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THE COLLECTED
MATHEMATICAL PAPERS
OF
JAMES JOSEPH ^SYLVESTER
F.R.S., D.C.L., LL.D., Sc.D.,
Honorary Fellow of St John s College, Cambridge;
Sometime Professor at University College, London ; at the University of Virginia ;
at the Royal Military Academy, Woolwich; at the Johns Hopkins University, Baltimore
and Savilian Professor in the University of Oxford
VOLUME I
(1837-1853)
Cambridge
At the University Press
1904
644 (A fM-u;
Cambridge :
PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.
T
AtATH.-
STAT.
LIBRARY
PREFATORY NOTE.
HE object aimed at in this volume has been to present a faithful record
of the course of the author s thought, without such additions as recent
developments of the subjects treated of might have afforded, and without any
alterations other than that considerable number involved in the attempt to
make the algebraical symbols read as the writer intended. While, for the
reader s convenience, the author s references to his own papers have been
accompanied by cross references to the pages of this volume, placed in square
brackets.
By far the longest paper in the volume is No. 57, " On the Theory of the
Syzygetic Relations of two Rational Integral Functions, comprising an
application to the Theory of Sturm s Functions," and to this many of the
shorter papers in the volume are contributory.
The volume contains also Sylvester s dialytic method of elimination
(No. 9, etc.), his Essay on Canonical Forms (No. 34), and early investigations
in the Theory of Invariants (Nos. 42, 43, etc.).
It contains also celebrated theorems as to Determinants (Nos. 37, 39, 48,
etc.) and investigations as to the Transformation of Quadratic Forms (the
Law of Inertia, No. 47, and the recognition of the Invariant factors of a
matrix, Nos. 22, 24, 36).
A full table of contents is prefixed.
H. F. BAKER.
ST JOHN S COLLEGE, CAMBRIDGE.
April, 1904.
M777301
TABLE OF CONTENTS
PAGES
1. Analytical development of FremeTs optical
theory of crystals . . . . . 1 27
(Philosophical Magazine 1837, 1838)
2. On the motion and rest of fluids . . 28 32
(Philosophical Magazine 1838)
3. On the motion and rest of rigid bodies . 33 35
(Philosophical Magazine 1839)
4. On definite double integration, supplementary
to a former paper on the motion and rest
of fluids 3638
(Philosophical Magazine 1839)
5. On an extension of Sir John Wilson s theorem
to all numbers whatever .... 39
(Philosophical Magazine 1838)
6. Note to the foregoing 39
(Philosophical Magazine 1839)
7. On rational derivation from equations of
coexistence, that is to say, a new and
extended theory of elimination, Part I. 40 46
(Philosophical Magazine 1839)
8. On derivation of coexistence, Part II., being
the theory of simultaneous simple homo
geneous equations 47 53
(Philosophical Magazine 1840)
9. A method of determining by mere inspection
the derivatives from two equations of
any degree 54 57
(Philosophical Magazine 1840)
10. Note on elimination ..... 58
(Philosophical Magazine 1840)
CONTENTS. vii
PAGES
11. On the relation of Sturm s auxiliary functions
to the roots of an algebraic equation . 59, 60
(Plymouth British Association Report 1841)
12. Examples of the dialytic method of elimina
tion as applied to ternary systems of
equations 61 65
(Cambridge Mathematical Journal 1841)
13. Introduction to an essay on the amount and
distribution of the multiplicity of the
roots of an algebraic equation . . 66 68
(Philosophical Magazine 1841)
14. A new and more general theory of multiple
roots 6974
(Philosophical Magazine 1841)
15. On a linear method of eliminating between
double, treble, and other systems of
algebraic equations ..... 75 85
(Philosophical Magazine 1841)
16. Memoir on the dialytic method of elimina
tion, Part L 8690
(Philosophical Magazine 1842)
17. Elementary researches in the analysis of
combinatorial aggregation . . . 91 102
(Philosophical Magazine 1844)
18. On the existence of absolute criteria for de
termining the roots of numerical equations 103 1 06
(Philosophical Magazine 1844)
19. An account of a discovery in the theory of
numbers relative to the equation
Ax 3 + y s +Cz 3 = Dxyz . . 107109
(Philosophical Magazine 1847)
20. On the equation in numbers
Ax* + Bif + Cz 3 = Dxyz,
and its associate system of equations . 110 113
(Philosophical Magazine 1847)
21. On the general solution, in certain cases, of
the equation x 3 + if + z 3 = Mxyz . 114 118
(Philosophical Magazine 1847)
viii CONTENTS.
PAGES
22. On the intersections, contacts, and other cor
relations of two conies expressed by
indeterminate coordinates . . . 119 137
(Cambridge and Dublin Mathematical Journal 1850)
23. An instantaneous demonstration of Pascal s
theorem by the method of indeterminate
coordinates . . . . . . 138
(Philosophical Magazine 1850)
24. On a new class of theorems in elimination
between quadratic functions . . . 139 144
(Philosophical Magazine 1850)
25. Additions to the articles On a new class of
theorems, and On Pascal s theorem, . 145 151
(Philosophical Magazine 1850)
26. On the solution of a system of equations in
which three homogeneous quadratic func
tions of three unknown quantities are
respectively equated to numerical multiples
of a fourth non-homogeneous function of
the same 152154
(Philosophical Magazine 1850)
27 . On aporismatic property of two conies having
with one another a contact of the third
order 155, 156
(Philosophical Magazine 1850) ,
28. On the rotation of a rigid body about a fixed
point 157161
(Philosophical Magazine 1850)
29. On the intersections of two conies . . 162 164
(Cambridge and Dublin Mathematical Journal 1851)
30. On certain general properties of homogeneous
functions 165 180
(Cambridge and Dublin Mathematical Journal 1851)
31. Reply to Professor Boole s observations on
a theorem contained in last November
number of this Journal. . . . 181 183
(Cambridge and Dublin Mathematical Journal 1851)
CONTENTS. ix
PAGES
32. Sketch of a memoir on elimination, trans
formation and canonical forms . . 184 197
(Cambridge and Dublin Mathematical Journal 1851)
33. On the general theory of associated alge
braical forms ,, 198 202
(Cambridge and Dublin Mathematical Journal 1851)
34. An essay on canonical forms, supplement to
a sketch of a memoir on elimination,
transformation and canonical forms . 203 216
(George Bell, Fleet Street, 1851)
35. Explanation of the coincidence of a theorem
given by Mr Sylvester in the December
number of this Journal with one stated
by Professor Donkin in the June number
of the same 217, 218
(Philosophical Magazine 1851)
36. An enumeration of the contacts of lines and
surfaces of the second order . . . 219 240
(Philosophical Magazine 1851)
37. On the relation between the minor deter
minants of linearly equivalent quadratic
functions 241 250
(Philosophical Magazine 1851) [See p. 647 below.]
38. Note on quadratic functions and hyper-
determinants 251
(Philosophical Magazine 1851)
39. On a certain fundamental theorem of de
terminants 252 255
(Philosophical Magazine 1851)
40. Extensions of the dialytic method of elimina
tion ; 256264
(Philosophical Magazine 1851)
41. On a remarkable discovery in the theory of
canonical forms and of hyper determinants 265 283
(Philosophical Magazine 1851)
42. On the principles of the calculus of forms . 284 327
(Cambridge and x Dublin Mathematical Journal 1852)
43. On the principles of the calculus of forms . 328 363
(Cambridge and Dublin Mathematical Journal 1852)
CONTENTS.
PAGES
44. Sur une propriete nouvelle de V equation qui
sert a determiner les inegalites seculaires
des planetes ; 364366
(Nouvelles Annales de Mathematiques 1852)
45. On a remarkable theorem in the theory of
equal roots and multiple points . . 367 369
(Philosophical Magazine 1852)
46. Observations on a new theory of multiplicity 370 377
(Philosophical Magazine 1852)
47. A demonstration of the theorem that every
homogeneous quadratic polynomial is
reducible by real orthogonal substitutions
to the form of a sum of positive and
negative squares . . . . . 378 381
(Philosophical Magazine 1852)
48. On Staudt s theorems concerning the contents
of polygons and polyhedrons, with a
note on a new and resembling class of
theorems 382 391
(Philosophical Magazine 1852)
49. On a simple geometrical problem, illustrating
a conjectured principle in the theory of
geometrical method .... 392 395
(Philosophical Magazine 1852)
50. On the expression of the quotients which
appear in the application of Sturm s
method to the discovery of the real roots
of an equation 396 398
(Hull British Association Eeport 1853)
51. On a theorem concerning the combination
of determinants ..... 399 401
(Cambridge and Dublin Mathematical Journal 1853)
52. Note on the calculus of forms . . . 402, 403
(Cambridge and Dublin Mathematical Journal 1853)
53. On the relation between the volume of a
tetrahedron and the product of the six
teen algebraical values of its superficies 404410
(Cambridge and Dublin Mathematical Journal 1853)
CONTENTS. xi
PAGES
54. On the calculus of forms, otherwise the theory
of invariants . . . . . 411 422
(Cambridge and Dublin Mathematical Journal 1853)
55. Theoreme sur les limites des r acmes reelles
des equations algebriques . . . 423
(Nouvelles Annales de Mathematiques 1853)
56. Nouvelle methode pour trouver une limite
superieure et une limite inferieure des
racines reelles dune equation algebrique
quelconque ...... 424 428
(Nouvelles Annales de Mathematiques 1853)
57. On a theory of the syzygetic relations of two
rational integral functions, comprising
an application to the theory of Sturm s
functions, and that of the greatest alge
braical common measure . . . 429 586
(Philosophical Transactions of the Koyal Society of London 1853)
58. On the conditions necessary and sufficient to
be satisfied in order that a function of
any number of variables may be linearly
equivalent to a function of any less
number of variables .... 587 594
(Philosophical Magazine 1853)
59. On Mr Cayley s impromptu demonstration
of the rule for determining at sight the
degree of any symmetrical function of
the roots of an equation expressed in
terms of the coefficients .... 595 598
(Philosophical Magazine 1853)
60. A proof that all the invariants to a cubic
ternary form are rational functions of
Aronholds invariants and of a cognate
theorem for biquadratic binary forms . 599 608
(Philosophical Magazine 1853)
61. On a remarkable modification of Sturm s
theorem 609 619
(Philosophical Magazine 1853)
xii CONTENTS.
PAGES
62. Note on a remarkable modification of Sturm s
theorem, and on a new rule for finding
superior and inferior limits to the roots
of an equation . . . . - . 620 626
(Philosophical Magazine 1853)
63. On the new rule for finding superior and
inferior limits to the real roots of any
algebraical equation . . . 627 629
(Philosophical Magazine 1853)
64. Note on the new rule of limits . . . 630 633
(Philosophical Magazine 1853)
65. The algebraical theory of the secular in
equality deter minantive equation gene
ralised . . . . . . . 634636
(Philosophical Magazine 1853)
66. On the explicit values of Sturm s quotients 637 640
(Philosophical Magazine 1853)
67. On a fundamental rule in the algorithm of
continued fractions 641 644
(Philosophical Magazine 1853)
68. On a generalisation of the Lagrangian
theorem of interpolation . . . 645, 646
(Philosophical Magazine 1853)
NOTE ON SYLVESTER S THEOREMS ON DETERMINANTS
IN THIS VOLUME. 647 650
1.
ANALYTICAL DEVELOPMENT OF FRESNEL S OPTICAL
THEORY OF CRYSTALS.
[Philosophical Magazine, XL (1837), pp. 461 469, 537541 ;
xii. (1838), pp. 7383, 341345.]
THE following is, I believe, the first successful attempt to obtain the
full development of Fresnel s Theory of Crystals by direct geometrical
methods. Hitherto little has been done beyond finding and investigating
the properties of the wave surface, a subject certainly curious and interesting,
but not of chief importance for ordinary practical purposes. Mr Kelland,
in a most valuable contribution to the Cambridge Philosophical Transactions*,
has incidentally obtained the difference of the squares of the velocities of a
plane front in terms of the angles made by it with the optic axes. I have
obtained each of the velocities separately, and in a form precisely the same
for biaxal as for uniaxal crystals.
I have also assigned in my last proposition the place of the lines of
vibration in terms of the like quantities, and that in a shape remarkably
convenient for determining the plane of polarization when the ray is given.
For at first sight there appears to be some ambiguity in selecting which of
the two lines of vibration is to be chosen when the front is known. If p be
the perpendicular from the centre of the surface of elasticity let fall upon
the front, i l} i 2 the angles made by the front with the optic planes, e lt e 2 the
angles between its due line of vibration and the* optic axes, I have shown that
- c 2 sin U V \a? - c 2 sin
so that all doubt is completely removed. The equation preparatory to
obtaining the wave surface is found in Prop. 6 by common algebra, without
any use of the properties of maxima and minima, and various other curious
relations are discussed.
Without the most careful attention to preserve pure symmetry, the
expressions could never have been reduced to their present simple forms.
* See Load, and Edinb. Phil. Mag. Vol. x. p. 336.
2 Analytical Development of [1
ANALYTICAL REDUCTION OF FRESNEL S OPTICAL THEORY OF CRYSTALS.
Index of Contents.
In Proposition 1, a plane front within a crystal being given, the two lines
of vibration are investigated.
In Proposition 2 it is shown that the product of the cosines of the inclina
tions of one of the axes of elasticity to the two lines of vibration, is to the
same for either other axis of elasticity in a constant ratio for the same crystal ;
and the two lines of vibration are proved to be perpendicular to each other.
In Proposition 3, a line of vibration being given, the front to which it
belongs is determined; and it is proved that there is only one such, and
consequently any line of vibration has but one other line conjugate to it.
In Proposition 4, certain relations are instituted between the positions of,
and velocities due to, conjugate lines.
In Proposition 5, the angles made by the front with the planes of
elasticity are found in terms of the velocities only.
In Proposition 6, the above is reversed.
In Proposition 7, the position of the planes in which the two velocities
are equal (viz. the optic planes) is determined.
In Proposition 8, the position of a front in respect to the optic axes is
expressed in terms of the velocities.
In Proposition 9, the problem is reversed, and it is shown that if v 1} v z
be the two normal velocities with which any front can move perpendicular
to itself, and i l} t z the angles which it makes with the optic planes, then
V = a? (si
\
sin - + c* cos
\
w 2 2 = a 2 (sin ^-5-^ ) + c 2 (cos 4l * 2 ) .
\ i ) \ 2i )
In the 10th the angle made by a line of vibration with the axes ot
elasticity is expressed in terms of the two velocities of the front to which it
belongs.
In the llth Proposition the velocity due to any line of vibration is ex
pressed in terms of the angles which it makes with the optic axes, viz.
v- 6 2 = (a- c 2 ) cos 6x cos e a .
In the 12th Proposition e 1; e 2 are separately expressed in terms of i 1} i 2 .
In the Appendix I have given the polar or rather radio-angular equation
to the wave surface, from which the celebrated proposition of the ray flows as
an immediate consequence.
1J Fresnel s Optical Theory of Crystals. 3
PROPOSITION 1.
If lx + my + nz = (a)
be the equation to a given front, to determine the lines of vibration therein.
It is clear that ir x, y, z be any point in one of these lines, the force
acting on a particle placed there when resolved into the plane must tend to
the centre. Consequently the line of force at x, y, z must meet the perpen
dicular drawn upon the front from the origin. Now the equation to this
perpendicular is
-=-=- m
I m n
and the forces acting at x, y, z are a 2 x, b 2 y, c 2 z parallel to as, y, z, so that the
equation to the line of force is
J 7 y = Z -T^- (2)
ax y c*z
From (2) we obtain
b*yX - o?x Y = (b- - a 2 ) xy (3)
c 2 z Y - b*yZ = (c 2 - b 2 ) yz (4)
u?xZ - c 2 zX = (a 2 - c 2 ) zx. (5)
Hence
(b 2 - a 2 ) xyn + (c 2 - b 2 ) yzl + (a 2 - c 2 ) zxm
= b 2 y (nX -lZ)+c 2 z(LY- mX ) + o?x (mZ -nY);
but by equations (1)
lZ-nX=Q, mX-lY=0, nY-mZ=0
therefore
nj 1 nyi
(b 2 -a 2 )- + (c 2 -b 2 Y- + (a 2 - c 2 ) - = 0. (b)
z x y
Also we have
nz + Ix + my = (a)
therefore
(b 2 - a 2 ) n 2 + (c 2 - 6 2 ) I 2 + nl ((c 2 -b 2 )- + (b 2 - a 2 ) -} = (a 2 - c 2 ) m 2
\ x z /
or
(c 2 - 6 2 ) (-Y + 1 {(c 2 - b 2 ) I 2 + (b 2 - a 2 ) n 2 - (a 2 - c 2 ) m 8 } - + (6 2 - a 2 ) = 0.
\SC / Tlv OG
And in like manner interchanging b, y, m with c, z,
n
ft* + , {(b 2 - c 2 ) I 2 + (c 2 - a 2 ) m 2 - (a 2 - b-) n 2 } % + (c 2 - a 2 ) = 0.
w j i/LL *K
12
Analytical Development of [1
Hence if ( ^ , -V^ , -} be the two systems of values of | , - then
\St/i &i/ \X% *"a/
Y _y l Z
~X = ^ X~xJ\3~x, X~ x,
are the two lines of vibration required.
PROPOSITION 2.
By last proposition it appears that
- c 2
(c)
therefore
therefore
/Y*
And therefore the two lines of vibration are perpendicular to each other.
KB. Equations (c) and (d) must not be overlooked.
PROPOSITION 3.
A line of vibration is given (that is ^ , - are given] and the position of
*? V OC\ OC\ i f
the front is to be determined.
Let Ix + my + nz = be the front required, then lx, + my, + nz v = 0, and
x\ y\ i
Eliminating n we get
therefore
m
, (F^V! 2 - (a 2 fe 2 ) 2
^ a 2 (^ 2 + 2/! 2 4- ^ 2 - a
y, 6 2 (^^T^+^iV 7 ^! 2
1] FresneTs Optical Theory of Crystals.
If now we make x? + y* + z? = 1
and therefore
l a> a 2 -
and in like manner
#! a 2
therefore
(a 2 - Vj 2 ) ! + (6 2 - Vj 2 ) y,y + (c 2 - v?) z^z =
is the equation required.
PROPOSITION 4.
- , having each only one value, shows that only one front corresponds
m n
to the given line of vibration. Let # 2 . 2/2, z 2 , v z correspond to x ly y l} z lt Vi for
the conjugate line of vibration, then the equation to the front may be
expressed likewise by
(a 2 - v 2 2 ) x& + (6 2 - w 2 2 ) y 2 y + (c 2 - u/) z& = 0,
so that
(avO] (& Wi a )yi (c 2 - vf} z l
PROPOSITION 5.
To find a), <j), -^r, the angles made by the front with the planes of elasticity
in terms of v l} v%.
By the last proposition
, 2 2 - 22
~
(a 2 - Vl 2 ) 2 x? + (6 2 - vff y? + (c 2 - vff z?
(a 2 vf) (a 2 fl 2 2 ) x-
_
(a 2 - ^ 2 ) (a 2 - v 2 2 ) x& + (6 2 - Wa 2 ) (6 2 - v 2 2 ) y^ + (c 2 - v?) (c- - v 2 2 )
Now, by Proposition 2,
6 Analytical Development of [1
therefore (cos to) 2
_
(a 2 - Vl 2 ) (a 2 -v 2 2 )(c 2 -6 2 )+(& 2 - vW - v^(a? - c 2 ) + (c 2 - v?) (c 2 - v 2 2 )(6 2 - a 2 )
(a 2 - v?) (a 2 - v 2 2 ) (c 2 - 6 2 )
a 4 (c 2 - 6 2 ) + 6 4 (a 2 - c 2 ) + c 4 (a 2 - 6 2 )
(a 2 - 6 2 ) (a 2 - c 2 )
Similarly,
PROPOSITION 6.
To find Vi, v z in terms of w, <, ^r.
By the last proposition
(cos a)) 2 a 2
a 2 - Vl 2 (a 2 - 6 2 ) (a 2 -"c 2 ) (a 2 - 6 2 ) (a 2 - c 2 )
_
2 - a 2 ) (6 2 - c 2 ) ! (6 2 - a 2 ) (6 2 - c 2 )
C 2 _ Vi ( c 2 _ 6 2 ) (c 2 - a 2 )
therefore
7 . 2 _ L_
(COS ft))" (COS <^>) 2 (COS T^r) 2 _
a 2 Vj 2 6 2 flj 2 c 2 i?! 2
Just in the same way
(COS O)) 2 (COS <) 2 (COS T/r) 2 _
7^ ITs + W^T^ + c* - V 2 ~
Cv "~ t/2 ^ 2 ^ ^2
so that Vj 2 , v 2 2 are the two roots of the equation
(cos <a) 2 (cos d>V (cos -\lr) -
-(- = U.
ft 2 fl2 ) 2 if C 2 y-
COR. Hence the equation to the wave surface may be obtained by making
(cos &)) x + (cos <) y + (cos ty)z = v,
1] FresneTs Optical Theory of Crystals.
or if we please to apply Prop. 5, we may make
/(a -VXo -qfl /(fr-t>i 1 )(fr -* a )
V (a 2 -6 2 )(a 2 -c 2 ) V (6 2 -a 2 )(6 2 -c 2 ) y
A*- * )(*-*. ) ,_
+ V 2 -a a )(c 2 -& 2 )
or, if we please *,
/(aht 2 - 1 ) (a*-^*) /(bW -
V (a 2 -ft 2 ) (a 2 ^c 2 ) X + V (& 2 - a 2
/(c 2 u 2 -
V (c 2 - a
2 ) (c 2 - 6 2 )
PROPOSITION 7.
To find when v l = v 2 .
By Prop. 4,
x z (v 2 2 - a 2 ) y, (v 2 2 - 6 2 ) ^ 2 (v 2 2 - c 2 )
Hence when v 1 = v. 2 we have, generally speaking,
&i _ y_ i _ fi
^2 2/2 ^2
Now ^#2 + 2/i2/2 + z \ z i j
therefore x* + yf + zf would = 0, which is absurd.
The only case therefore when v l can = v 2 is when one of those terms of
x z
equation (&} becomes ^ : thus suppose ^ = 6, then we have - = - = ^ , and
\J tf/ft &O \s
x l y l
we can no longer inter = -- .
x, y,
Let now (a) l , fa, ^i)((o. 2 , fa, fa) be the two systems of values which &&gt;, <f>,
ty assume when v l = v. 2 = b, then applying the equation of Prop. 5 we have
= . / COS ft)., = A I
V a 2 - c 2 V a ~ c
cos fa = Q cos fa =
COS ft)
COS J = \ tf3^ cos =
so that 6 must correspond to the mean axis.
[* See below, p. 27. ED.]
8 Analytical Development of [1
PROPOSITION 8.
i 1} L Z being the angles made by the front with the optic planes, to find
tj, t 2 in terms of v 1} v t .
By analytical geometry
COS tj = COS O) . COS ! + COS (f> . COS </>! + COS ^r . COS ^
/(Wi 2 - a 2 ) (vl - a 2 ) /a 2 - 6 2
= V (a 2 - & 2 ) (a - c 2 ) V a 2 -"c 2
/(y - c 2 ) (v 2 2 - c 2 ) /c 2 -_6_ a
+ y (c 2 ~-~a 2 ) (c 2 - 6 2 ) V c 2 -^"a 2
_ VK^ - a 2 ) fa 2 - a 2 )] + VjK - c 2 ) (fl, 2 - c 2 )}
a 2 - c 2
and similarly
cos / 2 = cos a) . cos &) 2 + cos <f> . cos <. 2 + cos ^r . cos i|r 2
^ ^1(^2 _ a ) ( Vj a _ a 2)| _ ^K^a _ C 2) (^2 _ C 2)|
tt 2 -C 2
PROPOSITION 9.
To find Vi, v z in terms of t 1} i 2 .
By the last proposition
(v, 2 - a 2 ) (v 2 2 - a 2 ) - (vj 2 - c 2 ) (v 2 2 - c 2 )
COS L! . COS to = x 2 _ 2 2
(a 2 -c 2 ) 2 "
(a 2 + c 2 ) - (^ 2 + w 2 2 )
(a 2 - c 2 )
therefore
v-f + vj = a 2 + c 2 (a 2 c 2 ) cos ^ cos t 2 .
Again, (sin ij) 2 . (sin t 2 ) 2 = 1 - (cos tj) 2 ~ (cos t 2 ) 2 + (cos tl ) 2 (cos i 2 ) 2
> (^i 2 ~ a 2 ) (^ 2 2 - a 2 ) + (fli 2 - c 2 ) (v, 2 - c 2 )
^ J_ ^i .
(a 2 - c 2 ) 2
(a 2 4- c 2 ) 2 - 2 (a 2 + c 2 ) (^ + v 2 2 ) + (i s + v 2 2 ) 2
(a 2 - c 2 ) 2
w/ - 2t>, V + w 2 4
(a 2 - c 2 ) 2
1]
FresneCs Optical Theory of Crystals.
therefore
but
therefore
^i 2 #2 2 = ( 2 c 2 ) sin ^ . sin t 2
^2 2 = ( 2 + c 2 ) (a 2 c 2 ) cos ^ cos t 2
a + c a-c
2 2
COS
= a" ( sin ---) +c 2 cos
^ / V ^
ft_ J
C/o
a 2 + c 2 a 2 c 2
T~ ~~2~
COS U,
Thus for uniaxal crystals where tj + * 2 = 180
= a 2
v 2 2 = a 2 (cos t) 2 + c 2 (sin t) 2 .
COR. Hence we may reduce the discovery of the two fronts into which
a plane front is refracted on enter
ing a crystal to the following trigo
nometrical problem.
Let a sphere be described about
any point in the line in which the
air front intersects the plane of in
cidence. Let the great circle PI
denote the latter plane, IF the
former, OA, OC also great circles,
the planes of single velocity. Sup
pose IGH to be one of the refracted
fronts intersecting OA, OC in G and H, then
Fig- 1.
c ) _ ( _ c a) cos (G + H) _ (sin PIF) 2
2 (vel. in air) 2
(sin PIGH ) 2
The double sign will give rise to two positions of the refracted front IGH.
The propositions which follow are perhaps more curious than immediately
useful.
10 Analytical Development of [1
PROPOSITION 10.
To determine the portion of a line of vibration in terms of the two
velocities of its corresponding front.
ni &
We have here to determine the quantities , (of Prop. 1) in terms of
1 1
v 1} v*, or on putting x l 2 + y l z + 2 1 ^=l, x 1} y l ,z l are to be found in terms of v lt v 2 .
By Prop. 3
I m n
and by Prop. 5
Z 2 : w 2 : ?i 2 : : (6 2 - c 2 ) (a 2 - ^ 2 ) (a 2 - v 2 2 )
: (c 2 - a 2 ) (6 2 - v, 2 ) (6 2 - v 2 2 )
: (a- - 6 2 ) (c 2 - ^ 2 ) (c 2 - v 2 2 ) ;
therefore ^ 2 : y? : z?
fit ,1 2 7,2 91 2 r 2 , 2
Let a, yS, 7 be the angles made by the given line of vibration with the
elastic axes, then
= (6 2 - c 2 ) (a 2 - v, 2 ) (6 2 - vx 2 ) (c 2 - v^)
divided by
(6 2 - c 2 ) (a 2 - v 2 2 ) (6 2 - v^) (c 2 - V) + (c 2 - a 2 ) (6 2 - v 2 2 ) (c 2 - Vl 2 ) (a 2 - v?)
+ (a 2 - 6 2 ) (c 2 - v 2 2 ) (a 2 - ^ 2 ) (6 2 - v/)
and therefore
_ (6 2 - c 2 ) (a 2 - v 2 2 ) (6 2 - Vi 2 ) (c 2 - Vj 2 )
~ K - 2 2 ) (a 2 - & 2 ) (& 2 ~ c 2 ) (c 2 - a 2 )
(where it is to be observed that the reduction of the denominator is simply
the effect of a vast heap of terms disappearing under the influence of
contact with the magic circuit (a 2 - 6 2 ), (Z> 2 - c 2 ), (c 2 - a 2 ), a simpler instance
of which was seen in Proposition 5).
In fact the coefficient of v 4 . v 2
= (&-> _ c a) + (c 2 - a 2 ) + (a 2 - & 2 )
=
1] Fresnels Optical Theory of Crystals. 11
that of v? . v? = (c" + & 2 ) (c 2 - & 2 )
+ (a 2 + c 2 ) . (a 2 - c 2 )
= ( C 4 _ 4) + ( a 4 _ C 4) + ^4 _ a 4)
= 0.
The term in which neither ^ nor v 2 enters
= a 2 6 2 c 2 {(6 2 - c 2 ) + (c 2 - a 2 ) + (a 2 - 6 2 )}
= 0.
The coefficient of
- Vl 2 = a 2 . (6 4 - c 4 ) + 6 2 . (c 4 - a 4 ) + c 2 . (a 4 - 6 4 )
and that of
v 2 2 = &v . ( C * _ 52) + c * a * _ ( a _ C 2) + a 5? . (&s _ a *
each of which
= (a 2 -6 ).(6 2 -c 2 ).(c 2 -a 2 ).
Hence
_ fli 2 - 6 2 (a 2 -^ 2 )(c 2 -^ 2 )
~ Vl 2 -v 2 2 (a 2 -6 2 )(a 2 -c 2 )
in like manner (cos /3) 2 = &c.
,, Vl 2 -6 2 ^- 2 - 2
and (cos 7 > = _ . ,_
PROPOSITION 11.
u e 2 em^ e aw^es between any line of vibration and the optic axes,
required the velocity due to that line in terms of e 1} e 2 .
By analytical geometry,
COS 6j = COS a . COS fa + COS 7 . COS A/T!
COS 2 = COS a . COS fa COS 7 . COS -V/T!
therefore cos Cl . cos e 2 = (cos a) 2 (cos fa) 2 - (cos 7 ) 2 (cos ^) 2
V? - 6 2 ((a 2 - ^ 2 2 ) . (c 2 - V) ~ (c 2 - V?) (a 2 ~
(a 2 -c 2 ) 2
^L^L 62 (a 8 ~ c 2 ) (X 2 - ^i 2 )
V^w 2 2 (a 2 -c 2 >
~ a 2 - c 2
Hence fli 2 = & 2 - (a 2 - c 2 ) cos e^ cos e 2 ,
and in like manner, for the conjugate line of vibration
2 2 = 6 2 (a 2 c 2 ) cos e/ cos e 2 .
12 Analytical Development of [1
PROPOSITION 12.
To find e 1} e 2 in terms of ti, to.
(cos ej) 2 + (cos e 2 ) 2 = 2 (cos a) 2 . (cos c^) 2 + 2 (cos <y) z . (cos i/^) 2
= 2 ^~
~ tvQ O 2 ~ ^ 2 ) + (c 2 - vj) . (a-
(a 2 - c 2 ) 2
but by Prop. 9
+ c- cos
therefore
2 ; v
y 2 2 = a 2 ( sin --0-- 2 ) + c 2 ( cos
\ ^ / \
1 2 7 ^
(a 2 c 2 ) sin ti . sin i 2
multiplied by
n ." - K sin 4 i) 2 + ( sin ^)
(a 2 c 2 ) sm 4j . sm t 2
and we have seen that
cos e, cos e. 2 =
therefore
cos 6j + cos e^ =
\ / I /** _ /d
therefore
sin i, sm i 9
COS 61 COS 6., =
~ a z - &
sin tj
COS ej = A / \ =7. .
a 2 c- sm i,
6 2 - vi 2 sin t,
cose2= V1"^^
and in like manner
/ (6 2 Va 2 sin tj
COS j = A / { -, - .
V (a c 2 sm i, 2
I (b 2 v 2 2 sin
COS 6 2 = A / ] 2 2" ~
where v 1} v z for the sake of neatness are left unexpressed in terms of i lt / 2 .
i]
FresneTs Optical Theory of Crystals.
13
This is the simplest form by which the position of the lines of vibration
can be denoted.
COR. From the last proposition it appears that
cos e!
cos e 2
sn L-L
sin t 2
Hence we may construct geometrically for the two planes of polarization.
Let /, K be the projections of
the two optic axes on a sphere, E
the projection of the normal to the
front, P the projection of one line
of vibration ; then
cosPK _ sinKE
cos PI ~ sin IE
Draw PEG the circle of which
P is the pole, meeting PK, PI pro
duced in G and F.
Then cos PK = sin KG,
and cos PI = sin IF,
therefore
am KG sinKE
sin FI sin IE
Fig. 2.
therefore
sin KG sin
smKE sin IE
therefore
sin KEG = sin IEF
therefore KEG=IEF or 180 - IEF. But PEF=PEG, therefore EP bisects
either the angle IEK or the supplement to it.
These two positions of EP give the two planes of polarization. The
construction is the same as that given in Mr Airy s tracts, and originally
proposed, I believe, by Mr MacCullagh.
14 Analytical Development of [1
ADDENDUM.
If in the equation of Prop. 6, viz.
(cos &&gt;) 2 (cos $) 2 (cos i/r) 2
a 2 - v* + >-w 2 H " c 2 ~^ =
we change a. 6. c, v into - . r . - , - . and consider v to be the length
a b c v
of a line drawn perpendicular to the plane
cos to . # + cos ( . y + cos -vjr . z = 0,
the equation to the extremity thereof must be
a 2 r 2 (cos a) 2 6 2 r 2 (cos <) 2 c 2 r 2 (cos ^
where &&gt;, $, ty denote the angles between the radius vector r, and the axes
of x, y, z, so that the equation may be written
,-2 /v2 A2 yy2 x^2 ^2
which is that of the wave surface.
But we have seen that
v- = CM cos
therefore the equation to the wave surface may be written
/ i *2\ 2 / h h
cos - [ sin
iv 2 y y 2
where / 1; t 2 denote the angles between the radius vector v and the two lines
which would be the optic axes if a, b, c were changed into - , r , - so that
CL C
if e be the inclination of either to the mean axis of elasticity
c
These lines I shall call by way of distinction the prime radii*.
* Upon the authority of Professor Airy I have appropriated the term optic axes to the lines
normal to the fronts of single velocity.
1] FresneTs Optical Theory of Crystals. 15
COR. 1. If T-L, r 2 be the two values of r corresponding to the same values
of i] , i 2 we have
1 1 1 (/ H-ii
= -M cos
~ 2 ~. 2 r 2 V 2
i FJ I/ ( \ *
which proves the celebrated problem of two rays having a common direction
in a crystal.
COR. 2. The intersection of any concentric sphere with the wave surface
is found by making r constant. Hence t t i 2 becomes constant, and there
fore rtj + rt 2 = constant. Hence the curve of intersection is the locus of
points, the sum or difference of whose distances from two poles when
measured by the arcs of great circles is constant ; the poles being the points
in which the prime radii pierce the sphere.
In three cases these spherico-ellipses or spherico-hyperbolas become great
circles :
(1) When ij + i 2 = the angle between the two poles, in which case the
curve of intersection is the great circle which comprises the two poles.
(2) When ^ ^=0, when the locus is a great circle perpendicular to
the former and bisecting the angle between the optic axes.
(3) When ^ + 1^ = 180, when the locus is a great circle perpendicular
to the two above, and bisecting the supplemental angle between the two
axes.
Various other properties may be with the greatest simplicity deduced
from the radio-angular equation. The hurry of the press leaves me time
only to subjoin the following
PROPOSITION.
To find the inclination of the radius vector to the tangent plane, in terms of
the angles which the radius vector makes with the prime radii.
Let be the centre of the wave surface, OA, OB the two prime radii,
OP any radius vector. Let OP^v, POA = i lt POB=t 2 , and let the in
clination of the planes POA,
sin
then - = -i -=-^- +
r 2 a 2 c 2
(taking only the positive sign for the sake of brevity).
16 Analytical Development of
Let OQ, OR be the two adjacent radii vectores, so assumed that
ROB = POB, ROA = POA + 8^,
and let p, q, r, a, b be the projections of P, Q, R, A, B on
a sphere of which is the centre, then it is clear that
qpa = 90, rpb = 90,
draw qm perpendicular to pb, then pm = 8i 2 , and therefore
pm pm Si.,
pq = -^^ - = -r^- = -r-2- .
sin p^m sin opo sm /i
In like manner
pr =
Now the angle QPO
r.POQ r.pq
= tan 1 . 7=^ ^ V, = tan * *-*-
sin
OQ-OP
dr
-j-
di 2
also
therefore
therefore
d
"V
-5
<u 2
dr
1
- - - COS -
c 2 a 2 /
l 1\ .
sm
cot QPO = V -. ( - - -3] sin (ti + *a) sin p.
4? \c a /
In like manner
r 2 /I 1\
cot EPO = r ) smfti + tj) sin /*,
4 \c 2 av
therefore
Also it is clear that rpq = apb = p. And
to find the inclination of OP to RPQ, we have
only to describe a sphere of which P is the
centre, and intersecting PQ, PR, PO in Q ,
R , .
Then R Q = ^ and
Fig. 4.
Q = O E = cot- 1 - 2 - s in (*i + ) sin
1] Fresnel s Optical Theory of Crystals. 17
Draw O ^V perpendicular to R Q , then ^ measures the inclination of
the radius vector to the tangent plane*.
And
therefore cos = tan O N . cot O Q ,
z
therefore
and therefore
cos-
/I 1\ . ii . ,
cot ON = ^r 2 . ( - - 1 sin ^ . sm (tj + t 2 ).
\CL C /
Let A OB the angle between the optic axes = 2e, then by mere trigonometry
to\ . / *l
2 -) sm ( e - "2
sin t x . sin 2
therefore the tangent of the inclination between the radius vector and
the normal
> >
i..
- sn
In like manner the tangent of the inclination between the same radius
vector and the normal at the other point of the wave-surface pierced by it
/sin \e
V J
- sin U,
sin tj . sin
We may, in the same way, find the inclination of the tangent plane
to either of the prime radii, and to the plane which contains them both,
in terms of ^ and t 2 ; the former by a remarkably elegant construction ;
but the final expressions do not present themselves under the same simple
aspect.
If we call <$> the angle between the ray and the front, we may still
further reduce by substituting for r 2 its values in terms of t lt i 2 and we
shall obtain
cot*. 2 _ (c -" > =-
( , li
Bin ( 0+ -- 1 sin [e ^ . cosec tj . cosec
\ / \ 2 /
* O is the projection of the ray and R O of the tangent plane. Therefore O N being
perpendicular to R Q represents their inclination.
S. 2
18 Analytical Development of [1
And if TTj, 7T 2 be the inclinations of the normal to the two prime radii,
it may be shown that
At
cos TTj = cos 9 sm ^ + sm 9 cos tj sin ~ ,
m
Li
cos 7T 2 = cos < sm < 2 + sm <f> cos t 2 sin ^ .
25
COR. 1. For uniaxal crystals ^ = 90 and 4 1 + t 2 =180, so that the
z
tangent of the inclination of normal to radius vector
= |r 2 . ( 2 j sin 2i for one point,
and = for the other.
COR. 2. For every point in the circular section which passes through
the poles sin ^ = 0, and for the other two circular sections tj + i 2 = or 180.
z
Therefore every point in the three circular sections is an apse.
COR. 3. When a nearly = c, - is very small ; and therefore the
a c~
normal and radius vector very nearly coincide.
COR. 4. Referring to fig. 4 we see that O N bisects the angle R O Q .
Now R O, Q O are respectively perpendicular to the planes passing through
and the optic axes ; and therefore the meridian plane as we may term it,
that is, the plane containing both the ray and the normal, always bisects
the angle formed by the two planes drawn through the ray and the two
optic axes.
r
COR. 5. When
tj or t, 2 = 0,
i 2 or tj = e.
And therefore </> assumes the form - , which indicates that the extremities
of the four prime radii are singular points.
In concluding for the present it behoves me to state that one step has
been omitted in the foregoing paper*, viz. the actual performance of the
eliminations which lead to, the rectilinear equation to the wave-surface.
But Mr Archibald Smith s elegant and brief Memoir in the Cambridge
Philosophical Transactions^ of last year leaves nothing to be desired further
on that head.
[* See below, p. 27. ED.] [t Vol. vi. Also Phil. Mag. April, 1838, p. 335. ED.]
1J FresneVs Optical Theory of Crystals. 19
That I have not exhibited it in its proper place (Prop. 6) arises only from
my respect to the principle of literary propriety. With this important blank
supplied the Analytical Theory may be pronounced to be complete.
For all errors and imperfections in what precedes my excuse must be
press of time and a total want of the materials to be derived from consulting
works of reference.
Since writing the above I have had an opportunity of reading the paper of our
living Laplace inserted as part of the Third Supplement to his System of Rays in
the Transactions of the Royal Irish Academy, in which the principal foregoing
results are obtained by aid of a more refined and transcendental analysis.
The nature of the four singular points is there discussed and the existence of
four circles of plane contact demonstrated.
The former may be very easily shown thus : when i x is very small ^ = 2e ij cos if/
very nearly, fy denoting the inclination of the plane in which e is reckoned to the
plane in which t a is reckoned.
Hence
/iy i/i i\ 1/1
(r) = 2U 2 + ^~2b-
1/1 1\ 1/1 1\ 1/1 1 .
" 5 * ^ ~ 2 V " cV COS2e ~ 2 t? ~ Sm 2e (C S *
therefore
Take \j/ constant and let the abscissae and ordinates be reckoned respectively along
and perpendicular to the prime ray.
Then h nearly, and r ,J(y 2 + x 2 ) = x,
OC
or, if we change the origin to the other extremity of the prime ray,
h. = 7 . r=b-x,
so that the equation becomes
-5-J<.*l) /l(l-^(-
y 2 V ^- \/ IV aVVc
2 2
20 Analytical Development of [1
Hence at each singular point the surface is touched by a cone, the equation to
the generating line of which is given by the above, the extreme angle between it
and the prime ray being
When b--a, \}/ always = ^ and the cone returns into a plane.
Again, let us suppose that the position of any perpendicular from the centre
is given, and that of the corresponding radius vector required.
Let OA, OS* denote what we have termed the optic axes, but which it will be
more agreeable to analogy to term the prime perpendiculars from centre, and let
OP be the given normal. Take OQ, OR contiguous perpendiculars from centre
in planes POQ, ROP, perpendicular to POA, FOB respectively, then the inclination
of the two former will be the same as that of the two latter, and may be
termed p.
Let i 1? 12 now denote the angles POA, FOB respectively, then
QOA = tl ,
The ray will be found by joining with the intersection of three planes drawn
at P, Q, , perpendicular to OP, OQ, OR, respectively.
