a
piano, and tho four other pointH W, A', H', 0' in the name or a dilVoront plane-
viz, for the new inter|)retation, u w ixhwI an denoting co-^iy, the linear function of
the coordinatCH x, y of tho point U, and the like iim regardw tho remaining ipiantitioN
o, 6, c. (/', a, b', c'.
Am in the homographic theory (but of noiiiw without the condition of being in
a line), we have A, A' ', B, B* \ 0, C given paii-H of oorroHponding pointH: the iM|iuition
now roproHontH two oquationH; and thoM((, when either of the pointH U, U' in given,
<leterinine the corroHponding point U' or U.
The homographic relation may hv written in I lie fornm
1, II, «', II a -0,
1, a, a', lut'
\, h, //, W
1, c, o', co'
ii — <i.b — c : u — b.c—ii : « — o . a — 6 ■ w' — a' ,b' — o' : a' — b'.c' — a : «' — o' . «' - b',
viz. thoHe arc forniH of one and tho Mime oiptation : and it may be lulded that, if m
in the firat nyNtem corrcHpondH to oo in tho neiwind Hyntem, atid m' in the Neoond
HyHtein to x in the firnt HyHtotn (of courae to, to are not c^orroHponding vahuiN in tho
two HyMtenm roBpeotively) ; ho that
I ,
1, a, a',
1. b, b'.
1, 0, a',
10 — u.b — c : (o —b .c— a : to ■
b — c : 0 -a : a — b~m'-a'.b' — o':to'~b'.o' — a':to' — o'.a'-b';
• fMlibinn, Oil. Wtrhe: t. ii., pp. 948 -HI4, nnil elNiiwhnrn.)
A>
-
0,
J,
to'
(»,«'
1, «,
a', ««'
bb'
1. b.
b'. bb'
cd
1. 0,
o'. oo'
-0.(
t-
-6-
b'-
-&
: o'-fi'
: ii'-b';
-0.
ON THE CIRCULAR RELATION OF MOBIUS.
[628
A =
1,
1,
1
a,
b.
c
a',
v.
c'
he' — h'c -^-ca — c'a + ah' — a'b,
614
whence also, if
then
— A.a) — a = b — c.c'—a'.a'—b', A . to' — a =b' — c' .c —a.a — b,
— A.a) — b = c—a.a' — b'.b'—c', A.w' —b' =c' —a' .a — b .b —c,
— A . eo — c = a — b . b' — c' . c' — a', A. eo' — c' = a' — b' .b — c .c —a.
Then, o», to' being thus determined, we have
ft>— a.ft)' — a' = a) — 6.0)' — 6' = «i)— c.w' — c' = ft) — M.fi)' — iif
b — c.c — a.a — b.b' — c'.c' — a'.a' — b' . .
= -^1 (=A suppose),
viz. we have to —u.o)' —n' = a given value ; which is the most simple form of the
relation between u, u'.
Interpreting everything in the first instance in regard to the homographic ranges,
the equations show that there is in the first range a point 0, and in the second
range a point (J, such that OA, &c. denoting distances, we have
OA .O'A' = OB.O'F = OC .O'C = 0U .O'U' (=A);
or, what is the same thing, if in the line of the first range we construct A^, Bi, 6',, Ui
by the formulae
OA . OA, = OB . OB, = OG. 00, = 0U. OU, = A,
that is, invert the first range in regard to the centre 0 and squared radius A, then
we have a range 0, ^i, B,, C„ U, equal to the range 0', A', E, C, U' ; viz. the
distances of corresponding points are equal in the two cases : or say a range
0, Au B„ C„ U, imposable upon 0', A', B', C, U'.
The like result holds for the circular relation, but the interpretation must be
explained more in detail. And, first, it is to be remarked that 0 in the first figure
is the point corresponding to any point whatever at infinity in the second figure;
viz. writing u' = ^ + irj', = oo , then, whatever value we give to the ratio of the two
infinite quantities ^', ij', we obtain the same complex value of <b, that is, the same
coordinates for the point 0. And, similarly, 0' in the second figure is the point corre-
sponding to any point whatever at infinity in the first figure.
To determine 0, we have the equation
a> — a b' — c c — a
w — b c' — a! ' a — b'
Any such equation gives at once the geometrical construction, viz. w — a = OA^^',
where OA is the distance of the points 0, A regarded as positive, and OAx is the
628] ON THE CIRCULAR RELATION OF MOBIUS. 615
inclination of the line OA regarded as drawn from ^ to 0 to the line Ax, such
angle being measured in the sense Ax to Ay ; where Ax, Ay are the lines drawn
from A in the senses x positive and y positive respectively : and so in other cases.