Now from Prop. 9 it appears that
using only one sign for the sake of simplicity, which we may do by throwing the
ambiguity upon the way in which ^ or t 2 is measured, also
OQ = OP + ^ Sta,
Q
Let Stj = Si.,, then it is clear that OQ = OR, and
the intersection of the two planes perpendicular to
OQ, OR is therefore a line perpendicular to the
plane QOR, and to the line which bisects the angle
QOR.
In fact if we draw QT, RT perpendicular to OQ, OR respectively in the plane
QOR, the intersection in question passes through T and is perpendicular to OT ;
also
to the first order of smallness.
* OA, OB are not expressed in the figure.
1] FresneTs Optical Theory of Crystals. 21
Now it is easy to see (just as on p. 16) that
*
sin /x
and also
sin /a
therefore ROP = QOP and therefore POT is perpendicular to QOR.
Hence the problem is reduced to finding L the intersection of two lines TL, PL
drawn in the same plane POT.
Now because OTL, OPL are each right angles, a circle may be made to pass
through L, T, P, 0.
Hence the angle
OP x 4^-cos
.. OP x POR . cos ip sin/*
- el. OP, =tan d.OP.
8l ~
and OL = OP. sec
Also the position of the plane P0Z is known, and therefore the radius is
completely determined in magnitude and position.
It may be worth while also to remark that the above constructions enable us
to form a series of equations between the magnitude of the radius and its incli
nations to the two prime perpendiculars.
In fact, if we call ir 1} ?r 2 the two inclinations in question
u,
cos T! = cos POL cos i! + sin POL sin ij . sin ^ ,
jU-
cos 7r 2 = cos POL cos ia + sin POL sin ^ . sm ^ ,
and of course if we call the angle between the two prime normals 2E
/ Ij+lA / I l
m (E + -i-jr- sin ( L --
\ / \
sn tj sn i a
COR. 1. When tl or ^ = 0, ten POL assumes the form ^ which may be
interpreted analogously to the method used in the reverse problem, but may be
more elegantly illustrated by
COR. 2. Which is that the meridian plane POT (that is, the plane in which
both normal and radius lie) bisects the angle formed by ROP, QOP, and therefore
22
Analytical Development of
[1
that formed by the planes drawn through the normal and the two prime normals
to which these two are perpendicular.
Now we have found (Cor. 4, page 18), that it also bisects the angle formed by
the two planes passing through the
radius and the two prime radii. Hence
when the ray is given, we may find
by the easiest geometry the normal
and the tangent plane, and vice versa.
Thus suppose (N, N ) (R, R ) to be
the projections of the prime perpen
diculars and prime radii on a sphere
concentric with the wave surface.
Let n be the projection of any
given perpendicular on the same
sphere ; join nN, nN ; bisect NnN by nM, which will be the meridian plane.
Draw from R , R TV perpendicular to nM and make R T=TV. Produce RV
to meet Mn in r, then Rr M - R rM, and
therefore r is the projection of the radius.
Just in the same way when r is given we
may find n.
Now suppose n to come to N, then
the position of the meridian plane nM
becomes indeterminate, and r from a point
becomes a locus, subject to the condition that R rN = RrN. From r draw rD
perpendicular to rN.
Then it is clear that because rN bisects RrR
sin RD sin Rr sin RN
smR D sinR r sinJt N
^\
and therefore D is a fixed point and ND a fixed length, and
cos r ND = tan rN . cot ND ;
therefore the projection of the locus of r upon a plane drawn at N perpendicular
to the line joining N with the centre is given by the equation
p = ON . cot ND . cos 0,
N being the origin and the projection of ND the prime radius ; which is the
equation to a circle passing through N, and whose diameter = ON cot ND.
Hence at the extremity of each prime perpendicular the tangent plane meets
the surface in a circle passing through that extremity and whose radius = |6 cot a,
a being to be found from the equation
sin (2E + a) sin (E + e)
Fig. 7.
that is
tan
sin (E e)
= (tan E}* cot e.
1] FresneTs Optical Theory of Crystals. 23
Just in the same way it may be shown that the trace of the perpendiculars to
the tangent planes of the surface at the point where it is pierced by any prime
radius upon a plane perpendicular to that radius at its extremity, is also a circle
passing through it, and curved in an opposite direction from the circle of plane
contact nearest to it.
Hence the enveloping cone at these points may be described as being perpen
dicular to the circular cone, formed by drawing lines from the centre to the above
described circle ; that is every generating line of the one will be perpendicular to
the generating line which it meets of the other.
More generally it easily appears from fig. 6 that if a series of great circles
(representing meridian planes) be taken intersecting the great circle NRR N
in a fixed point, a plane perpendicular to the radius passing through that
point, will intersect the cone of rays as well as the cone of perpendiculars corre
sponding to those meridian planes, in two circles. So that there exist an indefinite
number of circular cones of rays corresponding to circular cones of perpendiculars
touching each other in a line lying in the plane containing the extreme axes, and
having their circular sections perpendicular to that line.
The cusps are explained by the cone of rays degenerating into a right line, and
the circles of plane contact by the cone of perpendiculars so degenerating.
Furthermore I observe in conclusion that when a ray is given it follows from
the general geometrical construction above that there will be two meridian planes
according as we take R with JR , or with a point 180 from R , and consequently
these two planes will be perpendicular to each other.
And similarly when a normal is given there will be two meridian planes per
pendicular to each other.
Thus the planes passing through any radius and the two normals at the points
where it pierces the wave surface, are perpendicular to each other, as are also the
two planes passing through any normal and its two corresponding radii.
Moreover a glance at fig. 2 will show that the two lines of vibration
corresponding to any front lie respectively in the two meridian planes passing
through the perpendicular to that front or, in other words, the intersection
of a plane drawn through either ray belonging to a front perpendicular thereunto
is always a line of vibration in that front.
This has been noticed, I think, by Sir William Hamilton for the particular case
of the singular points.
As two fronts belong to every ray, so two rays pertain to every front. And
from what has been said above it appears that the two lines of vibration in any
front are the projections of its two rays upon its own plane.
24 Analytical Development of [1
NOTE 1.
In the paper above, it is shown that the meridian plane, that is,
the plane containing the ray and normal, always passes through a line
of vibration in the corresponding point. Now the line of force called into
action by a displacement in the line of vibration clearly lies in this very
plane ; for the resolved part of it lies in the line of vibration itself.
Harmony and analogy concur in suggesting that as two of these four
lines are perpendicular to each other, so are also the other two, or in
other words, that the ray is always perpendicular to the direction of
unresolved force.
The following investigation verifies this conjecture.
Let x, y, z be the coordinates of a point taken at distance unity from the
origin and in any line of vibration ; then the cosines of the angles made by
the line of force with the axes are as a?x : l>-y : tfz respectively.
Let a be the inclination between the line of vibration and the line of
force, then
a?x . x + fry . y + <?z . z a.-x i + fry + c% 2
)S W = *)} " V(a 4 ^ 2 + fry + cV)
Let V(a 4 # 2 +fry+cV) = P,
then P 2 =fl 4 (sec&&gt;) 2 .
Now let a, /3, 7 be the angles of inclination between the coordinate
planes and the front in which the line of vibration lies, and X some quantity
to be determined. I have shown in Prop. 3 that if
X cos a = (a 2 v 2 ) x,
then will X cos ft = (fr 2 - a 2 ) y,
and X cos 7 = (c 2 - a 2 ) z ;
therefore X 2 = V + Kf + c 4 * 2 - 2v 2 (a 2 ^ 2 + 6y + c 2 ^ 2 ) + v 4 = P 2 - v 4 .
Again,
1 (cos a) 8 (cos/3) 2 (cos 7 ) 2
p^4 - (ri^r v *)* + (& - tf>) 2 + (c 2 - v 2 ) 2
Nnw = - v - = . (cot ft)) 2 .
24
1] FresneVs Optical Theory of Crystals. 25
And in Mr Smith s investigation of the form of the wave surface (already
alluded to*) by great good fortune I find ready to my hand
(cos a) 2 (cosjS) 2 ^ (cos y) 2 _ 1
(a 2 ^ 2 ) 2 + (jp^tf)* (c 2 - t> 2 ) 2 ~ v*(r 2 - v 2 )
r being the radius vector to the point whose tangent plane is parallel to
the point in question.
Hence
V* v* p 1
~ ^TZ>~) ~ ^tf ~ jr~fi
p being the length of the perpendicular from the centre upon the tangent
plane, for p = v.
Hence (cot <w) 2 = the square of the cotangent of the angle between radius
vector and normal.
Or, in other words, the line of force is as much inclined to the line of
vibration as the ray is to the normal.
Now the normal is perpendicular to the line of vibration, and all four
lines lie in one plane.
Therefore the ray is perpendicular to the line of force. Q. E. D.
I may be allowed to conclude this long paper with a summary of some of
the most remarkable consequences which I have extricated from Fresnel s
hypothesis.
(1) The two meridian planes corresponding to any given radius are
perpendicular to each other -f.
(2) So are the two corresponding to any given normal.
(3) Every meridian plane bisects the angle formed by two planes drawn
through the radius and the two prime radii.
(4) It also bisects the angle formed by two planes drawn through the
normal and the two prime normals.
(5) Each meridian plane contains one line of vibration and the corre
sponding line of force.
(6) The ray is perpendicular to the line of force.
All these conclusions, except the fourth, are, I believe, original.
* See above, p. 18.
t I have defined the meridian plane to be that which contains radius vector and normal
belonging to the same point.
26 Analytical Development of [1
The theory of external and internal conical refraction follows immediately
as a particular consequence from the third and fourth combined as already
shown ; the same propositions also enable us to draw a tangent plane to any
point of the wave surface by mere Euclidean geometry. May not some of
these conclusions serve to suggest to physical inquirers the question, Has
the theory been started from the most natural point of view?
NOTE 2. Investigation* of the Wave Surface.
Since the appearance of the preceding parts, I have succeeded in com
pleting the self-sufficiency of my method by deducing the equation to the
wave surface from the expressions given in Prop. 5 for the angles between
a front and the principal planes in terms of its two velocities. If these
angles be &&gt;, <, ty, and the two velocities v lt v 2 we found
COS ft) =
(a 2 - 6 2 ) (a 2 - c 2 )
Let the tangent plane to the wave surface be written
COS 0) COS d>
then
, . ,
0)f
Let
* This investigation supplies the step which Mr Tovey was desirous should appear in the
Magazine. [Phil. Mag. March, 1838, p. 261. ED.]
t In lieu of v l we might write v 2 in the denominator without affecting the result.
Observe, that = %/ _ y) a . _ ^ } - ">* > - for tbe
1] FresneCs Optical Theory of Crystals. 27
then equation (7) becomes
A&+BTiy+C& = 0, (1)
and equation (ft)
Aa? Bb* Cc 2
and equation (a) may be written under two forms, viz.
(a 2 - w 2 2 ) 4 B + (6 2 - % 2 ) BT^ + (c 2 - v 2 2 ) C * = 1 , (3)
,,v
W-VT +-VS ?-* " 1 -
From (1)
Afc + Bqy = -%*. (5)
From (2)
Aa? Bb- Cc* ,.
- aH -- y ~-f- z -
From (3) and (1)
^(a 2 -c 2 )^ + J B(6 2 -c 2 )7/2/=l. (7)
From (2) and (4)
(8)
From (5) and (6)
C 2 c 2 * 2 _ ftfty-2 _ 4*0*0 = ABxy a^ + 6 2 \ . (9)
From (7) and (8)
C 2 _ & (b- - c 2 ) 2 y 2 - A n - (a 2 - c 2 ) 2 ^ 2 = ABxy ? + - x O 2 - c2 )( &2 ~ 2 )- ( 10 )
From (9) and (10)
AB (a 2 - 6 2 ) (a 2 - c 2 ) (6 2 - c 2 ) ^y - = a s c a - (a 8 - c 2 ) (6 2 - c 2 ) CV* 2
- {a 2 (6 2 - c 2 ) 2 - 6 2 (a 2 - c 2 ) (6 2 - c 2 )} B*f
- {a 2 (a 2 - c 2 ) 2 - a 2 (a 2 - c 2 ) (6 2 - c 2 )} A* a? = a 2 c 2 - cV - cy - a 2 ^ 2 . (11)
From (11), interchanging (a, a?, ) with (6, y, T;) we have
2 - a 2 ) (6 2 - c 2 ) (a 2 - c 2 ) xy = 6 2 c 2 - c 2 ^ 2 - c 2 ^ 2 - 6y. (12)
Finally, from (11) and (12) we have
{ 2 c 2 - (a 2 - c 2 ) a? - c 2 (.-r 2 + y 2 + z-}} {b~c~ - (b- - c 2 ) y- - c- (x- + f + z*)}
= (a- - c 2 ) (b- - c 2 ) a?f,
that is (# 2 + y- + z*) (d 2 x? + b 2 y 2 + c 2 ^ 2 ) - a 2 (b- + c-) x-
- b- (a- + c-) y- - c- (b- + a-) z- + a 2 6 2 c 2 =
the equation required.
2.
ON THE MOTION AND REST OF FLUIDS.
[Philosophical Magazine, XIIT. (1838), pp. 449 453.]
M. OSTROGRADSKY S memoir on this subject inserted in the Scientific
Memoirs seems to have excited much attention, and has been made the
occasion of some annotations* by a distinguished writer in the Philosophical
Magazine. Mr Ivory s recent papers in the same periodical must still more
tend to invest with a new interest all such speculations. It seems to me
desirable therefore to present the theory of fluids in all the simplicity of
which it is susceptible.
I consider a fluid as a collection of particles subject to some law of
relative position other than that of rigidity. These particles by their mutual
actions maintain the connections of the system. As to the law of force
between them we know nothing ; but I assume it is a general principle
of nature, that for each instant of time the sum of the internal actions
(reckoned by the product of each particle into the square of the space due
to the internal force acting on it) is a minimum. This in fact is Gauss s
principle of least restraint. We may if we please split this principle into
two parts ; that is to say, assume that the internal system of forces is always
such as if acting alone would keep the fluid at rest ; and then again assume
that any equilibrating system of forces must be subject to the law of virtual
velocities. I say assume, because it is impossible a priori to prove this.
Lagrange s so-called demonstration is unworthy of his name, and (albeit
sanctioned by the powerful oral authority of an ex-Cambridge Professor)
contrary alike to sense and honesty. It is better therefore at once to
proceed upon Gauss s principle. It might easily be shown that this is in
effect tantamount in all cases to D Alembert s and Lagrange s principles
combined.
Before entering upon the investigation I may call attention to one point
of great analytical interest, and relating to the difficult subject of the
algebraical sign, viz. that if the density of a point (a-, y) in any circumscribed
space be expressed by the quantity -y- + -j- so that the mass is
doc (-b
II dxdy ( -r-] + I dxdy [-*-
JJ 9 \dx) JJ 9 \dy
[* Phil. Mag. May, 1838, p. 385. ED.]
2] On the Motion and Rest of Fluids. 29
that is not equivalent to
I (udy + vdx),
that is if we please
[( dy dx\ j
-/ + v -T- ds,
J\ds dsj
(where s is for clearness sake and to avoid double limits taken an element of
the bounding curve) as at first sight it might appear to be, but is in fact
equal to
/7 dy dx\ ^
I M-/-0-T- ds.
J\ds dsj
I shall demonstrate this point in the next number* of the Magazine.
It at first caused me some trouble in conducting the annexed inquiry.
I shall also take occasion at some other time to revert to a new species
(as I believe) of partial differential equations ; that is to say, where there
are fewer of them than of the principal variables, which may be called
therefore Indeterminate Partial Differential Equations. A complete solution
of one of these appears in the subjoined
Investigation.
For the sake of simplicity I take an incompressible fluid. The method
is nowise different for a fluid of varying density.
Let A#, Ay, Az be any displacement undergone by a particle at the
point x, y, z parallel to the axes x, y, z respectively ; it is easily shown that
to satisfy the condition of invariability of mass we must have
_
dx dy dz
One relation between u, v, w the velocities parallel to x, y, z is obtained
immediately by putting u$t, v8t, w8t, for A#, Ay, A, which gives
du dv dw _
dx dy dz
as usual.
Again, if X, T, Z be the impressed forces, and X lt Y lt Z v the internal
forces acting on any particle parallel to the axes, we have
(2)
(3)
(4)
from the mere geometry of the question.
[* p. 36, below. ED.]
du
du
du
du
dt H
~dx U
f -j-v H
dy
dz
dv
dv
dv
dv
~dt^
dx
+ -j-v-
dy
dz
dw
dw
dw
dw
~dt~
dx
dy V
dz
30 On the Motion and Rest of Fluids. [ 2
Finally, Gauss s principle teaches us that
fjjdaidydz {X l &X 1 +Y l &Y 1 + Z 1 kZ 1 }=0. ()
. ,
Now j H 7 H
dx dy dz
(du\ 2 (dv\ 2 /dw\ 2 (dv dw dw du dudv,
~ \dx) \dy) \dz) " (dz dy dx dz dy dx*
as appears from the equations (1), (2), (3), (4); and hence
d&X,
dx dy dz
the complete solution of which, free from the sign of integration, is
rf^ d(j>
Al ~ dy~ dz>
- d ^
dz dx
-i -j ,
dx dy
, (f>, -^ being any three independent functions of x, y< z.
On substituting these values in equation (/9) we obtain
+///***{* aM}--
This may be put under the form
r
dy 1 1 dzdx
fff (dX-L dY-\
l\ dxdydz . T/T -j -r-
JJj \dy dxj
rr/- , , (dY, dz }]
II dxdydz . &&gt; -y ~ "3"
JJJ \dz dy)
III dxdydz . d> ( -r-^ -, -- ] = 0.
" \ Hjf rl 7 I
JJJ \ 1KB I** /
On the Motion and Rest of Fluids.
31
Here it must be remembered that o>, </>, ty are perfectly independent of each
other. Also the values of the three first written quantities depend upon the
values of X lt F 1( Z 1 at the bounding surface; the values of the three last-
written depend upon the general values of X 1} Y l} Z^. It is clear therefore
that each system of three equations and each member of each system must
be separately zero.
The three latter equations give
_
dy dx
_
dz dy
_
dx dz
(7)
The three former require that for each section of the surface parallel to
the plane xy
for each section parallel to yz
for each section parallel to zx
(By
and these equations are to hold good whatever ty, <, &&gt; may be. From the
equations (7) we derive
X,dx + Y,dy + Z,dz = df, (5)
from equations (8) we obtain
/ = constant for all points in any section of the bounding surface parallel
to the plane of xy,
/ = constant for all points in any section of the bounding surface parallel
to the plane of yz,
f = constant for all points in any section of the bounding surface parallel
to the plane of zx.
Now by drawing through all the points in a plane parallel to xy, planes
parallel to yz, we may cover the whole surface ; hence f is constant all over
the surface bounding the fluid.
See remark at introduction.
32 On the Motion and Rest of Fluids. [2
Therefore X l dx + Y l dy + Z l dz = 0, (6)
for all variations of dx, dy, dz taken upon the surface.
The equations (1, 2, 3, 4, 5, 6) are coincident with those obtained by the
usual method; with this difference, that X ly Y lt Z^ here take the place of
dp dp _ dp
dx dy dz
Thus then we have obtained all the conditions requisite for determining
the motion of fluids from the universal principle of least constraint conjoined
with the specific character of the system in question.
General Remarks.
In the case of equilibrium, that is in the case where no particle moves,
we have X 1 + X = 0, Y 1 + Y = 0,Z l + Z=Q. Hence Xdx+Ydy+Zdz is a
complete differential always and zero for the surface.
The above results have been obtained upon the principles of the differ
ential calculus, and the continuity of the forces has been tacitly assumed.
If now we were to suppose forces of finite magnitude (as compared with the
whole sum acting upon the entire system) to be applied to a layer of single
particles or to a layer of a thickness of the same order of magnitude as the
distances between the particles themselves, (which has been treated as an
infinitesimal) it would appear that our results would be no longer applicable,
just in the same manner as it would be erroneous to apply the principle
of vis-viva (for example) without modification, to the case of impulsive forces,
because we had deduced it by the calculus in the case of the motion
being continuous. Hence the above equations ought not strictly to apply
to the motion or rest of a fluid contained between physical surfaces ; for the
pressure afforded by these surfaces, whatever its actual value may be, we
know d priori is commensurable with the whole amount of force acting on
the fluid ; but the immediate application of this pressure (alias repulsive
force) is confined to the bounding layer of fluid particles, or at most extends
to a distance bearing a low ratio to the distances between the particles
themselves,
Accordingly, to the non-applicability of the equations for free fluids to the
case of fluids confined at the boundaries, and to an independent investigation
upon the minimum principle for this class of problems, it is, that I look for
the true explanation of the phenomena of capillary attraction (vulgarly so
called).
3.
ON THE MOTION AND REST OF RIGID BODIES.
[Philosophical Magazine, xiv. (1839), pp. 188190.]
IN the subjoined investigation, which, as far as I know, is my own,
I apply the same method to rigid as in the preceding paper I applied to
fluid systems.
Let x, y, z be the coordinates of any particle in a rigid body ; x , y , z 1 the
coordinates of some other particle, and let
x = x + h, y y + k, z = z + 1.
Call A#, Ay, A.Z the increments which x, y, z receive after the lapse of a
small interval of time ; so that terms in which they enter in two or more
dimensions may be neglected.
A-*
d&z , d&z j d&z , p
A (/) = bz + -3- h + -j k + -T- I + R,
dx dy dz
P, Q, R containing binary and higher combinations of h, k, I, which we shall
have no occasion to express.
At the commencement of the interval the squared distance of the two
particles was (x - #) 2 + (y - y) 2 + (z - zY ; at the end of the interval the
distance squared is
(x - x + A (x} - A*-) 2 + (y - y + A (y } - Ay) 2 + (z - z + A (z 1 ) - A*) 2 ,
and these two expressions must be the same by the conditions of rigidity
whatever h, k, and I may be ; that is
dec d dz
j , d&z j
+ j- h + -T- k + -r- 1 +
dx dy dz
for all values of h, k, and I.
34 On the Motion and Rest of Rigid Bodies. [3
Hence rejecting infinitesimals of the second order and equating to zero
separately the coefficients of A 2 , k z , P, and of kl, Ih, hk, we have
-*
(c)
By differentiating (d), (e), (/) with respect to 2, x, \j respectively, and
substituting from (a), (b), (c), we obtain
dz
_ _
~~ =:U ~ = -
By differentiating the same with respect to y, z, x respectively, and pro
ceeding as before, we have
dy 2
Thus, then, we have
_
=
dx dif- dz 2
dy dz 2 dx
dz dx 2 ~ dy 2
therefore A# = A + By + Cz, (o)
Ay = D + Ez + Fx, (p)
&z = G + Hx + Ky, (q)
A, B, C, D, E, F, being constant for a given instant of time ; between which
by virtue of the equations (d), (e\ (/), we have the relations
E + K = Q, H+C = 0, B + F = 0.
If we call n, v, w the three component velocities of the particles at x, y, z
parallel to the three axes, and X lt Y lt Z 1} the three internal forces, it is at
once seen that u, v, w, as also AJTj, AF 1; A^ must be subject to the same
equations as limit Aar, Ay, A^ ; so that
v = b + az yx, (2)
w = c + fix ay, (3)
A ^ = 6! +0^-7,*?, (j)
= c, + A - iy. (*)
3] On the Motion and Rest of Rigid Bodies. 35
Also if X, Y, Z be the impressed forces, we have
* + *-, (4)
.+ F-*.
(5)
(6)
And by Gauss s principle, calling m the mass of the particle at x, y, z,
Hence equating separately to zero the coefficients of a lt b 1} d and of
i> /3i> 7i m the quantity Sm(JT 1 AJT 1 + Y 1 ^.Y 1 + Z l ^Z 1 ) we have
2m Z x = Ov
2m Fj =
2m j =
(7-12)
2m (X,z - Z^os) =
2m ( YI# Zi2/) = 0-
Lastly, we have the equations
-I-
From the fifteen equations marked (1) to (15), the motion may be deter
mined by assigning the position of each particle at the end of the time t in
terms of its three initial coordinates, its three initial velocities, and the
initial values of the nine quantities
2m;?/,
In the case of rest X 1 = X, Y 1 = Y, Z l = - Z, and the equations (7)
to (12) inclusively taken, express the conditions of equilibrium.
The equations (o), (p\ (q), which have been obtained from conditions
purely geometrical, establish the well-known but interesting and not obvious
fact, that any small motion of a rigid body may be conceived as made up
of a motion of translation and a motion about one axis.
32
2m;y 2 ,
4.
ON DEFINITE DOUBLE INTEGRATION, SUPPLEMENTARY
TO A FORMER PAPER ON THE MOTION AND REST
OF FLUIDS.
[Philosophical Magazine, xiv. (1839), pp. 298 300.]
IN a paper on Fluids which appeared in the December Number of this
Magazine, I had occasion to remark, that the mass of an area having at the
point (x, y) a density ~r- + -r- could be expressed by the simple formula
f ( dy dx\ .
\u-f--v-j-\ds\
J i { as as)
I being the length, and ds an element of the bounding curve : this may be
thought to require some explanation.
(1) Let APBq represent any oval; PpL, QqM any two contiguous
ordinates cutting the curve in Pp, Qq
respectively, A C, BD the two extreme
tangents parallel to Oy, and p the
density at any point (x, y}. The ex
pression ffpdxdy will serve to denote
the mass of the oval area APBq, and
the limits may be twice taken, that is,
(i) the two values of y corresponding
to any one of x; and (ii) the two
values of x corresponding to C and D.
This method is in fact tantamount to
taking the sum of the columns Pp
qQ ; but this is not necessary, for
APBq may be considered as the algebraical sum of the mixtilinear area
APQBDC, and the mixtilinear area BDCApq, or (if any line O G D be
drawn parallel to OCLMD) of APQBD C and BD C Apq.
Thus then the mass = jdx ($pdy), fpdy being left indeterminate, and the
extremity of x travelled round from G to D, and back again from D to G.
L M
Fig. 1.
4] On Definite Double Integration. 37
This will be better expressed by transforming the variable, and summing
with respect to some quantity, such as the arc of the curve, which contin
uously increases, or if we please, with respect to 6, the angle subtending any
point taken within the curve.
The mass is then
= +
always remembering that no constant need be added to fpdy, and that the
doubtful sign arises from the choice of ways in which 6 may be measured
round. If the area be not included by one line ; but by several, as for
example, by a curve and a right line, the above integral, if broken up into
as many parts as there are breaches of continuity, will still apply.
(2) Let us suppose that we have two areas exactly coinciding with, and
overlapping one another ; but the density of the one at (x, y) to be p, and of
the other p .
Let the mass of the first be treated as the sum of columns parallel to Oy,
and that of the second as the sum of columns parallel to Ox.
The one will be represented by
f fir
de(fpdy)^,
Jz* dv
the other will be represented by
and the sum of the two, or the joint mass, by
ro
Zir
So long as these two operations are performed separately, the doubtful
signs may be preserved in each term, because s need not be travelled round
in the same direction for the two summations ; but if we perform the second
integration conjointly for the two masses, their sum
the mark of interrogation denoting that one or the other, but not either of the
signs + must be used, and the question is, which ?
This will be answered by taking different points in the bounding line
which may be continuous or not. Now every line returning into itself,
whether continuous or not, will naturally divide with respect of any given
38
On Definite Double Integration.
system of axes, into at most four parts, or sets of parts ; two in which dx and
dy both increase or both decrease, and two in which one increases and the
other decreases.
Take P lt P 2 , P 3 , P 4) any points in the four quadrants respectively, it
will be observed that,
At P! the p column enters ad-
ditively, and the p column subtrac-
tively.
At P 2 both columns are additive.
At P 3 the p column is additive
and the p column subtractive.
At P 4 both columns enter subtrac-
tively.
Again, reckoning round in the
Fig> 2- direction of the arrows,
At Pj, x and y are both increasing.
At P 2 , x is increasing and y decreasing.
At P 3 , x and y both decrease.
At P 4 , x is decreasing and y increasing.
Thus when fpdy and jp dx are affected with the same signs, dx and
dy are of opposite signs ; and when fpdy, jp dx are of opposite signs, dx and
dy are of the same sign.
Hence it appears that the mass of the area, whose density at (x, y} is
p + p, is capable of being represented by
5.
ON AN EXTENSION OF SIR JOHN WILSON S THEOREM
TO ALL NUMBERS WHATEVER.
[Philosophical Magazine, xm. (1838), p. 454.]
THE annexed original theorem in numbers will serve as a pendant to
the elegant discovery announced by the ever-to-be-lamented and com
memorated Horner*, with his dying voice, in your valued pagesf.
THEOREM.
If N be any number whatever and
Pi,p*,p 3 PC
be all the numbers less than N and prime to it, then either
Pl P-2 Ps Pc+1,
or else p\-p-ps p~lj
is a multiple of N.
6.
NOTE TO THE FOREGOING.
[Philosophical Magazine, xiv. (1839), pp. 47, 48.]
I HAVE to apologize for calling " original " (in the last Number of the
Magazine) the theorem of numbers which I termed " a pendant to Homer s
theorem." This Mr Ivory has done me the honour to inform me may be
found in Gauss s Disquisitiones Arithmetical, p. 76. As Horner s extension
of Format s theorem suggested this extension of Sir John Wilson s to me,
so I concluded that had this extension of Wilson s been known to the world
it would naturally have suggested his to Horner. No acknowledgment of
this kind having been made, I took it for granted that the theorem I gave
was new. Undoubtedly had Mr Horner been aware of Gauss s theorem
he would have made mention of it.
I take this opportunity of adding that my acquaintance with Gauss s
principle J has not been derived from the study of his works, but from a
casual statement of it in an English work, Dynamics, by Mr Earnshaw, of
St John s College, Cambridge.
* Homer s proof is highly valuable as a novel and highly ingenious form of reasoning, but
his theorem may be deduced with infinitely more ease and brevity from Fermat s than he seems
to have been aware of.
[t Phil. Mag. Vol. xi. p. 456. ED.] [J See p. 28 above. ED.]
7.
ON RATIONAL DERIVATION FROM EQUATIONS OF COEXIST
ENCE, THAT IS TO SAY, A NEW AND EXTENDED THEORY
OF ELIMINATION*. PART I.
[Philosophical Magazine, xv. (1839), pp. 428 435.]
ANY number of equations existing at the same time and having the
same quantities repeated, may be termed equations of coexistence : in the
present paper we consider only the case of two algebraical equations :
x m + a, x- 1 + a 2 x m - 2 + + a m = 0,
x n +b 1 x n - 1 + b 2 x n ~ 2 + + b n = 0.
The above being "equations of coexistence," x is called "the repeating term."
If we suppose the equation
C Q X r + C^- 1 + C 2 X r ~ 2 + + C r =
to be capable of being deduced from the two above, and, therefore, necessarily
implied by them, this will be called "a Particular Derivative" from the
equations of coexistence, of the rth degree, (r being supposed less than ra
and wf, and the coefficients being rational functions of the coefficients of the
equations of coexistence).
There will be an indefinite number in general of such derivatives, and
the form involving arbitrary quantities which includes them all is called
" the general derivative of the rth degree."
Any "Particular Derivative," in which the terms are all integral,
numerically as well as literally speaking, is called an " Integral Derivative."
That " Integral Derivative " of any given degree in which the literal
parts of the coefficients are of the lowest possible dimensions^, and the
numerical parts as low as they can be made, is called the " Prime Derivative "
[* The results of this and some following papers were repeated, with demonstrations, in the
paper "On a Theory of the Syzygetic Relations of two rational integral functions comprising an
application to the Theory of Sturm s Functions, and that of the greatest Algebraical Common
Measure," Phil. Trans. Royal Soc. Vol. CXLIII., Part i. pp. 407 -548, 1853. See below Section n.
Art. (16) of that paper. ED.]
t This restriction upon the value of r is not essentially requisite, and is only introduced
to keep the attention fixed upon the particular objects of this first Part.
Of course the dimensions of the coefficients in the equations of coexistence are to be
understood as denoted by the indices subscribed.
7] On Rational Derivation from Equations of Coexistence. 41
of that degree. So that there is nothing left ambiguous in the prime
derivative save the sign.
The " Derivative by succession " is that particular derivative which is
obtained by performing upon the equations of coexistence the process
commonly employed for the discovery of the greatest common measure, and
equating the successive remainders to zero.
To express the product of the sums formed by adding each of one row of
quantities to each of another row, we simply write the one row above the
other ; a notation clearly capable of extension to any number of rows, which
would not be the case if we spoke of differences instead of sums*.
THEOREM 1.
Let /ij, A 2 > h m , be the roots of one equation of coexistence, k lt & 2 , ... k n ,
the roots of the other. The general derivative of the rth degree is repre
sented by
2 (SR(hi, h h 3 ... hr) {(x-h l )(x-h^... (x-h r )} x f T h *^ ~ h I) = 0,
\ ^ KI , ft7 2 K n ) /
SR (h 1} h 2 , h s ... h r } denoting any symmetrical rational (integral or fractional)
function of h 1} h 2 . . . h r ;
("T+I "r+2 ""in
If If - If \
A/! , ft/2 A ?*,)
being to be interpreted as above explained, and S of course including as
many terms as there are ways of putting n things r and r together f.
A form tantamount to the above, and which may be substituted for it,
is its analogue,
V/CfD/7 7 i \ </ 7 \ / 7\ / T\I r^r+ij n/ r +2 ""n \ r\
2, ( SR(fc i , k 2 ... k r ) {(x kj (x & 2 ) ... (x kr)} x < , * H=0.
When r = the theorem gives simply
, /lj /?m
7 7 7 (
( KI , K 2 ... K n }
and is coincident with that given by Bezout in his Theory of Elimination.
* The wider views which I have attained since writing the above, and which will be developed
in a future paper, lead me to request that this notation may be considered only as temporary.
It would have been more in accordance with these views to have used the two rows to denote
products of differences than of sums. But a change now in the text would be very apt to cause
errors in printing.
t The general derivative may clearly be expressed also by the sum of any two particular
derivatives affected respectively with arbitrary rational coefficients. The equivalency of an
arbitrary function to two arbitrary multipliers is very remarkable, and analogous to what occurs
in the solution of certain differential equations.
42 On Rational Derivation [7
Subsidiary Theorem (A).
If h l} h 2 ... h m be the roots of the equation
x m + a^" 1 - 1 + a 2 x m ~ z + ...... + a m = 0,
and if e m + a^- 1 + a. 2 e m ~* + ...... + a m -w = 0,
then 2 /T -- rwi --- r\ --- TL -- r~\ = -- 1 j ^ ( er+1 )>
(hi ~ AB) (^i - h 3 ) (hi ~ hm) r+1 du
u being made zero after differentiation.
COR. If R(h^ denote any integral rational function of h 1} then
__
l - h s ) ...(h,- h m )
is always integral and is zero when the dimensions of R(h^ fall short of
(m-1).
Subsidiary Theorem (B).
i, h z ...h r
[ "r+i> "r+2
can be expressed by the sum of terms, each of which is the product of series
of the form
it is always integral, and when the dimensions of the numerator fall short of
(m r) r it vanishes*.
Subsidiary Theorem (C).
The only modes of satisfying the equation
2 {f(hi, A 2 . . . M xSRfa.h,... h r )} = 0,
for all forms of the latter factors short of (m -r)(n- r) dimensions, are to
h 2 ...h r ) = 0, or else
constant
f(h,, /to ... hr) =
* It may be remarked also in passing, that any term in the numerator which contains any
one power not greater than m-2r may be neglected and thrown out of calculation. Moreover,
an analogous proposition may be stated of fractions in the denominators of which any number
of rows are written one under the other ; see the first note, page 41.
7] from Equations of Coexistence. 43
THEOREM 2.
By virtue of the subsidiary theorem (B), the two equations
,v
1, _ L- t \
= 0,
s
r & 2 ...-
are each integer derivatives of the rth degree.
THEOREM 3.
And by virtue of the subsidiary theorem (C), the two above equations
are the "Prime Integer Derivatives," and are exactly identical with each
other.
COR. 1. The leading coefficient of the "prime derivative" of the rth
degree is always of (m r)(n r) dimensions.
COR. 2. If P r be the prime derivative of the rth degree and if
(Z = 0, 7=0) be the two equations of coexistence, and \ r , jj, r the two "prime
constituents of multiplication" to the said derivative, that is if \ r and p r
satisfy the equation \ r X + n r Y=P r , then the coefficient of the leading
terms in \ r and in p, r is of (m -r-l)(n-r-l) dimensions.
THEOREM 4.
The " Prime Derivative " of any given degree is an exact factor of the
" derivative by succession," of the same degree. The quotient resulting from
striking out this factor is called " the quotient of succession."
THEOREM 5.
If A, L z , L 3 , &c., be the leading coefficients of the derivatives occurring
first, second, third, &c., in order after the equations of coexistence, and if
Qi, Qz, Qa, & c -> represent the first, second, third, "quotients of succession"
reckoned in the same order, then
44 On Rational Derivation [7
and in general
Q_ -L/2 -^4 J- 2n4 -L^l 2 *
2n ~ 4 r
/} _ -"1 -^3 -^2713 -"2M 1 i
Vm+1 ~ /> 4 L * L y 2 "
-L/ 2 -L/4 ... Jv 2 W 2 -^271
COR. Hence, in place of Sturm s auxiliary functions, we may substitute
the functions derived from the equations of coexistence f/# = 0, -^- O
y Ct-5/
according to Theorem 2, due regard being had to the sign.
Scholium. Hitherto it has been supposed that the values of the coeffi
cients in the equations of coexistence are independent of one another, but
particular relations may be supposed to exist which shall cause the leading
terms given by Theorem 2 to vanish, giving rise to anormal or singular
primes, as they may be called, of the degree r of fewer than (ra r) (n r)
dimensions. The theory of this, the failing case (so to say), is highly interest
ing, and I have already discovered the law of formation for the quotients of
succession on the supposition of any number of primes vanishing consecutively;
but I forbear to vex the patience of my reader further, the more so, as I
hope soon to be able to present a complete memoir, with all the steps here
indicated filled up, and numerous important additions, (the perfect image
of which this is but a rough mould), as homage to the learned and illustrious
society which has lately done me the honour of admitting me into its ranks.
Why this has not already been done must be excused, by the fact of the
theory having suggested itself abroad in the intervals of sickness J. Yet thus
much will I add in general terms, namely, that as many primes as vanish
consecutively, so many units must be added to the index 2 of the accessions
* That the appearance of the index 4 may not startle, let my reader bear in mind that there
are what may be termed secondary derivatives of succession for every degree appearing in the
process of successive division.
t The prime derivatives must be capable of yielding an internal evidence of the truth of
Sturm s theorem. In fact, for the case of all the roots being possible, a little consideration will
serve to show that the leading term of each prime derivative of the equation ijfe-r | =0 will
( (Ix )
consist of a series of fractions, each of which fractions is, numerically speaking, of the same sign.
+ The reflections which Sturm s memorable theorem had originally excited, were revived by
happening to be present at a sitting of the French Institute, where a letter was read from the
Minister of Public Instruction, requesting an opinion upon the expediency of forming tables of
elimination between two equations as high as the 5th or 6th degree containing one repeating
term. The offer was rejected, on the ground of the excessive labour that would be required.
I think that this has been very much overrated ; and probably many will be of the same opinion
who have dwelt upon the fact that no numerical quantity will occur in the result higher than
the highest index of the repeating term. Would it not redound to the honour of British science
that some painstaking ingenious person should gird himself to the task? and would not this be
a proper object to meet with encouragement from the Scientific Association of Great Britain ?
7] from Equations of Coexistence. 45
received in the numerator and denominator of the subsequent quotient ;
and in the quotient after that, it is not the square of the leading term of
the penultimate prime, but the product of this term by the leading term
of that anormal prime of the same degree which has the lowest dimen
sions, that finds its way into the numerator. The rest of the formation
remaining undisturbed, unless and until a new failure have taken place.
NOTE ON STURM S THEOREM.
When one of the equations of coexistence is the differential coefficient
with respect to the repeated term of the other, the prime derivatives given
in Theorem 2 which coincide in this case with Sturm s auxiliary functions
reduced to their lowest terms, may be exhibited under an integral aspect.
Let 8PD intimate that the squared product of the differences is to be
taken of the quantities which follow it.
Let Si indicate the sum of the quantities to which it is prefixed.
S 2 the sum of the binary products.
S s the sum of the ternary products, and so on
Let A! , h z ...h n be the roots of any equation.
Then Sturm s last auxiliary function may be replaced by
SPD(h 1 ,h,...h n ).
The last but one may be replaced by
2SPD (h lt h, . . . h^) x + 2$ (h 2 ,h 3 ... h n ^) SPD (h^hz... h n ^).
The one preceding by
2&PD (A! , h, . . . A n _ 2 ) a? + 2$ (A x , A 2 . . . A ?l _ 2 ) SPD (h, , A, . . . A_ 2 ) x
+ 2 &(/*!, A 2 ... A n _ 2 ) SPD (/>! , A 2 . . . A w _ 2 ),
and so on.
Thus then Sturm s rule for determining the absolute number of real
roots in an equation is based wholly and solely upon the following
ALGEBRAICAL PROPOSITION.
If there be n quantities, real and imaginary, the imaginary ones entering
in pairs, as many changes of sign as there are in the terms
ly A 2 ),
, A 2 ,A 3 ),
so many in number are these pairs.
46 On Rational Derivation from Equations of Coexistence. [7
Query (1). Is there no proposition applicable to any n quantities
whatever ?
Query (2). Is there no faintly analogous proposition applicable to higher
powers than the squares ?
Query (3). Seeing that in forming the coefficients in the equation of
the squares of the differences, we pass from n functions of the roots to
n ? and not n functions, of their squared differences, does not a natural
tt
passage to the former lie through n functions of the squared differences ?
In other words, may not the quantities ZSPD^, h 2 ... /i n ), &c., serve as
natural and valuable intermediaries between the coefficients of an equation
involving simple quantities and the coefficients of the equation involving the
squares of their differences ?
P.S. In the next part I trust to be able to present the readers of this
Magazine with a direct and symmetrical method of eliminating any number
of unknown quantities between any number of equations of any degree, by
a newly invented process of symbolical multiplication, and the use of com
pound symbols of notation.
I must not omit to state that the constituents of multiplication X,. and
p r explained in Cor. 2 to Theorem 3 are equal to the expression
_,. , ,
Z(x- k,) (x - fc a ) . . . (x - k n -r-, , ,
/Ki, At-2 . . . li n r i
\ Knr "*n
and its analogue respectively.
8.
ON DERIVATION OF COEXISTENCE. PART II. BEING THE
THEORY OF SIMULTANEOUS SIMPLE HOMOGENEOUS
EQUATIONS.
[Philosophical Magazine, xvi. (1840), pp. 3743.]
Art. (1). We shall have constant occasion in this paper to denote
different quantities by the same letter affected with different subscribed
numerical indices.
Such a letter is to be termed a " Base."
Every character consisting of a base and an inferior index, this index
is called an argument of the base, namely, the first, second, or nth
argument, according as 1, 2, or in general n, be the number subscribed.
Art. (2). I use the symbol PD to denote the product of the differences
of the quantities to which it is prefixed (each being to be subtracted from
each that follows); thus
PD (a, b, c) indicates (6 a) (c a) (c b).
PD (0, a, b, c) indicates abc (b a) (c a) (c b).
PD (0, a, b, c ... I) indicates abc ... I x PD (a, b, c ... I).
Art. (3). For want of a better symbol I use the Greek letter to denote
that the product of factors to which it is prefixed is to be effected after a
certain symbolical manner. This I shall distinguish as the zeta-ic product.
The symbol will never be prefixed except to factors, each of which is
made up of one or more terms, consisting solely of linear arguments of
different bases, that is, characters bearing indices below but none above.
I am thereby enabled -to give this short rule for zeta-ic multiplication:
" Imagine all the inferior indices to become superior, so that each argument
is transformed into a power of its base ; multiply according to the rules of
ordinary algebra ; after the multiplication has been done fully out depress
all the indices into their original position ; the result is the zeta-ic product*."
* It is scarcely necessary to add that an analogous interpretation may be extended to any
zeta-ic function whatever. Thus
2 = 2 + 2ffl A + & 2
f cos (>) = !- - +
48 On Derivation of Coexistence. [8
Thus for example (a r> 6,) is the same as simply a r b s , but (a r , a,)
represents not a r a g but a r+s .
So in like manner
a h b rn b k ai+ b m+k ,
= the depressed product of (a 6) (a - c) (6 c)
= the depressed value of a 2 (6 - c) + 6 2 (c - a) + c 2 (a - b),
that is, = aA - a 2 d + 6 s d - b^a l + c^ - c-A-
Art. (4). We shall have occasion in this part to combine the two symbols
, PD : thus we shall use
PD(aA) to denote (6 X -aO,
PD (aAci) to denote {(^ - Oj) (d - a,) (d - 60}.
Art. (5). For the sake of elegance of diction I shall in future sometimes
omit to insert the inferior index when it is unity; but the reader must
always bear in mind that it is to be understood though not expressed.
I shall thus be able to speak of the zeta-ic product of such and such bases
mentioned by name.
Art. (6). We are not yet come to the limit of the powers of our notation.
The zeta-ic product of the sum of arguments will consist of the sum of
products of arguments, each argument being (as I have defined) made up of
a base and an inferior index. Now we may imagine each index of every term
of the zeta-ic product after it is fully expanded to be increased or diminished
by unity, or each at the same time to be increased or diminished by 2, or each
in general to be increased or diminished by r. I shall denote this alteration
by affixing an r with the positive or negative sign to the Thus
% ( ftl - 6j) (i - d) being equal to a 2 - c^d + &id -Mi,
+i ( ai _ fcj) ( ai - d) is e q ual to a s ~ a * c * + ^c 2 - M 2 >
^ ( Oj _ fc^ ( ai _ Cl ) is equal to a l -a c + b c -b a .
In like manner %PD (a, b, c) indicating
b 2 a 1 - 6 2 C! + cj>i - 02^ + 2 Ci - aj>i,
% r PD (a, b, c) indicates
b 2 r a lr - 6 2r c 1r + c 2r 6ir - C 2r a 1r + a 2 rC 1 r - a 2 Ar-
I shall in general denote $+,PD (a, b,c...l) actually expanded as the
zeta-ic product of a, b, c, .. .1 in its rth phase.
8] On Derivation of Coexistence. 49
Art. (7). General Properties of Zeta-ic Products of Differences.
If there be made one interchange in the order of the bases to which
is prefixed, the zeta-ic product, in whatever phase it be taken, remains-
unaltered in magnitude, but changes its sign.
If in any phase of a zeta-ic product two of the bases be made to coincide,
the expansion vanishes.
Let /! be used, agreeably to the ordinary notation, to denote the sum of
the quantities to which it is prefixed, / 2 to denote the sum of the binary
products, J 3 of the ternary ones, and so on.
Thus let /j (aj)^]) or /j (a, b, c) indicate a l + b l + c 1 ,
and / 2 (a^Ci) or / 2 (a, b, c) indicate a^ + a^ + b^,
and / 3 (a ACi) or J 3 (a, b, c) indicate a&d ,
we shall be able now to state the following remarkable proposition connecting
the several phases of certain the same zeta-ic products.
Art. (8). Let a, b, c, ... I, denote any number of independent bases, say
(n 1); but let the arguments of each base be periodic, and the number of
terms in each period the same for every base, namely n, so that
d r df-^-n = d,n , Cl n = tt(, = ft_ n ,
Of O r j rn == O r n , O n = = n ,
C r C r + n = C) n , C n = GO = C_jj,
i r ir+n n > "n o n >
r being any number whatever. Then
(0, a, b, c ... 0= H/i(, M... I) SPD(0, a, b,c... I)},
(0, a,b,c...l) = [f, (a, b,c...l) PD (0, a, b, c . . . 0},
(0, a, b, c . . . 1) = [f r (a,b,c... I) %PD (0, a, b, c ... I)}.
This proposition admits of a great generalization*, but we have now all that
is requisite for enabling us to arrive at a proposition exhibiting under one
coup d oeil every combination and every effect of every combination that cau
possibly be made with any number of coexisting equations of the first degree,
containing any number of repeated, or to use the ordinary language of
analysts, (variable or) unknown quantities.
* See the Postscript to this paper for one specimen.
50 On Derivation of Coexistence. [8
For the sake of symmetry I make every equation homogeneous ; so that
to eliminate n repeated terms, no more than n equations will be required.
In like manner the problem of determining n quantities from n equations
will be here represented by the case in which we have to determine the
ratios of (n + 1) quantities from n equations.
Art. (9). Statement of the Equations of Coexistence.
Let there be any number of bases (a, b, c ... I), and as many repeated
terms (x,y, z ... t), and let the number of equations be any whatever, say n.
The system may be represented by the type equation
a r x + b r y + c r z + . . . + l r t = 0,
in which r can take up all integer values from - oo to + cc . The specific
number of equations given will be represented by making the arguments of
each base periodic, so that
a r = tt^n+r, b r = bft. n+r , C r = C^ n+r , IT = ^n+r>
/A being any integer whatever.
Art. (10). Combination of the given Equations. Leading Theorem.
Take / g, ...k as the arbitrary bases of new and absolutely independent
but periodic arguments, having the same index of periodicity (n) as a, b, c . . .1,
and being in number (n - 1), that is, one fewer than there are units in that
index.
The number of differing arbitrary constants thus manufactured is
n (n 1 ).
Let Ax + By + Cz + . . . + Lt = be the general prime derivative from the
given equations, then we may make
, c,f,g...k),
Art. (11). COR. 1. Inferences from the Leading Theorem.
Let the number of equations, or, which is the same thing, the index of
periodicity (n), be the same as the number of repeated terms (x, y, z ...t},
then one relation exists between the coefficients : this is found by making
the (7i-l) new bases coincide with (n-1) out of the old bases. We get
accordingly, as the result of elimination,
8] On Derivation of Coexistence. 51
Art. (11). COR. 2. Let the number of equations be one more than that
of the given bases, there will then be two equations of condition. These
are represented by preserving one new arbitrary base, as X. The result of
elimination being in this case
, a, b, c . . .1, X) = 0.
Example. The result of eliminating between
!* + hy = 0,
a^x + b 2 y = 0,
a s x + b s y = 0,
is %PD (0, a, b, X) = 0, that is
from which we infer, seeing that X 3 , X 2 , A,! are independent,
ftjCtg 63^ = 0,
any two of which imply the third.
In like manner, in general, if the number of equations exceed in any
manner the number of bases or repeated terms, the rule is to introduce so
many new and arbitrary bases as together with the old bases shall make up
the number of equations, and then equate the zeta-ic product of the differ
ences of zero, the old bases and the new bases, to nothing.
Art. (12). COR. 3. Let the number of equations be one fewer than the
number (n) of bases or repeated terms ; the number of introduced bases in
the general theorem is here (n 2). Make these (n 2) bases equal severally
to the bases which in the type equation are affixed to z, u ... t, then
(7=0,
D-0,
JD-0,
and we have left simply
PD(0, a,c, d...kl)a;+!;PD(0,b, c,d...kl)y = 0.
In like manner we may make to vanish all but A and C, and thus get
, a, M ... kl)x+^PD (0, c, b, d ... M)z=0,
42
52
On Derivation of Coexistence.
[8
and similarly
Hence
PD (0, a,b...k)a; +
x
y
are severally as
,b, c ... l)t = 0.
,b,c...l)
%PD(a, 0, O...I)
PD(a,b, 0...0
\PD(a,b, C...O).
This is the symbolical representation as a formula of the remarkable
discovered by Cramer, perfected by Bezout and demonstrated by
Laplace for the solution of simultaneous simple equations.
Art. (13). COR. 4. In like manner if the number of repeated terms
be two greater than the number of equations, we have for the relation
between any three of them, taken at pleasure, for instance, x, y, z,
PD(0, a, d ... l)x+^PD(Q, b, d... l)y + PD(Q, c, d ... l)z = 0.
And in like manner we may proceed, however much in excess the number
of repeated terms (unknown quantities) is over the number of equations.
Art. (14). Subcorollary to Corollary 3.
If there be any number of bases (a, b,c...l), and any other two fewer in
number (/, g ... k)
PD(a,f,g ... A;)* PD(b,c...l)
+ PD(6,/, g ... k) x PD(a, c ... I)
(c,f,g...K)x
+ PD (l,f, g ... k} x PD (a, 6, c ... ) =
a formula that from its very nature suggests and proves a wide extension
of itself.
In conclusion I feel myself bound to state that the principal substance
of Corollaries (1), (2) and (3) may be found in Garnier s Analyse Algebrique,
in the chapter headed "Developpement de la Theorie donnee par M. Laplace,
&c." But I am not aware of having been anticipated either in the fertile
notation which serves to express them nor in the general theorems to which
it has given birth.
P.S. I shall content myself for the present with barely enunciating
a theorem, one of a class destined it seems to the author to play no secondary
part in the development of some of the most curious and interesting points,
of analysis.
* The cross is used to denote ordinary algebraical multiplication.
8] On Derivation of Coexistence. 53
Let there be (n - 1) bases a,b,c...l, and let the arguments of each be
" recurrents of the nth order*," that is to say let
, / 2-7TA , . / 27TA / 27TA , I 2iri\
0, = <> COS ------ , 6 t = "b COS - , C t = V COS , ...... i t = ft) COS - .
r V / V n J ^ \ n J \ n ]
Let R r denote that any symmetrical function of the rth degree is to be
taken of the quantities in a parenthesis which come after it, and let ^
indicate any function whatever. Then the zeta-ic product
{R r (a, b, c ... I) x SfiPD (0, a,b,c... I)}
is equal to the product of the number
multiplied by the zeta-ic phase
* I am indebted for this term to Professor De Morgan, whose pupil I may boast to have been.
I have the sanction also of his authority, and that of another profound analyst, my colleague
Mr Graves, for the use of the arbitrary terms zeta-ic, zeta-ically. I take this opportunity of
retracting the symbol SPD used in my last paper, the letter S having no meaning except for
English readers. I substitute for it QDP, where Q represents the Latin word Quadratus. On
some future occasion I shall enlarge upon a new method of notation, whereby the language of
analysis may be rendered much more expressive, depending essentially upon the use of similar
figures inserted within one another, and containing numbers or letters, according as quantities
or operations are to be denoted. This system to be carried out would require special but very
simple printing types to be founded for the purpose.
In the next part of this paper an easy and symmetrical mode will be given of representing any
polynomial either in its developable or expanded form.
9.
A METHOD OF DETERMINING BY MERE INSPECTION THE
DERIVATIVES FROM TWO EQUATIONS OF ANY DEGREE.
[Philosophical Magazine, xvi. (1840), pp. 132 135.]
LET there be two equations, one of the nth, the other of the mth degree
in x\ let the coefficients of the first equation be a n , a n - l} a n _ 2 ... a Q , each
power of x having a coefficient attached to it, a n belonging to x n and a to
the constant term.
In like manner let b m , b m ^ ... b be the coefficients of the second equation.
I begin with
A Rule for absolutely eliminating x.
Form out of the (a) progression of coefficients m lines, and in like
manner out of the (6) progression of coefficients form n lines in the follow
ing manner :
1. (a) Attach (m 1) zeros all to the right of the terms in the
(a) progression ; next attach (m 2) zeros to the right and carry over to the
left ; next attach (m 3) zeros to the right and carry over 2 to the left.
Proceed in like manner until all the (m 1) zeros are carried over to the
left and none remain on the right.
The m lines thus formed are to be written under one another.
1. (6) Proceed in like manner to form n lines out of the (6) progression
by scattering (n 1) zeros between the right and left.
2. If we write these n lines under the m lines last obtained, we shall
have a solid square (m + n) terms deep and (m + n) terms broad.
3. Denote the lines of this square by arbitrary characters, which write
down in vertical order and permute in every possible way, but separate the
permutations that can be derived from one another by an even number of
interchanges (effected between contiguous terms) from the rest ; there will
thus be half of one kind and half of another.
9] On Elimination and Derivation by mere Inspection. 55
4. Now arrange the (m + n) lines accordingly, so as to obtain
squares of one kind which shall be called positive squares, and an equal
number of the opposite kind which shall be called negative.
Draw diagonals in the same direction in all the squares; multiply the
coefficients that stand in any diagonal line together : take the sum of the
diagonal products of the positive squares, and the sum of the diagonal
products of the negative squares; the difference between these two sums
is the prime derivative of the zero degree, that is, is the result of elimination
between the two given equations reduced to its ultimate state of simplicity,
there will be no irrelevant factors to reject, and no terms which mutually
destroy.
Example. To eliminate between
ax 2 + bx + c = 0,
lx 2 + mx + n = 0,
I write down
a, b, c, 0,
0, a, b, c,
1, m, n, 0,
0, I, m, n.
(1)
(2)
(3)
(4)
I permute the four characters (1), (2), (3), (4), distinguishing them into
positive and negative ; thus I write together
Positive Permutations.
1
2
3
1
2
3
2
1
3
4
4 4
2
3
1
4
4
4
1
3
2
2
1 3
3
1
2
2
3
1
4
4
4
1
3 2
4
4
4
3
1
2
3
2
1
3
2 1
and again
Negative Permutations.
1
2
3 4
4
4
2
1
3
2
1
3
2
3
1 1
2
3
4
4
4
1
3
2
4
4
4 2
3
1
1
3
2
3
2
1
3
1
2 3
1
2
3
2
1
4
4
4
56
On Elimination and Derivation
I reject from the permutations of each species all those where 1 or 3
appear in the fourth place, and also those where 2 or 4 appear in the first
place, for these will be presently seen to give rise to diagonal products
which are zero.
The permutations remaining are
Positive effectual permutations.
1
3
3
1
2
1
4
3
3
2
1
4
4
4
2
2
Negative effectual permutations.
3
1
1
3
1
4
3
2
4
3
2
1
2
2
4
4
I now accordingly form four positive squares, which are
a, b, c, 0, /, in, n, 0, I, m, n, 0, a, b, c, 0,
0, a, b, c, a, b, c, 0, 0, I, m, n, I, m, n, 0,
1, m, n, 0, 0, a, b, c, a, b, c, 0, 0, I, m, n,
0, I, m, n, 0, I, m, n, 0, a, b, c, 0, a, b, c.
Drawing diagonal lines from left to right, and taking the sum of the
diagonal products, I obtain d*n 2 + Wn + 2 c 2 + am-c. Again, the four negative
squares
I, m, n, 0,
0, a, b, c,
a, b, c, 0,
0, I, m, n,
give as the sum of the diagonal products
Ibmc + alnc + ambn + lacn,
that is, Ibmc + ambn + 2acln.
Thus the result of eliminating between
aa? + bx + c = 0,
la? + mx + n = 0,
ought to be, and is
a 2 w 2 + Z 2 c 2 Zacln + Ib 2 n + am?c Ibmc ambn = 0.
I, m, n, 0,
a, b, c, 0,
0, I, m, n,
0, a, b, c,
a, b, c, 0,
0, I, m, n,
I, m, n, 0,
0, a, b, c,
a, b, c, 0,
I, m, n, 0,
0, a, b, c,
0, I, m, n,
9] by a Process of mere Inspection. 57
Rule for finding the prime derivative of the first degree, which is
of the form Ax B.
Begin as before, only attach one zero less to each progression; we
shall thus obtain not a square, but an oblong broader than it is deep, con
taining (m + n2) rows, and (m + n-l) terms in each row: in a word,
( m 4. n _ 2) rows, and (m + 11 1) columns.
To find A reject the column at the extreme right, we thus recover
a square arrangement (m + n 2) terms broad and deep.
Proceed with this new square as with the former one ; the difference
between the sums of the positive and negative diagonal products will give A.
To find B, do just the same thing, with the exception of striking off not
the last column, but the last but one.
Rule for finding the prime derivative of any degree, say the rth, namely,
A r x r - A r _ l x r -^ + A .
Begin with adding zeros as before, but the number to be added to the
(a) progression is (m r) and to the (6) progression (n r).
There will thus be formed an oblong containing (m + n 2r) rows, and
( m + n r ) terms in each row, and therefore the same number of columns.
To find any coefficient as A 8 , strike off all the last (r+ 1) columns except
that which is (s) places distant from the extreme right, and proceed with the
resulting squares as before.
Through the well-known ingenuity and kindly preferred help of a dis
tinguished friend, I trust to be able to get a machine made for working
Sturm s theorem, and indeed all problems of derivation, after the method
here expounded ; on which subject I have a great deal more yet to say, than
can be inferred from this or my preceding papers.
10.
NOTE ON ELIMINATION.
[Philosophical Magazine, xvn. (1840), pp. 379, 380.]
THE object of this brief note is to generalise Theorem 2 in my paper on
Elimination* which appeared in the last December number of this Magazine.
The theorem so generalised presents a symmetry which before was wanting.
Here, as in so many other instances, the whole occupies in the memory a
less space than the part.
To avoid the ill-looking and slippery negative symbols, I warn my reader
that I now use two rows of quantities written one over the other, to denote
the product of the terms resulting from taking away each quantity in the
under from each in the upper row.
Let Aj, h 2 ... h m be the roots of one equation of coexistence,
&j , k, . . . k n of the other,
and let the prime derivative of the degree r be required. Take any two
integers p and q, such that p + q = r. The derivative in question may be
written
_l_1 /t_J_9 ftf
... k q
. . . h m \
... k n ]
T , ,
n 2 . . . ftp\ I A/I
l k 2 . . . h
N.B. Whatever p and q be taken, so long only as p + q = r, the above
expression changes nothing but its sign ; which, therefore, upon transcendental
grounds, it is easy to see is of one name or another, according as p is odd
or even.
In the original paper, I asserted this theorem only for the case of p = 0,
or 5=0.
[* p. 43 above. ED.]
11.
ON THE RELATION OF STURM S AUXILIARY FUNCTIONS TO
THE ROOTS OF AN ALGEBRAIC EQUATION.
[Plymouth British Association Report 1841, (Ft n.), pp. 23, 24.]
THE author availed himself of the present meeting of the British
Association to bring under the more general notice of mathematicians his
discovery, made in the year 1839, of the real nature and constitution of the
auxiliary functions (so-called) which Sturm makes use of in locating the roots
of an equation : these are obtained by proceeding with the left-hand side of
the equation and its first differential coefficient as if it were our object to
obtain their greatest common factor; the successive remainders, with their
signs alternately changed and preserved, constitute the functions in question.
Each of these may be put under the form of a fraction, the denominator of
which is a perfect square, or in fact the product of many: likewise the
numerator contains a huge heap of factors of a similar form.
These therefore, as well as the denominator, since they cannot influence
the series of signs, may be rejected ; and furthermore we may, if we please,
again make every other function, beginning from the last but one, change its
sign, if we consent to use changes wherever Sturm speaks of continuations
of sign, and vice versa.
The functions of Sturm, thus modified and purged of irrelevancy, the
author, by way of distinction, and still to attribute honour where it is really
most due, proposes to call " Sturm s Determinators " ; and he proceeds to lay
bare the internal anatomy of these remarkable forms.
He uses the Greek letter " " to indicate that the squared product of the
differences of the letters before which it is prefixed is to be taken.
Let the roots of the equation be called respectively a, b, c, e...l, the deter-
minators taken in the inverse order are as follows :
(, b, c, e ... I).
, c, e . . . I) x 2a (b, c, e ... I).
******
2 {(k, I) (x -a)(x-b)(x-c)(x-e)...(x- h}}.
60 Sturm s Auxiliary Functions. [11
It may be here remarked, that the work of assigning the total number of
real and of imaginary roots falls exclusively upon the coefficients of the
leading terms, which the author proposes to call " Sturm s Superiors " : these
superiors are only partial symmetric functions of the squared differences, but
complete symmetric functions of the roots themselves, differing in the former
respect from those other (at first sight similar-looking) functions of the
squared differences of the roots, in which, from the time of Waring downwards,
the conditions of reality have been sought for. It seems to have escaped
observation, that the series of terms constituting any one of the coefficients
in the equation of the squares of the differences (with the exception of the
first and last) each admit of being separated and classified into various
subordinate groups in such a way, that instead of being treated as a single
symmetric function of the roots, they ought -to be viewed as aggregates of
many. In fact, Sturm s superior No. 1 is identical with Waring s coefficient
No. 1 ; Sturm s superior No. 2 is a part of Waring s coefficient No. 3 ; Sturm s
superior No. 3 is a part of Waring s coefficient No. 6 ; and so forth till we
come to Sturm s final superior, which is again coextensive and identical with
the last coefficient in the equation of the squares of the differences. The
theory of symmetric functions of forms which are themselves symmetric
functions of simple letters, or even of other forms, the author states his belief
is here for the first time shadowed forth, but would be beside his present
object to enter further into. He would conclude by calling attention to the
importance to the general interests of algebraical and arithmetical science
that a searching investigation should be instituted for showing, a priori, how,
when a set of quantities is known to be made up partly of possible and partly
of pairs of impossible values, symmetrical functions of these, one less in
number than the quantities themselves, may be formed, from the signs of the
ratios of which to unity and to one another the respective amounts of possible
and impossible quantities may at once be inferred : in short, we ought not to
rest satisfied, until, from the very form of Sturm s Determinators, without
caring to know how they have been obtained, we are able to pronounce upon
the uses to which they may be applied.
12.
EXAMPLES OF THE DIALYTIC METHOD OF ELIMINATION
AS APPLIED TO TERNARY SYSTEMS OF EQUATIONS.
[Cambridge Mathematical Journal, n. (1841), pp. 232 236.]
THIS method is of universal application, and at once enables us to reduce
any case of elimination to the form of a problem, where that operation is to
be effected between quantities linearly involved in the equations which
contain them.
As applied to a binary system, fx = 0, <# = 0, the method furnishes
a rule by which we may unfailingly arrive at the determinant, free from every
species of irrelevancy, whether of a linear, factorial, or numerical kind.
The rule itself is given in the Philosophical Magazine (London and
Edinburgh, Dec. 1840). The principle of the rule will be found correctly
stated by Professor Richelot, of Kb riigsberg, in a late number of Crelles
Journal, at the commencement of a memoir in Latin bordering on the same
subject (" Nota ad Eliminationem pertinens ").
My object at present is to supply a few instances of its application to
ternary systems of equations.
Ex. 1. To eliminate x, y, z, between the three homogeneous equations
Aif- 2C xy + Ba? = Q, (1)
Bz- - ZA yz + Cif = 0, (2)
Co? - ZB zx + Az- = 0. (3)
Multiply the equations in order by z 2 , x 2 , y 2 , add together, and divide
out by 2xy ; we obtain
C z* + Cxy - A xz - B yz = 0. (4)
By similar processes we obtain
A x- + Ayz-B yx- C zx = 0, (5)
E f -f Bzx - G zy - A xy = 0. (6)
62 Examples in Dialytic Elimination. [12
Between these six, treated as simple equations, the six functions of
x, y, z, namely, a?, y", 2*, xy, xz, yz, treated as independent of each other, may
be eliminated ; the results may be seen, by mere inspection, to come out
ABC (ABC - AB * - BC 2 -CA * + ZA B C ) = 0,
or rejecting the special (N.B. not irrelevant} factor ABC, we obtain
ABC - AB - - BC 2 -CA 2 + 2A B C = 0.
I may remark, that the equations (1), (2), (3), or (4), (5), (6), express the
condition of
Ax* + By 2 + Cz* + ZA yz + ZB zx + Wxy,
having a factor \x + p,y + vz ; a general symbolical formula of which I am in
possession for determining in general the condition of any polynomial of
anv degree having a factor, furnishes me at once with either of the two
systems indifferently. The aversion I felt to reject either, led me to employ
both, and thus was the occasion of the Dialytic Principle of Solution mani
festing itself.
Ex.2. Ax 2 + ayz + bzx + cxy = 0, (1)
My 1 + lyz + mzx + iixy = 0, (2)
Rz- + pyz + qzx + rxy = 0. (-S)
Multiply equation (1) by Py + yz, equations (2) and (3) by vz and icy
respectively, and add the products together, we obtain terms of which y*z
and yz"- are the only two into which x does not enter.
Make now the coefficients of each of these zero, and we have
ay + lv+ RK = 0,
ic = 0.
Let v = a, K = a, then y = -(l + R), (3 =
Hence, multiplying as directed, and then dividing out by x, we obtain
(mv + by) z 1 + (TK + c/3) y- + (b/3 +cy + nv + q/c) yz + Aftxy + Ayasz = 0,
or by substitution,
[ra - c (M + p)} f + {ma - b (I + R)} z 2 + {an + aq - b (M + p) - c (I + R)} yz
- A (M +p)xy- A (M+p)xz = 0. (4)
Similarly, by preparing the equations so as to admit in turn of y and z
as a divisor, we obtain
[ma -l(R + b)} z- + [mr -n(A + q)} x 2 + {me + mp - n (R + b} - I (A + y)} xz
-M(R + b)yz-A(A+q)xy=0, (5)
[rm -q(A + n)} a? + {ra-p(M + c)} f+{rl + rb-p(A+n)-q(M + c)} xy
-R(A+n)xz-R(M+c)yz=0. (6)
12] Examples in Dialytic Elimination. 63
Between the six equations (1), (2), (3), (4), (5), (6), x 2 , y-, z 2 , xy, xz, yz, may
be eliminated ; the result will be a function of nine letters [three out of each
equation (1), (2), (3)} equated to zero. Perhaps the determinant may be
found to contain a special factor of three letters ; and if so, may be replaced
by a simpler function of six letters only.
>
Ex. 3. To eliminate between the three general equations
Ax- + By- + Cz 2 + Wyz + 2Ezx + 2Fxy = 0,
Lx- + My- + Nz 2 + 2Pyz + 2Qzx + 2Rxy = 0,
fa + gy + hz = o.
By virtue of one of the two canons which limit the forms in which the
letters can appear combined in the determinant of a general system of
equations, we know that the determinant in this case (freed of irrelevant
factors) ought to be made up in every term of eight letters (powers being
counted as repetitions), namely, (A, B, C, D, E, F) must enter in binary com
binations, (L, M, N, P, Q, R) the same, whereas /, g, h must enter in quaternary
combinations.
To obtain the determinant, write
A x- + By 2 + Cz 2 + Dyz + Ezx + Fxy = 0, (1)
Lx 2 + My 2 + Nz- + Pyz + Qzx + Rxy =0, (2)
fa? + gyx + hzx = 0, (3)
fxy+gy 2 + hzy = 0, (4)
fxz + gyz + hz 2 = 0. (5)
We want one equation more of three letters between x 2 , y 2 , z 2 , xy, xz, yz.
To obtain this, write
(Ax + Ez + Fy) x, + (By + Fx + Dz) y l + (Cz+Dy + Ex) z l = 0,
(Lx +Qz + Ry) x 1 + (My + Rx + Pz) y 1 + (Nz + Py + Qx) z 1 = 0,
fai + gyi + hz, = 0.
Forget that x 1 =x, y l =y, z 1 = z, and eliminate x 1} y lf z lt we obtain
, | (Ax + Ez + Fy) (My + Rx + Pz)}
1 \- (By + Fx + Dz) (Lx + Qz + Ry)}
(Cz + Dy+Ex) (Lx + Qz + Ry) }
_
J - (Cz + Dy + Ex) (My + Rx + Pz)}
64 Examples in Dialytic, Elimination. [12
This may be put under the form
ax 2 + @y 2 + jz 2 + a yz + ft zx + y xy = 0, (6)
where the coefficients are of the first order in respect to /, g, h, L, M, N,
P, Q, R, A, B, C, D, E,F; in all of the third order.
Between the equations marked from (1) to (6), the process of linear
elimination being gone through, we obtain as equated to zero a function of
5 _j_ 3^ or O f eight letters, two belonging to the first equation, two to the
second, and four to the third ; so that the determinant is clear of all factorial
irrelevancy.
Ex. 4. To eliminate x, y, z between the three equations
Ax 2 + By 2 + Cz 2 + ZA yz + ZB zx + ZC xy = 0,
Lx* -f My- + Nz z + ZL yz + -iM zx + ZN xy = 0,
Px 2 + Qy 2 + Rz 2 + ZP yz + 2Q zx + ZR xy = 0.
Call these three equations U=0, V= 0, W= 0, respectively. Write
X U=Q, (1) yU=0, (2) zU = 0, (3)
(4) yV=0, (5) zV=0, (6)
(7) yW=0, (8) zW=0. (9)
We have here nine unilateral equations : one more is wanted to enable us
to eliminate linearly the ten quantities
X s , y 3 , z\ x 2 y y x-z, xif, xz 2 , xyz, y 2 z, yz 2 .
This tenth may be found by eliminating x, y, z between the three equations
x (Ax + B z + G y) + y(By+ C x + A z) + z (Cz + A y + B x) = 0,
x (Lx + M z + N y) + y(My + N x + L z) + z (Nz + L y + M x) = 0,
x (Px + Q z + R y) + y(Qy + R x + P z) + z(Rz + Py + Q x) = 0;
for, by forgetting the relations between the bracketed and unbracketed letters,
we obtain
+ L z} (Rz
4-
+ &c. + &c. = 0,
which may be put under the form
ax 3 + frf + yz 3 + 8x 2 y + ...... =0*. (10)
* We might dispense with a 10th equation, using the nine above given, to determine the
ratios of the ten quantities involved to one another ; and then by means of any such relations as
X 3 yxxy s = x 2 y 2 xx 2 y 2 , or x 3 x y s =x 2 y x xy 2 , &c.
obtain a determinant. But it is easy to see that this would be made up of terms, each containing
literal combinations of the 18th order.
Again, we might use five out of the nine equations to obtain a new equation free from
3) y 2 Zj y^ t Z 3 . that is, containing x in every term : which being divided by x, and multiplied
12] Examples in Dialytic Elimination. 65
By eliminating linearly between the equations marked from (1) to (10),
we obtain as zero a quantity of the twelfth order in all, being of the fourth
order in respect to the coefficients of each of the three equations, which is
therefore the determinant in its simplest form.
I have purposely, in this brief paper, avoided discussing any theoretical
question. I may take some other opportunity of enlarging upon several
points which have hitherto been little considered in the theory of elimination,
such as the Canons of Form, the Doctrine of Special Factors, the Method
of Multipliers as extended to a system of any order, the Connexion between
the method of Multipliers and the Dialytic Process, the Idea of Derivations
and of Prime Derivatives extended to ultra-binary Systems. For the present
I conclude with the expression of my best wishes for the continued success of
this valuable Journal.
by y, or by z, would furnish a 10th equation no longer linearly involved in the 9 already found.
The determinant, however, found in this way, would consist of 14-ary combinations of letters.
Finally, we might, instead of a system of ten equations, employ a system of 15, obtained by
multiplying each of the given three by any 5 out of the 6 quantities x 2 , j/ 2 , z 2 , xy, xz, yz ; but the
determinant, besides being not totally symmetrical, would contain combinations of the 15th
order.
I may take this opportunity of just adverting to the fact, that the method in the text does in
fact contain a solution of the equation
where r + s + t = l, and X, /*, v are functions of the second degree in regard to x, y, z to be
determined.
13.
INTRODUCTION TO AN ESSAY ON THE AMOUNT AND DIS
TRIBUTION OF THE MULTIPLICITY OF THE ROOTS OF
AN ALGEBRAIC EQUATION.
[Philosophical Magazine, xvm. (1841), pp. 136 139.]
I USE the word multiplicity to denote a number, and distinguish between
the total and partial multiplicities of the roots of an algebraic equation.
There may be r different roots repeated respectively h lt h 2 ... h r times.
r is the index of distribution.
A!, ^2 ... h r are the partial multiplicities, and if h = h^ + h, 2 + ... + h r
h is the total multiplicity.
The total multiplicity it is clear may be denned as the difference between
the index of the equation and the number of its roots distinguishable from
one another.
In this Introduction, I propose merely to consider how existing methods
may be applied to determine the amount and distribution of multiplicity
in a given equation, and conversely, how equations of condition can be
formed which shall imply a given distribution aud amount.
Let the greatest common factor between fx (the argument of the pro-
dfx r
posed equation) and - be called /#.
df-^x
And in like manner, let the greatest common factor of/! a? and -,-- be
CLOG
called f z a; and so on, till in the end we come to f r x, which has no common
df r x
factor with -*; .
dx
Let k lt k,...k r denote the degrees in x of fx, f^x ...f r x respectively.
It is easy to see that
61 - &a, partial multiplicities, are less than 2, that is, are each units.
6 2 - & 3 , partial multiplicities, will be less than 3, and therefore either 1
or 2 in value respectively, and so on till we come to
k r _^ k r which will severally be between zero and r 1, and
Jc r of values intermediate between zero and r.
13] On the Multiplicity of an Algebraic Equation. 67
Hence there will be
&! 2k 2 + k s multiplicities each of the value 1,
k 2 - 2& 3 + k 4 2,
& r _i 2k r ... of the value
and k r of the value
T i c s -xt. dfx . , , . dfx . d 2 fx
In place ot fx with -5- we might employ -y with -/ and so on for
dx dx ax 2
the rest; the values of k 2 ,k 3 ...k r will remain unaffected by this change;
but the former method would be more expeditious in practice.
The total multiplicity is, of course, =k l .
Suppose now that we propose to ourselves the converse problem to
determine the conditions that an algebraic equation may have a given
amount of multiplicity distributed in a given manner.
If A 1; h. 2) h s ... h r be used to denote the given number of partial multi
plicities which are respectively of the values 1 , 2, 3 ... r, it is easy to see
that the quantities derived above by ^ , k 2 . . . k r are respectively equal to
A! + 2A 2 + +rh r ,
h r .
dfx
Now from j having a factor of the degree ^ common with/a; we obtain
dx
df^x
&j conditions, from - J , 1 having a factor of the degree k 2 common with/, x we
obtain k 2 more, and so on. So that altogether we obtain in this way
&i + k 2 + + k r conditions.
But it may easily be seen that the total multiplicity being k 1} the number
of conditions need never to exceed k^ in number, no matter what its distri
bution may be. Hence, besides the enormous labour of the process, and the
extreme complexity of the results, we obtain by this method more equations
by far than are necessary, and it requires some caution to know which to
reject.
In my forthcoming paper (to appear in Philosophical Magazine of next
month) I shall show, by a most simple means, how without the use of derived or
other subsidiary functions, to obtain the simplest equations of condition which
correspond to a given distribution of a given amount of multiplicity.
The total multiplicity, say ra, being given in as many ways as that
number can be broken into parts, so many different systems of m equations
can be formed differing each from the other in the dimensions of the terms.
52
68 On the Multiplicity of an Algebraic Equation. [13
These systems may be arranged in order so that each in the series shall
imply all those that follow it, and be implied in all those that go before,
without the converse being satisfied.
The subject of the unreciprocal implication of systems of equations is
a very curious one, upon which the limits assigned to me prevent me from
enlarging at present. It is closely connected with a part of the theory of
elimination, which, as far as I am aware, has either been overlooked, or has
not met with the attention which it deserves ; I mean the theory of Special
Factors.
An example may make what I mean by these clear.
Let C be a function (if my reader please) void of x, which equivalent to
zero implies two given equations in x having a common root.
Let C be rid of all irrelevant factors, that is, let C be the simplest form
of the determinant, when the coefficients of the two equations are perfectly
independent qualities. Now suppose, as is quite possible in a variety of ways,
that such relations are instituted between the coefficients alluded to as make
C split up into factors, so that C = LxMxN=0.
Only one of the factors L, M, N will satisfy the condition of the co
existence of the two given equations : the others are clearly, however, not to
be confounded with factors of solution, or irrelevant factors, as they are
termed, but are of quite a different nature, and enjoy remarkable properties,
which point to an enlarged theory of elimination, and constitute what I call
special or singular factors.
I shall feel much obliged to any of the readers of your widely circulated
Journal, interested in the subject of this paper, who would do me the honour
of communicating with me upon it, and especially if they would (between
now and the next coming out of the Magazine) inform me whether any
thing, and if so how much, different from what is here stated has been done
in the matter of determining the relations between the coefficients of an
equation corresponding to a given amount and distribution of multiplicity in
its roots.
I ought to add, that my method enables me not merely to determine
the conditions of multiplicity, but also to decompose the equations con
taining multiple roots into others free of multiplicity, that is, to find,
a priori, the values of the several quantities
f
Moreover, other decompositions, not necessary to be enlarged upon in this
place, may be obtained with equal facility.
14.
A NEW AND MORE GENERAL THEORY OF MULTIPLE ROOTS.
[Philosophical Magazine, xvm. (1841), pp. 249 254.]
I SHALL begin with developing the theory of polynomials containing
perfect square factors, one or more.
First, let us proceed to determine the relations which must exist between
the coefficients of such polynomials, and afterwards show how they may be
broken up into others of an inferior degree.
A parallelogram filled with letters standing in one row is intended to
express the product of the squared difference of the quantities contained.
Thus (06) indicates (a - 6) 2 , (abc) is used to indicate (a - &) 2 (a - c) 2 (b - c) 2 ,
and so forth.
Suppose now that two of the roots e 1} e 2 ... e n belonging to the equation
fx = are equal to one another, it is clear that (e ly e 2 ... e M ) = ; and more
over is a symmetric function, and can be calculated in terms of the coefficients
of/a;.
Next let us suppose that we have two couples of equals (as for instance
a and 6, two of the roots equal, as also c and d two others), it is clear, that on
leaving any one of the roots out, the (n 1) that are left will still contain
one equality, and therefore we have
, e 3 .. ._en) = 0, (e 1} e s ...e n ) = . . . (e ly e z ...e n ^) = 0.
None of the parallelogrammatic functions above taken singly, are symmetric
functions of the coefficients, but their sum is; so also is the sum of the
product of each into the quantity left out.
Now in general, suppose that the polynomial fx contains r perfect square
factors, so that we have r couples of equal roots belonging to the equation
fi (n _ 1) (n _ r + 2)
fx = 0, it is clear that (e r , e r+l ... e n \ and all the other - - - " ,
1 . . . . \r 1 )
functions of which it is the type are severally zero. Moreover, the sum of
70 On a new and more general [14
these or the sum of the products of each by any symmetrical function of the
(r 1) letters left out will be a symmetrical function of the coefficients of
the powers of x in fx. To express now the affirmative* conditions corre
sponding to the case of there being r pairs of equal roots, we might employ
the r equations,
. e 2 ... e n ) = 0,
2 (e 3 ... e n ) = 0,
2 (e r ,e r+1 ...e n ) = 0.
But these, except the last, are not the simplest that can be employed ; that
is to say, we can write down r others, the terms of which shall be of lower
dimensions in respect to the roots.
Let fp denote that any rational symmetrical function of the yath degree
is to be taken of the quantities which it precedes.
Then the r equations in question are all contained in the general equation
N* f C / \ / \1 A
^/ ] J u. \^l j ^2 * * ^?* 1 ) \ ^f y ^y-f-1 C/TI / r ~~ v 5
/A being taken from up to (r 1) we obtain r equations, which in respect
to the roots are respectively of all degrees between
n (n 1) ... (n r + 2) . n (n - 1) ... (n r + 2) , nN
7T2 fr~^17~ " ~T2 (> -T)~ +( r ~ T )
reckoned inclusively.
Now at this stage it is important to remark that the above r equations,
although necessary, are not sufficient; and indeed, no mere affirmations of
equality can be sufficient to ensure there being r pairs of equal roots.
To make this manifest, suppose r = 2. Then in order that an equation
may have two pairs of equal roots, we must have by the above formula
2 (eg, e s ... e n ) = 0, 2 {e L (e z , e s ... e n )} = 0.
But if instead of there being two perfect square factors there be one
perfect cube factor in fx, it may be shown by the same reasoning as above, that
t he very same two equations apply. In fact, it may be shown in general
that no such equations as those given above can be affirmed in consequence
of there being an amount r of multiplicity consisting of unit parts which
may not be affirmed with equal truth as necessary consequences of the same
* The importance of the restriction hinted at by the use of the word affirmative will appear
hereafter.
14] Theory of Multiple Roots. 71
amount distributed in any other manner whatever. How to obtain affirma
tive equations sufficient as well as necessary (under certain limitations) will
appear at the close of this present paper.
It is worthy of being remarked, that if we make / M denote the sum of
the products of the quantities to which it is prefixed, taken /A and p together,
the equations of affirmation become identical with those obtained by elimin-
dfx
ating between fx and ; *.
doc
It can scarcely be doubted that the illustrious Lagrange, had he chosen
to perfect the incomplete theory of equal roots given in the Resolution
Numerique, by applying to it his own favourite engine of symmetric func
tions, could scarcely have failed of stumbling by a back passage upon Sturm s
memorable theorem.
Let us now proceed to show how a polynomial known to contain one or
more perfect square factors may be decomposed.