The equation is therefore equivalent to the two equations
qA__B'G' CA^
0B~ C'A'- AB'
and
Z OAx - I. OBx = z RG'x - z O'A'x + z CAx - z ABx.
The former of these expresses that 0 is in a certain circle which, having its centre
on the line AB, cuts AB and AB produced in the one or the other sense ; the latter
that it is in the segment described on a determinate side of AB and containing
a given angle : hence 0, as the intersection of the segment with the first-mentioned
circle, is a uniquely determined point. Similarly (7 is a uniquely determined point.
It is not obvious how to construct A, from its original value as given above (but,
CO being known, we can without difficulty construct it from the value
— A .to — a = b — c.c' — a'.a'— b'),
nor consequently A from its expression in terms of A : but, <o and w' being known, we
can construct A from the expression tu — a . <u' — a' = A ; supposing it thus constructed,
= ke'^ suppose, then if, with centre 0 and squared radius k, we invert the first figure,
thereby obtaining the points A], B,, C,, Ui such that
OA.OA, = OB.OB,='OC.OC, = OU.OU,=^k,
(the points Ai, 5,, 0,, fT, being on the lines OA, OB, 00, OU respectively,) then the
equations
a — a. w' — a' = m — b . m' — b' = fo — c . co' — c' = to — u . to' — u = ke^*
give
CO — a.co' — a' = OA . OA^e^,
that is,
OA . 0'A'= OA . 0A„ or, simply, O'A' = OA,,
and
^AOx + ^AO'x' = d,
or, what is the same thing,
/:A,Ox+ ^A'Ox' = e,
and so for the other letters, viz. we have
O'A', O'B', O'C, 0'U':=OAu 0B„ 0G„ 0U„
respectively; and further
Z's A,Ox. B,Ox, Gfix, Ufix = e-A'0'x', O-B'O'x, 0-C'O'x', e-U'O'x',
616 ON THE CIRCULAR RELATION OF MOBIUS. [628
respectively: viz. the sj-stem of points 0, Ai, B,, C,, Uj is equal to the system
C, A', R, C, U', that is, the distances of corresponding points and magnitudes of
coiresponding angles are severally equal — but the angles AiOx and A'C^x', &c. are in
opposite senses, as appears by the just mentioned equations Afix — Q — A'0'x, &c. ;
that is, the two figures are symmetrically equal : but the one of them is not, except
by a turning over of its plane, imposable upon the other.
The conclusion is, the two figures A, B, G, U and A', B', C, U' are each of
them equal by symmetry, but not superimposably, to a figure which is the inverse
of the other of them ; viz. there exists in the first figure a point 0, and in the
second figure a point 0', such that, inverting say the first figiire, with centre 0 and
a squared radius of determinate magnitude, we obtain the points Ai, fi,, Cj, Ui,
forming with 0 a figure equal by symmetry, but not superimposably, to the second
figure A', B', C, U', (/. Hence also to any line in the first figure corresponds in the
second figure a circle through (7, and to any line in the second figure there corre-
sponds in the first figure a circle through 0 ; or, more generally, to any circle in either
figure there corresponds a circle in the other figure.
There is a particular case of peculiar interest, viz. writing for greater convenience
d, d' as corresponding values in place of u, it', the system a, b, c, d corresponds
homographically to itself in three different ways; that is, we may have
(«', b', c', d') = (b, a, d, c), (c, d, a, b) or (d, c, b, a).
To fix the ideas, attending to the first case, we have thus the range of points
(A, B, G, D) corresponding homographically to {B, A, D, G), viz. here w'=a>, and
<o — a.co — b = a3— c.a>— d, that is, the corresponding points U, U' belong to the in-
volution where A and B and also C and D are corresponding points. The like theory
applies to the circular transformation: viz. the points {A, B, G, D) may correspond to
{B, A, D, G), viz. there exists a point 0 (or say 0,) and squared radius ^,, such that,
inverting the figure and marking the inverse points of A, B, G, D as Bi, Ay, D^, Ci
respectively, the new figure Oi, Ai, B^, 0,, A is equal by symmetry, but not super-
imposably, to the original figure OABGD. The equation w, — a . a>, — 6 = », — c . Wj — d
gives the geometrical definition of the point 0,, viz. this is a point such that
OiA . OiB = OiG . OiD and further that AB and GD subtend at 0, equal angles : we
have (»i= r -j, giving for toi — a, ta, — 6, Wi — c, w, — d convenient expressions
the first of which is a>i — a = — -r- r . We hence obtain a convenient construction
c+d—a—b
for 0, viz. taking M for the middle point of AB and N for the middle point of
CD, and drawing from A in the sense M to A^ & line AP, = 2MN, then this equation
may be written w, — a = '■ (^ the function x + iy which belongs to the
point P) ; thence OiA = — ^-j- and
/lOtAx = Z.CAx + ^DAx-zPAx,
conditions which determine uniquely the position of 0,.