Let us begin with supposing that it contains but one such factor ; so
that fx = <j)X (x a) 2 .
I shall show how to obtain the equations
C(x-a) = 0, D(f>x(x-a) = > E(x-a) 2 = Q, F((f)x) = 0,
each in its lowest terms.
1. To form the equation Lx + M = 0, where x = a, it is easy to see that
if we write down in general the expression (x e^ (e z , e 3 ... e n ) this will
become zero whenever the root e a left out is not one of the equal roots (a) :
so that in fact (calling the two equal roots e ly e 2 respectively)
2 {(x - e^ x (02, e 3 . . . e n )} = (x - gj) x (e 2 , e s ... e n ) +(x- e 2 ) x (e lt e s ...e n ),
or simply = 2 (x a) (e^, e 3 ... e n }.
Hence by making
a;2 (e z , e 3 ...e n ) - 2 {e, x (e z , e 3 ... e n )} = 0,
we have an equation for finding the equal roots e lt e z .
Again, it is easily seen upon the same hypothesis, that
2 {(x - eo) (x - e 3 ) (x - e t ) ... (x- e n ) x (e 2 , e s ... e n )}
= 2(x- e 2 ) (x - e s ) . . . (x - e n ) x (e 2 , e 3 ... e n ).
* See my note on Sturm s Theorem, Phil. Mag., December, 1839 [p. 45 above. ED.].
72 On a new and more general [14
Hence, to form the equation having the same roots as (so a) </>#, we have
only to make
x n ~ l 2 (e 2 ,e 3 ... e n ) - x n ~ 2 2 {(e 2 + e 3 + . . . e n ) x (e 2 , e 3 ... e n )} ......
+ 2 {(e 2 e 3 ... e n ) x (e 2 , e 3 ... e n )} = 0.
Suppose now in general that we have r perfect square factors, so that
fx = (f)x(x dj) 2 (x a 2 ) 2 ...(# a r ) 2 -
To form the equation G (x aj (x a 2 ) . . . (x a r ) = 0, we have only to
make
2 {(SB - ej (x - e a ) ... (x - e r ) x (e r+l , e r+2 ...e n )} = 0.
And to obtain
D(f>x x (x di} (x a 2 ) . . . (x a r ) = 0,
we must make
2 {O - e r +i) - e r+2 ) . . . O - e n ) x (e r+l , e r+2 ... e n )} = 0.
The theory of perfect square factors is not yet complete until it has been
shown how to obtain constructively <f>x, and, as analogy suggests, the com
plementary part D (x a^ 2 (x a 2 ) 2 ...(x a r ) 2 , each in its lowest terms.
To effect the latter it might be said that it is only necessary to take the
square of C (x a^ (x a 2 ) ...(x a r }. It is true the polynomial so formed
would contain every pair of equal factors, but not in the lowest terms as
regards the coefficients (as we shall presently show).
To solve this last part of the problem, let it be agreed that two rows of
letters inclosed in a parenthesis shall indicate the product of the squares
of the differences got by subtracting each in the row from each in the other,
so that
= (a - &)*, = (a - by (a - c)\ = (a - c) (a - <2) 2 (6 - c) 2 (6 - d}\
Let us begin with supposing that fx has one pair only of equal roots ;
to form the simplest quadratic equation containing this pair, write down
(x - e,} (x - e 2 ) x (e 3 , e t ...e n ) x r 1 ^ ) .
\e 3 , 64 ... 6 n /
Now if e l and e z are the two equal roots in question neither of the
multipliers of (x BI) (a e 2 ) vanishes.
If e l and e 2 are neither of them equal roots (e 3 , e 4 . . . e n } = 0.
If one of the two only belong to the pair of equal roots
= 0.
&3>
14] Theory of Multiple Roots. 73
Hence it is clear that
- e 2 x *...., x _ =
is the equation desired.
In like manner if there be r pairs of equal roots the equation of the
(2r)th degree which contains them all may be written
... e n )
x [ "
\
The coefficient of af in this equation is clearly of
(n - 2r) (n - 2r - 1) + 4r (TO - 2r),
that is, of (n + 2r 1) (n 2r) dimensions. The coefficient of x r in the equa
tion which contains the r equal roots unyoked together is of (71 r) (n r 1)
dimensions, and consequently the coefficient of o? r in the square of this
equation would be of 2(n r) (n r 1) dimensions, that is, would be
n 2 + 6r 2 (4r + 1) n dimensions higher than needful.
Finally, to obtain an equation clear of simple as well as double appear
ances of the equal roots, we have only to write the complementary form
2 JO - e.zr +1 ) (oc - e 2r+2 ) . . . (x - e n ) x (e^. +1 + e n ) x ( 2 " ") [ = 0.
(. V #2r+l l / )
Let us, now that we are more familiarized with the notation essential to
this method, revert to the question with which we set out, and endeavour to
obtain r such equations as shall imply unambiguously the existence of r pairs
of equal roots.
The existence of r such pairs enables us to assert the following disjunc
tive proposition, which cannot be asserted when the same amount of multi
plicity is distributed in any other way.
To wit, on selecting any r roots out of the entire number, either these
r will all be found again in those that are left, or those that are left will
contain inter se, one repetition at least ; so that except on the latter supposi
tion any (r 1) may be absolutely sunk out of those that are left, and there
will still be one root common to the (n 2r+l) remaining, and to the r
originally selected to be left out.
Wherefore calling the roots e 1} e z ... e n , and giving p any value whatever,
we have
\ / \ v / c i
l} e 2 ...e r )x (e r+l , e r+2 ... e n ) x 2
> \ t/01"
^2T>
74 Theory of Multiple Roots. [14
Hence the simplest distinctive equations indicative of the existence of r
pairs of equal roots are to be found by putting p equal in succession to all
values from up to (r 1).
For instance, if we require that an equation of the seventh degree shall
have three pairs of equal roots, we need only to call the seven roots respec
tively a, b, c, d, e, f, g, and then our type equation becomes
-0.
a b cj \a b c) \a b
From this it appears that the r distinctive equations for r pairs of equal
roots are of different dimensions from the r general or overlying ones corre
sponding to the multiples r, anyhow distributed; the lowest of the latter
being of (n r + 1) (n r), the lowest of the former of
(n -r)(n-r-l) + 2r (n-2r+ 1),
that is, of n(n 1) 3r (n 1) dimensions. In general we shall find that
the more unequally distributed the multiplicity may be the lower are the
dimensions of the distinctive equations, and are accordingly lowest when the
multiplicity is absolutely undistributed *.
* It must not, however, be overlooked, that the equations above given, although decisive as
to the existence of r pairs of equal roots when the multiplicity is known to be not greater than r,
do not enable us to affirm with certainty their existence when this limitation is absent : for
should the multiplicity exceed r, then inevitably (no matter how it may be distributed)
(e r+] , e r+2 ... e n j is always zero, and consequently nullifies each term of every one of the equa
tions in question. In fact (repugnant as it may appear to be to the ordinary assumptions of
analytical reasoning), it is not possible to express with absolute unambiguity the conditions of
there being a multiplicity (r) distributed in any assigned manner by means of r affirmative
equations alone.
15.
ON A LINEAR METHOD OF ELIMINATING BETWEEN DOUBLE,
TREBLE, AND OTHER SYSTEMS OF ALGEBRAIC EQUATIONS.
[Philosophical Magazine, xvni. (1841), pp. 425 435.]
PART I. BINARY SYSTEMS.
LET U and V be two integer complete homogeneous functions of x and
y, one of the with, the other of the nth degree ; and let it be required to
express the condition of the coexistence of the two equations 7=0, V=0
by means of the equation (7=0, where C is free from all appearances of
x or y.
This equation, according to the system of notation developed in a pre
ceding paper, and which has been since adopted and sanctioned by the high
authority of M. Cauchy, I call the final derivative : the quantity C is desig
nated the final derivee : and it is our present object to show how this may
be obtained in a prime form, that is to say, divested of irrelevant factors :
in this state it must consist of terms, each containing m + n letters, of which
n belong to the coefficients of U, and m to those of V.
Of course in applying this rule it is to be understood that every combina
tion of powers in U or V has a single letter prefixed for its coefficient,
and that in the final derivee powers are represented by repetitions of the
same character.
Every term in U or V being of the form Cx*y*, x*-\p is called an argu
ment, C its prefix.
Assume two integer positive numbers r and r , and also two others
s and s, such that r + r = n l,s + s = mI, and form from 7=0, V=Q
two new equations,
x r y r U = 0, a?y* 7=0.
Such equations are termed the augmentatives of the two given ones respec
tively ; also x r y r V and its fellow are termed the augmentees of U and V.
76 On a linear Method of Eliminating betiveen [15
r and r are termed the indices of augmentation belonging to U, s and s
the same belonging to V.
Finally, it will be useful hereafter to call the given polynomials U and V
themselves the proposees, and the given equations which assert their nullity,
the propositive equations, or, briefly, the propositives.
Now as many augmentees of either proposee can be formed as there are
ways of stowing away between two lockers (vacancies admissible) a number
of things equal to the index of the other* ; hence we shall have n aug
mentees of U, and m of V: thus there will be m +n augmentatives each of
the degree m + n 1, and the number of arguments is clearly m + n also,
so that they can be eliminated linearly, and the final derivee thus found,
containing m + n letters (properly aggregated) in each term, will be in
its prime form, that is, incapable of further reduction, and void of irrelevant
factors.
It is worthy of remark, that the final derivee obtained by arranging in
square battalion the prefixes of the augmentees, permuting the rows or
columns, and reading off diagonal products, affected each with the proper-
sign (according to the well known rule of Duality), will not only be free
from factorial irrelevancy, but also of linear redundancy, which latter term
I use to signify the reappearance of the same combination of prefixes, some
times with positive and sometimes with negative signs : furthermore, it
follows obviously from the nature of the process that no numerical quantity
in the final derivee will be greater than the higher of the indices of the two
given polynomials.
PART II. TERNARY SYSTEMS.
r *
CASE A. Indices all equal.
Method 1.
Let there be now three proposees, U, V, W, integer complete homo
geneous functions of x, y, z, each of the degree n : let
r+r+r"=n - 1, s + s + s" =n - 1, t + t + t"=n-l,
x r y r z r "U, x s y s z 8 "V, at-fz^W,
will, as above, be called the augmentees of U, V, W, and every other part of
the notation previously described is to be preserved.
* "Tot Augments utriusvis ex oequationibus propositis formari possunt quot modi sint inter
duo receptacula (utrivis vel ambobus omnino vacare licet) rerum, quarum numerus indicem
alterius tequat, distributionem faciendi."
15] double, treble, and other Systems of Algebraic Equations. 77
Suppose now
J7=0, F=0, F = 0,
we shall have as many augmentative equations formed from each proposee
as there are ways of stowing away n things between three lockers (vacancies
admissible)*, that is, n _ of each kind ; in all, therefore, 3 ^ , and
every one of these will be of the degree 2n 1, so that the number of
arguments to be eliminated is equal to the number of ways of stowing
away 2n 1 things between three lockers (empty ones counting), that is
2
As yet, then, we have not enough equations for eliminating these linearly.
Make, however,
a + /3 + 7 = n+ 1,
and write
W = x*H+ yPH + zyH"= 0,
it will always be possible to make the multipliers of of-, if, z* integer
functions : for if we look to any argument in U, V, or W, it is of the form
n M y b ^ t and one of the letters a, b, c must be not less than its correspondent
a, /3, 7, for otherwise a + b + c would be not greater than a + & + 7 - 3,
that is, n would be not greater than (n + 1) 3, or n 2, which is absurd:
if now any one, as a, be equal to or greater than a, it may be made to
supply an integer part to the multiplier of x a .
Here it may be asked what is to be done with such terms as Ka^y b z e ,
when two letters a, b are each not less than their correspondents a, /3 : the
answer is, such terms may be made to enter under the multiplier of #*,
or of #0, or to supply a part to both in any proportion at pleasure f.
From the equations above we get, by linear elimination,
FG H" + GH F" + HF G" - GF H" - HG F" - FH G" = 0.
This may be denoted thus : II (a, /3, 7) = 0, which equation I call a secondary
derivative, and the left side of it a secondary derivee ; a, @, 7 may likewise
be termed the indices of derivation (as r, s, t, &c. are of augmentation).
Now since a + fi + y = n + l, it is clear that the index of II (a, /3, 7)
is always n + n + n (w+1); that is, 2n 1.
* See for Latin translation the preceding note.
t The prefixes of any such terms (say K) may be conceived as made up of two parts, an
arbitrary constant, as e and (K- e) ; e will disappear spontaneously from the final derivee.
78 On a linear Method of Eliminating between [15
1st. Let any two of the indices of derivation be taken zero, then it is
easily seen that all the terms in H (a, ft, 7) vanish, and consequently the
secondary derivative equations obtained upon this hypothesis become mere
identities, and are of no use.
2nd. Let any one of them become zero.
It is manifest, from the doctrine of simple equations, that H (a, ft, 7) may
be made equal to
(
or
upon the understanding that
\ = G H" - G"H , p = H F" - H"F , v = F G" - F"G ,
V = G"H - GH", // = H"F - HF", v =F"G- FG",
\" = GH - G H, p" = HF - H F, v" = FG - F G.
The three rows of coefficients will be respectively of the degrees
(n ft) + (n 7), (n 7) + (n a), (n a.) + (n ft).
Thus if any one of the indices a, ft, 7 be zero, H (a, ft, 7) becomes
identical with A/ U + i*? V + v ? W, where the multipliers of U, V, W are of
<2 n (a. + ft + 7) dimensions, that is of (n 1) dimensions, and may accord
ingly be put under the form
>^ A wf/tfl *yf II 1 ? r\nf*8ll& *?& v I . 7 f ;O"^7/^ &* \W
j^Y _\_ ik (J & \J ^^ duw J J vU U 4f V \^ ^ Vy ft/ U .v r r *
that is to say, becomes a linear function of the augmentatives, and therefore
if combined with them in the process of linear elimination would give rise
to the identity = 0.
Hence we must reject all such secondary derivatives as have zero for one
of the indices of derivation. But all others, it may be shown, will be linearly
independent of one another, and of the augmentees previously found. Hence,
besides 3 ^ ^ equations of augment of the degree 2?i 1, we shall have
of the same degree so many equations of derivation as there are ways of
stowing away between three lockers (n+l) things, under the condition that
no locker shall ever be left empty, that is ^
n-l _n(w + l)
Thus, then, in all we have n ^ + 3-^ -= *-- equations,
which is exactly equal to the number of arguments to be eliminated. Hence
* Vide page 76 for the Latin version.
15] double, treble, and other Systems of Algebraic Equations. 79
the final derivee can be obtained by the usual explicit rule of permutation,
and moreover will be its lowest form, for it will contain in each term
ifl (yi 4-1)
prefixes belonging to the augmentatives of U, and a like number
2
711
pertaining to those of V and of W, as well as n ~ belonging to the
2
secondary derivatives, each prefix in any one of which is triliteral, containing
a prefix drawn out of those belonging to each of the proposees.
77 t |_ 77 1
Thus every member containing n _ h n ~^~ , that is n z of the original
prefixes belonging to U, V, W, singly and respectively, the final derivee
evolved by this process will be in its lowest terms ; as was to be proved.
CASE A. Indices all equal.
Method 2.
It is remarkable that we may vary the method just given by making
r + r + r" = n - 2, s + s + s" = n - 2, t + t + t" = n-2.
The augmentatives will thus be of the degree 2n 2.
Furthermore, we must make a + /3 + 7 = n + 2. It will still be possible
to satisfy by integer multipliers the equations
[these it will be useful in future to term the equations, x a , y$, z** being the
arguments, and F, G, H, &c. the factors of decomposition] for otherwise
calling the indices of x, y, z in any original argument a, b, c, their sum or n
would be not greater than (n + 2) 3, that is (n 1), which is absurd.
For the same reasons as in the last case no index of augmentation must
be made zero : the degree of each will be (n - a) + (n /3) + (n 7), that is
(2w-2), and their number -^ ; the number of augmentatives will be
2 - linearly uninvolved, each of the degree 2n 2, and therefore
^ . . (2n-l)2n
containing - - arguments.
Now * * |
3 (n- l)n (2n - 1) 2n
80 On a linear Method of Eliminating between [15
Hence the final derivee may be found, and it will be in its lowest terms,
for every member will contain letters due to the augmentative,
z
and ^ - due to the partial derivative equations; in all then there will
m
be 3?i 2 letters in each term.
This second method being applied to three quadratic equations of the
most general form, leads to the problem of eliminating between six simple
equations which lies within the limits of practical feasibility, and it is my
intention to register the final derivee upon the pages of some one of our
scientific Transactions as a standing monument for the guidance of hereafter
coming explorers*.
SCHOLIUM TO CASE A.
If we attempt to carry forward these processes to quaternary systems, it
becomes necessary to make
a + /3 + v + S = (r-2)n+l
or else a + /3 + 7 + S = (? 2)n + %,
where r is the number of proposees.
Now if the factors in the equations of decomposition are all integer,
one of the indices of derivation must be not greater than the corresponding
index in any of the original arguments, which may easily be shown to be
always impossible for a system of equations, complete in all their terms,
whenever their number r is greater than three, ifa + (3 + <y+S = ( > 2) w + 2;
but if a + /3+7+S = (r 2) n + 1 only possible for the case of n = 2.
PARTICULAR METHOD APPLICABLE TO FOUR QUADRATICS.
Let 17=0, F=0, W=0, Z = 0, be four quadratic equations existing
between x, y, z, t.
Make xU=0, acV=Q, xW = Q, xZ=0,
yU=0, yV=0, yW=Q, yZ=0,
zU = 0, zV=0, zW = 0, zZ=0,
tU=0, tV=0, tW=0, tZ = 0.
* Elimination between two quadratics leads to a final derivee made up of seven terms only ;
the final derivee of three quadratics is made up of at least several thousand ; nay, I believe I may
safely say, several myriads of terms !
15] double, treble, and other Systems of Algebraic Equations. 81
Also write U = a?F + yF + zF" + tF " = 0,
F= x*G + yG + zG" + tG" = 0,
W = a?H + yH + zH" + tH " = 0,
Z = o?K + yK + zK" + tK " = 0.
By eliminating linearly we get
2 [FZG (H"K " - H "K"}\ = 0,
which will be of the third degree, since the factors represented by the
unmarked letters F, G, H, K are of zero, and all the rest of unit dimensions.
Similarly we may obtain other equations, so that besides the sixteen
augmentatives already written down, we have four secondary derivatives,
namely,
n(2iii)=o, n(i2in = o, 11(11 2i) = o, n(iii2) = o.
Thus we have twenty equations and as many arguments to eliminate, since
a perfect cubic function of four letters contains twenty terms.
The final derivee will contain 16 + 4 . 4 letters, that is 32, 8 or 2 3 belonging
to each system of original prefixes in each member, and will therefore be in
its lowest terms : for one of the canons of form teaches us, a priori, that
every member of the derivee deduced from any number of assumed equations
must contain in each member as many prefixes belonging to one equation of
the system as there are units in the product of the indices of all the rest
taken toether.
COROLLARY TO CASE A.
Either of the two methods given as applicable to this case enables us to
determine integer values of X, Y, Z, which shall satisfy the equation
XU+YV+ZW = FxPyOz*,
where F is the final derivee and p + q + r = 3n 2. For by the doctrine of
simple equations we know how to express F in terms of the linear functions,
out of which it is obtained by permutation, that is we are able to assign
values of A, B, C, and their antitypes, as also of L and its antitype, which
shall satisfy the equation
(1)
where A, B, C, as well as L and all the quantities formed after them, are
made up of integer combinations of the original prefixes.
Now the functions II (a, /3, 7) may be expressed in three ways in terms of
U, V, W, as has been already shown.
s. 6
82 On a linear Method of Eliminating between [15
We may therefore suppose these functions to be divided into three
groups, and make
su+tr+irw
sin (a*)- s
yf- or
^
~lfl
And it is evident that the equations (1) and (2) lead immediately to the
equation
XU+
if we call a, b, c the greatest values attributed respectively to a, ft, 7.
Now if we suppose the first method to be followed,
And it will always be possible to make a, b, c of what values we please
subject to the condition of a + b + c = n - 1 ; for one at least of the indices
of derivation in II (a, ft, 7) must be not greater than its correspondent
among a, b, c ; otherwise a + ft + 7 would be not less than (a + 6 + c) + 3; but
a+ b + c = n 1,
which is absurd.
Hence we can satisfy XU + YV+ ZW = Fx y* z r , p, q, r being subject to
the condition of p + q + r = 3?i - 2, but otherwise arbitrary.
Moreover, we can not do so if p + q+r be less than 3/^ 2, for that
would require a + b + c to be less than n - 1. Now if two of the indices
of derivation, as a and /3, be made equal to a + 1, 6 + 1 respectively, the
third 7 = (n+l)-(a+6 + 2) = (w- l)-(a + 6), and is therefore greater
than c : so that a + @ + 7 for this case becomes greater than a + b + c, and
the method falls to the ground.
In fact, I have discovered a theorem which lets me know this, a priori,
a law which serves as a staff to guide my feet from falling into error in
devising linear methods of solution, and the importance of which all candid
judges who have studied the general theory of elimination cannot fail to
recognize. To wit, if X,, X 3 , X 3 ... X n be n integer complete polynomial
functions of n letters a^, x 2 ... a? n , and severally of the degree b lt b t ,b 3 ... b n ;
then it is always possible to satisfy the identity
15] double, treble, and other Systems of Algebraic Equations. 83
if a : + 2 + a s + + w be equal to or greater than b 1 + b 2 + b 3 + ... +b n n+l,
but otherwise not*.
This again is founded immediately upon a simple proposition, of which
I have obtained a very interesting and instructive demonstration, shortly to
appear, and which may be enumerated thus : " The number of augmentees
of the same degree that can be formed, linearly independent of one another,
out of any number of polynomial functions of as many variables, may be
either equal to or less than the number of distinct arguments contained in such
augmentees, but never greater. The latter will be the case when the index
of the augmentees diminished by unity is less than the sum of the indices of
the original unaugmented polynomials each so diminished ; the former, when
the aforesaid index is equal to or greater than the aforesaid sum."
To return to the particular case of finding X, Y, Z to satisfy
XU+
This has been already done according to the first method ; if we employ
the second method of elimination we shall have
But, now since a + /3 + 7 = n + 2, we shall easily see by the same method
as above, that the least value of a + b + c {where a, b, c denote respectively
the greatest values of a, /3, 7, appearing in the denominator of the fractional
forms used to express II (a, ft, 7)}, will be one greater than before, or n ; so
that f+g + h + a + b + c will still be equal to 3n 2, as we might, d priori,
by virtue of our rule, have been assured.
TERNARY SYSTEMS.
CASE B. Two of the indices equal ; the third less by a unit.
Let U=0, V = 0, W=0, be the three given equations severally of the
degree n, n, (n 1).
* Hence it is apparent, that in applying the method of multipliers, a curious and important
distinction exists between the cases of there being two equations, and there being a greater
number to eliminate from : for in the first case the element of arbitrariness needs never to appear ;
in the latter it cannot possibly be excluded from appearing in the multipliers.
This will explain how it comes to pass that the method of the text may be employed to give
various solutions of the XU + YV+ ZW=Fx p yiz r ; thus not only can p, q and r be variously
made up of (/ + ), (g + b), (h + c), but also II (a, /3, -y) when two of the indices (a, /3 suppose) are
each not greater than the assigned greatest values a, b may be made to figure indifferently either
under the form
.
or that of
3
62
84 On a linear Method of Eliminating between [15
Make r + r + r" = n 2, s + s + s" = n 2, t + t + t" = n l,
by multiplying U into x r y r z r ", V into x*y* z s ", W into off V", we obtain
augmentees each of the same, namely, the (2n 2)th degree.
The number of these is
+ - -g- + - 2 -
Again, make a + /3 + <y= n + I.
It will still be possible, as before, to form equations of decomposition in
which x*, y, zt are the arguments, and affected with integer factors. For if
we look to W even, all its arguments are of the form & M y b z c , where
a + b + c=(n- 1), and each of these cannot be less than its correspondent,
for that would be to say that (n - 1) is not greater (n + 1) - 3, a fortiori,
U and V can be decomposed in the manner described. Thus, then, we
shall obtain as many secondary derivees as in the last case (Method 1),
that is, n ^ ~ - (since a + /3 + 7 is still equal to (n + 1)}, as before. More
over, each of these will be of (n - a) + (n - ) + (w - 1 - 7), that is of 2?r - 2
dimensions.
Altogether, therefore, we have
(n-l)n (n l)n n(n + l)} (n l)n
^ - rr- 2 h *-= 1 H
2 j 2
linear independent equations of the degree 2n 2, and the number of
arguments to eliminate is ^ ]> ~ - . Now these two numbers are equal.
Thus we obtain a final derivee containing of U s coefficients ^, + 2
n ( n j_ i) (n 1)n
an equal number of F s, but of W s - ^ - + -^ J ~ , now n(n-l\
n(n-l) and n 2 exactly express the number that ought to appear of each
of these respectively: hence the final derivee is clear of irrelevant factors.
TERNARY SYSTEMS.
CASE C. Two of the indices equal ; the third one greater by a unit.
Here, calling n the highest index, the augmentees must each be made
of the degree (2n - 3), their number will evidently be
(n-2)(re-l) (n-l)n (n-l)n
^ T ,- c
15] double, treble, and other Systems of Algebraic Equations. 85
making the sum of the indices of derivation now, as before, equal to (n + 1);
it will be still possible to form integer equations of decomposition, which will
give rise to augmentatives of the degree (n a) + (n 1) /3 + (n 1) 7,
that is, of (2/i 3) dimensions. The total number of equations, what with
augmentatives and secondary derivatives, will be
2)(n 1) (n \}n (n \)n\ n(n 1) _ 4n 2 4?i + 2 _ (2?i 2)(2n 1)
~~2~~ ~2~~ I"}" 1 2 ~2~ ~1T~
that is, is equal to the exact number of distinct arguments contained between
them.
Also the final derivative will contain in each member
(n-2)(n-l) n(n-l)
2 2
that is, (n l)(w 1), letters belonging to the first equation, and
(n I)n n(n 1)
that is, ?r (w 1) belonging to those of the second and of the third, and will
therefore be in its lowest terms.
COROLLARY TO CASES B AND C.
It is not necessary, after all that has been already said, to do more than
just point out that the processes applicable to these cases enable us to deter
mine X, Y, Z, which satisfy the equation
XU+YV+ZW = Fxfyvz h ,
where /+ g + h = 3w 3 for Case B,
and f + g + h = 3n 4 for Case C.
16.
MEMOIR ON THE DIALYTIC METHOD OF ELIMINATION.
PART I.
[Philosophical Magazine, xxi. (1842), pp. 534 5-39*.]
THE author confines himself in this part to the treatment of two equations,
the final and other derivees of which form the subject of investigation.
The author was led to reconsider his former labours in this department
of the general theory by finding certain results announced by M. Cauchy in
L Institut, March Number of the present year, which flow as obvious and
immediate consequences from Mr Sylvester s own previously published
principles and method.
Let there be two equations in x,
U = ax n + bx n ~ l + cx n ~ 2 + ex n ~ 3 + &c. = 0,
V = ax m + j3x m ~ l + 7 wl - a + &c. = 0,
and let n m + t, where t, is zero or any positive value (as may be).
Let any such quantities as x r U, X s V, be termed augmentatives of U or V.
To obtain the derivee of a degree s units lower than V, we must join s
augmentatives of U with s + 1, of V. Then out of 2s + i equations
we may eliminate linearly 2s + i 1 quantities.
Now these equations contain no power of x higher than m + t + s 1 ;
accordingly, all powers of x, superior to ms, may be eliminated, and the
derivee of the degree (m s) obtained in its prime form.
Thus to obtain the final derivee (which is the derivee of the degree zero),
we take m augmentatives of U with n of V, and eliminate (m+n 1)
quantities, namely,
x, x 2 , x 3 , ...... up to x m+n ~ 1 .
* Eeprinted from Proc. Roy. Irish Acad., Vol. n. (1840 1844), p. 130.
16] On the Dialytic Method of Elimination. 87
This process, founded upon the dialytic principle, admits of a very simple
modification. Let us begin with the case where i = 0, or m = ??. Let the
augmentatives of U be termed U , U l , U 2 , U s , ... and of V, F , V l} F 2 , F 3 , ...,
the equations themselves being written
U= ax n + bx 11 1 + cx n - 2 + &c.
V = a a; n + b x n ~ l + c x n ~ 2 + &c.
It will readily be seen that
(b U -bV )+(a U l -aV l ),
(c U - cF ) + (b U, -bV l ) + (a U, - aF 2 ), &c.
will be each linearly independent functions of x, x-, ... x m ~\ no higher power
of x remaining. Whence it follows, that to obtain a derivee of the degree
(m - s) in its prime form, we have only to employ the s of those which occur
first in order, and amongst them eliminate x rnr ~ l , x m ~\ ... x m - s+l . Thus,
to obtain the final derivee, we must make use of n, that is, the entire number
of them.
Now, let us suppose that i is not zero, but m = n - i. The equation V
may be conceived to be of n instead of m dimensions, if we write it under
the form
Ox n + Ox" 1 + Qx n ~ 2 + ... + Ox m+l + ax + fix- 1 + &c. = 0,
and we are able to apply the same method as above ; but as the first L of the
coefficients in the equation above written are zero, the first t of the quantities
(a U - a Fo), (b U -b F ) + (a U,-a V,), &c.
may be read simply
-eiF , -6F,-aF,, -cFo-fcFj-aF^&c.
and evidently their office can be supplied by the simple augmentatives
themselves,
F =0, F 1 = 0, F 2 =0... F t _ l5 =0;
and thus i letters, which otherwise would be irrelevant, fall out of the several
derivees.
The author then proceeds with remarks upon the general theory of simple
equations, and shows how by virtue of that theory his method contains a
solution of the identity
X r U+ Y r V=D,.
where D r is a derivee of the rth degree of U and F, and accordingly, X r of
the form
I + mx + . . . + tx n
88 On the Dialytic Method of Elimination. [16
and accounts a priori for the fact of not more than (n r) simple equations
being required for the determination of the (m + n 2r) quantities X, p, v, &c.
I, m, n, &c., by exhibiting these latter as known linear functions of no more
than (n r) unknown quantities left to be determined.
Upon this remarkable relation may be constructed a method well adapted
for the expeditious computation of numerical values of the different derivees.
He next, as a point of curiosity, exhibits the values of the secondary
functions,
c U - cV. + b U.-bV^a U.-aV^&c.
under the form of symmetric functions of the roots of the equations U=0,
V = 0, by aid of the theorems developed in the London and Edinburgh
Philosophical Magazine, December 1839*, and afterwards proceeds to a more
close examination of the final derivee resulting from two equations each of
the same (any given) degree.
He conceives a number of cubic blocks each of which has two numbers,
termed its characteristics, inscribed upon one of its faces, upon which the
value of such a block (itself called an element) depends.
For instance, the value of the element, whose characteristics are r, s, is the
difference between two products: the one of the coefficient ?-th in order
occurring in the polynomial U, by that which comes 5th in order in V\ the
other product is that of the coefficient sth in order of the polynomial U, by
that rth in order of V; so that if the degree of each equation be n, there will
be altogether \n(n + 1) such elements.
The blocks are formed into squares or flats (plafonds) 6f which the
T3 (Yl -L ^
number is _or , according as n is even or odd. The first of these contains
2* 2.
n blanks in a side, the next (n 2), the next (n 4), till finally we reach a
square of four blocks or of one, according as n is even or odd. These fiats
are laid upon one another so as to form a regularly ascending pyramid, of
which the two diagonal planes are termed the planes of separation and
symmetry respectively. The former divides the pyramid into two halves,
such that no element on the one side of it is the same as that of any block
in the other. The plane of symmetry, as the name denotes, divides the
pyramid into two exactly similar parts ; it being a rule, that all elements
lying in any given line of a square (plafond} parallel to the plane of separation
are identical; moreover, the sum of the characteristics is the same, for all
elements lying anywhere in a plane parallel to that of separation.
[* p. 40 above. ED.]
16]
On the Dialytic Method of Elimination.
89
All the terms in the final derivee are made up by multiplying n elements
of the pile together, under the sole restriction, that no two or more terms of
the said product shall lie in any one plane out of the two sets of planes
perpendicular to the sides of the squares. The sign of any such product is
determined by the places of either set of planes parallel to a side of the
squares and to one another, in which the elements composing it may be
conceived to lie.
The author then enters into a disquisition relating to the number of terms
which will appear in the final derivee, and concludes this first part with the
statement of two general canons, each of which affords as many tests for
determining whether a prepared combination of coefficients can enter into
the final derivee of any number of equations as there are units in that
number, but so connected as together only to afford double that number, less
one, of independent conditions.
The first of these canons refers simply to the number of letters drawn out
of each of the given equations (supposed homogeneous) ; the second to what he
proposes to call the weight of every term in the derivee in respect to each of
the variables which are to be eliminated.
The author subjoins, for the purpose of conveying a more accurate
conception of his Pyramid of derivation, examples of the mode in which it is
constructed.
When n = 1 there is one flat, viz.
When n = 2 there is one flat, viz.
1, 2
2, 3
2, 4
2, 4
3, 4
Let n = 3, there will be two
flats:
2, 3
Let n = 4, there will still be two
flats only :
2, 3
2, 4
2, 4
3, 4
1, 2
1, 3
1, 4
1, 3
1, 4
2, 4
1, 4
2, 4
3, 4
1, 2
1, 3
1, 4
1, 5
1, 3
1, 4
1, 5
2, 5
1, 4
1, 5
2. 5
3, 5
1, 5
2, 5
3, 5
4, 5
90 On the Dialytic Method of Elimination.
Let n = 5, there will be three flats :
[16
2, 3
2, 4
2, 5
2, 4
2, 5
3, 5
2, 5
3, 5
4, 5
1, 2
1, 3
1, 4
1, 5
1, 6
1, 3
1, 4
1, 5
1, 6
2, 6
1, 4
1, 5
1, 6
2, 6
3, 6
1, 5
1, 6
2, 6
3, 6
4, 6
1, 6
2, 6
3, 6
4, 6
5, 6
Let n = 6, there will be three flats :
2, 3
2, 4
2, 5
2, 6
2, 4
2, 5
2, 6
3, 6
2, 5
2, 6
3, 6
4, 6
2, 6
3, 6
4, 6
5, 6
1, 2
1, 3
1, 4
1, 5
1, 6
1, 7
1, 3
1, 4
1, 5
1, 6
1, 7
2, 7
1, 4
1, 5
1, 6
1, 7
2, 7
3, 7
1, 5
1, 6
1, 7
2, 7
3, 7
4, 7
1, 6
1, 7
2, 7
3, 7
4, 7
5, 7
1, 7
2, 7
3, 7
4, 7
5, 7
6, 7
Thus the work of computation reduces itself merely to calculating
n - elements, or the n(n + 1) cross-products out of which they are con-
tt
stituted, and combining them factorially after that law of the pyramid, to
which allusion has been already made.
17.
ELEMENTARY RESEARCHES IN THE ANALYSIS OF
COMBINATORIAL AGGREGATION.
[Philosophical Magazine, xxiv. (1844), pp. 285296.]
THE ensuing inquiries will be found to relate to combination-systems,
that is, to combinations viewed in an aggregative capacity, whose species
being given, we shall have to discover rules for ranging or evolving them in
classes amenable to certain prescribed conditions. The question of numerical
amount will only appear incidentally, and never be made the primary object
of investigation*.
The number of things combined will be termed the modulus of the system
to which they belong. The elements taken singly, or combined in twos,
threes, &c., will be denominated accordingly the monadic, duadic. triadic
elements, or simply the monads, duads, or triads of the system.
Let us agree to denote by the word synthemef any aggregate of com
binations in which all the monads of a given system appear once, and
once only.
It is manifest that many such synthemes totally diverse in every term
may be obtained for a given system to any modulus, and for any order of
combination.
Let us begin with considering the case of duad synthemes. Take the
modulus 4 and call the elements a, b, c, d.
(ab, cd), (ac, bd), (ad, c6) constitute three perfectly independent
synthemes, and these three synthemes include between them all the duad
elements, so that no more independent synthemes can be obtained from them.
* The present theory may be considered as belonging to a part of mathematics which bears
to the combinatorial analysis much the same relation as the geometry of position to that of
measure, or the theory of numbers to computative arithmetic ; number, place, and combination
(as it seems to the author of this paper) being the three intersecting but distinct spheres of
thought to which all mathematical ideas admit of being referred.
t From ffvv and ridrifju.
92 Elementary Researches in the Analysis of [17
Again, let a, b, c, d, e,fbe the monads; we can write down five independent
synthemes, to wit,
ab, cd, ef
ad, cf, eb
ac, de, fb
af, bd, ce
ae, df, be
We can write no more than these without repeating duads which have
already appeared*.
We propose to ourselves this problem : A system to any even^ modulus
being given, to arrange the whole of its duads + in the form of synthemes ; or in
other words, to evolve a Total of daad synthemes to any given even modulus .
When the modulus is odd, as before remarked, the formation of a duad
syntheme is of course impossible, for any number of duads must necessarily
contain an even number of monadic elements ; but there is nothing to
prevent us from forming in all cases what may be termed a bisyntheme or
diplotheme, that is, an aggregate of combinations, where each element occurs
twice and no more.
For instance, if the elements be called after the letters of the alphabet,
we have ( ab b C f *7 *?} , the bisynthematic total to modulus 5 ; and in
\ac, ce, eb, bd, da/
* Such an aggregate of synthemes may be therefore termed a Total.
t The modulus must be even, as otherwise it is manifest no single syntheme can be formed.
We shall before long extend the scope of our inquiry so as to take in the case of odd moduli.
J Triadic systems will be treated of hereafter.
It is scarcely necessary to advert here to the fact of the problem being in general indeter
minate and admitting of a great variety of solutions. ,
When the modulus is four there is only one synthematic arrangement possible, and there is
no indeterminateness of any kind ; from this we can infer, a priori, the reducibility of a biquad
ratic equation ; for using 0, /, F to denote rational symmetrical forms of function, it follows that
tf{<t>(a, b), 0(c,d)}J
F ]/{< (a, c), $ (b, d)}\ is itself a rational symmetric function of a, b, c, d.
(/{</> (a, d), 0(6, c)})
Whence it follows that if a, 6, c, d be the roots of a biquadratic equation, f {<j> (a, b), <p (c, d)} can
be found by the solution of a cubic: for instance, (a + b) x (c + d) can be thus determined, whence
immediately the sum of any two of the roots comes out from a quadratic equation.
To the modulus 6 there are fifteen different synthemes capable of being constructed ; at first
sight it might be supposed that these could be classed in natural families of three or of five each,
on which supposition the equation of the sixth degree could be depressed ; but on inquiry this
hope will prove to be futile, not but what natural affinities do exist between the totals ; but in
order to separate them into families each will have to be taken twice over, or in other words,
the fifteen synthemes to modulus 6 being reduplicated subdivide into six natural families of five
each. Again, it is true that the triads to modulus 6 (just like the duads to modulus 4) admit of
being thrown into but one synthematic total, but then this will contain ten synthemes, a number
greater than the modulus itself.
17] Combinatorial Aggregation. 93
like manner
ab, be, cd, de, ef, fg, ga\
ac, ce, eg, gb, bd, df, fa\ the total to modulus 7.
ad, dg, gc, cf, fb, be, ea)
nl
In general, if n be the modulus, the number of duads is n = , n being
even, - duads go to each syntheme, and therefore the total contains (n 1)
Z
of these. If n be odd, then, since always n duads go to a bisyntheme, the
n l
number of such in the total is - .
Before proceeding to the solution of the problem first proposed, let us
investigate the theory of diplothematic arrangement. Here we shall find
another term convenient to employ. By a cyclotheme, I designate a fixed
arrangement of the elements in one or more circles, in which, although for
typographical purposes they are written out in a straight line, the last term
is to be viewed as contiguous and antecedent to the first ; the recurrence
may be denoted by laying a dot upon the two opened ends of the circle ;
a.b.c.d.e will thus denote a cyclotheme to modulus 5 ; a . b . c . d . e ./. g.h.k
the same to modulus 9; so also is d.b.c, d.e.f, g.h.k a cyclotheme of
another species to the same modulus. In general the number of terms will
be alike in each division of a cyclotheme.
Now it is evident that every cyclotheme, on taking together the elements
that lie in conjunction, may be developed into a diplotheme. Thus
1.2.3 = 12, 23, 31,
1.2.3.4 = 12, 23, 34, 41,
712, 23, 31\
(1.2.3; 4.5.6; f.8.9)^ 45, 56, 64 J.
\78, 89, 97/
Hence we shall derive a rule for throwing the duads of any system into
bisynthemes.
Let m = 3, we have simply dbc,
m = 5, we write a.b.c.d.e,
d . c .e.b.d,
the second being derived from the first by omitting every alternate term ;
similarly below, the lines are derived each from its antecedent.
m = 7, we have a .b.c .d . e.f. g,
d .c . e. g . b . d.f,
d. e . b .f.c.g .d.
94 Elementary Researches in the Analysis of [17
A very little consideration will serve to prove that in this way, m being
{Y\-l I
a prime number, = cyclothemes may be formed, such that no element
m
will ever be found more than once in contact on either side with any other ;
whence the rule for obtaining the diplothematic total to any prime-number
modulus is apparent.
For example, to modulus 7 the total reads thus :
1st. ab, be, cd, de, ef, fg, ga\
2nd. ac, ce, eg, gb, bd, df, fa
3rd. ae, eb, bf, fc, eg, gd, da)
and no more remains to be said on this special case.
Let us now return to the theory of even moduli, and show how to apply
what has been just done to constructing a synthematic total to a modulus
which is the double of a prime number.
Suppose the modulus to be six, the number of synthemes is five. Let the
six elements, a, b, c, d, e,f, be taken in three parts, so that each part contains
two of them ; let these parts be called A, B, C, where A denotes ab, B, cd,
and C, ef.
Now the duads will evidently admit of a distinction into two classes,
those that lie in one part, and those that lie between two ; thus ab, cd, ef
will be each unipartite duads, the rest will be bipartite.
The unipartite duads may be conveniently formed into a syntheme by
themselves; it only remains to form the four remaining bipartite duad
synthemes.
Write the parts in cy cloth em atic order, as below :
ABC.
It will be observed that each part may be written in two positions ; thus
i -i a i. b
A may be expressed by , or by ,
c d
B d " c
c e f
f e
Now we may form a cyclic table of positions as below :
ABC
1 1 1
122
212
221
17] Combinatorial Aggregation. 95
Here the numbers in each horizontal line denote the synchronic positions
of the parts.
On inspection it will be discovered that A will be found in each of its
two positions, with B in each of its two ; similarly B with C, and C with A.
In fact the four permutations, 11, 12, 21, 22, occur, though in different
orders, in any two assigned vertical columns.
Now develope the preceding table, and we have
dee adf bcf bde,
bdf bee ade acf;
and these being read off (the superior of each antecedent with the inferior
of each consequent*) must manifestly give the four independent bipartite
synthemes which we were in quest of, videlicet
(ad, cf, eb), (ac, de, fb), (bd, ce, fa), (be, df, ea) :
these four, together with the syntheme first described (ab, cd, ef), constitute
a duad synthematic total to modulus 6.
Before proceeding further let us take occasion to remark that the fore
going table of positions may evidently be extended to any odd number
of terms by repetition of the second and third places, as seen in the annexed
tables of position.
1. 1.1. 1.1 1.1.1:1.1.1.1,
1.2. 2.2.2 1.2.2.2.2.2.2f,
2.1.2.1.2 2.1.2.1.2.1.2,
2.2.1.2.1 2.2.1.2.1.2.1.
Now let 10 be the modulus.
As before divide the elements into five parts, which call A, B, C, D, E.
The unipartite duads fall into a single syntheme ; the eight remaining
bipartite synthemes may be found as follows :
Arrange in cyclothemes (
A,B,C, D, E. We have thus
Arrange in cyclothemes (- - in number) the odd modulus system
ABODE,
ACEBD.
* Any other fxed order of successive conjunction would answer equally well.
t It will not fail to be borne in mind that in operating with these tables only contiguous
elements are taken in conjunction : the first with the second, the second with the third, the
third with the fourth, &c., and the last with the first ; no two terms but such as lie together are
in any manner conjugated with one another.
96 Elementary Researches in the Analysis of [17
Let each cyclotheme be taken in the four positions given in the table
above, we have thus 2x4, that is, eight arguments.
db cde . d /3 7 8 e . abyde . a/3 cB e,
3/378 e, abode, a /3 cSe, abyde,
dcebd . dye/38 . etc eb8 . aye(3d,
aye/3 8, acebd, ayejSd, ace b 8.
And each of these arguments will furnish one bipartite syntheme, by reading
off, as before, the superior of each antecedent with the inferior of each
consequent ; and the least reflection will serve to show that the same duad
can never appear in two distinct arguments.
In like manner, if the modulus be 14 and seven parts be taken, the
bipartite synthemes, twelve in number, may be expressed symbolically thus :
I- 1 - 1 1 - - 1 ! f A.B.C.D.E.F.G
1.2.2.2.2.2.2 A.C.E.G.B.D.F
+ 2.1.2.1.2.1.2
+ 2.2.1.2.1.2.1
.......
(+A.E.B.F.C.G.D)
A B
Nay more, from the above table, if we agree to name the elements . J * , &c.,
Xl 2 -/5o
we can at once proceed to calculate each of the twelve synthemes in question
by an easy algorithm. For instance,
(1 . 2 . 2 . 2 . 2 . 2 . 2) x (A . C. E. G . B. D . F)
And again
(2 . 1 . 2 . 1 . 2 . 1 . 2) x (A . E. B. F. C. G . D)
each figure occurring once unchanged as an antecedent and once changed
as a consequent.
If it were thought worth while it would not be difficult, by using numbers
instead of letters, to obtain a general analytical formula, from which all
similarly constituted synthemes to any modulus might be evolved.
But the rule of proceeding must be now sufficiently obvious ; the modulus
being 2p, we divide the elements into p classes ; these may be arranged into
^^ distinct forms of cyclothematic arrangement, and each of the cyclo-
22
1
themes taken in four positions, thus giving 4 x * , that is, 2p - 2 bipartite
synthemes, the whole number that can be formed to the given modulus 2p.
17] Combinatorial Aggregation. 97
I shall now proceed to the theory of bipartite synthemes to the modulus
2m x p, by which it is to be understood that we have p parts each containing
2m terms, and p is at present supposed to be a prime number ; the total
number of synthemes to the modulus 2mp being 2mp 1, and 2m 1 of
these evidently being capable of being made unipartite ; the remainder,
2mp 2m, that is, (p 1) 2m, will be the number of bipartites to be
obtained*:
v- 1
2m (p 1) = x 4m ;
z
1 .
- denotes the total number of cyclothemes to modulus p ; 4m, as will be
Z
presently shown, the number of lines or syzygies in the Table of position.
To fix our ideas let the modulus be 4 x 3, and let A, B, C be three parts:
4J
b l 6 2 b 3 6 4 > their constituents respectively.
Cj C 2 Cy C 4 /
Give a fixed order to the constituents of each part, then each of them may
be taken in four positions ; thus A may be written
0,4
Assume some particular position for each, as, for instance,
and read off by coupling the first and third vertical places of each ante
cedent with the second and fourth respectively of each consequent ; we have
accordingly,
A, b^, da,,
a 3 b i} 6 3 c 4 , c s a 4 .
It is apparent that the same combinations will recur if any two contiguous
parts revolve simultaneously through two steps ; or in other words, that
A r B s A r+2 B s+2> where p is any number, odd or even.
* In general, if there be ?r parts of /j. terms each, and /JLTT be even, the number of bipartite
synthemes is (IT - 1) p, as is easily shown from dividing the whole number of bipartite duads
by the semi-modulus.
s. 7
98 Elementary Researches in the Analysis of [17
Symbolically speaking, therefore, as regards our table of position,
r:s = ?*+2:s + 2,
or more generally,
-r + 24: + 24.
So that
1:1 = 3:3, 2:1 = 4:3,
1:2 = 3:4, 2.2 = 4:4,
1:3 = 3.1, 2.3 = 4:1,
1:4 = 3.2, 2.4 = 4:2.
There are therefore no more than eight independent unequivalent permuta
tions to every pair of parts. Now inspect the following table of position :
1.1.1, 2.1.2,
1.2.3, 2.2.4,
1.3.2, 2.3.1,
1.4.4, 2.4.3.
It will be seen that in the first and second, second and third, third and
first places, all the eight independent permutations occur under different
names ; the law of formation of such and similar tables will be explained
in due time ; enough for our present object to see how, by means of this
table, we are able to obtain the bipartite synthemes to the given modulus
4x3; the number according to our formula is 2 x 4 x -- = 8, and they
may be denoted symbolically as follows :
1. 1.1 + 1, 2. 8 + 1. 3. 2 + 1. 4. 4\
2. 1.2 + 2. 2. 4 + 2. 3. 1 + 2. 4. 3/
Each of the eight terms connected by the sign of + gives a distinct syntheme;
for example, let us operate on
A . B . C x (2 . 3 . 1).
2.3.1 denotes 2.3, 3.1, 1.2.
2 . 3 gives rise to 2 (3 + 1) + (2 + 2) . (3 + 3) = 2 . 4 + 4 . 2.
3 . 1 gives rise to 3 (1 + 1) + (3 + 2) . (1 + 3) = 3 . 2 + 1 . 4.
1 . 2 gives rise to 1 (2 + 1) + (1 + 2) . (2 + 3) = 1 . 3 + 3 . 1.
The syntheme in question is therefore
A,B 4t A 4 B 2 , jB,C a , B&, CU 3 , C,A lt
and so on for all the rest, the rule being that
r : s = r (s + 1) + (r + 2) (s + 3).
(A.B.C)^
17] Combinatorial Aggregation. 99
Now, as before, it is evident that if we look only to contiguous terms, the
above table of position may be extended to any number of odd terms, simply
by repetition of the second and third figures in each syzygy ; and hence the
rule for obtaining the bipartite synthemes to the modulus 4 x p is apparent.
7 1
For instance, let p = 7, there will be 8 x , that is, 8 x 3 of them denoted
Z
as follows :
.D.E.F.G Ito l ot 1 * 2 - 1 - 2 1 - 2 - 1 - 2
1-3-3-2. 3- 2. 3 + 2. 2. 4. 2. 4. 2. 4
+A.E.B.F.C.G.D +1-3-2.3.2.3.2 + 2.3.1.3.1.3.1!
1.4.4.4.4.4.4 + 2.4.3.4.3.4.3
As an example of the mode of development, let us take the term
2. 4. 3. 4. 3. 4. 3 = (2: 4, 4:3, 3:4, 4:3, 3:4, 4:3, 3:2)
= / 2.1) 4.4) 3.1) 4.4) 3.1) 4.41 3 . 3) \
" U 4 . 3) + 2 . 2} + 1 . 3) + 2 . 2j + 1 . 3j + 2 . 2) + 1 . 1 j )
A.E.B.F.C.G.D=A.E, E.B, B.F, F.C, C.G, G.D, D.A,
and the product
(A Z/ 7 3T*D D Ti It 1 d n f*< n T\ T\ A \
AZ&!, A 4 1J 4 , -DS-PI, ^4^4, ^(T!, (jr 4 -U 4 , -UsA 3 \
At/ Jj* D 7?37 r t 7 /^ Cl r^ C* T\ T\ A I *
^1-4 J , J&2 J *- ? -*-^l ^3j * 9^8i ^1 ^"3 ) \JTv-LSf) , - - I -^J- 1/
Let the modulus be 6x3; as before, give a fixed cyclic order to the
constituents of each part, and each will admit of being exhibited in six
positions.
Write similarly as before,
and take the odd places of each antecedent with the even places of each
consequent ; it will now be seen that
n />
and the number of independent permutations is ^- =2.6; and so in
o
general, if there be 2m constituents in a part, the number of independent
. 2ra . 2m
permutations is - = 4m.
m
2
72
100
Elementary Researches in the Analysis of
[17
The rule for the formation of the table will be apparent on inspection.
I suppose only three parts, as the rule may always be extended to any
number by reiteration of the second and third terms. The table will be
found to resolve itself naturally into four parts, each containing m lines.
Let m = 1, we have
m = 2, we have
m = 3, we have
m = 4, we have
1
.1.
1
2.
1
.2
1
.2.
2
2.
2
.1
1
.1.
1
2.
1
.2
1
.2.
3
2.
2
.4
1
.3.
2
2.
3
.1
1
.4.
4
2.
4
.8
1
.1.
1
2.
1
.2
1
.2.
3
2.
2
.4
1
.3.
5
2.
3
.6
J
.4.
2
2.
4
.1
1
. 5.
4
2.
5
.3
1
.6.
6
2.
G
. 5
1
.1.
1
2.
1
.2
1
.2.
3
2.
2
.4
1
.3.
5
2.
3
.6
1
.4.
7
2.
4
.8
1
.5.
2
2.
5
.1
1
.6.
4
2.
(i
.3
1
.7.
6
2.
7
. .">
1
.8.
8
2.
8
.7
So that x, going through all its values from 1 to m, the general expression
for the four parts is
To show the use of this formula, let us suppose that we have seven parts,
each containing ten terms, the general expression for the bipartite duad
synthemes is
A.B.C.D.JS.F.M
A . C.E. G.B.D.F\ x + l(5 + X j 2 x(5 + x) 2x (5 + x) 2
(+A.8.B.F.C.G.D) +2(5+*)(2-l)(5+a)(2-l)(5+*)(2-l)
17]
Combinatorial Aggregation,
101
Make, for example, x = 3, one of the synthemes in question out of the
twelve corresponding to this value will be
Here
3.6.3.6.3
6 =
. j ~ j
= 2.4
3.7
6.4 I
3.7^
6.4 ,
3.7
6.3
+ 4.6
+ 5.9
+ 8.6
+ 5.9
+ 8.6
+ 5.9
+ 8.5
+ 6.8
. + 7.1
+10.8
.+7.1
, + 10.8
, + 7.1
+ 10.7
+ 8.10
, +9.3
+ 2. 10
+ 9.3
+ 2.10
+ 9.3
+ 2.9
+ 10.2
1 +1.5
+ 4.2 ,
+ 1.5
+ 4.2 ,
+ 1.5
+ 4.1,
and the product
= A,G 4 , C S E 7 , E.G., G 3 B 7) B 6 D 4 , D 3 F 7 , F 6 A 3
A t C 6 , C 5 E 9 , E S G 6 , G 5 B 9 , B S D 6 , D 5 F 9 , F 8 A 5 ,
&c. &c. &c.
To prove the rule for the table of formation, it will be sufficient to show
that no two contiguous duads ever contain the same or equivalent permutations;
the equation of equivalence it will be remembered is
r : s = r + 2i 2m :s + 2i 2m.
Now, as regards the first and second terms, it is manifest that 1 : x cannot be
equivalent, either to 1 : x nor to 2 : x, nor to 2 : x, where x is any number
differing from x.
Similarly, as regards the last and first terms, x : 1 cannot be equivalent to
x : 1, nor to x : 2, nor to x : 2 ; therefore there is no danger as far as the first
term is concerned, either as antecedent or consequent.
Again, it is clear that x : (2ac - 1) cannot interfere with x : 2x, nor
(m + x) : 2x with (m + x) : (Zx - 1) ; neither can (2x - 1) : x with 2a? : x , nor
2# : (m + x) with (2a/ - 1) : (m + x ).
Again, if possible, let
x : (2a? - 1) = (m + x } : (2x - 1);
then
and
therefore
or
m + x x =
2m = 2i
which is impossible, since +i is the difference between two indices, each
less than m.
102 Researches in Analysis of Combinatorial Aggregation. [17
Similarly,
ra + x : 2x cannot = x : 2a; ,
and vice versa with the terms changed
2x : (m + x) cannot = 2x : x ,
and
(2x 1) : x cannot = (2x I):(m + x ),
which proves the rule for the table of formation.
So much for the bipartite duad synthemes. As regards the unipartite
synthemes little need be said, for every part may be treated as a separate
system, and as each will produce an equal number of synthemes, these being
taken one with another, will furnish just as many unipartite synthemes of
the whole system as there are synthemes due to each part. Thus then the
synthematic resolution of the modulus 2m x p may be made to depend on
the synthematization of 2m and the cyclothematization of p. This has been
already shown (whatever m may be) for the case of p being a prime number ;
but I proceed now to extend the rule to the more general case of p being
any number whatever.
18.
ON THE EXISTENCE OF ABSOLUTE CRITERIA FOR DETER
MINING THE ROOTS OF NUMERICAL EQUATIONS.
[Philosophical Magazine, xxv. (1844), pp. 442 445.]
I WISH to indicate in this brief notice a fact which I believe has escaped
observation hitherto, that there exist, certainly in some cases, and probably
in all, infallible criteria for determining whether a given equation has all its
roots rational or not.
In the equation of the second degree it is enough, in order that this may
be the case, that the expression for the square of the difference of the roots
shall be a perfect square; in other words, if ac 2 px + q = have its roots
rational, p*4q must be not only a positive number (the condition of the
roots being real), but that number must also be a complete square. In this
case it is further evident that p must be either prime to q, or if not, the
greatest common measure of p 2 and q must be a perfect square ; but this
condition is contained in the former, which is a sufficient criterion in itself.
If we now consider the equation of the third degree,
a? px- + qx r = 0,
one condition is, that the product of the squared differences shall be a perfect
square ; in other words, the equation cannot have all its roots rational
unless
pY - 4>q 3 - ISpqr - 4pV - 27 r 2
be a positive square number.
This remark is made at the end of the second supplement of Legendre s
Theory of Numbers, and is indeed self-evident ; and in like manner one
condition may be obtained for an equation of any degree which is to have
all its roots rational ; but this is far from being the sole condition required.
104 On Absolute Criteria for determining [18
In the equation of the third degree, however, one other condition, conjoined
with that above expressed, will serve to determine infallibly whether all the
roots are rational or not.
To obtain this condition, let us suppose that by making 3x = y+p we
obtain the equation
Calling the three roots of this new equation a, /3, 7 (all of which it
is evident must be rational if those of the first equation are so), we have
a + /3 + 7 = 0,
Q = - (a/3 + ay + 7) = a 2 + a/3 + #-,
R = a/3y.
From the last two equations it is easily seen that if k be any prime factor
common to Q and R, k 2 will be contained in Q, and A^ in R ; or, in other
words, k will be a common measure of a, ft, 7.
We have therefore a second condition, that 9q 3p 2 shall be a negative
quantity, which is either prime to 2p 3 9qp + 27r, or else so related to it,
that the greatest common measure of the cube of the first and the square
of the second is a perfect sixth power.
I now proceed to show the converse, that if these two conditions be both
satisfied (and it will appear in the course of the inquiry that the first does
not involve the second), the roots cannot help being all rational.
It is evident that the two conditions in question are tantamount to
supposing that the roots of the proposed equation are linearly connected
with those of another z? Qz R = (by virtue of the assumption 3# = kz +p),
where Q may be considered as prime to R ; and where 4Q 3 27E 2 is a
perfect square.
Let now 4Q 3 - 27 2 = D\ then D 2 + 27 2 = 4Q 3 , or Z> 2 + 3 (3) 2 = 4Q 3 .
Here, as Q is prime to R, D can have no common measure but 3,
with 3R.
Firstly, let Q be prime to 3R.
Then putting / 2 + 3# 2 = Q 3 , the complete solution of the equation im
mediately preceding is contained in the two systems :
1st. jD=2/ 3R = 2g.
2nd. D=(f3g), 3R=f+g,
and for both systems,
18] the Roots of Numerical Equations. 105
The second system must therefore be rejected, for g evidently contains 3,
and therefore /= 3E + g will contain 3, and therefore D and therefore Q will
do the same, contrary to supposition.
Hence
2 - 2 V V 27
J^
27
and the three roots of the equation being
will evidently be all rational, which of course includes the necessity of their
being also integer.
Again, secondly, if we suppose that Q does contain 3, D 2 will contain 27,
and consequently D will contain 9 ; and we shall have
Here R being prime to -^ , it may be shown, as in the last case, that the
complete solution is
R. D ,,
7T "T^ ^ ^ ~~ O ) I / V
iO
consequently
and the three roots of the equation are
respectively, and are therefore all rational.
Here it may be observed that the condition of R being an even number,
which we know, d priori, is the case when all the roots are rational, is
106 On Absolute Criteria of Numerical Equations. [18
involved in the two more general conditions already expressed. It will now
be evident that the first condition by no means involves the second, as it is
perfectly easy to satisfy the equation f- + %g- = Q? without supposing anything
relative to k, the common measure of/, g, Q, except that it be itself of the
form X 2 + 3//r, which will give
an equation which can be solved in rational terms for all values of \ p, r, s;
and consequently the product of the squares of the differences of the roots
may be a square, and at the same time the roots themselves may be
irrational*.
I believe it will be found on inquiry that the equation x 11 qx + r =
will always have two rational roots if
w-"-. q n -n
be a complete square, provided that q be prime to r.
Furthermore, viewing the striking analogy of the general nature of the
conditions of rationality already obtained, to those which serve to determine
the reality of the roots of equations, I am strongly of opinion that a theorem
remains to be discovered, which will enable us to pronounce on the existence
of integer, as Sturm s theorem on that of possible roots of a complete equation
of any degree : the analogy of the two cases fails however in this respect,
that while imaginary roots enter an equation in pairs, irrational roots are
limited to entering in groups, each containing two or MORE.
* Thus then it appears that the total rationality of the roots of the equation x s -qx-r=Q
may be determined by a direct method without having recourse to the method of divisors to
determine the roots themselves ; the two conditions being that iq 3 27r 2 shall baa perfect square,
and the greatest common measure of q 3 and r- a perfect sixth power.
19.
AN ACCOUNT OF A DISCOVERY IN THE THEORY OF
NUMBERS RELATIVE TO THE EQUATION Aa? + By* + Cz* = Dxyz.
[Philosophical Magazine, xxxi. (1847), pp. 189191.]
FIRST GENERAL THEOREM OF TRANSFORMATION.
IF in the equation
Aa? + By 3 + Cz s = Dxyz, (1 )
A and B are equal, or in the ratio of two cube numbers to one another, and
if 27 ABC -D 3 (which I shall call the Determinant) is free from all single or
square prime positive factors of the form Qn+l, but without exclusion of
cubic factors of such form, and if A and B are each odd, and C the double or
quadruple of an odd number, or if A and B are each even and C odd, then,
I say, the given equation may be made to depend upon another of the form
A u s + B tf + C iv 3 = D uvw ;
where
A B G = ABC,
D = D,
uviv = some factor of z.
The following are some of the consequences which I deduce from the
above theorem. In stating them it will be convenient to use the term Pure
Factorial to designate any number into the composition of which no single or
square prime positive factor of the form 6n + 1 enters.
The equations
a? + y 3 + 2z 3 = Dxyz,
x 3 + y 3 + 4<z 3 = Dxyz,
Zx 3 + 2y 3 + z 3 = Dxyz,
are insoluble in integer numbers, provided that the Determinant in each case
is a Pure Factorial.
108 On a Discovery in the Theory of Numbers. [19
The equation
is insoluble in integer numbers, provided that the Determinant, for which in
this case we may substitute A 27 B 3 , is a pure factorial whenever A is of
the form 9n 1, and equal to 2p 3il or 4p 3<1 , p being any prime number
whatever.
I wish however to limit my assertion as to the insolubility of the
equations above given. The theorem from which this conclusion is
deduced does not preclude the possibility of two of the three quantities
x, y, z being taken positive or negative units, either in the given equation
itself or in one or the other of those into which it may admit of being
transformed. Should such values of two of the variables afford a particular
solution, then instead of affirming that the equations are insoluble, I should
affirm that the general solution can be obtained by equations in finite
differences*.
SECOND GENERAL THEOREM OF TRANSFORMATION.
The equation
/W + g A f + fcz 3 = Kxyz (2)
may always be made to depend upon an equation of the form
Au 3 + Bv 3 + Cw 3 = Duvw,
where
ABC = R 3 - S 3 ,
D = ZR;
and uvw = some factor of fx + gy + hz.
R representing K + Gfgh,
S K-3fffh.
* Take for instance the equation x 3 + y s + 2z 3 = 3xyz. The Determinant 27.25 is a Pure
Factorial : consequently if the solution be possible, since in this case the transformed must
be identical with the given equation, this latter must be capable of being satisfied by making x
and y positive or negative units. Upon trial we find that x = l, y = l, z = 2 will satisfy the
equation. I believe, but have not fully gone through the work of verification, that these are the
only possible values (prime to one another) which will satisfy the equation. Should they not be
so, my method will infallibly enable me to discover and to give the law for the formation of all
the others.
Here, then, under any circumstances, is an example, the first on record, of the complete
resolution of a numerical equation of the third degree between three variables.
19] On a Discovery in the Theory of Numbers. 109
I have not leisure to show the consequences of this theorem of trans
formation in connexion with the one first given, but shall content myself
with a single numerical example of its applications:
a? + y 3 -f z 3 = Qxyz
may be made to depend on the equation
u 3 + v 3 + w 3 = 0,
and is therefore insoluble.
It is moreover apparent that the Determinant of equation (2) trans
formed is in general - 27J? 3 , and is therefore always a Pure Factorial, and
consequently the equation
f 3 x 3 + g 3 y 3 + h 3 z* = Kxyz
will be itself insoluble, being convertible into an insoluble form, provided that
K+Qfgh is divisible by 9, and provided further that (K + 6fgh) 3 - (K - Sfgh} 3
belongs to the form m 3 Q, where Q is of the form 9?i + l,and also of one or the
other of the two forms 2p si1 , 4p 3i1 , p being any prime number whatever.
Pressing avocations prevent me from entering into further developments
or simplifications at this present time.
It remains for me to state my reasons for putting forward these dis
coveries in so imperfect a shape. They occurred to me in the course of
a rapid tour on the continent, and the results were communicated by me to
my illustrious friend M. Sturm in Paris, who kindly undertook to make them
known on my part to the Institute.
Unfortunately, in the heat of invention I got confused about the law of
oddness and evenness, to which the coefficients of the given equation are in
the first theorem generally (in order for the successful application of my
method as far as it is yet developed) required to be subject. I stated this
law erroneously, and consequently drew erroneous conclusions from my
Theorems of Transformation, which I am very anxious to seize the earliest
opportunity of correcting. I venture to flatter myself that as opening out
a new field in connexion with Fermat s renowned Last Theorem, and as
breaking ground in the solution of equations of the third degree, these
results will be generally allowed to constitute an important and substantial
accession to our knowledge of the Theory of Numbers.
20.
ON THE EQUATION IN NUMBERS Aa? + Ef + Cz s = Dxyz, AND
ITS ASSOCIATE SYSTEM OF EQUATIONS.
[Philosophical Magazine, xxxi. (1847), pp. 293 296.]
IN the last Number of this Magazine I gave an account of a remarkable
transformation to which the equation
Aa? + By 3 + Cz 3 = Dxyz
is subject when certain conditions between the coefficients A, B, C, D are
satisfied ; which conditions I shall begin by expressing with more generality
and precision than I was enabled to do in my former communication.
1. Two of the quantities A, B, G are to be to one another in the ratio of
two cubes.
2. 27ABC D 3 must contain no positive prime factor whatever of the
form 6n + 1. I erred in my former communication in not excluding cubic
factors of this form.
3. If 2 m is the highest power of 2 which enters into ABC, and 2 n the
highest power of 2 which enters into D, then either in must be of the form
3n + 1, or if not, then m must be greater than 3n.
These three conditions being satisfied, the given equation can always be
transformed into another,
A u 3 + B tf + C w s = D uvw,
where
A B C = ABC, D = D, uvw = a factor of z.
The consequence of this is, as stated in my former paper, that wherever
A, B, C, D, besides satisfying the conditions above stated, are taken so as
likewise to satisfy the condition, firstly, of ABC being equal to 2 3m1 , or
secondly, of ABC being equal to 2 3w1 .^p 3n1 , provided in the second case
that ABC is of the form 9m 1, and that D is divisible by 9, p being in
20] On the Equation in Numbers Ax 3 + By* + Cz 3 = Dxyz. Ill
both cases a prime, then the given equation will be generally insoluble. And
I am now enabled to add that the only solution of which it will in any case
admit, is the solitary one found by making two of the terms Ax 3 , By 3 , Cz 3
equal to one another ; so that, for instance, if the given equation should be of
the form
a? + y 3 + ABCz 3 = Dxyz,
then the above conditions being satisfied, the one solitary solution of which
the equation can possibly admit, is x 1, y = 1,
Az 3 - Dz + 2 = 0,
which may or may not have possible roots. I call this a solitary or singular
solution, because it exists alone and no other solution can be deduced from
it ; whereas in general I shall show that any one solution of the equation
A a? + By 3 + Cz 3 = Dxyz
can be made to furnish an infinity of other solutions independent of the one
supposed given, that is, not reducible thereto by expelling a common factor
from the new system of values of x, y, z deduced from the given system.
The following is the Theorem of Derivation in question :
Let
Aa 3 + B/3 3 + 6y = DoL/3j.
Then if we write
and make
x = F*G + G*H + H*F - 3FGH,
T?r* i r 1 U" i TJ PO o EVV TJ
y = (JT - + Cr/l - + ti r ~ 6f (jrtl ,
D
or
= 3/87 \F~ + G" 2 + H- FG FH GH],
we shall have
x 3 + y 3 + ABCz 3 = Dxyz.
I am hence enabled to show that whenever x 3 + y 3 + Az 3 = Dxyz is
insoluble, there will be a whole family of allied equations equally insoluble.
For instance, because x 3 + y 3 + z 3 = is insoluble in integer numbers, I know
likewise that
/>>6 I /j6 I ^.6 Af**4j9 I /y*i^f> I -J/3^-3
w ^^ U i^ & i/. U ~f^ vL 4i ~\ U &
are each equally insoluble.
112 On the Equation in Numbers [20
In fact
(x 3 + y* + z 3 ) x (x 6 + y s + z 6 x 3 y 3 x 3 z 3
x (X K + y 6 + z e - y s z 3 - y 3 x 3
x (of + y 6 + z 6 x 3 z 3 z 3 y 3 + 2?/ 3 # 3 )
= u 3 + if + w 3 ,
where u, v, w are rational integral functions of x, y, z.
Hence each of the factors must be incapable of becoming zero *.
As a particular instance of my general theory of transformation and
elevation, take the equation
y? + y 3 + 2z 3 = Mxyz.
Then, with the exception of the singular or solitary solution x=l, y = l, of
which I take no account, I am able to affirm that for all values of M between
7 and 6, both inclusive, with the exception of M = 2, the equation is
insoluble in integer numbers.
Take now the equation where M = 2, namely
a? + y 3 + 2z 3 + 2xyz = 0.
One particular solution of this is
x=\, y = I, z=l.
Another, which I shall call the second f, is
x = 1, y = 3, z = 2.
From the first solution I can deduce in succession the following :
x =11, y = 5, z=-7,
x = -793269121, y= 1179490001, z = - 1189735855,
&c. &c. &c.
From the second,
x = - 10085, y = 8921, z = - 8442,
x = &c. y = &c. z = &c.
As another example, take the equation
a? + y 3 + Qz 3 = Qxyz.
* It is however sufficiently evident from their intrinsic form, which may be reduced to
+ 3A T2 ), that this impossibility exists for all the factors except the first.
t See Postscript.
20] Ax" + By* + Cs 8 = Dxyz. 113
One solution of the transformed equation
u 3 + 2^ + Sw 3 = 6uwu
is evidently
w = 1, y=. 1, w = 1.
Hence I can deduce an infinite series of solutions of the given equation, of
which the first in order of ascent will be
x=o, y = 7, z = 3.
Again, the lowest possible solution in integers of the equation
x 3 + y 3 4- Gz 3 =
will be
__ -1 hr Ohr _ _ n~l
The equation
a? + f + 9z 3 =
admits of the solutions
#=-271, y = 919, * = -438.
I trust that my readers will do me the justice to believe that I am
in possession of a strict demonstration of all that has been here advanced
without proof. Certain of the writer s friends on the continent have, in their
comments upon one of his former papers which appeared in this Magazine,
complimented his powers of divination at the expense of his judgment, in
rather gratuitously assuming that the author of the Theory of Elimination
was unprovided with the demonstrations, which he was too inert or too beset
with worldly cares and distractions to present to the public in a sufficiently
digested form. The proof of whatever has been here advanced exists not
merely as a conception of the author s mind, but fairly drawn out in writing,
and in a form fit for publication.
P.S. It must not be supposed that the two primary or basic solutions
above given of the equation
x 3 + y 3 + 2z 3 + 2xyz = 0,
namely, x=l, y = l, z=l,
are independent of one another. The second may be derived from the first,
as I shall show in a future communication. In fact there exist three inde
pendent processes, by combining which together, one particular solution may
be made to give rise to an infinite series of infinite series of infinite series of
correlated solutions, which it may possibly be discovered contain between
them the general complete solution of the equation
x 3 + y 3 + Az 3 = Dxyz.
s. 8
21.
ON THE GENERAL SOLUTION (IN CERTAIN CASES) OF
THE EQUATION a? + f + A? = Mxyz, &c.
[Philosophical Magazine, xxxi. (1847), pp. 467 471.]
I SHALL restrict the enunciation of the proposition I am about to
advance to much narrower limits than I believe are necessary to the
truth, with a view to avoid making any statement which I may hereafter
have occasion to modify. Let us then suppose in the equation
P + 2^ + A& = Mxyz
that A is a prime number, and that 27 A M 3 is positive, but exempt from
positive prime factors of the form 6i + 1. Then I say, and have succeeded
in demonstrating, that all the possible solutions in integer numbers of the
given equation may be obtained by explicit processes from one particular
solution or system of values of x, y, z, which may be called the Primitive
system.
This system of roots or of values of x, y, z is that system in which the
value of the greatest of the three terms as, y, A* .z (which may be called the
Dominant) is the least possible of all such dominants. I believe that in
general the system of the least Dominant is identical with the system of the
least Content, meaning by the latter term the product of the three terms out
of which the Dominant is elected. I proceed to show the law of derivation.
To express this simply, I must premise that I shall have to employ such
an expression as S = </> (S) to indicate, not that a certain quantity, 8 , is
a function of S, but that a certain system of quantities disconnected from
one another, denoted by S , are severally functions of a certain other system
of quantities denoted by S; and, as usual, I shall denote </><$ by <f>-S,
<JxjrS by (f> 3 S, and so forth.
Let now P be the Primitive system of solution of the equation
x 3 + y 3 + Az 3 = Mxyz,
P denoting a certain system of values of and written in the order of the
21] On the Equation x* + y* + Az z = Mxyz, &c. 115
letters x, y, z, which may always be found by a limited number of trials
(provided that the equation admits of any solution). That this is the case is
obvious, since we have only to give the Dominant every possible value from
the integer next greatest to A* upwards, and combine the values of a?, y 3 ,
Az z so that none shall ever exceed at each step the cube of such dominant,
and we must at last, if there exist any solution, arrive at the System of the
Least Dominant.
Now, every system of solution is of one or the other of two characters.
Either x and y must be odd and z even, or x and y must be one odd and the
other even and z odd. That all three should be odd is inconsistent with
the given conditions as to A being odd and M even ; and if all three were
even, by driving out the common factor we should revert to one or the other
of the foregoing cases.
The systems of solution where z is even may be termed Reducible, those
where z is odd Irreducible. Let (/> denote a certain symbol of transformation
hereafter to be explained.
Then the Reducible systems of the first order may be expressed by
(f>P, (frP, <f) 3 P, ad infinitum ;
or in general by (/> n P, n v being absolutely arbitrary. I will anticipate by
stating that the function (f> involves no variable constants ; that is to say,
<f> (S) may be found explicitly from S without any reference to the particular
equation to which S belongs. Let now -fy denote another symbol of trans
formation, also hereafter to be defined, and differing from < insofar as it does
involve as constants the three values of x, y, z contained in P : then the
general representations of Irreducible systems of the first order will be
denoted by i/r< n P.
It is proper to state here that the symbol i/r is ambiguous ; and -ty-p^P,
when P and n v are given, will have two values, according to the way in
which the terms represented by P are compared with x, y, z in the given
equation
a? + y 3 + Az s = Mxyz ;
for it is obvious that if x = a, y = b, z = c satisfies the equation, so likewise
will
x = b, y = a, z = c.
Each however of these values of -v/r^P gives a solution of the kind above
designated.
Proceeding in like manner as before, the Reducible system of the second
order may be designated by < n * . ^^ . P, the Irreducible by *^$ n * . ^</>" . P ;
and in general every possible system of values of x, y, z satisfying the proposed
equation, in which z is even, is comprised under the form
l . -vjr. . .<* . ^(/>"> . P ;
116 On the General Solution of [21
and every possible system of such values, in which z is odd, is comprised
under the form
the quantities ^,w 2 ...w r being of course all independent of one another,
and unlimited in number and value.
Thus then we may be said to have the general solution of the given
equation in the same sense as an arbitrary sum of terms, each of a certain
form, is in certain cases accepted as the complete solution of a partial
differential equation.
As regards the value of the symbols ^ and <, $ indicates the process by
which a, b, c becomes transformed into a, ft, 7, the relations between the two
sets of elements being contained in the following equations :
a = a 2 b + V*c + cV - Sa b c,
ft = a b * + b c- + c a* - Sa b c ,
7 = abc (a 2 + b" 2 + c" 2 - a b - ac - b c }.
Next, as to the effect of the Duplex symbol ^. Let e, g, t be the
elements of the Primitive system P : i being the value of z and e, g of
x and y taken in either mode of combination, each with each, which satisfy
the proposed equation
ar* + y 3 + Az 3 = Mxyz.
Let I, TO, n represent any system S,
\, jju, v represent any system -^r(S),
^S has two values, which we may denote by ^ S, ^S respectively, and
accentuating the elements X, /*, v accordingly to correspond, we shall have
V = 3#m (gl - em) + 3 A in (d - en) - M(gd* - e-lm),
v = %el (en - d) + 3gm (gn - tni) - M(egn 2
we have then
and in like manner
= X >, v,
^S being derived from ^ S by the mere interchange of e and g one with the
other.
21] the Equation x* + y 3 + Az s = Mxyz, &c. 117
I have stated that every possible solution of the proposed equation comes
under one or the other of the orders, infinite in number and infinite to the
power of infinity in variety of degree, above given : this is not strictly true,
unless we understand that all systems of solution are considered to be
equivalent which differ only in a multiplier common to all three terms
of each ; that is to say, which may be rendered identical by the expulsion
of a common factor. So that ma, mft, my as a system is treated as identical
with a, ft, 7, which of course substantially it is ; and it should be remarked
that there is nothing to prevent the operations denoted by </> and ^ intro
ducing a common factor into the systems which they serve to generate, and
the latter in particular will have a strong tendency so to do.
I believe that this theorem may be extended with scarcely any modifica
tion to the case where A, instead of being a prime, is any power of the same,
and to suppositions still more general. I believe also that, subject to certain
very limited restrictions, the theorem may prove to apply to the case where
the determinant 27JL - M 3 becomes negative.
The peculiarity of this case which distinguishes it from the former, is
that it admits of all the three variables x, y, z in the equation
a? + y* + Az 3 = Mxyz
having the same sign, which is impossible when the determinant is positive ;
or in other words, the curve of the third degree represented by the equation
F 3 + X 3 + I = , XY (in which I call the coefficient of XT the character-
A.*
istic), which, as long as the quantity last named is less than 3, is a single
continuous curve extending on both sides to infinity, as soon as the
characteristic becomes equal to 3 assumes to itself an isolated point, the
germ of an oval or closed branch, which continues to swell out (always lying
apart from the infinite branch) as the characteristic continues indefinitely
to increase.
I ought not to omit to call attention to the fact that the theorem above
detailed is always applicable to the case of the equation
x* + f + Az 3 = 0,
when A is any power of a prime number not of the form 6i + 1 ; in other
words, the above always belongs to the class of equations having Mouogenous
solutions, which for the sake of brevity may be termed themselves Mono-
genous Equations*.
* Thus the equation x 3 + y 3 + 9z*=Q alluded to by Legendre is Monogenous, and the Primitive
system of solution is x=l, y = 2, z= -1, from which every other possible solution in Integers
may be deduced.
118 On the Equation x 3 + y 3 + Az 3 = Mxyz, &c. [21
On the probable existence of such a class of equations I hazarded
a conjecture at the conclusion of my last communication to this Magazine.
As I hope shortly to bring out a paper on this subject in a more complete
form, I shall content myself at this time with merely stating a theorem
of much importance to the completion of the theory of insoluble and of
Monogeuous equations of the third degree; to wit, that the equation in
integers
a (a? + if + 2 s ) + c (x-y + y*z + z*x + xy* + yz 2 + zx 2 ) + exyz =
may always be transformed so as to depend upon the equation
fu 3 + gv 3 + hw 3 = (6a e) wow,
wherein fgh = ae? - (c 2 + 3a s ) e + 9a 2 - 3ac 2 - 2c :i .
By means of the above theorem, among other and more remarkable
consequences, we are enabled to give a theory of the irresoluble and
monogenous cases of the equation
a? + y 3 + m 3 z 3 = Mxyz,
when m is some power of 2, or of certain other numbers.
22.
ON THE INTERSECTIONS, CONTACTS, AND OTHER CORRE
LATIONS OF TWO CONICS EXPRESSED BY INDETER
MINATE COORDINATES.
[Cambridge and Dublin Mathematical Journal, v. (1850), pp. 262 282.]
LET /"=(), V=0 be two homogeneous equations of the second degree
with real coefficients, between the same three variables , 77, .
The direct and most general mode of determining the intersections of
the conies expressed by these equations would be to make
= t,
a g + b rj + c =u :
eliminating , 77, between the four equations in which they appear, there
results a biquadratic equation between t and u. The nature of the inter
sections will depend upon the nature of the roots of this biquadratic ; and
thus the conditions may be expressed analytically, which will represent
the several cases of all the intersections being real or all imaginary, or one
pair real and the other imaginary. These analytical conditions will depend
upon the signs of certain functions of the coefficients of the given and the
assumed equations being of an assigned character ; my endeavour has been
to obtain conditions of a character perfectly symmetrical and free from the
coefficients arbitrarily introduced.
In this research I have only partially succeeded, but the method
employed, and some of the collateral results, will, I think, be found of
sufficient interest to justify their appearance in the pages of this Journal.
Adopting Mr Cayley s excellent designation, let the four points of inter
section of the two conies be called a quadrangle. This quadrangle will have
three pairs of sides; the intersections of each pair, from principles of
analogy, I call the vertices of the quadrangle. Then, inasmuch as the four
120 On the Correlations of two Conies [22
sets of ratios f : 17 : f , corresponding with the four sets of the ratio t : u,
must be so related that we may always make
we may easily draw the following conclusions.
If all the four points of the quadrangle of intersection are real, the three
vertices and the three pairs of sides are all real. If only two points of the
quadrangle are real, one vertex and one of the three pairs of sides will be
real ; the other two vertices and two pairs of sides being imaginary. If all
four points of the quadrangle are unreal, one pair of sides will be real
and the other two pairs imaginary, as in the last case ; but all the three
vertices will remain real, as in the first case. Hence we have a direct and
simple criterion for distinguishing the case of mixed intersection from inter
section wholly real or wholly imaginary ; namely, that the cubic equation
of the roots of which the coordinates of the vertices are real linear functions
shall have a pair of imaginary roots. This is the sole and unequivocal
condition required.
The equation in question is, or ought to be, well known to be the deter
minant in respect to ,77, f of \U + pV. In fact, if we write , -
U = a!>* + by*
F = af
= (a\ + ap) p + &c. =
the ratios of the coordinates f, 77, of the vertex of \U + fiV may easily be
shown to be identical with
AB-C 2 : C A - B B : B C - A A,
and will be real or imaginary as \ : p is one or the other.
If then the cubic equation in X : p, namely, n (xZ7 + /*F) = 0, has a pair of
imaginary roots, that is, if nn a (XZ7+ /iF) is a positive quantity, the inter
sections of U and F are of a mixed kind, that is, the two conies have two
real points in common.
22] expressed by Indeterminate Coordinates. 121
I may remark here, en passant, that if we form the biquadratic equation
in t and u, $ (t, u) = from the equations
U=0,
F=0,
a% + bf] + c = t,
a +b v + c!;=u,
and if any reducing cubic of this equation be P (6, CD) = 0, the determinant
of P (0, &&gt;) must, from what has been shown above, be identical with
a n (X U + /u, V) multiplied by some squared function of the extraneous
^M f 1 ?^
coefficients
a,b,c; a, b , c.