628]
ON THE CIRCULAR RELATION OF MOBIUS.
617
We may have (A, B, C, D) corresponding to (C, D, A, B) and (Z), G, B, A), the
inversions for these depending on the points 0^ and O3 respectively: I annex a figure
showing the three inversions of the same four points A, B, G, B.
C IX.
78
618 [629
ON THE LINEAR TRANSFORMATION OF THE INTEGRAL [
629.
du
[From the Proceedings of the London Mathematical Society, vol. viii. (1876 — 1877),
pp. 226—229. Read April 12, 1877.]
The quartic function U is taken to be =e.u — a.u — h.u — c.u — d, where a, b, c, d
are imaginary values represented in the usual manner by means of the points A, B, C, D;
viz. if a = 0(1 + a^i, then A is the point whose rectangular coordinates are a^, Oj ; and
the like as regards B, C, D. And I consider chiefly the definite integrals such as
f * du
I jrj where the path is taken to be the right line from A to B. There is here
nothing to fix the sign of the radical; but if at any particular point of the path we
assign to it at pleasure one of its two values, then (the radical varying continuously)
this determines the value at every other point of the path; and the integral defined
as above is completely determinate except as to its sign, which might be fixed as
above, but which is better left indeterminate. The integral, thus determinate except
as to its sign, is denoted by {AB).
I wish to establish the theorem that, if the points A, B, C, D taken in this
order form a convex quadrilateral, then
(AB) = ± (CD). {AD) = ± (BC), but not (AC) = + (BD) ;
whereas, if the four points fonn a triangle and interior point, then the three equations
all hold good. I regard the theorem as the precise statement of Bouquet and Briot's
theorem, A-B+C-D = 0, or say (OA)-(OB) + (OC)-(OD) = 0, where the four terms
are the rectilinear integrals taken from a point 0 to the four points A, B, C, D
respectively. The two cases may be called, for shortness, the convex and the reentrant
cases respectively.
629] ON THE LINEAB TBANSFORMATION OF THE INTEGRAL \ rjf . 619
To prove in the case of a convex quadrilateral that (AC) is not = + (BD), it is
■ , where A, B, C, D are the points (1, 0),
Vm*- 1
(0, 1), (—1, 0), and (0, —1) respectively, and where, writing v = iu, it at once appears
that we have
P du _, ■(' du
J-i^K* — l J-i's/u* — !'
that is,
(AO) = + i (BD), not (AG) = + (BD).
But I consider the general question of the linear transformation. If a', b', c', d'
correspond homographically to a, b, c, d, then to represent these values a, b', c', d' we
have the points A', E, C, D', connected with A, B, C, D according to the circular
relation of Mobius; and then, making u, a, b', c', d' correspond homographically to
M, a, b, c, d, and representing in like manner the variables u, u' by the points U, JJ'
respectively, we have the circular relation between the two systems U, A, B, C, D
and U', A', R, C, U.
Before going further I remark that the distinction of the convex and reentrant
cases is not an invariable one ; the figures are transformable the one into the other.
Thus, taking 0 on the line BD (that is, between B and Z), not on the line produced),
there is not this relation between B, C, D', and the figure A'RC'D" is convex or
reentrant as the case may be. Giving to G an infinitesimal displacement to the one
side or the other of the line BD, we have in the one case a convex figure, in the
other case a reentrant figure ABGD; but the corresponding displacement of C' being
infinitesimal, the figure A'RG'U remains for either displacement, convex or reentrant,
as it originally was ; that is, we have a convex figure ABGD and a reentrant figure
ABGD, each corresponding to the figure A'BG'U (which is convex, or else reentrant,
as the case may be).
Writing for convenience
a, b, c, f, g, h =6 — c, c —a, a —b, a —d, b —d, c —d,
a', b', c', f, g', h' = 6' - c', c' - a', a' - b', a'-d', b' - d', c' - d',
80 that identically
af+bg + ch = 0, a'f' + b'g' + c'h' = 0,
then the homographic relation between (a, b, c, d), (a', b', c', d') may be written in the
forms
af : bg : ch = a'f' : b'g' : cTi',
or, what is the same thing, there exists a quantity iV such that
aT^bV_cJi^^^^,_
af bg ch
78—2
620 ON THE LINEAR TRANSFORMATION OF THE INTEGRAL \-jjf' [^^9
The relation between u, u' may be written in the forms
u' — a' _ pU — a n' — b' _ ^u — b u —c'_rfii-c
u' —d' u — d' u' — d' u — d' u' — d'~ ii — d'
and then, writing for it, u' their corresponding values, we find
b1i_cg ci_o^_^ p_a'g_b'f
bh'~cg" ^~cf'~ah" '"~ag'"bf"
giving
tgh
Differentiating any one of the equations in (u, u), for instance the first of them,
we find
f'du' _ (Pdii
{u'-dj~{u-df'
then, forming the equation
•Je.u'-a' .u'-b' . u' -c'.u'-d' _ s/PQR sle.u-a.u-b .u-c.u- d
{u'-dj ~* {u-dy
and attending to the relation PPN'' = r^QR, we obtain
Ndu _ du
which is the differential relation between u, u'.
We have in connection with A, B, G, D the point 0, and in connection with
A', R, C, ly the point 0'. As U describes the right line AB, V describes the
arc not containing 0' of the circle A'B'O' ; for observe that 0' corresponds in the second
figure to the point at infinity on the line AB, viz. as U passes from A to B, not
passing through the point at infinity, U' must pass from A' to F, not passing through
the point 0', that is, it must describe, not the arc A'O'B', but the remaining arc
27r- A'O'B", say this is the arc A'B'. The integral in regard to ii' is thus not the
rectilinear integral (A'B'), but the integral along the just-mentioned circular ai-c, say
this is denoted by (A'B'); and we thus have
(AB) = ± N(A^).
But we have (A'R) = or not = (A'B'), according as the chord A'R and the arc
A'B do not include between them either of the points C, D', or include between
them one or both of these points ; and in the same cases respectively
(AB) = or not = + N(A'B').
Of course we may in any way interchange the letters, and write under the like
circumstances
(AC) = or not =±N(A'G'), &c.
f du
]7u-
629] ox THE LINEAR TRANSFORMATION OF THE INTEGRAL I ^ryr . 621
Suppose now that ABCD is a convex quadrilateral, and consider first in regard
to {AB), and next in regard to {AG), the three transformations A'B'C'D' = BADG,
= GDAB, and =DCBA, respectively. We have here a figure as in the paper "On
the circular relation of Mobius," [628], p. 617 of this volume, the points Oj, 0.,, O3
belonging to the three cases respectively. It will be observed in the figure, and it is
easy to see generally, that the points Oj and O3 are interior, the point 0, exterior.
We have N=l, and therefore
(AB) = or not = + (AB), = or not = ± (CD), = or not = + (GD),
according as
(1) the chord AB and the arc AB of ABOi do not or do inclose C and D or
either of them ;
(2) the chord CD and the arc GD of GDOn do not or do inclose A and B or
either of them;
(3) the chord GD and the arc GD of GDO3 do not or do inclose A and B or
either of them.
The first test gives merely the identity {AB) = ± {AB) ; the other two each of them
give (AB) = ±{GD), as is seen from the positions of the points 0,, Oj, O3.
Next, apply the test to ^10; we have
{AG) = or not = + {BD), = or not = ± {AG), = or not = ± {BD),
according as
(1) the chord AG and the arc AG of AGOi do not or do inclose B and D or
either of them ;
(2) the chord BD and the arc BD of BDO3 do not or do inclose A and G or
either of them;
(3) the chord BD and the arc BD of BDO, do not or do inclose A and G or
either of them.
In the second case, neither A nor G is inclosed, but we have merely the identity
(AG)=±{AG); in the first case, B is inclosed and, in the third case, G is inclosed;
and the tests each give (AG) not = ± (BD).
I have not taken the trouble of drawing the figure for a reentrant quadrilateral
ABGD; the mere symmetry is here enough to show that, having one, we have all
three, of the relations in question
{AD)=±{BG), {BD)=±{GA), {GD)=±{AB).
END OF VOL. IX.
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