If aa(\U + /j,V) is a negative quantity, it remains to distinguish
between the cases of the conies intersecting really in four points or not at all.
The most obvious mode of proceeding to distinguish between purely real
and purely imaginary intersections would be as follows. Let \ l} /^j X>> fa ,
X 3 , fj, 3 , be the three sets of values of X, //, which satisfy the equation
and make
G! = c\i + 7/u.j , (7 2 = c\ 2 + 7^2, (7 3 = c\ 3 + 7/1x3 ,
/ = b\ + /3>: , B z = b \ 2 + /3> 2 , 5/ = b \ 3 + fffj* ,
A 1 C l -B 1 z = e 1 , A&-B 3 * = e,, A 3 G 3 - B 3 * = e 3 .
Now if the equation
A? + Brf + C? + 2A ^+ 2B g + ZC fr =
represent a pair of straight lines, it may be thrown into the form
AC-B *
-- A~~ ~ v
where u and v are linear functions of f, 77, ^, and the straight lines will be
real or imaginary, according as B 2 AC is positive or negative ; hence one or
else all of the quantities e lt e 2 , e 3 , will be necessarily negative, and the inter
sections will be all real or all imaginary, according as all three are negative
or only one is so. A cubic equation in e may be formed containing e lt e. 2) e 3
as its roots by eliminating between the equations
and the conditions for the reality of the intersections will be that all four
coefficients of this cubic shall be of the same sign, which in reality amount
only to two, since the first and last must in all cases have the same sign.
122 On the Correlations of two Conies [22
The same objection however of want of symmetry and consequent
irrelevancy and complexity attaches to this as much as to the method
originally proposed. The following treatment of the question relieves the
objection of want of symmetry as far as the coefficients of the same equation
are concerned, but in its practical application necessitates an arbitrary and
therefore unsymmetrical election to be made between the two sets of coeffi
cients appertaining to the two equations. It is however, I think, too curious
and suggestive to be suppressed.
I observe that if the four intersections are all real, an imaginary conic
cannot be drawn through them ; for the equation to an imaginary conic may
always be reduced to the form Ax 2 + By z + Cz 2 = 0, where A, B, C are all
positive and can therefore have at utmost one real point. Consequently
the case of total non-intersection is distinguishable from that of complete
intersection by the peculiarity that in the one case //, may be so taken that
U+fAV=Q shall represent an imaginary conic, that is, U + pV will be a
function whose sign never changes for real values of , 77, whereas in the
latter case no value of p will make U + ^ V = the equation to an imaginary
conic, and therefore U + pV will have values on both sides of zero. On the
other hand, it is obvious that an infinite number of real as well as unreal
conies may be drawn through four imaginary points of intersection. Con
sequently if we make J7+/u,F=0 (supposing the intersections of U and V
to be imaginary), there will be a range or ranges of values of p consistent,
and another range or ranges of values of p inconsistent with real values of
, 17, ; in other words, U/j,V treated as an equation between the four
variables , rj, , /A, will give one or more maxima or minima values of /u,
in the case supposed, but no such values when the intersections are two or
all of them real.
To determine these values of /JL, let dp = ; then we have
that is Ps(U fJ> V) = 0.
In order that any value of p found from this equation may be a maximum
or minimum, Lagrange s condition requires that
d , d , d\*
-TT. + K -j h I -ji,} p
dg drj d<s/
may be a function of unchangeable sign.
22] expressed by Indeterminate Coordinates. 123
dU dV Jr dp
Now -jTT = A 4 ~Jt + v jfc
d a ?
therefore since rf/u, = 0,
d^7_ d 2 F F d>
dp -^dp d?-
Hence
similarly
rf rf _ 1 r/T yi
2? ^~
&c. &c. &c.
Making now as before
U = at; 2 + brj" + &c.,
a fia = A, b fM@ = B, &c.,
the condition for /*, a root of a{U-pV}=0, giving /A a maximum or mini
mum, may be expressed by saying that
Atf + fife + Cl* + ZA kl + ZB hl + Whk
shall be unchangeable in sign for all real values of A, k, I.
The above quantity, by virtue of the equation a = 0, is always the
product of two linear functions. Hence we see, as above indicated, that if
all these pairs are real, that is, if all the points of intersection of V and V
are real, there is no maximum or minimum value of p, ; but if only one pair
be real and the other two pairs be imaginary, that is, if all the four inter
sections are imaginary, then two of the values of /A, namely those correspond
ing to the imaginary pairs, are real maxima or minima values of /A, but the
third is illusory.
Now I shall show that if 7=0 is a real conic, but the intersections of
[/"and Fare all unreal, the value of fi which makes U + pV the product of
real linear functions of , 77, t, is always one or the other extreme of the three
values of p, which satisfy the equation
Assume as the three axes of coordinates the three lines joining the
vertices of the quadrangle each with each, the two non-intersecting conies
may evidently be written under the form
U= G (x- + 7/ 2 ) - e (y- + z~) = 0,
F=- 7 (> 2 + / 2 ) + e(/ a + z-)= ;
124 On the Correlations of two Conies [22
these equations being only other modes of writing
U=
in which A, B, C; A , B , C will be real, because by hypothesis a((7+/&F) =
has all its roots real.
Hence x, y, z are linear functions of f, 77, , and consequently, by a simple
inference from a theorem of Prof. Boole*, the roots of p [U + pV} are
identical with those of
f* f* - f>
These latter are evidently -, -, - ; the third of which is the one
7 6 7 e
which makes U + /j.V the product of two real linears, for we have
<yU + cV= (ce - ye) (y- + z 2 ),
eU+ eV = (ec - ey} (x z + y"),
(7 - e) U+ (c - e) V- (ce - 67) (z* - Of.
Now
c c e _ ey ce
7 7 - e 7 (7 - e )
e c e _ ey ce
e 7 e e (y e)
and e, 7 are supposed to have the same sign, as otherwise V would be an
unreal conic ; hence the ascending or descending order of magnitudes of
the three values of X follows the scale - , - , , as was to be shown.
767-6
Imagine now lengths reckoned on a line corresponding to all values of
/u, from oo to + oo , and mark off upon this line by the letters A, B, C,
the lengths corresponding with the three roots of a (U+pV) = 0. Then
observing that when /u, = +oo, U + fj-V is of the same nature as F, and is
therefore a possible conic by hypothesis, and agreeing to understand by a
possible and impossible region of //,, a range of values for which U + p.V
corresponds to a possible and impossible conic respectively, one or the other of
the annexed schemes will represent the circumstances of the case supposed :
Poss. Reg. A Imposs. Reg. B Poss. Reg. C Poss. Reg.
1 i i
Poss. Reg. A Poss. Reg. B Imposs. Reg. C Poss. Reg.
i i i
But in either scheme it is essential to observe that the middle root of
Q(7-t-/iF) = divides a possible from an impossible region; and therefore
* See Postscript. f ^ 2 - x 2 = of course represents a real pair of lines.
22] expressed by Indeterminate Coordinates. 125
if we can find n, v, any two values lying between the first and second and
second and third roots of the above equation arranged in order of their
magnitude, one of the two equations U+vV=0, U+nV=0, will represent
a possible and the other an impossible conic: one such couple of values
may always be found by taking the roots of the quadratic equation
Hence calling the two roots thereof in and M, we see (which is in itself
a theorem) that one at least of the conies U+mV=0, U + MV=0, must
be a possible conic, provided only that F=0 be a possible conic: if both
U+mV and U+MV are possible conies, the intersections of U and Fare
all real, and if not, not*. The criteria for distinguishing possible from
impossible conies being well known need not be stated in this place.
We may of course proceed analogously by forming the two conies IU + V,
LU + V, where I and L are roots of -7- n [\U + V] = upon the supposition
ttA,
of U = being a possible conic.
If either of the two U and V be not possible, their intersections are of
course impossible, and the question is already decided.
It will be seen as pre-indicated that this method only fails in symmetry
because of the choice between the couples m, M, and I, L. But moreover
a perfect method for the discrimination of the two cases of unmixed inter
section one from the other should (perhaps ?) require the application of only
a single test (in lieu of the two conditions which the above method supposes),
over and above the condition which expresses -the fact of the intersections
being so unmixed. Such more perfect method I have not yet been able
to achieve.
Another interesting question of intersections remains to be discussed,
namely, supposing the two conies are known to be non-intersecting, how are we
to ascertain if they are external to one another, or if one contains the other ?
In order to settle this point we must first establish a criterion for determin
ing whether a given point is internal or external to a given conic ; the point
being in general said to be external when two real tangents can be drawn
from it to the curve, and internal when this cannot be done.
* It must be well observed however that the possibility of the conies U+mV&nd of U+MV
does not imply the reality of the intersections unless the conic V is known to be possible.
/ _ a
For if V be impossible e and y have opposite signs, and therefore --- is intermediate between
C y y
- and - , and the scheme for u, will be as here annexed :
e f
Impossible. A Possible. B Possible. C Impossible, tig? +
1 1 1
so that U+mV and U+MV will both represent possible conies.
126
On the Correlations of two Conies
[22
Let now
</> O, y, z) = aa? + by- + cz- + 2a yz + 2b zx + Zc xy = 0,
be the equation to any conic : I, m, n the coordinates of any point. Let
A = bo - a 2 , B = ca- b 2 , C = ab - c 2 ,
A = aa-b c, B = bb -c a f , C = cc -a V.
Then the reciprocal equation to the conic is
A? + Br)- + C? + 2Ar,t;+ 25 g + ZC fr = 0,
and in making 1% + mr) + n%= 0, the ratios of , 77, must be real if the
tangents drawn from I, m, n are real : this will be found to imply that the
determinant
A, C , B , I
C", B, A , m
B , A , C, n
I, m, n,
shall be negative*. This determinant may be shownf to be equal to the
product of the determinant
a, c, b
by the quantity
c, b, a
b , a , c
ul 2 + bin- + en 2 2a mn 2b ln 2c lm,
that is, equal to < (I, m, n) x D.
Hence I, m, n is internal or external to </> (x, y, z) according as <f>(l, m, n)
and n< have the same or contrary sign.
If $ (I, m, n) = 0, the point lies on the conic, and the point is neither
internal nor external ; if D $ = 0, the conic becomes a pair of straight lines,
and no point can be said either to be within or without such a system.
Hence our criterion fails, as it ought to do, just in the very two cases where
the distinction vanishes. I believe that this criterion is here given for
the first time.
* See theorem of the " Diminished Determinant " in Postscript to this paper.
t As we know a priori by virtue of a theorem given by M. Cauchy, and which is included as
a particular case in a theorem of my own, relating to Compound Determinants, that is, Deter
minants of Determinants, which will take its place as an immediate consequence of my funda
mental Theorem given in a Memoir about to appear. The well-known rule for the multiplication
of Determinants is also a direct and simple consequence from my theorem on Compound
Determinants, which indeed comprises, I believe, in one glance, all the heretofore existing
Doctrine of Determinants.
22] expressed by Indeterminate Coordinates. 127
To return to the two non-intersecting conies. Let us again throw them
under the form
F = k O 2 + f-) - fee 2 (z 2 + 7/ 2 ),
e and e being real, that is, U and F being both functions corresponding to
possible conies. Suppose U external to F; then any point in U is an
external point to F.
Take in U either of the two points represented by the equations y = 0,
x 2 = eV ; substituting these values of y and x, V becomes k (e 2 e 2 ) 2 ,
and oF becomes -frV(l-e 2 ); therefore (1 - e 2 ) (e 2 - e 2 ) must be positive,
that is, e 2 must be one of the extremes of the three values 1, e 2 , e 2 . In like
manner, if F is external to U, e will be also one of the extremes of the same
three quantities ; and hence, if the two conies are mutually external, unity
will be the middle magnitude of the group e 2 , I, e 2 .
Now the three roots of n (F + \U) = 0, are
e 2 , 1 e 2
X = K. X = K . A . = K - .
e 2 1 - e-
Hence if U and F be without one another, or, as it may be termed, are
extra-spatial, the third value of X will be of a different sign from the first
two ; but if the two conies be co-spatial, that is, if one includes the other, all
the three values of X will have the same sign. Hence we have the following
elegant criterion of co-spatiality of two possible conies expressed by the
equations U = Q, F= 0, between indeterminate coordinates , ?;, ; the
coefficients of the cubic function a ( X ?7 + /it F) must give only changes or
fif
only continuations of sign.
If this test be not satisfied, it will remain to determine which of the two
conies contains, and which is contained by the other. Let U contain F,
1 e 2
then the order of magnitudes will be 1, e 2 , e 2 ; therefore k _ , is greater
1 e 2
than k, and therefore k -, which is that root of the equation D( F+X?7) =
1 - e-
which is always one or the other of the extremes, is the greatest of the three.
Hence the scheme for the impossible and possible regions of X will be as
below :
Poss. A Imposs. B Poss. C Poss. g5f +00
Hence if the two roots of ^- {V + \U} = be I and L, and of the two
conies V+IU = Q, F+ LU=Q, the former be the possible, and the latter the
impossible one, U contains F or is contained in it according as I is greater
or less than L.
128 On the Correlations of two Conies [22
Observe that if U and V be non-cospatial, so that the three values of
/i in a (U+fj,V) = have not all the same sign and consequently zero lies
between the greatest and least of them, it will not be necessary to make
trial of the characters of the two curves ?7+raF=0, and U+MV=Q,
in order to ascertain whether U and V intersect or not ; for it will be
sufficient to find which of the two quantities ra and M substituted for fj.
in o(7-f /u,F) causes it to have the opposite sign to o(f + OF), that is,
nil, and this one of the two it is, if either, which will make U + fiV an
impossible conic, and will thus alone serve to determine whether the inter
sections of U and V are unreal, or the contrary.
It might be a curious question to consider whether, in a certain sense,
conies not both possible may not be said to lie one within or without the
other. Upon general logical grounds, I think it not improbable that two
impossible conies might be discovered each to contain the other; but this is
an inquiry which I have not had leisure to enter upon.
I have thus far supposed the roots of n(\U+ F)=0 to be all distinct
from one another. I now approach the discussion of the contact of two
conies, in which event two or more of the roots will be equal. The condition
for simple contact is evidently a p (XZ7 + /*F) 0.
" AM rt
The unpaired value of X in a(\U + V) makes \U + V an impossible
pair of lines, and therefore, in the scheme for X drawn as above, will separate
the possible from the impossible region.
Whether the conies intersect in two real or two unreal points, besides the
point of contact, will be known at once by ascertaining whether E7 +/iF=0
represents two real or two imaginary lines. If the latter, the two curves lie
dos-d-dos or one within the other, according as the successions of sign in
D (\U + F) are all of the same kind or not ; if they be all of the same kind,
one will include the other, namely, U will include F if the equal roots are
greater, and be included in it if they be less than the unequal one. This last
conclusion however, it should be observed, is inferred upon the principle of
continuity, by making two values of X approach indefinitely near to one
another, but cannot be strictly deduced from the equations given for U and
F applicable to the general case, in which the axes of coordinates are the
three axes joining the vertices ; since these latter, in the case supposed,
reduce to two only, and consequently such representation of U and F
becomes illusory.
If all three values of X, are equal, the three vertices come together,
and hence the two conies will have three consecutive points in common,
that is, will have the same circle of curvature. On this supposition the two
curves cut at the point of contact, and all four points of intersection are
of course real.
22] expressed by Indeterminate Coordinates. 129
The classification of contacts between two conies maybe stated as follows:
Simple contact = one case.
Second degree contact = two cases, namely, common curvature or double
contact.
Third degree contact = one case, namely, contact in four consecutive points.
These four cases of course correspond to the several suppositions of there
being two equal roots, three equal roots, two pairs of equal roots, or four
equal roots in the biquadratic equation obtained between two variables by
elimination performed in any manner between the given equations in the
two conies.
. The first species and the first case of the second species have been already
disposed of. I proceed to assign the conditions appertaining to the second
case of the second species, when U and V have a double contact.
Let A, A , B, E be the two pairs of coincident points in which the
conies are supposed to meet; either pair of lines AB, A B , and AB , A B,
becomes a coincident pair. Hence such a value of yu, can be found as will
make U + fiV the square of a linear function of , 77, ". If therefore we
make U + fj,V= W, and form the determinant
d*W d*W_ d-W_
~d?
d*W
d?
p, q, r, o
= A p- + B(f + Or 2 + ZFqr + 2Grp + 2Hpq,
where all the coefficients are quadratic functions of //,, and make
A = Q, 5 = 0, C = 0, F=Q, G = 0, H = 0,
each of these six equations in fj, will have one and the same root in common.
It is, however, enough to select any three ; if these vanish together
for any value of //,, the remaining three must also vanish. This is a simple
application of a general law * which will appear in a forthcoming memoir on
"Determinants and Quadratic Forms," of which this paper is to be considered
as an accidental episode.
* For statement of this law called the Homaloidal Law, see Philosophical Magazine of this
month "On Certain Additions, &c." [p. 150 below. ED.]
s. 9
130 On the Correlations of two Conies [22
Take now any three of the six equations which for the sake of generality
call P = 0, Q = 0, R = 0. The hypothesis of double contact requires that
P and Q, Q and R, R and P shall have a factor in common; but these
conditions are not sufficiently explicit for our present object, since P, Q, R
might be of the form
K(\-O)(\- b), K (\ -b)(\- c), K" (\ - c) (X - a),
and would thus satisfy the conditions above stated, without P, Q, R having
a common factor. A sufficient criterion is that fQ + gR and P shall have
a common factor for all values of/ and g.
Let then the resultant of fQ + gR and P be
we must have
L=0, iM=0, N = Q,
where L is the resultant of P and Q,
N R and Q ;
and M is a new function, which if we call Q = <f> (\), R=^ (X), and suppose
a and 6 to be the two roots of P = 0, is easily seen to be equal to
<f)d . tyb + <f>b . ^ra.
This I call the connective of P . Q and P . R.
L, M, N may conveniently be denoted by the forms
P.Q, P.R, Q.P.R.
We may now take more generally
aP + bQ + cR,
aP+PQ + yR,
which will have a factor in common for all values of a, b, c, a, ft, y.
I am indebted to Mr Cayley for the remark that the resultant of these
two functions is a new quadratic function, which, according to my notation
just given, may be put under the form
PQ (a/3 - 6a) 2 + QR (by - c/3) 2 + RP (ca - ay)*
+ PRQ (by - c/3) (ca- 07) + QPR (ca - ay)(a/3 - la) + RQP (a/3 - ba)(by - c/3).
Ternary systems of the six coefficients formed upon the type of (PQ,
PQR, QR), I call complete systems, because the three functions included in
such a system equated severally to zero, imply that the remaining three
coefficients are all zero. Such a system as (PQ, QR, RP) I term an incom
plete ternary system as not drawing with it the like implication. Probably (?)
-we should find on investigation that P^Q, QPR, RQP, would also be an
22] expressed by Indeterminate Coordinates. 131
incomplete system, but that systems formed after the type of PRQ, RQ, RQP
are complete. This however is only matter of conjecture, as I have been too
much occupied with other things to enter upon the inquiry. The distinct
types of ternary systems are altogether six in number, namely, four of a
symmetrical species,
PQ, QR, RP,
PRQ, QPR, RQP,
PQ, PQR, QR,
PRQ, RQ, RQP;
and two of an unsyrnrnetrical species, namely,
PQ, PQR, PR,
PRQ, RQ, QPR*
If instead of confining ourselves to three out of the six original quantities,
A, B, C; F, G, H, we take them all into account, and write down the
resultant of
a A + bB + cC +fF + gG + hH,
we shall obtain a quadratic function of 15 variables (not however all indepen
dent) having 120 coefficients, all of which must be zero. It would be
extremely interesting to determine how many complete ternary groups can
be formed out of these 120 terms.
It will be recollected that we have assigned as the condition of contact
in three consecutive points, that a certain cubic equation shall have all its
roots real. Now, as well remarked by Mr Cayley, we cannot express this
fact by less than three equations in integral terms of the coefficients. Thus
if the cubic be written
aX 3 + 3b\- + 3cX + d = 0,
we have as one of such ternary systems,
U=ac-b- = 0, V=ld-c" = 0, W=bc-ad = 0.
The significant parts of these equations are of course, however, capable of
being connected by integral multipliers U , V, W, such that
U U+V V+ W W = 0.
* PQ, QR, RP, may be compared in a general way with the angles, and PRQ, QPR, RQP,
with the sides of a triangle.
92
132 On the Correlations of two Conies
Any number of functions U, V, W so related, I call syzygetic functions,
and U , V, W I term the syzygetic multipliers*. These in the case
supposed are c, a, b, respectively.
In like manner it is evident that the members of any group of functions,
more than two in number, whose nullity is implied in the relation of double
contact, whether such group form a complete system or not, must be in
s y z ygy-
Thus PQ, PQR, QR, must form a syzygy ; nor is there any difficulty
in assigning a system of multipliers to exhibit such syzygy. Galling
P=<(X), R=ilr(\), a and 6 the two roots of Q = 0, I have found that
{(^-a) 3 + (i/rfc) 3 } PQ - (<j>a . ^a + </>> . +1) PQR + {(</>) 4- (<&) } QR = 0.
Again, if we take the incomplete system
(PQ), (QR), (RP),
it will be found that
L (QR) + M (RP) + N (PQ) = 0,
provided that, calling a, b; c, d; e, f, the roots of P = 0, Q = 0, R=0,
respectively, we make
7- /7 , 7 ,7 ./I? ,\( a - c }( a -d)(a- e)(a-f)
L (KQ + fc 1 a + k 2 a? + k s a 3 + & 4 a 4 ) ~
+ (* + k lb + tjf + kjt + W - --- ,
~ CL
MT // 7 ,\ (c a) (c b) (c d) (c e) (c f)
M = (& + &! c + /t-oC 2 + 7t 3 c 3 + k^) - -~f- ^-
*~~" Ct
c
, T . 7 7 7 7 7 ,. (e a) (e b) (e c) (e d)
N = (k + l\e + & 2 e 2 + k 3 e 3 + k 4 e 4 ) N - -^
e ~
k , k lt k 2 , k 3 , /t* 4 being quite arbitrary, and L, M, N, although presented in
a fractional form, being essentially integral.
This fact of L, M, N constituting a system of multipliers to the syzygy
QR, RP, PQ, is easily demonstrated ; for
QR=(c-e)(c-f)(d-e)(d-f),
RP = (e-a)(e-b)(f-a)(f-b),
PQ = ( a -c)(a- d) (b -c)(b- d).
* There will be in general various such systems of multipliers.
22] expressed l>y Indeterminate Coordinates. 133
Hence L (QR) + M (RP) + N(PQ)
= (a _ c)(o _ rf) (a - e )(o -f)(b - c)(b - d)(b - e )(b-f)(c-e)(c-f)(d-e)(d-f)
k + k^a + k 2 a" + k s a? + & 4 a 4
(a &X c) (a d) (a e) (a /)
My theory of elimination enables me to explain exactly the nature of
L, M, N, and the reason of their appearance as syzygetic factors.
Let L r , M. r , N,. signify what L, M, N become, when all the k s except k r
are taken zero. Then the theory given by rne in the Philosophical Magazine
for the year 1838, or thereabouts^, shows that L \ + L^ is the prime derivee
of the first degree between the two equations P and Q x R, or, in other
Off
words, will be the remainder integralized of p .
In like manner M \ + M 1} N \ + N l are the integralized remainders of
RP f PQ
j=r- and or -^ respectively.
Q R
If now the resultant of P, Q and of Q, R are each zero, but the resultant
of P and R is not zero, it will be evident that P, Q, R must be of the form
/(X + a)(X + c), g(\ + c)(\ + d), h(\ + d) (\ + b);
and therefore P x R will contain Q, and consequently we must have
More generally, if we write
Q-0,
P x R = 0,
and eliminate dialytically, that is, treating X 4 , A, 3 , X 2 , X as distinct quantities,
we shall obtain *
X 4 : X 3 : X 2 : X : 1 : : M t : M 3 : M 2 : M, : M ;
and therefore when P x R contains Q,
* This cannot be obtained directly from what is stated in the paper referred to, although
contained in the general theory of derivation there given. The arbitrary functions which enter
into the expression for the general derivees have been in that paper evaluated only for the prime
derivees, which however are only particular phenomena, with reference to the general results of
Dialytic Elimination. Hereafter I may give a more general exposition of this remarkable,
although ignored or neglected theory. The prime derivees of fx and f x are Sturm s Functions,
cleared of quadratic factors, and are expressed by virtue of the general theorems there laid down
as functions of .c and of symmetrical functions of the roots of fx. [t p. 40 above. ED.]
134 On the Correlations of two Conies [22
In like manner, when Q x P contains R,
N = Q, ^ = 0, .iV 2 = 0, N 3 = 0, iV 4 =0;
and when R x Q contains P,
Z = 0, A = 0, 2 = 0, Z 3 = 0, Z 4 = 0.
Accordingly, we see from the equation
L(QR) + M(RP) + N(PQ) = 0,
that if QR = 0, JRP = ; but PQ not = 0, then JV= ; and therefore
^ = 0, iV 1 = 0, iV 2 =0, iV 3 = 0, J\ r 4 = 0,
and so in like manner for the remaining corresponding two suppositions*.
Before proceeding to consider the remaining case of the highest species
of contact, I must observe that besides the equations involved in the condi
tion that A, B, C ; F, G, H, or, which is the same thing, that any three
of them shall all have a factor in common, we must have D (U + \V) con
taining the square of such common factor. In the memoir before adverted
to a general theorem will be given and proved, which shows that this latter
condition is involved in the former one ; in fact, more generally (but still
only as a particular case) that when U and V are quadratic functions of n
letters, but U + eF admits of being represented as a complete function of
(n 2) quantities only, which are themselves linear functions of the n letters,
then D([/ + XF), which is of course a function of X of the ?ith degree, will
contain the factor (X e) 2 .
When the two conies have four consecutive points in common, the
characters of double-point contact and of contact in three consecutive points
must exist simultaneously; and consequently the factor common to A, B, C;
F, G, H, will enter not as a binary but as a ternary factor into n (U+ \V).
This gives the extra condition required. As an example take the two conies,
U = ^-, + a? - z* = 0,
J. A*
V = f + a? - Zkxz + ( 2k - 1) z- = 0,
* Since we are able to assign the values of the syzygetic multipliers in the equations
L (PQ) + M (QR) +X
L (PQ) + 3I (PQR) + N
L" (QR) + M" (QEP) + N"
L " (IIP) + J/ " (BPQ) +N " (PQ) = 0,
it follows that we may eliminate between these four equations any three of the six quantities
(PQ), (P-RQ), &c., and thus express any one of them in terms of any two others : this method,
however, is not practically convenient. I may probably hereafter return to this subject.
22] expressed by Indeterminate Coordinates. 135
The complete determinant of U + XF is then
{1 +(1 -k) X} {(1 +X) 2 - 2&X(1 + X) + kW] -- l {!+(!
A, B, C are the determinants of 7+ XV, when # = 0, y = 0, s = 0, respectively.
Thus
C = k 2 \- - (1 + X) (1 + X (1 - 2&)} = X 2 (1 - fc) a - 2X (1 - k) - I ;
X = , , makes A = 0. B = 0. C = 0, and the factor X + - y- enters cubed
1 JB 1
into a (U + \V).
Hence the two conies have a contact of the third order.
This is easily verified ; for if we pass from general to Cartesian and
rectangular coordinates, and make z unity; 7=0 will represent an ellipse
with centre at the origin, eccentricity \/k, and mean focal distance 1, and
V = the circle of curvature at the extremity of the axis major*.
I had intended to have added some other remarks connected with the
present discussion, and also to have appended an a posteriori proof of the
propositions relative to the reality and otherwise of the vertices and chordal
pairs of intersection which I have, at the commencement of this paper,
deduced quite legitimately, but in a manner not at first sight perhaps easily
intelligible, from the general principles of conjugate forms ; but this dis
cussion has run on already to a length so much greater than I had
anticipated and than the importance of the inquiry may seem to justify,
that I must reserve for a future number of the Journal what further matter
I may have to communicate concerning it.
POSTSCRIPT. As I have alluded to Professor Boole s theorem relative to Linear
Transformations, it may be proper to mention my theorem on the subject, which
is of a much more general character, and includes Mr Boole s (so far as it refers to
Quadratic Functions) as a corollary to a particular case. The demonstration will
be given in the forthcoming memoir above alluded to.
Let U be a quadratic function of any number of letters x 1} x. 2 ... x n , and let
any number r of linear equations of the general form
I a r x 1 + . 2 a r x.,+ ...... + n a,.x n = 0,
* We have thus discussed all the four cases of biconical contact: for an exactly parallel
discussion of the theory of contact of a plane with the curve of double curvature in which two
surfaces of the second order intersect, see the paper in the Philosophical Magazine for this month,
before referred to. [p. 148 below. ED.]
136
On the Correlations of two Conies
[22
be instituted between them : and by means of these equations let U be expressed
as a function of any (n ?) of the given letters, say of x r+lf x r+2 aj, t , and let
V, so expressed, be called M. Let
jOty ^1 ~~t~ owj.&ij *r~ i Tjwj.tt/^
be called L r . Then the determinant of M in respect to the (n r) letters above
given is equal to the determinant of
U + Lix n+1 + L. 2 x ll+2 + + L,.x n+r ,
considered as a function of the (n + r) letters
X^X-2 X H + r ,
divided by the square of the determinant
1 /*) o r j tv
This I call the theorem of Diminished Determinants.
If now we have U a function of r letters, and V of r other letters, and V is
derived from U by linear transformations, that is, by r equations connecting the
2r letters ; then, since U may be considered as a function of all the 2r letters with
abortive coefficients for all the terms where any of the second set of r letters enter,
we may apply our theorem of diminished determinants to the question so con
sidered, and the result may be found to represent Mr Boole s theorem in a form
rather more general and symmetrical, but substantially identical with that given
by Mr Boole.
Thus suppose %ax* + bxy + %cy 2 say P, and |aw 2 + ftuv + yv 2 say Q, are mutually
transformable by virtue of the linear equations
Ix + my = At* + ju.v,
I x + m y = X u + fj. v,
P may be considered as a function of x, y, u, v, and Q as the value of P, when we
eliminate x and y by virtue of the two linear equations
.Z/j = Ix + my \u p.v = 0,
L 2 = I x + m y - \ u p. v = ;
we have therefore by our theorem the determinant of Q equal to the squared
reciprocal of the determinant
I, m
I , m
multiplied
by the determinant
, b, 0, 0,
I, I
,
b, c, 0, 0,
m, m
o,
0, 0, 0, -
-A, - A
o,
0, 0, 0, -p., -//
I, m,
A, -M,
0,
I , m, -A , - p,
0,
22]
expressed by Indeterminate Coordinates.
137
which last determinant is evidently equal to the determinant of P multiplied by
A, ft
the square of the determinant
divided by the square of
the square of
I, m
A,
X ,
A ,
Whence we see that the determinant of Q
, is equal to the determinant of P divided by
There is also another way more simple, but less direct, by
means of which the theorem of diminished determinants may be made to yield
Mr Boole s theorem of transformation*. Some unavowed use has been made
in the foregoing pages of this former theorem, one of the highest importance in
the analytical and geometrical theory of quadratic functions. It has been nearly
a year in my possession, and I trust and believe that I am committing no act of
involuntary misappropriation in announcing it as a result of my own researches.
* Namely, by considering P and Q as each derived from some common function of x, y, u, v,
w, by means of the equations L 1 = 0, L 2 = 0; the law of Diminished Determinants will then indicate
the determinants of P and Q, each under the form of fractions having the same numerator, but
whose denominators will be
and
Z, m
I , m
respectively.
23.
AN INSTANTANEOUS DEMONSTRATION OF PASCAL S
THEOREM BY THE METHOD OF INDETERMINATE
COORDINATES.
[Philosophical Magazine, xxxvu. (1850), p. 212.]
THE new analytical geometry consists essentially of two parts the one
determinate, the other indeterminate.
The determinate analysis comprehends that class of questions in which
it is necessary to assume independent linear coordinates, or else to take
cognizance of the equations by which they are connected if they are not
independent. The indeterminate analysis assumes at will any number of
coordinates, and leaves the relations which connect them more or less
indefinite, and reasons chiefly through the medium of the general pro
perties of algebraic forms, and their correspondencies with the objects of
geometrical speculation. Pascal s theorem of the mystic hexagon, and the
annexed demonstration of its fundamental property, belong to this branch of
the subject, and afford an instructive and striking example of the application
of the pure method of indeterminate coordinates.
Let x, y, z, t, u, v be the sides of a hexagon inscribed in the conic U. Let
the hexagon be divided by a new line $ in any manner into two quadri
laterals, say xyzfy, tuv<f>.
Then ay$ + l>xz = U CLU^> + /3tv ;
therefore (ay au) <f> = {3tv bxz ;
therefore ay in and <f> are the diagonals of the quadrilateral txvz.
By construction, <J> is the diagonal joining x, v (that is, the intersection of
x and v) with z, t\ and thus we see that ay au is the line joining t, x with
v, z\ but this line passes through y, u. Therefore x, t ; y, u ; z, v lie in one
and the same right line. Q.E.D.
24.
ON A NEW CLASS OF THEOREMS IN ELIMINATION
BETWEEN QUADRATIC FUNCTIONS.
[Philosophical Magazine, xxxvu. (1850), pp. 213 218.]
IN a forthcoming memoir on determinants and quadratic functions, I have
demonstrated the following remarkable theorem as a particular case of one
much more general, also there given and demonstrated.
Let U and V be respectively quadratic functions of the same 2n letters,
and let it be supposed possible to institute n such linear equations between
these letters as shall make U and V both simultaneously become identically
zero*. Then the determinant of \U + /j,V, which is of course a function of
X and fju of the 2nth degree, will become the square of a function of X and /u
of the nth degree ; and conversely, if this determinant be a perfect square, U
and V may be made to vanish simultaneously by the institution of n linear
equations between the 2n letters f.
Let now P and Q be respectively quadratic functions of three letters only,
say x, y, z ; and let
U = P + (lx + my + nz) t,
V= Q + k (Ix + my + nz) t.
The determinant of \U+ p.V in respect to x, y, z, t is easily seen to be
(\ + kfj,)- x the determinant of
\P + fiQ + (Ix + my + nz} t
in respect to x, y, z, t. Hence if we call
\P + pQ + (Ix + my + nz) t = W,
and make ^ IF a squared function of X. 11 or which is the same thing, if
xyzt
A/it xyzt
* In the more general theorem above alluded to, the number of letters is any number m, the
number of linear equations being any number not exceeding .
2i
t When u=l, we obtain a theorem of elimination between two quadratics, which has been
already given by Professor Boole.
140 On a new Class of Theorems [24
U and V will vanish simultaneously when two linear relations are instituted
between the quantities (all or some of them) x, y, z, t.
In order that this may be the case, it will be seen to be sufficient that
P = 0, Q = 0, (lx + my + nz) = 0,
shall coexist ; for then two equations between x, y, z of which lx+my+nz = Q
will be one, will suffice to make U and V each identically zero. Hence we
have the following theorem :
n n {\tr+u,V+{la;+my+nz)t}
A/u xyzt l
is a factor of the resultant of
P = 0, Q = 0, lx + my + nz = 0.
A comparison of the orders of the resultant and the determinant shows
that they must be identical, d-ci-pres, of a numerical factor, which, if the
resultant be taken in its general lowest terms, may no doubt be easily shown
to be unity.
As an illustration of our theorem, let
P = xy + yz + zx,
Q = cxy + ayz + bzx.
Then
0, X + en, ^ + b/A, <
{XP + /J.Q + (lx + my + nz) t} =
X + c/i, 0, X + a/ji, m
X + afj,, 0, n
I, m, n,
n"
- 2lm (\ + In) (X + ap) - 2mn (X + c/*) (X + bp) - 2nl (\ + ap) (X + cp)
= \- {n y + m- + I- - Zlm - 2mn - 2nl]
+ 2X/Lt {en 2 + bm- + al- - Im (a + b)- mn (b + c) - nl (c + a)}
-\- n" {c 2 n 2 + 6 2 ??i a + a-l- 2ablm Zbcmn
And we thus obtain, finally,
= (n 2 + m 2 + 1- 2lm Imn 2nl)
x (c 2 w 2 + b"m- + a?l- 2ablm 2bcmn - 2canl)
- {(en 2 + bm- + al* - Im (a + b)- mn (b + c)-nl(c + a)} 2
= - 4lmn {(a - b) (a - c) I + (b - a) (b - c) m + (c-a)(c- b) n}.
24] in Elimination between Quadratic Functions. 141
Now to obtain the resultant of
xy + yz + zx = 0,
cxy + azy + bxz = 0,
Ix 4- my + 112 = 0,
we need only find the four systems in their lowest terms of x : y : 2, which
satisfy the first two equations, and multiply the four linear functions obtained
by substituting these values of x, y, z in the fourth : the product will contain
the resultant of the system affected with some numerical factor. In the
present case, the four systems of x, y, z are
2 = 0, # = 0, y = 1,
# = ( a -&)(- c), y = (b-a}(b-c\ z = (c - a) (c - 6),
and accordingly the product of
IX-L + iny 1 + nz l ,
lx% + my r 2 + nz z ,
Ix 3 + my 3 + nz 3 ,
lx + my 4 + nz 4 ,
becomes
Imn {(a - 6) (a - c) I + (b - a) (6 - c) m + (c- a) (c - 6) n],
agreeing with the result obtained by my theorem, a special numerical
factor -i, arising from the peculiar form of the equations, having disappeared
from the resultant.
A geometrical demonstration may be given of the theorem which is
instructive in itself, and will suggest a remarkable extension of it to
functions containing more than three letters; the equation
a {\U + fiV + (Ix + my + nz) t] = 0,
xyzt
which is a quadratic equation in X : /z, may easily be shown to imply that the
conic \U + pV is touched by the straight line
Ix + my + nz = 0.
And we thus see that in eneral two conies,
passing through the intersections of two given conies,
cr = o, F=O,
On a new Class of Theorems
r-24
may be drawn to touch a given line. If, however, the given line passes
through any of the four points of intersection, in such case only one
conic can be drawn to touch it ; accordingly
n n (X U -f /JL V + (lx + my + nz) t}
must be zero when I, m, n are so taken as to satisfy this condition, that is, if
lx l + my l + nz 1 = 0,
or lx., + my., + nz., = 0,
or Ix 3 + my 3 + nz 3 0,
or lx + my 4 + nz 4 = 0,
whence the theorem.
Now suppose U and V to be each functions of four letters, x, y, z, t;
when
o (\ U + u V + (lx + my + nz + pt) u} = 0,
xyztu l
the conoid \U + pV touches the plane
lx + my + nz + pt = ;
and D = being a cubic equation, in general three such conoids can be drawn.
Considerations of analogy make it obvious to the intuition, that in the
particular case of two of these becoming coincident, the given plane
lx + my + nz + pt
must be a tangent plane to those two coincident conoids at one of the points
where it meets the intersections of (7=0, V = ; that is
lx + my + nz + pt =
will pass through a tangent line to, or in other words, may be termed
a tangent plane to the intersections. Hence the following analytical
theorem, derived from supposing q, r, s, t to be proportional to the areas
of the triangular faces of the pyramid cut out of space by the four
coordinate planes to which x, y, z, t refer. As these planes are left
indefinite, q, r, s, t are perfectly arbitrary.
Theorem. The resultant of
1.
2.
3.
4.
U=0
V
I , where U and V are functions of x, y, z, t ;
lx + my + 1.
dU
dU
dU
dU
dec
dy
~dz
dt
dV
dV
dV
dV
dx
dy
dz
dt
I,
m,
n,
P
q>
r,
s,
t
= 0;
24] in Elimination between Quadratic Functions. 143
which system, it will be observed, consists of three quadratic functions, and
one linear function of x, y, z, t, contains the factor
pt)u}.
o \\U+ uV +
xyzt l
+ my + nz
This last quantity is of the 4 x oth, that is, the 12th order in respect of
the coefficients in U and F combined ; of the 4 x 2th, that is, the 8th order
in respect of I, m, n, p ; and of the zero order in respect of q, r, s, t.
The resultant which contains it is of the (4 + 4 -f 2 . 4)th, that is, 16th
order in respect to the coefficients in U and F; of the (4 + 8)th, that is, the
12th, in respect of I, m, n,p\ and of the 4th in respect of q, r, s, t. Hence
the special (and, as far as the geometry of the question is concerned, the
unnecessary, I may not say extraneous or irrelevant) factor which enters into
the resultant is of the 4th order in respect to the combined coefficients of U
and F* ; and of the same order in respect to I, m, n, p, and in respect to
q, r, s, t.
I have not yet succeeded in divining its general value.
In the very particular example, of the system,
aa? + &y* = 0,
c* 2 + dt* = 0,
Ix + my + nz + pt = 0,
j ouv, /3y, 0,
0, 0, cz,
1, m, n,
q, 0, 0,
I find that the double determinant is
c 2 rf 2 2 /3 2 (cp 2 + dn*y* (m 2 a + l 2 /3)-,
and the resultant is
2 4 c 2 d 2 a 2 /3 4 (cp 2 + d /t 3 ) 4 (m 2 a + 1-/3)-,
giving as the special factor
q 4 ^- (cp 3 + dn-)\
I believe that the theorem which I have here given for determining the
condition that lx+ my + nz+pt shall be a tangent plane to the intersection
of two conoids U and F, namely, that the determinant of
\U+ /j-V + (lx + my + nz + pt)u
shall have two equal roots, is altogether novel.
* And consequently of the second in respect to the separate coefficients of each.
dt
p
= 0,
144 On a new Class of Theorems in Elimination. [24
What is the meaning of all three roots of this determinant becoming
equal, that is, of only one conoid being capable of being drawn through the
intersection of U and V to touch the plane
Ix + my + nz +pt ?
Evidently (ex vi analogue) that this plane shall pass through three con
secutive points of the curve of intersection, that is, that it shall be the
osculating plane to the curve.
If we return to the intersection of two co-planar conies, and if we suppose
a line to be drawn through two of the points of intersection, the conies
capable of being drawn through the four points of intersection to touch the
line, besides becoming coincident, evidently degenerate each into a pair
of right lines. It would seem, therefore, by analogy, that if a plane be
drawn including any two tangent lines to the curve of intersection of two
surfaces of the second degree, this should be touched by two coincident
cones drawn through the curve of intersection, and consequently every such
double tangent plane to the intersection of two conoids (and it is evident
that one or more of these can be taken at every point of the curve) must
pass through one of the vertices of the four cones in which the intersection
may also be considered to lie ; and it would appear from this, that in general
four double tangent planes admit of being drawn to the curve, which is the
intersection of two conoids, at each point thereof. At particular points
a tangent plane may be drawn passing through more than one of the
vertices, and then of course the number of double tangent planes that can
be drawn will be lessened. These results, indicated by analogy, become
immediately apparent on considering the curve in question as traced upon
any one of the four containing cones. For the plane drawn through a
tangent at any point, and the vertex of the cone being a tangent plane
to the cone, must evidently touch the curve again where it meets it. We
thus have an additional confirmation of the analogy between a point of
intersection of two curves and the tangent at any point of the intersection
of two surfaces.
I might extend the analytical theorems which have been established for
functions of three and four to functions of a greater number of variables;
but enough has been done to point out the path to a new and interesting
class of theorems at once in elimination and in geometry, which is all that
I have at present leisure or the disposition to undertake.
25.
ADDITIONS TO THE ARTICLES*, "ON A NEW CLASS OF
THEOREMS," AND "ON PASCAL S THEOREM."
[Philosophical Magazine, xxxvu. (1850), pp. 363370.]
FIRST addition. I have alluded in the second of the above articles to a
more general theorem, comprising, as a particular case, the theorem there
given for the simultaneous evanescence of two quadratic functions of
2;i letters, on n linear equations becoming instituted between the letters.
In order to make this generalization intelligible, I must premise a few
words on the Theory of Orders, a term which I have invented with particular
reference to quadratic functions, although obviously admitting of a more
extended application. A linear function of all the letters entering into a
function or system of functions under consideration I call an order of the
letters, or simply an order. Now it is clear that we may always consider
a function of any number of letters as a function of as many orders as there
are letters; but in certain cases a function may be expressed in terms of
a fewer number of orders than it has letters, as when the general character
istic function of a conic becomes that of a pair of crossing lines or a pair of
coincident lines, in which event it loses respectively one and two orders, and
so for the characteristic of a conoid becoming that of a cone, a pair of planes
or two coincident planes, in which several events, a function of four letters
becomes that of only three orders, or two orders, or one order, respectively.
When a function may be expressed by means of r orders less than it contains
letters, I call it a function minus r orders. I now proceed to state my
theorem.
Let U and V be functions each of the same m letters, and suppose that
the determinant in respect of those letters of U + pV contains i pairs of
[* pp. 138, 139 above. ED.]
o IV
146 On a new Class of Theorems. [25
equal linear factors of /A ; then it is possible, by means of i linear equations
instituted between the letters, to make U and V each become functions of
the same m 2i orders ; and conversely, if by i equations between the letters
U and V may be made functions of the same m 2i orders, the determinant
of U+ /jiV considered as a function of p will contain i square factors.
Thus when m = 2/i and i = n, U and V will each become functions of
zero orders, that is, will both disappear, provided that on the institution of
a certain system of n linear equations, among the letters of which U and V
are functions, the determinant of (U + ^V} is a perfect square, which is
the theorem given in the article referred to.
So for example if U and V be quadratic functions of four letters, and
therefore the characteristics of two conoids, a(U+fJbV) being a perfect square,
expresses that these conoids have a straight line in common lying upon each
of their surfaces.
If U and F be quadratic functions of three letters only, and admit there
fore of being considered as the characteristics of two conies, a ( U + //. V}
containing a square factor, is indicative of these conies having a common
tangent at a common point, that is, of their touching each other at some
point ; for it is easily shown that the disappearance of two orders from any
quadratic function by virtue of one linear function of its letters being zero,
indicates that the line, plane, &c. of which the linear function is the
characteristic is a tangent to the curve, surface, &c. of which the quadratic
function is the characteristic.
I pass now to a generalization of the theorem which shows how to
express, under the form of a double determinant, the resultant of one linear
and two quadratic homogeneous functions of three letters (which I should
have given in the original paper, had I not there been more .intent upon
developing an ascending scale than of expatiating upon a superficial ramifi
cation of analogies), and which constitutes my Second addition to that paper,
to wit
If U and F be homogeneous quadratic, and L lt L 2 ... L n homogeneous
linear functions of (n + 2) letters ac lt 4... x n+ *, the determinant of the entire
system of n + 2 functions is equal to
(Hi i i (XJ7 + //.F + jMi + L n L + ... +L n t n ] ;
\, I* X t , X% ...Xn+ztu t 2 ... t n
the demonstration is precisely similar to the analytical one given in the
September Number* for the particular case of ?i = l.
When n = 0, we revert to Mr Boole s theorem of elimination between U
and F already adverted to. The proof, it will be easily recognized, does not
require the application of the more general theorem relative to the simul-
[* p. 140 above.]
25] On a new Class of Theorems. 147
taneous depression of orders of two quadratic functions, but only the limited
one before given, which supplies the conditions of their simultaneous
disparition. I now proceed to develope more particularly certain analogies
between the theory of the mutual contacts of two conies, and that of the
tangencies to the intersection of two conoids.
But here again I must anticipate some of the results which will be given
in my forthcoming memoir on Determinants and Quadratic Functions,
by explaining what is to be understood by minor determinants, and the
relation in which they stand to the complete determinant in which they are
included. This preliminary explanation, and the statement of the analogies
above alluded to, will constitute my Third and last addition.
Imagine any determinant set out under the form of a square array of
terms. This square may be considered as divisible into lines and columns.
Now conceive any one line and any one column to be struck out, we get
in this way a square, one term less in breadth and depth than the original
square ; and by varying in every possible manner the selection of the line
and column excluded, we obtain, supposing the original square to consist of
n lines and n columns, n 2 such minor squares, each of which will represent
what I term a First Minor Determinant relative to the principal or complete
determinant. Now suppose two lines arid two columns struck out from the
( / 1 \"\
original square, we shall obtain a system of j 2 j squares, each two
terms lower than the principal square, and representing a determinant of
one lower order than those above referred to. These constitute what I term
a system of Second Minor Determinants; and so in general we can form
a system of rth minor determinants by the exclusion of r lines and r
columns, and such system in general will contain
1.2 ... r
v
distinct determinants.
I say " in general " ; because if the principal determinant be totally or
partially symmetrical in respect to either or each of its diagonals, the
number of distinct determinants appertaining to each system of minors will
undergo a material diminution, which is easily calculable.
Now I have established the following law :
The whole of a system of rth minors being zero, implies only (r + 1) 2
equations, that is, by making (r + 1) 2 of these minors zero, all will become
zero ; and this is true, no matter what may be the dimensions or form of the
complete determinant. But furthermore, if the complete determinant be
formed from a quadratic function, so as to be symmetrical about one of its
diagonals, then l(r + l)(r+ 2) only of the rth minors being zero, will serve
102
148 On a new Class of Theorems. [25
to imply that all these minors are zero. Of course, in applying these
theorems, care must be taken that the (r+1) 2 or |(r + l)(r + 2) selected
equations must be mutually non-implicative, and shall constitute indepen
dent conditions.
In the application I am about to make of these principles, we shall have
onlv to deal with a system of first minors and of a symmetrical determinant.
If three of these properly selected be zero, from the foregoing it appears
that all must be zero.
Now let U and F be characteristics of two conies, that is, let each be
a function of only three letters, it may be shown (see my paper* in the
Cambridge and Dublin Mathematical Journal for November, 1850) that the
different species of contacts between these two conies will correspond to
peculiar properties of the compound characteristic U + n V.
If the determinant of this function have two equal roots, the conies
.simply touch ; if it have three equal roots, the conies have a single contact
of a higher order, that is, the same curvature; if its six first minors
become zero simultaneously for the same value of p, the conies have a double
-contact. If the same value of /*, which makes all these first minors zero,
be at the same time not merely a double root (as of analytical necessity
it always must be) but a treble root of
then the conies have a single contact of the highest possible order short of
absolute coincidence, that is, they meet in four consecutive points.
The parallelism between this theory and that of two quadratic functions
P, Q, and one linear function f f four letters, say x, y, z, t, is exact J.
For let P + Lu + pQ be now taken as our compound characteristic (a func
tion, it will be observed, of five letters, as, y,z,t,u); if its determinant have
two equal roots, L has two consecutive points in common with the inter
section of P and Q, that is, passes through a tangent to that intersection ;
if it have three equal roots, L has three consecutive points in common with
the said intersection, that is, is an osculating plane thereto ; if its fifteen
first minors admit of all being made simultaneously zero, L has a double
contact with the intersection of P and Q, that is, it is a tangent plane to
some one of the four cones of the second order containing this intersection ;
[* p. 119 above.]
f Observe that P=0, Q=0, L = now express the equations to two conoids and a plane
respectively.
This parallelism may be easily shown analytically to imply, and be implied, in the
geometrical fact, that the contact of the plane L with the intersection of the two surfaces P and
Q, is of exactly the same kind as the contact (which must exist) between the two conies which
.are the intersections of P and Q respectively with the plane L.
25] On a new Class of Theorems. 149
if the same linear function of jj, which enters into all these first minors be
contained cubically in the complete determinant, then the plane L passes
through four consecutive points of the intersection of P and Q, and the
points where it meets the curve will be points of contrary plane flexure ;
and, as it seems to me, at such points the tangential direction of the curve
must point to the summit of one or other of the four cones above
alluded to*. In assigning the conditions for L being a double tangent
plane to the intersection of P and Q, we may take any three independent
minors at pleasure equal to zero. One of these may be selected so as to be
clear of the coefficients of L ; in fact, the determinant of P + /j,Q will be
a first minor of P + p,Q + Lu ; p, may thus be determined by a biquadratic
equation ; and then, by properly selecting the two other minors, we may
obtain two equations in which only the first powers of the coefficients of
x, y, z, t in L appear, and may consequently obtain L under the form of
(ae + a.) x + (be + /3) y + (ce + 7) z + (de + 8) t,
where a, a. ; b, /3 ; c, 7 ; d, 8 will be known functions of any one of the four
values of /A. The point of contact being given will then serve to determine
e, and we shall thus have the equation to each of the four double tangent
planes at any given point fully determined.
In the foregoing discussions I have freely employed the word character
istic without previously defining its meaning, trusting to that being apparent
from the mode of its use. It is a term of exceeding value for its significance
and brevity. The characteristic of a geometrical figure ! i ^ ne function
which, equated to zero, constitutes the equation to such figure. Plticker,
I think, somewhere calls it the line or surface function, as the case may
be. Geometry, analytically considered, resolves itself into a system of rules
for the construction and interpretation of characteristics. One more remark,
and I have done. A very comprehensive theorem has been given at the
commencement of this commentary, for interpreting the effect of a complete
determinant of a linear function of two quadratic functions (U+ A 1 ^) having
* If this be so, then we have the following geometrical theorem : " The summit of one of the
four cones of the second degree which contain the intersections of two surfaces of the second order
drawn in any manner respectively through two given conies lying in the same plane, and having
with one another a contact of the third degree, will always be found in the same right line, namely
in the tangent line to the two given conies at the point of contact."
t More generally, the characteristic of any fact or existence is the function which, equated to
zero, expresses the condition of the actuality of such fact or existence.
Perhaps the most important pervading principle of modern analysis, but which has never
hitherto been articulately expressed, is that, according to which we infer, that when one fact of
whatever kind is implied in another, the characteristic of the first must contain as a factor the
characteristic of the second ; and that when two facts are mutually involved, their characteristics
will be powers of the same integral function.
The doctrine of characteristics, applied to dependent systems of facts, admits of a wide
development, logical and analytical.
150 On a new Class of Theorems. [25
one or more pairs of equal factors (e 4- e/i). But here a far wider theory
presents itself, of which the aim should be to determine the effect and
meaning of this determinant, having any amount and distribution of multi
plicity whatsoever among its roots. Nor must our investigations end at
that point; but we must be able to determine the meaning and effect of
common factors, one or more entering into the successive systems of minor
determinants derived from the complete determinant of U + /j,V.
Nor are we necessarily confined to two, but may take several quadratic
functions simultaneously into account.
Aspiring to these wide generalizations, the analysis of quadratic functions
soars to a pitch from whence it may look proudly down on the feeble and
vain attempts of geometry proper to rise to its level or to emulate it in its
nights.
The law which I have stated for assigning the number of independent,
or to speak more accurately, non-coevanescent determinants belonging to
a given system of minors, I call the Hoinaloidal law, because it is a corollary
to a proposition which represents analytically the indefinite extension of
a property common to lines and surfaces to all loci (whether in ordinary
or transcendental space) of the first order, all of which loci may, by an
abstraction derived from the idea of levelness common to straight lines and
planes, be called Homaloids. The property in question is, that neither two
straight lines nor two planes can have a common segment ; in other words,
if n independent relations of rectilinearity or of coplanarity, as the case
may be, exist between triadic groups of a series of n + 2, or between tetradic
groups of a series of n + 3 points respectively, then every triad or tetrad
of the series, according to the respective suppositions made, will be in
rectilinear or in plane order. So, too, if n independent relations of coincidence
exist between the duads formed out of n+l points, every duad will con
stitute a coincidence.
This homaloidal law has not been stated in the above commentary in
its form of greatest generality. For this purpose we must commence, not
with a square, but with an oblong arrangement of terms consisting, suppose,
of m lines and n columns. This will not in itself represent a determinant,
but is, as it were, a Matrix out of which we may form various systems of
determinants by fixing upon a number p, and selecting at will p lines and p
columns, the squares corresponding to which may be termed determinants
of the pih order. We have, then, the following proposition. The number of
uncoevanescent determinants constituting a system of the pth order derived
from a given matrix, n terms broad and m terms deep, may equal, but can
never exceed the number
25] On Pascal s Theorem. 151
Remark on PASCAL S and BRIANCHON S Theorems.
I omitted to state, in the September Number of the Journal *, that the
demonstration there given by me for Pascal s, applied equally to Brianchon s
theorem. This remark is of the more importance, because the fault of the
analytical demonstrations hitherto given of these theorems has been, that
they make Brianchon s consequence of Pascal s, instead of causing the two
to flow simultaneously from the application of the same principles. No
demonstration can be held valid in metJiod, or as touching the essence of the
subject-matter, in which the indifference of the duadic law is departed from.
Until these recent times, the analytic method of geometry, as given by
Descartes, had been suffered to go on halting as it were on one foot. To
Pliicker was reserved the honour of setting it firmly on its two equal
supports by supplying the complementary system of coordinates. This
invention, however, had become inevitable, after the profound views pro
mulgated by Steiner, in the introduction to his Geometry, had once taken
hold of the minds of mathematicians. To make the demonstration in the
article referred to apply, totidem literis, to Brianchon s theorem (recourse
being had to the correlative system of coordinates), it is only needful to
consider U as the characteristic of the tangential envelope of the conic,
x, y, z, t, u, v as the characteristics of the six points of the circumscribed
hexagon, <j> the characteristic of the point in which the line x, v meets the
line z, t ; ay au will then be shown to characterize the point in which
t, x meets v, z ; and thus we see that y, u; t, x: v, z, the three pairs of
opposite sides of the hexagon, will meet in one and the same point, which
is Brianchon s theorem.
[*p. 138 above.]
26.
ON THE SOLUTION OF A SYSTEM OF EQUATIONS IN WHICH
THREE HOMOGENEOUS QUADRATIC FUNCTIONS OF THREE
UNKNOWN QUANTITIES ARE RESPECTIVELY EQUATED
TO NUMERICAL MULTIPLES OF A FOURTH NON-HOMO
GENEOUS FUNCTION OF THE SAME.
[Philosophical Magazine, xxxvu. (1850), pp. 370 373.]
LET U, V, W be three homogeneous quadratic functions of x, y, z, and
let eo be any function of x, y, z of the nth degree, and suppose that there
is given for solution the system of equations
U=Aa>,
V=Ba>,
W = Ca>.
Theorem. The above system can be solved by the solution of a cubic
equation, and an equation of the nth degree.
For let D be the determinant in respect to x, y, z of
/U+gV+hW,
then D is a cubic function of/, g, h. Now make
D=0, Af+Bg+Ch = 0;
the ratios of f:g : h which satisfy the last two equations can be determined
by the solution of a cubic equation, and there will accordingly be three
systems of y, g, h which satisfy the same, as
fi, #]> ^i,
/ 2 , ffa, h 2 ,
/ 3 , g a , h*>
Now D implies that/7 +gV+ hW breaks up into two linear factors;
accordingly we shall find
(l^x + n^y + n^z) (X^ + ^y + v l z} = 0,
(I 2 x + m z y + n 2 z) (X^a; -I- //. 2 y + v 2 z) = 0,
(I 3 x + m 3 y + n 3 z) (\ 3 x + /j, 3 y + v 3 z) = 0,
26] On the solution of a System of Equation.
c 7 \ i i, ran be expressed without diffi-
in which the several sets of I, TO, n , X,f*. v can I
culty in terms of the several values of V/> V0, V^
Let the above equations be written under the form
PP = 0,
RR = 0.
Since the given equations are perfectly general, it is readily seen that
the equations _
(P Q P = 0), (Q = 0, Q = 0). (^ = ^> ^ = *v
will severally represent pairl of opposite sides of a 4-drangle expressed by
win j r i , fnnftmns H. li will oe a
linear function of P and Q and also of
In order to solve the equations, we need only consider two such pan
as PP = 0, QQ = ; we then make
or
F-0, Q = o,
P = 0, Q = 0.
Any one of these four systems will give the ratios of J : .^>
<f n .1 .,,-.-T/-vT>C! \H7(3 Anira.in I) UC
by substitution in anyone of the given equaUons, we obta n the va
Jy z by the solution of an ordinary equation of the nth degree.
number of systems a, y, z is therefore always 4.
The equations connected with the solution of ^
problem, "In a given triangle to inscribe three circles such hat each c
rf - 3
Ztn Mathematical Journal, to wit,
6y 2 + c0 2 + 2/>2 = &-a (be -/ 2 ) = ^ ,
C 2 2 + aa; 2 + Zgzx = ^ 2 ^ ( ca T 9*) = 5
a^ 2 + bif + Zhxy = ^ 2 c (ab - Ii 2 ) = G,
come under the general form which has just been solved. It so Happens,
however, that in this particular case
for this being the case? thetuJr of solutions
154 On the solution of a System of Equations. [26
become respectively
1 o x
-
C
and the cubic equation is resolved without extraction of roots.
It follows from my theorem that the eight intersections of three con
centric surfaces of the second order can be found by the solution of one cubic
and one quadratic equation ; and in general, if we have </>, ^, any three
quadratic functions of x, y, z, and </> = 0, i/r = 0, 6 = be the system of
equations to be solved, provided that we can by linear transformations
express <f>, ^r, 6 under the form of
U aw,
V-bw,
W-cw,
U, V, W being homogeneous functions, and w a non-homogeneous function
of three new variables, x, y , z, we can find the eight points of intersection
of the three surfaces, of which U, V, W are the characteristics, by the
solution of one cubic and one quadratic. But (as I am indebted to Mr Cayley
for remarking to me) that this may be possible, implies the coincidence
of the vertices of one cone of each of the systems of four cones in which the
intersections of the three surfaces taken two and two are contained.
I may perhaps enter further hereafter into the discussion of this elegant
little theory. At present I shall only remark, that a somewhat analogous
mode of solution is applicable to two equations,
U=aP\
V=bP>,
in which U, V are homogeneous quadratic functions, and P some non-homo
geneous function of x, y.
We have only to make the determinant of fU + gV equal to zero, and we
shall obtain two systems of values of/ g, wherefrom we derive
l^ + 111^ = V(
I 2 x + m. 2 y = V(
from which x and y may be determined.
27.
ON A PORISMATIC PROPERTY OF TWO CONICS HAVING
WITH ONE ANOTHER A CONTACT OF THE THIRD ORDER.
[Philosophical Magazine, xxxvir. (1850), pp. 438, 439.]
IF two conies have with one another a contact of the third order, that is,
if they intersect in four consecutive points, it will easily be seen that their
characteristics referred to coordinate axes in the plane containing them must
be of the relative forms x- + yz, k (y 2 + x- + yz) respectively, y characterizing
their common tangent at the point of contact*.
Hence if we take planes of reference in space, and call t the characteristic
of the plane of the conies, the equations to any two conoids drawn through
them respectively will be of the relative forms
V = y 2 + x 1 + yz + tv = 0.
Using W to denote V U, and (W) to denote what W becomes when ey
is substituted for t, we see that W and (W) are of the respective forms
y- + tw and yO ; showing that the former is the characteristic of a cone which
will be cut by any plane t ey drawn through the line (t, y) in a pair
of right lines ; or, in other words, that one of the cones containing the inter
section of the two variable conoids (Fand U) will have its vertex in the
invariable line which is the common tangent to the two fixed conies : this
proves the theorem stated by me hypothetically in a foot-note in one of my
papers in the last number of the Magazine^, The steps of the geometrical
proof there hinted at are as follows.
* These relative or conjugate forms are taken from a table which I shall publish in a future
number of this Magazine, exhibiting the conjugate characteristics in their simplest forms,
correspondent to all the various species of contacts possible between lines and surfaces of the
second degree. This table is as important to the geometer as the fundamental trigonometrical
formula to the analyst, or the multiplication table to the arithmetician ; and it is surprising
that no one has hitherto thought of constructing such.
[t p. 149 above.]
156 On a Porismatic Property of two Conies. [27
The four consecutive points in which the two conies intersect will be
consecutive points in the curve of intersection of the two variable conoids.
This curve lies in each of four cones of the second degree. Every double
tangent plane to it passes through the vertex of one amongst these. The
plane containing four, that is, two (consecutive) pairs of consecutive points,
is a double tangent plane, and will therefore pass through a vertex ; but four
consecutive points of a curve of the fourth order described upon a cone,
and lying in one tangent plane thereto, can only be conceived generally
as disposed in the form of an /, of which the belly part will point to the
vertex; or, in other words, at any point where two consecutive osculating
planes coincide so that the spherical curvature vanishes, the linear curvature
will also vanish, that is, there will be a point of inflexion at which, of course,
the tangent line must pass through the vertex of the cone. This is the
assumption felt to be true, but stated by me hypothetically in the paper
referred to, because a ready demonstration did not at the moment occur
to me. The legitimacy of this inference is now vindicated by the above
analytical demonstration.
The methods of general and correlative coordinates and of determinants
combined possess a perfectly irresistible force (to which I can only compare
that of the steam-hammer in the physical world) for bringing under the
grasp of intuitive perception the most complicated and refractory forms of
geometrical truth.
28.
ON THE ROTATION OF A RIGID BODY ABOUT A
FIXED POINT.
[Philosophical Magazine, xxxvir. (1850), pp. 440 444.]
IN the Cambridge and Dublin Mathematical Journal for March 1848,
an article by Professor Stokes, of the University of Cambridge, is ushered in
with the words following :
"The most general instantaneous* motion of a rigid body moveable in
all directions about a fixed point consists in a motion of rotation about an
axis passing through that point. This elementary proposition is sometimes
assumed as self-evident, and sometimes deduced as the result of an analytical
process. It ought hardly perhaps to be assumed, but it does not seem
desirable to refer to a long algebraical process for the demonstration of
a theorem so simple. Yet I am not aware of a geometrical proof anywhere
published which might be referred to."
The learned and ingenious professor is indubitably right, and might have
trusted himself to assert less hesitatingly the necessity of demonstrating
this proposition, which possesses none of the characters of a self-evident
truth ; but it is to be regretted that he should have stated it in such a form
as naturally to lead the incautious reader to mistake the nature and grounds
of its existence, which consist in this fact that any kind of displacement
of a body moveable about a fixed axis, whether instantaneous and infini
tesimal, or secular and finite, is capable of being effected by a single rotation
about a single axis.
The annexed simple proof of this capital law has the advantage of afford
ing a rule for compounding into one any two (and therefore any number of)
rotations given in direction, magnitude and order of succession.
* The italics do not exist in the original.
158 On the Rotation of a [28
It will somewhat conduce to simplicity if we fix our attention upon a
spherical surface rigidly connected with the rotating body, and having its
centre at the fixed point thereof. When the positions of two points in
this are given, the position of the body is completely determined.
Now evidently two points A, B may be brought respectively to A B
(if AB = A B ) by two rotations ; the first taking place about a pole situated
anywhere in the great circle bisecting AA at right angles, the second about
A , the position into which it is brought by the first rotation. This view
leads us to consider the effect of two rotations taking place successively
about two axes fixed in the rotating body. Or again, we may make the plane
A B revolve into the position AB round a pole taken at the node in which
the two planes intersect, and then the points A, B swing into their new
positions A , B by means of a rotation about the pole of the great circle,
of which A B forms a part. This mode of effecting the displacement
naturally suggests the consideration of the effect of rotations taking place
successively about two axes fixed in space.
First, then, let us study the effect of the combination of a rotation
(a) having P for its pole, followed by another (/3), of which Q is the pole,
P and Q being points in the surface of the revolving sphere.
In drawing the annexed figure, I have supposed that the two rotations
r are of the same kind, each tending, when a
spectator is standing with his head to the
respective poles and his feet to the centre, to
make a point at his right-hand pass in front of
his face towards his left-hand. Let now PQ
revolve through - positively into the position of
PR, and through negatively into that of QR.
Then I say that the two impressed rotations a
and ft about P and Q will be equivalent to a single rotation about R, equal
to twice the acute angle between QR, RP.
Let the first rotation about P bring Q to Q and R to R ; it is clear that
QPR, Q PR, Q PR are all equal triangles. Therefore R Q R = 2PQR = /3.
Consequently the positive rotation /3 about Q (the new position of Q) will
carry R back again to R, its original position. Hence the actual motion
which results from the successive rotations combined being consistent with
R remaining at rest, must be equivalent to a single rotation about R.
To find its magnitude, let the second rotation carry P to P *; then the
angular displacement PRP (which is the required rotation of the whole
* The reader is requested to fill in the point P and join P R.
28] Rigid Body about a Fixed Point. 159
body) is equal to twice the acute angle between Q R, RP, which is the same
as that between QR, RP, as was to be shown. Thus we see that the semi-
rotations about three poles (considered as the angular points of a spherical
triangle), which, taken in order, would bring the sphere back to its first,
undisturbed position, are equal to the included angles at such poles respec
tively.
If in our figure the order of the rotations had been reversed, PQr, QPr
would have been taken respectively equal to PQR, QPR, but on the opposite
side of PQ, and r would have been the resultant pole, the resultant rotation
remaining in amount the same as before.
If either of the rotations had been negative, the resultant pole would be
found in QR produced, namely, at the intersection of rQ or rP with PQ.
Calling the resultant rotation 7, we have always
sin - : sin - : sin ^ :: sin QR : sin RP : sin PQ.
25 25 ^
When the component rotations are infinitesimal in amount, R and r will
come together in QP ; the order of succession of the rotations will be
indifferent, and we shall have
a : /3 : 7 :: sin ^ : sin ^ : sin ^ :: sin QR : sin RP : sin PQ,
which gives the rule for the parallel >grammatic composition of two simul
taneously impressed rotations*.
If, next, we consider the effect of rotations about two poles, P and Q,
fixed in space (supposing, as above, that they take place first about P and
then about Q), we must take QPr equal to half the contrary of the rotation
about P, and PQr to half the direct rotation about Q (the angle being now
taken positive which was on the first supposition negative, and vice versa} ;
so that, retaining the original figure, the first rotation will bring r to R^
and the second carry R back to r ; showing that r is the resultant pole,
and thatf PVP, the resultant rotation, will be double the acute angle
between Qr, rP, as in the former case.
To popular apprehension the important doctrine of uniaxial rotation
may be made intelligible by the following mode of statement. Take a
pocket-globe, open the case and roll about the sphere within it in any
manner whatever ; then closing the case, there will unavoidably remain two
points on the terrestrial surface touching the same two points on the celestial
surface as they were in apposition with before the sphere was so turned about
in its case.
* Compare Mr Airy s Tracts, Art. "On Precession and Nutation.."
t P is not expressed in the figure given.
160 On the Rotation of a [28
It is right to bear in mind that the whole of this doctrine is comprised
iin, and convertible with, the following easy geometrical proposition relative
to arcs of great circles on any spherical surface, including the plane as an
extreme case.
" The arcs joining the extremities (each with each in either order) of two
other equal ares, subtend equal angles at either of the points of intersection
of two great circles bisecting at right angles the first-named connecting
arcs*."
The spherico-triangular mode of compounding rotations given in the
above simple disquisition may easily be made the parent of a whole brood of
geometrical consequences, which, however, I must leave to the ingenuity
.and care of those who have a turn for this kind of invention.
But I ought not to omit to invite attention to a remarkable form, which
.may be imparted to the theorems above stated for the composition of finite
rotations, or rather to a theorem which may be derived from them by an
obvious process of inference.
Let P, Q, R . . . X, Z be any number of points on a sphere capable of
moving about its centre, joined together by arcs of great circles so as to form
.a spherical polygon. Imagine any number of rotations to take place about
these points in succession as poles. It matters not which is considered the
first pole of rotation, but the order of the circulation must be supposed given,
as, for instance, PQR...XZ, or QR...XZP, or R ... XZPQ, &c. This
will be one order; the reverse order would be PZX ...RQ, or QPZX ...R, &c.
I shall suppose the circulation to be of the kind first above written.
Now we may make two hypotheses:
1. That the poles are fixed in space.
2. That they are fixed in the rotating body.
In the first case, let the rotations about the given poles P, Q,R,S ... X, Z
be double the amounts which would serve to transport PQ to QR, QR to
R8 ... XZ to ZP respectively.
In the second case, let the rotations be double the amounts which would
carry PZ to ZX ... SR to RQ, RQ to QP respectively. Then, on either
supposition, the sum of the combined rotations is zero; or, to use a more
convenient and suggestive form of expression, if the poles of rotation form a
closed spherical polygon whose angles are respectively equal to the semi-
rotations about the poles, the resultant rotation is zero.
* This proposition will be seen to be immediately demonstrable, by the comparison of equal
triangles, when viewed as the converse of this other. "The arcs (or right lines) joining the
.correspondent extremities of .the bases of two similar isosceles spherical (or plane) triangles
having a common vertex, Are equal to each other."
28] Rigid Body about a Fixed Point. 161
This proposition is immediately derivable from the fundamental one
relative to three poles, given above, by dividing the polygon into triangles
by arcs, joining any one of the poles with all the rest, or (as pointed out to
me by my eminent friend Prof. W. Thomson) it becomes apparent as a
particular case of a more general proposition, on representing the motion
about the successive axes as effected by two equal pyramids having a
common vertex at the centre of motion, of which the one is fixed in
space, and the other is fixed in the revolving body and rolls over the
first, so that the corresponding equal faces are successively brought into
coincident apposition.
P.S. To find the pole of rotation whereby PQ may be brought into the
position P Qf, we may use the following simple construction.
Measure off from the node of the great circles (or right lines) con
taining PQ and P Q , two distances in the proper direction upon each (four
distinct assumptions may be made), say OR and OS equal to one another
and to the difference between OP and OP , then the pole of rotation required,
say E, is the centre of the circle described about ROS, and the amount of
rotation is the angle subtended by OR or OS at E. The writer of this paper
suggests that axis of displacement would be a convenient term for designating
the line whereby any finite change in the position of a body moveable about
a fixed centre may be brought about ; a geometrical theory of rotation leading
to the investigation of a very curious species of correlation, now opens upon
the view, the general object of which may be stated as follows :
" Given upon a sphere or plane any curve considered as the locus of
successive poles of instantaneous rotation, and the ratio of the rotation
about each pole to its distance from the one that follows*, to construct
the curve of the poles of displacement, and to determine the amount of
rotation corresponding to each such pole."
The discussion of this question offers a fine field for the exercise of
geometrical taste and skill.
* Which by analogy may be termed the " density of rotation."
11
29.
ON THE INTERSECTIONS OF TWO CONICS.
[Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 18 20.]
LET the two conies be written
U = act? + by 2 + cz 2 + Za yz + Zb zx + Zc xy = 0,
V = ax* + $y* + 7^ 2 + 2ayz + 2/3 zx + 2y xy = 0,
and make
U + XF = Aa? + Bf + Cz 2 + ZA yz + ZB zx + 2C xy.
In my paper in the last number of the Journal*, I showed that the case of
intersection of the two conies in two points was distinguishable from all other
cases by the equation a(U + \V) = Q having two imaginary roots. When
all the roots are real, the curves either intersect in four points or not at all.
On the former supposition,
which are quadratic functions of X, will be negative for all three values of X.
On the contrary supposition, one value of X will make all f these three
quantities negative, but the other two values with each make them all
three positive.
Hence we obtain a symmetrical criterion (which I strangely omitted to
state in my former paper) by forming the quantity
A 2 + B * + C" 2 - AB - AC- BC.
A cubic equation
may be then constructed, of which the three values of the above function
corresponding to three values of X will be the roots.
The condition for real intersection is that L, M, N, P should be all of the
same sign. The conies being supposed real, L and P are necessarily in both
cases of the same sign. The condition is therefore satisfied if either L, M,
[* p. 119 above.]
29] On the Intersections of Two Conies. 163
N, or M, N, P be of the same sign, and is consequently equivalent to the
M N N M
condition that - f and -^ shall be both positive, or -^ and -~ both positive.
Li Li Jr
It does not appear to be possible in the nature of the question to find a
criterion for distinguishing between the two cases, dependent on the sign
of one single function of the coefficients.
The case of double contact, abstraction being made of binary intersection,
is a sort of intermediary state between intersection in four points and non-
intersection ; and accordingly, as shown in my former paper for this case, the
two equal values of X will make the three quantities
AB- G \ EG - A *, CA - B 2
all real ; so that two of the values of y corresponding to the equal values of X
are zero, and the criterion becomes nugatory as it ought to do.
Again, when the two conies do not intersect, I distinguished two cases
according as they lie each without, or one within the other, that is, according
as they have four common tangents or none.
But, as Mr Cayley has well remarked to me, a similar distinction exists
when the conies intersect in four points; in that case also they may have
four common tangents or not any : when they intersect in two points they
have necessarily two and only two common tangents. There is no difficulty
in separating these four cases.
Let the conies be written
(17) and (F) being what V and F become when the coordinates are changed
from x, y, z to , 77, f,
A, B, C are the three values of X in the equation
If the curves intersect A C,B must have different signs, that is, C
must be an intermediary quantity between A and B.
Again, the tangential equations to the conies expressed by the correlative
system of coordinates will be
__ _ _ _
A B C~
and that these may have four real systems of roots,
!_! 1_I
A C C B
must have the same sign ; and consequently, as A C and C B are
112
164 On the Intersections of Two Conies. [29
supposed to have the same sign, A and B, and therefore all three A, B, C,
have the same sign. We have therefore the following rule:
Let the equation in X, namely, a (U + XF) = 0, be called = 0, and the
equation in y, above given, &&gt; = 0. By an equation being congruent or
incongruent, understand that its roots have all the same sign or not all
the same sign.
Then &&gt; congruent, 6 congruent, implies that the intersections and
common tangents are both real ; &&gt; congruent, 6 incongruent, implies that
the intersections are real, but the common tangents imaginary ; &&gt; incon
gruent, 9 congruent, implies that the intersections and common tangents
are both imaginary ; eo incongruent, incongruent, implies that the inter
sections are imaginary, but the common tangents real.
In like manner, as the cases of contact of lines are limiting cases to those
which relate to the relative configurations of their points of intersection, so
the cases of contact of surfaces are limiting cases in which the characters
which usually separate the different forms of their curve of intersection exist
blended and indistinguishable. The first step therefore to the study of the
particular species of the curve of the fourth degree*, in which two surfaces
of the second degree intersect, is to obtain the analytical and geometrical
characters of their various species of contact. Accordingly I have made an
enumeration of these different species, no less than 12 in number, many of
them highly curious and I believe unsuspected, which the reader may consult
in the Philosophical Magazine for February, 185 If.
By the aid of these landmarks, I have little doubt, should time and leisure
permit, of mapping out a natural arrangement of the principal distinctions of
form between that class at least of lines in space of the fourth order which
admit of being considered the complete intersection of two surfaces.
* I have found that the 16 points of spherical flexure in this curve are the four sets of four
points in which it meets the four faces of the pyramid whose summits are the vertices of the four
cones of the second degree in which the curve is completely contained, which 16 points reduce to
4 when the two surfaces have an ordinary contact, and to 1 when they have a cuspidal contact :
of course in the case of contact the pyramid above described in a manner folds up and vanishes,
as there are no longer 4 distinct vertices. I have found also that when the factors of D (U + \V),
(U and V being the characteristics of the two surfaces) are all unreal, the points of flexure are all
unreal. When two factors are real and two imaginary, two of the faces of the pyramid (namely,
its two real faces) will each contain one (and only one) pair of real points of flexure, and the other
two planes none ; and lastly, when the factors of D (U + XV) are all real, then either all the points
of flexure are imaginary, or else all the eight contained in a certain two of the pyramidal faces
are real : and these two cases admit of being distinguished by a method analogous in its general
features to that whereby I have shown in the text above how to distinguish between the cases of
4 real and 4 imaginary points of intersection of two conies. Where the two surfaces have an
ordinary contact, the curve of intersection, it is well known, has a double point ; and where the
surfaces have a higher contact, the curve has a cusp. Thus in the fact of the 16 flexures reducing
to 4 and to 1 in these respective cases, we see a beautiful analogy to what takes place with the 9
flexures of a plane curve of the third degree, which contract to 3 and 1, according as the curve
has a double point or a cusp.
[t p. 219 below.]
30.
ON CERTAIN GENERAL PROPERTIES OF HOMOGENEOUS
FUNCTIONS.
[Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 1 17.]
LET x denote the operation
and A the operation
d d d
\ ~; T #2 ~j T T &n ~j >
aa 1 aa 2 aa n
d d d
- -- h a -v -+...+ a n -
and now suppose that o>, a homogeneous function of i dimensions of
Ou a 2 ...a n , and not of any of the quantities x i} x% ... oc n , is subjected to
the successive operations indicated by A*x r .
We have
d d d \ / d d d \
^ + ^ 2 da* + - + ^ daj
d d d \
j -- h a 2 j h . . . + a n -j )
da l da 2 da n j
for % r ~ 1 &&gt; is of (r - 1) dimensions, lower than &&gt; (which is of i dimensions) in
a 1; a 2 ...a n .
Hence
= &c. = {r(r-l)...(r-s+l)}
-. (1)
166 On certain General Properties [30
Now in the expression
X r &&gt;(!, a 2 ... a n ),
suppose that we write
we have, by Taylor s theorem,
where ?7 r &&gt; denotes what ^ r o> becomes, on substituting u s for # s, and J. now
represents
d d d
u ij 1" U 2 J~ + + u n j
da l aa 2 da n
This expansion stops spontaneously at the (r + l)th term, because % r &&gt; is
only of r dimensions in x lt x z ... x n .
Applying now theorem (1), we obtain
X r o> = U r a> + r(i-r + I) U^toe + %r(r- 1) {(t - r + 1) (i - r + 2)} U^ax?
+ ... + {( l -r+l)(i-r+2)...i}a, r . (2)
In using this theorem in the course of the ensuing pages, it will be found
convenient to assign to e a specific value, and I shall suppose it equal to
^n 1 1
; this gives
u n x n
a n
= 0.
And inasmuch as the U symbol now contains a ls a 2 , ...a n , so that UU r no
longer equals U r+ \ I shall write U r for U r . Theorem (2) will thus assume the
form
30] of Homogeneous Functions. 167
where U r for all values of r denotes what
d d d
becomes, on substituting w n w 2 , ... ^n-\ for #1, # 2 x n-\> after the processes
of derivation have been completed : this it is essential to observe, because
Wj, u 2 , ... < n _i now involve a n a 2 , ... a n _!, a n . The term # w -= is omitted from
aa n
the symbol of linear derivation, because in the substitutions x n will be
replaced by zero.
As an example of this last theorem, take
&) = a 3 + b 3 + c 3 + kabc ;
then
A;6c + kcay + kabz,
66?/ 2 + 6c^ 2 + 2kcxy + Zkayz + Zkbzsc,
3 a> = Gar 5 + Qy* + 62? + Qkxyz.
, 07 o 77
-- 1 + 36 2 1 y -- I + A;6c a? --
C / \ C / \ C
a ( aZ Y , CL / M 2 , 07 / ^ / 1>*\
a3 = Galas -- + oo { -- + 2*0 ( a - } (-
V c / V c / V c/ V c /
and it will be found that the equations given by theorem (3) are satisfied,
namely
&
%a> = Uo) + 3 - to,
v 2 w = U 2 a) + 2 . 2 - Ua> + 2 . 3 Z - &&gt;,
c c 2
y 3 o) = U s a> + 3 - U 2 w + 3 . 1 . 2 i 2 f7o> + 1 . 2 . 3 ^ w.
c c a c 3
Probably, as this theorem is of rather a novel character, the annexed
sketch of a somewhat different course of demonstration may be not un
acceptable to my readers.
We have
/ d d d \
yco = #j -5 -- 1- # 2 -j -- 1- + x n 7 < ;
\ rfttj rfa 2 daj
and by the well-known law for homogeneous functions,
d d d
iw=a l -j h 2 7 - +...+ T .
aaj aa 2 daj
168 On certain General Properties [30
Hence
( x ^\ - ( d d d
Hence
&c. = &c.
But in performing the process indicated by the several factors it must be
carefully borne in mind that UU r is not = U r+l ; this would be the case were
it not for the terms x n , 2 x n , &c., which enter into u lt u^ ... w n _j. But
Q/ n d n
on account of these terms, we have
rrTJ ( d d d W d d d
U U r o) = {u l -j \-Uzj h . . . + u n -i -j Nij H w 2 T h . . .
7 ~ i " ni j I \ t i T <*2 7 T T " ni 7
aa 2 da n iJ V ai aa 2 cla n - l
x n ( d d d
(J"fl I W/Cvj ct t/ 2 CLCL/ft.
f d d d x, }
f QJ- 1/= It = M
rfdj c?ti 2 da n 1 a n
Hence
/>
Let be called e ; we find
O
X =U+ie,
= UU 2 + 2 (t - 1) eUU + (i - 1) ie*U
+ (t-2)eU a + 2(t- 2)(i-I) e 2 U+(i - 2) (t -
= tT, + 3 (t - 2) eCT"., + 3 (t - 2) (t - 1) e-U + (t - 2) (t - 1) te 3 .
30] of Homogeneous Functions. 169
The same process being continued will lead to results identical with those
previously obtained and expressed in theorem (3).
The expansion of x r > treated according to this second method, appears to
require the solution of the partial equation in differences
ttr+i, s+i == Qr, *+i T \l AT) &r, s>
a 0j , being given as unity for s = 1 and as zero for all other values of s.
It is probable however that the solution of this equation might be evaded
by some artifice peculiar to the particular case to be dealt with. I do not
propose to dwell upon this inquiry, which would be foreign to the object
of my present research. It may however not be out of place to make the
passing remark, that the equations expressing x r in terms of powers of U
admit easily of being reverted, as indeed may the more general form
1
which becomes the equation of formula (3), on making
/v
r = r (i -t- 1 - r) , Xr = x r , and u r = U r (o ;
d n
for let u r = ej e 2 . . . e r v r ,
then y r = v r + v r
d_
whence v r = e dr y r
and therefore u r = %r *rXr-i + Je r e r _ 1 ^ r _ 2 + &c.
Thus we obtain, from equation (3),
sr
l <a + &c.
As a first application of theorem (3), I shall proceed to show how
Joachimsthal s equation to the surface drawn from a given point (a, ^, 7, 8)
through the intersection of two surfaces < (#, y, z, t) = 0, 6 (x, y, z, t) = 0, may
be expressed under the explicit form of the equation to a cone.
The equation in question is obtained by eliminating X between
1 + (>xm ~ 2 + &c - = >
r^> % 2 ^ m ~ 2 + i- r
JL Z JL i Z
170 On certain General Properties [30
where
* = *(,& 7 ,s), * = *(,& 7, ), x-**9&+*&*&
By theorem (3), these two equations, on writing = e, become
\rn-2
+ {U 2 <j> + 2 (m -l)U(j>6 + (m - 1) mfc 2 } ~ + &c. = 0,
1 . u
n0e] V*- 1 + (U*0 + &c.) ^ + { 7 3 + 3 (w - 2) U-6e
i. .
Now on writing \ = /j, e, these equations take the forms
&c. = 0,
-
1 Z
0/4" 4- tffl/A"- 1 + U 2 6 ^-. 2 + &c. = 0,
-L Zi
as is easily seen by substituting back X + e in place of /*. Consequently
no longer appears in the coefficients of the terms of the equations between
which the elimination is to be performed, and the resultant will accordingly
come out as a function only of <, U(f>, U 2 (f>, &c., that is, of a, @, 7, 8,
and of
T __ f u t 7 t
B J 8 8
showing that the equation in x, y, z, t, is of the form of that to a cone, as we
know , priori it ought to be. Precisely a similar method may be applied to
the elucidation of the corresponding theorem for a system of rays drawn from
a given point through the locus of the intersection of two curves.
Before entering upon some further and more interesting applications of
theorem (3), it will be convenient to explain a nomenclature which has been
employed by me on another occasion, and which is almost indispensable in
inquiries of the nature we are now engaged upon. Homogeneous functions
may be characterized by their degree, by the number of letters which enter
into them, and lastly, by the lowest number of linear functions of the letters
which may be introduced in place of the letters to represent such functions.
Any such linear function I designate as an order, and am now able to dis
criminate between the number of letters and the number of orders which
enter into a given function. The latter number, generally speaking, is the
same as the former ; it can never exceed it, but may be any number of units
ess than it.
30]
of Homogeneous Functions.
171
I need scarcely observe that a pair of points becoming coincident, a conic
becoming a pair of lines, a conoid becoming a cone, and so forth, for the
higher realms of space, will be expressed by the homogeneous function of the
second order which characterizes such loci*, losing one order, that is, having
an order less than the number of letters entering therein. Calling such
characteristic $(#, y, z ... i), it is well known that the condition of such loss
of an order is the vanishing of the determinant
da? dxdy " dxdt
_
dydx dy 2 " dydt
dtdx dtdy " dP
A conoid becoming a pair of planes, a cone becoming a pair of coincident
lines, a pair of points becoming indeterminate, will, in like manner, be
denoted by their characteristic losing two orders, and so forth, for the
higher degrees of degradation. In like manner, in general, a homogeneous
function of three letters of any degree losing an order, typifies that the
locus to which it is the characteristic will break up into a system of
right lines.
Now let w be a homogeneous function of a, /3, 7 ... 8 : and suppose that we
have the equations to = 0, %a) = 0, ^ 2 <o = 0, where ^ as above
d d
da
+ -r- +
I say that on eliminating any of the variables x, y, z . . . t between the second
and third of the above equations, the resulting equation will be of one order
less than the number of letters, that is, the expulsion of one letter will be
attended by the expulsion of two orders.
For we have, by theorem (3),
Yo> = Ua> + 2 Xn o> = 0,
and by hypothesis
Hence we have also
y 2 o) = U,a> + 2 -^ Ua> + 2 p w = 0,
* a*. \cu
= 0.
and since Uw, U 2 a) contain one order less than the number of letters in
* If U=0 is the equation to any locus, U may be said to characterize the same, or to be its
characteristic.
/ d d d d\
(x-f- + y + z-^- + -y&&gt;, that is, v&&gt; = 0,
\ do. J df3 dy d&
172 On certain General Properties [30
, the resultant of the elimination between them will contain two orders
less than the number of letters in &&gt; ; and consequently, whichever of the
letters x, y, z ...t we eliminate between ^<u = and ^ 2 o = 0, provided that
<a = 0, the resultant equation will contain one order less than the number of
letters remaining.
Thus we see how it is that the tangent line to a conic meets it in two
coincident points, the tangent plane to a conoid in two intersecting lines,
and so forth, for the higher regions of space*. For instance, if we take
G> (x, y, 2, t) = 0, the equation to a conoid, and a, ft, 7, S, the coordinates to
any point therein, we shall have W(OL, ft, 7, 8) = 0,
d d d d
-^-
dy
and &) (x, y, z, t), that is, ^ 2 &&gt; = 0,
x, y, z, t representing the coordinates of any point in the intersection of the
conoid by the tangent plane.
Consequently, by what has been shown above, on eliminating any one of
the four letters x, y, z, t, the resultant function of three letters will contain
only two orders, and will thus represent a pair of lines, real or imaginary,
intersecting one another at a, ft, 7, 8.
The fact which has just been demonstrated (that the resultant of ^<w = 0,
% 2 o = 0, loses an order if &&gt; = 0), indicates that on expressing one of the
quantities x, y, z ...t in terms of the others, by means of the first equation,
and then substituting this value in the second, the determinant of the
equation so obtained must be zero.
Now by virtue of a theorem which was given by me in a notef to my
paper in the preceding number of this Journal, this determinant will be equal
to the squared reciprocal of the coefficient in the equation %<B = of the
letter eliminated multiplied by the determinant in respect to x, y, z ... t, X of
This latter determinant is therefore zero ; but this determinant is the
resultant of the equations
d_f d d V did d
~j~ \ X -y T y -jr + &C. I 03 H j I X j T V ~JT
ax V da do J dx \ da do
*L( A. A gy V d ( <L A
dy \ da db / dy\ da db
&c. &c. &c. &c.
d
vw = 0, that is, [ T- + tf -55 +...)&&gt; = 0,
\ da * db )
* Thus a tangential section of a hyperlocus of the second degree at any point cuts it in
two cones.
[t p. 135 above.]
30]
of Homogeneous Functions.
173
Thus we obtain the singular law, that the symmetrical determinant
d d
da da
d d
da db
d d a>
da dl
d
&,
da
d d ^
db da
d d
db db "
d d
db dl i
d
db W
d d
dc da
d d
dc db
d d
dc dl
d
dc
d d
dl da
d d
dl db 03 ""
d d
dl dl 3
d
di"
d
da"
d
d
is zero when &&gt; is zero.
This is easily shown independently by means of a remarkable and
I believe novel theorem, relative to homogeneous functions.
If &) be any homogeneous function of t dimensions of a, b, c ... I, we
have (by Euler s theorem already repeatedly applied), remembering that
da) dm da>
<= , -rr--- -,-,- are all homogeneous,
da db dl
J J J X
= 0,
+ (a d +1
d
7 d \
+ I -T7
dm
(t 1) -T-
da
-( t -l} d(0
\ da
f d d
+ (a-j- -r-
\ da da
f d d
+ \ a
*db +
, d d
+ b-r- -TJ- +
da db
dl)
d d\
da dl/
l d d ]
> db
( L n rf&)
\ db da
fee,
f d d
4- a ~^r -TT-
&c.
+ ..
db dl)
&c.
d d\
-T, T,
dl al
= 0.
Between these equations we may eliminate all the letters, a, b, c ... I, and we
obtain the equation
d d
da da
d d
da db
d d
da dl
d
da
d d
db da
d d
db db W
d d
" db dl <U)
d
db W
d d
dc da
d d
dc db
d d
" dc dl
d
dc
d d
dl da
d d
dl db W>
d d
" dl dl W>
d
dl W
d
d
d
i
da
db w "
dl
t-l a
174
On certain General Properties
[30
As a corollary to this theorem, we see that if <y = the determinant
obtained in the previous investigation becomes zero, agreeing with what
has been already shown; in fact the last-named determinant is always
equal to
i-l
d d
d d
d d
da da
da db
da dl
d d
d d
d d
dl da
dl db w "
dldl
a) x
This remarkable theorem, which I have communicated to friends nearly
a twelvemonth back, is here, I believe, published for the first time*.
Suppose next that w (x, y, z) is the characteristic of a line of any degree,
to which a tangent is drawn at the point a, /3, 7, using U in a manner corre
spondent to its previous signification to denote
d , / & \ d
j-+[y *) 33i
da \ y 7 Jd/3
and understanding <a (a, /3, 7) by w, we have for determining the point of
intersection, o> = 0, %&&gt; = 0, y^w = ; and consequently, by aid of our theorem
(3), we shall obtain
U n co + n U n ^ o}+ ... =0.
7
By means of the two latter equations, we obtain
Thus let z be a homogeneous function in x and y of t dimensions, and let
dx* dxdy
be called p, q, r, s, t; we shall find
r, *, p
s, t, q
= 0,
that is,
rt -
30] of Homogeneous Functions. 175
where F and G are each of only (n 2) dimensions, and serve to determine
the intersections of the tangent with the curve, extraneous to the two
coincident ones at the point of contact.
Again, suppose that &&gt; is a function of any degree of any number of
letters a, /3, 7, &c., and that we have given o> = 0, %&&gt; = 0, ^ 2 <o = 0, . . . ^ w &&gt; = ;
it is evident from our fundamental theorem that these equations may be
replaced by
and consequently that the expulsion of (m 1) letters, by aid of the last ra
of the given equations, will be attended by the disappearance of m orders, or,
in other words, the resultant will be minus an order, that is, will have one
order less than the number of letters remaining in it.
In applying to space conceptions the preceding theorem, it will be con
venient to use a general nomenclature for geometrical species of various
dimensions.
Thus we may call a line a monotheme, a surface a ditheme, the species
beyond a tritheme, and so on, ad infinitum.
A system of points according to the same system of nomenclature would
be called a kenotheme.
An ?i-theme has for its characteristic a homogeneous function of (n + 2)
letters.
Again, it will be convenient to give a general name to all themes ex
pressed by equations of the first degree. Right lines and planes agree in
conveying an idea of levelness and uniformity ; they may both be said to be
homalous. I shall therefore employ the word homaloid to signify in general
any theme of the first degree.
Now let w (x, y, z . . . t) be the characteristic to an w-theme of the nth
degree.
The number of letters x, y, z . . . t is (n + 2).
As usual, let o> represent &&gt; (a, /3, 7 . . . 8), and suppose
G) = 0, ^o> = 0, ^ 2 &&gt; = ... ^ w o) = 0,
and consequently
t7 1 w = 0, U 2 a) = ... U n u = Q.
On eliminating (n 1) letters between the n last equations, the resulting
function will be of three letters but of only two orders, and of the 1 . 2 . 3 . . . n
degree. Hence we see that at every point of an ?i-theme of the nth degree,
176 On certain General Properties [30
and lying in the tangent homaloid thereto, 1.2....n right lines may be
drawn coinciding throughout with the w-theme.
Thus one right line can be drawn at each point of a line of the first
order lying on the line ; two right lines at each point of a surface of the second
order lying on the surface; six right lines at each point of a hyperlocus of the
third degree, and so forth.
It is obvious that a surface may be treated as the homaloidal section of
a tritheme, just as a plane curve may be regarded as a section of a surface.
I shall proceed to show upon this view, how we may obtain a theorem given
by Mr Salmon for surfaces of the third degree of a particular character from
the law just laid down, according to which a tritheme of the third degree
admits of six right lines being drawn upon it at every point*.
Let (o (a, y, z, t, u) be the characteristic of any tritheme of the third
degree ; a, ft, 7, 8, e, coordinates to any point in the same. Then
ft> (a, ft, 7, 8, e) = 0, and the equation to the tangent homaloid will be
yea (a, /3, 7, 8, e) = 0, and the equation to the polar of the second degree
to the given tritheme in relation to the assumed point as origin, (that is,
the infinite system of homaloids that may be drawn from the point to
touch the tritheme), will be % 2 a> (a, ft, 7, 8, e) = 0.
But the section of any polar through its origin is the polar of the section
to the same origin ; hence the polar to the intersection of a> (x, y, z, t, u) = 0,
with x<u (a, ft, 7, 8, e) = 0, is the intersection of %&&gt; = with yj w = 0.
The projections of these intersections upon the space x, y, z, t will be
found by eliminating u, and getting the correspondent two equations
between x, y, z, t. Hence we see that the projection of the latter inter
section upon any space x, y, z, t is a cone ; or, in other words, this
intersection itself, that is, the polar to the intersection of the tritheme
with its tangent homaloid, is a cone ; that is to say, the -Surface of the
third degree formed by cutting a tritheme of the third degree by any
tangent homaloid has a conical point at the point of contact ; so that
every surface of the third degree with a conical point may be considered
as the intersection of a tritheme of the third degree with any tangent
homaloid thereto f.
* The reduction of any equation of the sixth degree to depend upon one of the fifth may be
shown by Mr Jerrard s method to be equivalent to drawing a straight line upon a tritheme of the
third degree, just as the reduction of the equation of the fifth degree to a trinomial form may be
shown to be dependent upon our being able to draw a straight line upon a ditheme of the second
degree. Now at every point of a tritheme straight lines may be drawn, but as they keep together
in groups of sixes they cannot be found in general at a given point without solving an equation
of the sixth degree.
t So in like manner a surface of the third degree with more than one conical point may be
generated by the intersection of the tritheme with a pluri-tangent plane ; and so too we may get
other varieties by taking homaloidal sections of trithemes whose characteristics are minus one or
more orders.
30] of Homogeneous Functions. 177
Hence then we see, as an instantaneous deduction from our general
theorem, that at any conical point (when one exists) of a surface of the
third degree six right lines may be drawn lying completely upon it. This
theorem is thus brought into an immediate and natural connexion with the
well-known one, that at every point in a surface of the second degree, two
right lines can be drawn lying wholly upon the surface*.
The last geometrical application of the theorem (3) which I shall make,
refers to the equations employed by Mr Salmon in No. xxi. (New Series) of
this Journal, to obtain the locus of the points on any surface at which
tangent lines can be drawn passing through four consecutive points. I may
remark in passing that these equations may be obtained by rather simpler
considerations than Mr Salmon has employed so to do, and without any
reference to Joachimsthal s theorem ; for if we take , rj, 0, as the co
ordinates of any point in one of the tangent lines above described, and if we
take the first polar to the surface with this point as origin, three out of the
four original points will be found in such polar consecutive but distinct ; and
consequently in the second polar, referred to the same origin, two will con
tinue consecutive but distinct, and consequently one will remain over in the
third polar.
Hence writing the equation to the surface o (x, y, z, t) = 0, and using D to
denote :r + 7 ?J- + ^ + ^ we sha11 evidentl y have
0, (1)
Dm = 0, (2)
D 2 o> = 0, (3)
D*a> = 0, (4)
as obtained by Mr Salmon. And the same kind of reasoning precisely
applies to the theory of points of inflexion in curves; three consecutive
points in a right line in this case corresponding to four such in the case
above considered.
If now we make |r T $ = u >
v
* If we have an indeterminate system of algebraical equations consisting of one quadratic and
another n c function of three variables, this may be completely resolved by considering the first as
an equation to a surface of the second degree, finding at any point thereof the two lines which
lie upon the surface, and determining their respective intersections with the surface represented
by the second equation. This will require therefore the solution only of a quadratic and an n c
equation. In like manner an indeterminate system of two equations of four variables, one of the
third and the other of the nth degree, may be completely resolved (with the aid of the theorem in
the text) by means of two equations, one of the sixth and the other of the nth degree.
12
178 On certain General Properties [30
the equations (2), (3), (4), by our theorem, may be expressed in terms of
u, v, w, which being eliminated we obtain an equation between x, y, z, t,
which will express the surface whose intersection with the given surface
<y = serves to determine the locus of the points in question.
Hence if we proceed in the ordinary manner to eliminate two of the four
letters, as and rj, between the equations (2), (3), (4), the resultant will be of
the form M x <j>(, 6), where M does not contain 77, f or 0, and where by
the general laws of elimination <f> (, 0) will be an integral function of the
sixth degree in respect to ,0: and it is manifest that M x $ ( 0) will be
identical with the resultant of (2), (3), (4) expressed in terms of u, v, w,
when u and v are eliminated cy-pres of an integralizing factor, showing that
< ( 0) is w 6 integralized, that is, is equal to (t-z0)<. Consequently as M<f>
is of the order (n- 1) 2 . 3 + (n - 2) 1 . 3 + (n - 3) 1 . 2, that is, llw - 18 in
respect to x, y, z, t, it follows that M = Q, the equation to the second surface
spoken of above, will be of the order lift- 24, agreeable to Mr Salmon s
showing.
I shall conclude this paper by showing the application of our theorem to
the subject propounded by Mr Jerrard and Sir William Hamilton, of systems
of equations containing a sufficient number of variable letters for effecting
the solution without elevation of degree.
If we have n homogeneous equations containing a sufficient number of
letters a l} a 2 ...a m to enable us to express the solution of (n - 1) of the
equations under the form
,! =a 1 + X&,
o, 2 = <z 2 + X/3 2 ,
Q"m ^m "T- m i
where a,, 2 ... a m , &, & ... m are supposed known, and X is indeterminate,
it is evident that by substituting these values in the nth equation, X may be
found by solving an equation of the same degree as that equation contains
dimensions of a l} a^ ... a m .
Let us then propose this question: how many letters a ly a 2 ...a r are
needed to obtain a linear solution of a system of n equations
< 1 = 0, < 2 =0,...4> n = 0,
of the several degrees t 1( (,...*, without elevation of degree; by a linear
solution being understood a solution under the form
where X is left indeterminate.
30 J of Homogeneous Functions. 179
Let us suppose that a lt Oo...a r , substituted respectively for a lf a 2 ...a r ,
satisfy the given system of equations. The determination of these values
without elevation of degree will, from what has been said before, depend
upon the linear solution of a system of equations differing from the given
system by the omission of any one of them at pleasure.
Now make
-n d d d
D = j -j- + a., -, + . . . + a r -j >
aaj aa 2 da r
and then write
D<k = 0, D 2 ^, = 0... D"<k =
D< 2 = 0, D 9 ^ = . . . D"</> 2 =
The values of a lt a 2 ... a r derived from this system, say (a^, (a) 2 ... (a),,
give
a 1 = 1 + \(a) 1 , a 2 = a 2 + A,(a) 2 , ... a r = a r + X(a) r ,
a solution under the required form, where A, is left indeterminate.
The solution of this new system without elevation of degree depends on
the linear solution of all but one of them ; this excepted one may be taken
the one whose dimensions i r are the highest or as high as any of the
quantities i ly t 2 ... i n .
Consequently, if we use the symbol (k 1} & 2 ... k r ) to denote the number of
letters required for the linear solution (without elevation of degree) of k r
equations of the first degree, k 2 of the second, k 3 of the third, ..., k r of
the rth, it would at first sight appear from the preceding reduction that
we must have
QkA-.JW-K,X*...>Z t *i,-K r l,
where K 1 = k l + k z + ... + k r ^ + k r ,
K z = k 2 + . . . + & r _i + k r ,
tl-ri for 1 ~l~ k r ,
K r = k r - 1.
But now steps in our theorem (3), and shows that the system (0) may be
superseded by another, in which the variables, instead of being a 1} a 2 ... a n ,
will be
a i 2 _
a i~ ~ Un, Q-i -- a n , ... a n ^. l --- a n ;
a n On Cl n
consequently the number of really independent variables is only (n 1) ; we
must therefore have
122
180 General Properties of Homogeneous Functions. [30
Since the introduction of a new simple equation is equivalent to the
requirement of one more disposable letter, we may write the above more
symmetrically under the form
where K l = 1 + ^ + & 2 + . . . + k r ,
K r = k r -l.
By means of this formula of reduction (k lt k 2 ... k r ) may be finally brought
down to the form (L), and the value of (L) being the number of letters
required for the linear solution of a system of L linear equations is evidently
L + 2.
Thus, to determine the number of letters required for the linear solution
of a single quadratic, we write
(0,1) = (2) = 4.
For two quadratics, we write
(0,2) = (3,1) = (5) = 7;
for a quadratic and a cubic,
(0,1,1) = (3, 2) = (6,1) = (8) =10;
for two cubics,
(0, 0, 2) = (3, 2, 1) = (7, 3) = (11, 2) = (14, 1) = (16) = 18.
These results coincide with those obtained by Sir William Hamilton in
his Report on Mr Jerrard s Transformation of the Equation of the Fifth
Degree in the Transactions of the British Association. I have much more
to say on the subject of the linear solution of a system of indeterminate
equations, and am, I believe, able to present the subject in a more general
light than has hitherto been done ; but my observations on this matter must
be deferred until a subsequent communication.
31.
REPLY TO PROFESSOR BOOLE S OBSERVATIONS ON A
THEOREM CONTAINED IN THE LAST NOVEMBER
NUMBER OF THE JOURNAL.
[Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 171 174.]
THE restricted space that can be spared for discussion in these pages,
necessitates me to compress within the narrowest limit the remarks which
I feel bound to make on Mr Boole s extraordinary observations* in the
February number of this Journal, on my theorem contained in the ante
cedent number thereoff, which statements I cannot, in the interests of truth
and honesty, suffer to pass unchallenged. The object of that theorem was
to show how the determinant of the quadratic function resulting from the
elimination of any set of the variables between a given quadratic function
and a number of linear functions of the same variables, could be represented
without performing the actual elimination by a fraction, of which the nume
rator would be constant whichever set of the variables might be selected for
elimination, and the denominator the square of the determinant corresponding
to the coefficients of the variables so eliminated. The numerator itself is a
determinant, obtained by forming the square corresponding to the determinant
of the given quadratic function, and bordering it horizontally and vertically
with the lines and columns corresponding to the coefficients of all the variables
in the given linear equations. An immediate corollary from this theorem leads
to Mr Boole s. Conversely upon the principle that "tout est dans tout"
Mr Boole devotes a page and a half of close print merely to indicate the
steps of a method by which from his theorem mine is capable of being
deduced, ending with the announcement, that the numerator in question
is equal to the quantity
(the symbols above employed being Mr Boole s own), and concludes with
assuring his readers that " he has ascertained that Mr Sylvester s result is
reducible to the above form." Mr Sylvester would be very sorry to put his
[* Cambr. and Dublin Math. Jour. vi. (1851), pp. 90, 284.] [t p. 135 above.]
182
Reply to Professor Boole s Observations
[31
result under any such form. Mr Boole could scarcely have reflected upon the
effect of his words when he indulged in the remark which follows " there
cannot be a doubt that for the discovery of the actual relation in question, the
above theorem is far more convenient than Mr Sylvester s." Of the value to
be attached to this assertion the annexed comparison of results is submitted
as a specimen.
Let the quadratic function be
aa? + by 2 + cz 2 + dt 2 + 2exy -4- 2ezt + Zgxz + tyyt + 2hyz + 2tjxt,
and the linear functions (taken two in number)
Ix + my + nz + pt,
I x + m y + n z+p t.
My numerator will be the determinant (hereinafter cited as the extended
determinant),
a
e
9
ri
I
r
e
b
h
7
m
m
9
h
c
e
n
n
n
7
d
P
P
I
m
n
P
V
m
n
P
To find the numerator of
symbolical operator
Mr Boole s fraction, we must form the
d
d
j- + m -TL + n j + P jj
da do dc da
-r
de
d_
dc
A
de
2ln
-j-
dy
-ji
dh
+ 2m p
r
and after expanding the determinant hereunder written
^ g rj
, b h 7
< h c e
) <y e d
~
dh
perform the operations above indicated upon the result so obtained.
These are the operations and processes which, on Professor Boole s
authority, we are to accept "as without doubt far more convenient" than
the one simple process of forming, and when necessary, calculating the
31] on a Theorem of Mr Sylvester s. 183
extended determinant above given. Here for the present I leave the
case between Mr Boole and myself to the judgment of the readers of
this Journal.
In the April Number of the Philosophical Magazine*, I have shown
that the extended determinant serves, not only to represent the full and
complete determinant of the reduced quadratic function, but likewise all
the minor determinants thereof; the last set of which will be evidently no
other than the coefficients themselves. For instance, in the example above
given, if we wish to find the coefficient of x* after z and t have been
eliminated, we have only to strike out the line and column e b h 7 m ni from
the extended determinant ; if we wish to find the coefficient of y z , we must
strike out the line and column a e g 77 IV ; to find the coefficient of xy, we
must strike out the line a eg rj 1 1 and the column ebh<y m m, or vice versa.
In each of these cases the determinant so obtained is the numerator of
the equivalent fraction ; the denominator remaining always the same function
of the coefficients of transformation as in the original theorem.
Again, if there be taken only one linear equation, and by aid of it x is
supposed to be eliminated ; and if the reduced quadratic function be called
Lf + Mz* + Nt* + 2Pzt + IQyt + 2Rzy,
the same extended determinant as before given will serve, when stripped of
its outer border, consisting of the line and column I m n p , to produce
the various equivalent fractions : thus form the square
L R Q
R M P
Q P N.
L R
The numerator of the fraction equivalent to
may be found by striking out from the form of the extended determinant the
L Q
line and column rj 7 e dp; that corresponding to
RM
that is, to LM-R*,
RP
, that is, LP-RQ, will
be found by striking out the line ghcen and the column rjyedp, or vice
versa ; and so forth for all the first minor determinants ; and similarly the
second minors, that is, L, M, N, P, Q, R, may be obtained by striking out in
each case a correspondent pair of lines and pair of columns. Thus, to find
the numerator of >the same pair of lines and columns, namely, (ghee n),
(77 7 e d p), must be elided. To find the numerator of R, the pair of lines
(ghee n), (77 7 e d p), and the pair of columns (ebhj m), (77 7 e d p), or vice
versa, will have to be elided ; and so forth for the remaining second minors.
I may conclude with observing, that the theorem contested by Mr Boole is
an immediate corollary from the general Theory of Relative Determinants
alluded f to in the " Sketch " inserted in the present number of the Journal.
[* p. 241 below.] [t P- 188 below.]
32.
SKETCH OF A MEMOIR ON ELIMINATION, TRANSFORMATION,
AND CANONICAL FORMS.
[Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 186 200.]
THERE exists a peculiar system of analytical logic, founded upon the
properties of zero, whereby, from dependencies of equations, transition may
be made to the relations of functional forms, and vice versa : this I call the
logic of characteristics.
The resultant of a given system of homogeneous equations of as many
variables, is the function whose nullity implies and is implied by the possi
bility of their coexistence, that is, is the characteristic of such possibility ;
but inasmuch as any numerical product of any power of a characteristic is
itself an equivalent characteristic, in order to give definiteness to the notion
of a resultant, it must further be restricted to signify the characteristic taken
in the lowest form of which it in general admits.
The following very general and important proposition for the change of
the independent variables in the process of elimination, is an immediate
consequence of the doctrine of characteristics.
Let there be two sets of homogeneous forms of function ;
the 1st, $1, $2 <f>n,
the 2nd, fa, fa... fa.
Let the results of applying these forms to any sets of n variables be
called
(<fc), (<*)... (</>),
(fa), (fa)... (fa);
then will the resultant (in respect to those variables) of
^{(fa), (fa)...(+ n )},
(fa)--- (fa)},
32] Elimination, Transformation, and Canonical Forms. 185
be the product of powers (assignable by the law of homogeneity) of the
separate resultants of the two systems,
By means of the doctrine of characteristics the following general problem
may be resolved.
Given any number of functions of as many letters, and an inferior number
of functions of the same inferior number of letters, obtained by combining,
inter se, in a known manner, the given functions, to determine the factor by
which, the resultant of the reduced system being divided, the resultant of the
original system may be obtained.
If in the theorem for the change of the independent variables both sets
of forms of functions be taken linear, we obtain the common rule for the
multiplication of determinants : if we take one set linear and the other not,
we deduce two rules, namely, That the resultant of a given set of functional
forms of a given set of variables, enters as a factor into the resultant,
1st, of linear functions of the given functions of the given variables ;
2nd, of the given functions of linear functions of the given variables :
the extraneous factor in each case being a power of what may be con
veniently termed the modulus of transformation, that is, the resultant of
the imported linear forms of functions.
From the second of these rules we obtain the law first stated I believe
for functions beyond the second degree by Mr Boole, to wit, that the deter
minant of any homogeneous algebraical function (meaning thereby the
resultant of its first partial differential coefficients) is unaltered by any
linear transformations of the variables, except so far as regards the intro
duction of a power of the modulus of transformation. This is also
abundantly apparent from the fact, that the nullity of such determinant
implies an immutable, that is, a fixed and inherent, property of a certain
corresponding geometrical locus.
There exist (as is now well known) other functions besides the deter
minant, called by their discoverer (Mr Cayley) hyperdetermiuants, gifted
with a similar property of immutability. I have discovered a process for
finding hyperdeterminants of functions of any degree of any number of
letters, by means of a process of Compound Permutation. All Mr Cayley s
forms for functions of two letters may be obtained in this manner by the aid
of one of the two processes (to wit, that one which will hereafter be called
the derivational process), for passing from immutable constants to immutable
forms. Such constants and forms, derived from given forms, may be best
186 Sketch of a Memoir on Elimination, [32
termed adjunctive ; a term slightly varied from that employed by M. Hermite
in a more restricted sense.
The two processes alluded to may be termed respectively appositional and
derivational. The appositional is founded upon the properties of the binary
function x^ + yrj + z+ ... ; in which, whether we substitute linear functions
of x, y, z, fec., or linear functions of , V, & & c -> in P lace of x > V> z > & c -> or
, 77, , &c., the result is the same.
Consequently, if we apply the form < to , TJ . . . , and take any constant
(in respect to ,??... ) adjunctive to
calling this quantity ^(x,y ...z, t\ the form -v/r is evidently adjunctive to the
form (f> : and if we expand so as to obtain
ty(x, y ...z, t) = ^r l (x, y...z)t" + ^(x, y ...z)tP + toe.,
it is evident ^r l , -v|r 2 , toe. will be each separately adjunctive to <. These
forms, when ty is obtained by finding the determinant in respect to , 77 ...
of S, are, in fact, identical with Hermite s " formes adjointes."
The derivational mode of generating forms from constants depends upon
the property of the operative symbol
fc d _, d j i.r d
X = tdx + *dy + - + Sdz>
applied to ^ a function of x,y...z; namely, that if in <j>, in place of these
letters, we write linear functions thereof, to wit x , y ... z , we may write
where , rj ... ? will be the same functions of 77 ... that a? , y ... z are of
x, y . . . z.
Suppose now, in the first place, that in regard to 77 ... ^(x,y...z) is
adjunctive to % r </>(tf, y *)5 then is the form ^ ad J unctive to the form ^ for
on changing x, y ... z to x, y ... z ,
becomes f + V , + - + ?
and consequently ^ (, y . . . z) becomes + (x , y ... z), multiplied by a power
of the modulus of transformation, the modulus of that transformation, be it
well observed, whereby of, y ... * would be replaced by x,y...z, and not as
in the appositional mode of that converse transformation according to which
32] Transformation, and Canonical Forms. 187
x, y ... z would be replaced by x , y ... % . It is on account of this converse-
ness of the modes of transformation that the appositional and derivational
modes of generating forms cannot except for a certain class of restricted
linear transformations be combined in a single process. More generally, if
instead of a single function -^ r <^> (x, y . . . z}, we take as many such with
different indices to % as there are variables, and form either the resultant
in respect to , TJ ... , or any other immutable constant in regard to those
variables, (presuming in extension of the hyperdeterminant theory and as no
doubt is the case, that such exist), every such resultant or other constant will
give a form of function of x, y ... z adjunctive to the given form <.
It may be shown that every such resultant so formed will contain <f> as a
factor.
Again, in the former more available determinant mode of generation, if
we take the determinant in respect to , 77 ... , it may be shown that all the
adjunctive functions so obtained will be algebraical derivees of the partial
differential coefficients of </> in respect to x, y ... z: that is to say, if these be
respectively zero, all such adjunctive functions so derived, as last aforesaid,
will be zero, or in other words, each such adjunctive is a syzygetic function of
the partial differential coefficients of the primitive function.
To Mr Boole is due the high praise of discovering and announcing, under
a somewhat different and more qualified form and mode of statement, this
marvel-working process of derivational generation of adjunctive forms. I was
led back to it, in ignorance of what Mr Boole had done, by the necessity
which I felt to exist of combining Hesse s so-called functional determinant,
under a common point of view with the common constant determinant of a
function ; under pressure of which sense of necessity, it was not long before
I perceived that they formed the two ends of a chain of which Hesse s end
exists for all homogeneous functions, but the other only when such functions
are algebraical.
In fact, if we give to r every value from 2 upwards, the successive
determinants in respect to , 77 ... of
4- -\-t
ix dy <
will produce the chain in question, which, when < is algebraical and of
n dimensions, comes to a natural termination when r = n 1. The last
member of and the number of terms in this chain are identical with the
last member of and the number of terms in Sturm s auxiliary functions,
when the variables are reduced to two. There is some reason to anticipate
that this chain of functions may be made available in superseding Sturm s
chain of auxiliaries ; and if so, then the fatal hindrance to progress, arising
from the unsymmetrical nature of the latter, is overcome, and we shall be
188 Sketch of a Memoir on Elimination, [32
able to pass from Sturm s theorem, which relates to the theory of Keno-
themes, or Point-systems, to certain corresponding but much higher theories
for lines, surfaces, and w-themes generally.
The restriction of space allowed to me in the present number of the
Journal will permit me only to allude in the briefest terms to the theory
of Relative Determinants, which, as it will be seen, plays an important part
in the effectuation of the reductions of the higher algebraical functions to
their simplest forms. Nor can the effect of the processes to be indicated be
correctly appreciated without a knowledge of the circumstances under which
the resultant of a given system of equations can sink in degree below the
resultant of the general type of such system. Abstracting from the case
when the equations separately, or in combination, subdivide into factors, this
lowering of degree, as may be shown by the doctrine of characteristics, can
only happen in one of two ways. Either the particular resultant obtained is
a rational root of the general resultant, or the general resultant becomes zero
for the case supposed, and the particular resultant is of a distinct character
from the general resultant, being in fact the characteristic of the possibility
not of the given system of equations being merely able to coexist (for that is
already supposed), but of their being able to coexist for a certain system of
values other than a given system or given systems. Such a resultant may be
termed a Sub-resultant ; the lowest resultant in the former case may be
termed a Reduced-resultant. The theory of Sub-resultants is one alto
gether remaining to be constructed, and is well worthy equally of the
attention of geometers and of analysts.
As to the theory of Relative Determinants, the object of this theory
is to obtain the determinant resulting from eliminating as many variables
as can be eliminated, chosen at pleasure from a set of variables greater in
number than the equations containing them ; and the mode of effecting
this object is through the method of the indeterminate multiplier. To avoid
the discussion of the theory of sub-resultants and other particularities, I shall
content myself with giving the rule applicable to the case (the only one of
which as yet a practical application has offered itself to me in the course of
my present inquiries) when all but one of the functions are linear.
If U, A, LZ ... L m be the first an n c and the others linear functions of n
variables, and it be desired to find the determinant of the resultant arising
from the elimination of any m out of the n variables, the following is the
rule :
Find the determinant, that is, the resultant of the partial differential
coefficients in respect to the given variables, and of X 1} \. 2 ...\ m of
U -}- LI\I
32] Transformation, and Canonical Forms. 189
This resultant, in its lowest form, will be always a rational (n l)th root of
the resultant of the homogeneous system of equations to which the system
above given can be referred as its type ; and this reduced resultant divided
by a power (deterrninable by the law of homogeneity) of the resultant of
L-i, L z ...L m , when all but the selected variables are made zero, will be the
resultant determinant required*. As regards what has been said concerning
the reducibility of the general typical resultant in the case before us, this is
a consequence of, and may be brought into connexion with, the following
theorem, which is easily demonstrable by the theory of characteristics. If
Qi Q-2 Qm be m homogeneous functions of m variables of the same
degree, r of which enter in each equation only as simple powers uncom-
bined with any of the other variables, then the degree of the reduced
resultant is equal to the number of the equations multiplied by the
(m r l)th power of the number of units in the degree of each,,
subject to the obvious exception that when r is m, (there being in fact
but one step from r = m 2 to r = m), instead of r, (r 1) must be em
ployed in the above formula. As an example of a sub-resultant as
distinguished from a reduced-resultant, I instance the case of three
quadratics U, V, W, functions of x, y, z, in each of which no squared
power of z is supposed to enter : it may easily be shown by my dialytic
method that instead of six equations, between which to eliminate a?, y 2 , z 2 ,
xy, xz, yz, we shall have only 5, the three original ones and two instead of
three auxiliaries between which to eliminate x 2 , y 2 , xy, xz, yz, the apparent
resultant is accordingly of the 9th instead of the 12th degree. But this is
not the true characteristic of the possibility of the coexistence of the given
systems, which in fact is zero, as is evidenced by the fact that they always do
coexist, since they are always satisfiable by only two relations between the
variables, to wit # = 0, y = 0. Tbe apparent resultant is then something
different, and what has been termed by the above a Sub-resultant.
I take this opportunity of entering my simple protest against the appro
priation of my method of finding the resultant of any set of three equations
of degrees equal or differing only by a unit, one from those of the other two,
by Dr Hesse, so far as regards quadratic functions, without acknowledgment,
four years after the publication of my memoir in the Philosophical Magazine :
the fundamental idea of Dr Hesse s partial method is identical with that of
my general one. Still more unjustifiable is the subsequent use of the dialytic
principle, by the same author, equally without acknowledgment, and in cases
where there is no peculiarity of form of procedure to give even a plausible
ground for evading such acknowledgment. It is capable of moral proof that
* The same method applies not only to the Final or Constant Determinant, but likewise to
all the Functional Determinants in the chain above described, extending upwards from this to-
the Hessian, or as it ought to be termed, the first Boolian Determinant.
190 Sketch of a Memoir on Elimination, [32
what I had written on the matter was sufficiently known in Berlin and at
Konigsberg, at each epoch of Dr Hesse s use of the method.
I now proceed to the consideration of the more peculiar branch of my
inquiry, which is as to the mode of reducing Algebraical Functions to their
simplest and most symmetrical, or as my admirable friend M. Hermite well
proposes to call them, their Canonical forms. Every quadratic function of
any number of variables may always be linearly transformed into any other
quadratic functions of the same, and that too in an infinite variety of ways ;
but in every other instance there will be only a limited number of ways,
whereby, when possible, one form will admit of being transmuted into any
other : and with the sole exception of a cubic function of two letters, such
transmutation will never be possible, unless a certain condition, or certain
conditions, be satisfied between the constants of the forms proposed for
transmutation. The number of such conditions is the number of para
meters entering into the canonical form, and is of course equal to the
number of terms in the general form of the function diminished by the
square of the number of letters. Thus there is one parameter in the
canonical form for the biquadratic function of two and the cubic function
of three letters, and no parameter in the cubic function of two letters.
Hitherto no canonical forms have been studied beyond the cases above
cited, but I have succeeded, as will presently be shown, in obtaining
methods for reducing to their canonical forms functions with two and four
parameters respectively. Owing to what has been remarked above, the
theory of quadratic functions is a theory apart. Simultaneous transforma
tion gives definiteness to that theory, but has no existence for any useful
purpose for functions of the higher degrees. Where the theory of simul
taneous transformation ends, that of canonical forms properly .begins ; and
in what follows, the case of quadratic forms is to be understood as entirely
excluded. Such exclusion being understood, there is no difficulty in assigning
the canonical, that is, the simplest and most symmetrical general, form to
which every function of two letters admits of being reduced by linear trans
formations. If the degree be odd, say 2m + 1, the canonical form will be
Ml n+i + Wj aH-i + . . . + u +* ;
if the degree be even, say 2m, the canonical form will be
all the u s being linear functions of the two given variables. It is easy to
extend an analogous mode of representation to functions of any number of
letters. From the above we see that for cubic, biquadratic, and quintic
functions of two letters, the canonical forms will be respectively
with a linear relation in the last-named case between u, v, w.
32 J Transformation, and Canonical Forms. 191
First as to the reduction of any 4 C function to Cayley s form
u* + v* + Ku 2 v 2 .
This may be effected in a great variety of ways, of which the following is not
the simplest as regards the calculations required, but the most obvious. Let
the modulus of transformation, whereby the given biquadratic function, say
F(x, y), becomes transmuted into its canonical form, be called M ; let the
determinant of F be called D lt and the determinant of the determinant in
respect to and 77 of
d
which latter, for brevity s sake, may be termed the Hessian of F, (although
in stricter justice the Boolian would be the more proper designation), be
called D 2 . Then, by examining the canonical form itself (which is as it
were the very palpitating heart of the function laid bare to inspection),
we shall obtain without difficulty the two equations
m (1 - 9m 2 ) 2 (m 2 - I) 2 = MD
Eliminating the unknown quantity M, we obtain
m 2 (m 2 1 ) 2
/
"
(l-9m 2 ) 2 " l-9m 2
where c is a known quantity.
This cubic equation for finding m is of a peculiar form ; it being easy to
show d priori, by going back to the canonical form, that its three roots are
m, B(m), &*(m}, where
ra-1
6 (m) =
3m + 1
6 being a periodical form of function such that 3 (m) = m.
This it is which accounts for the simple expression for m, that may be
obtained by solving the cubic above given. A better practical mode is to
take, instead of the determinant of the given function and its Hessian, the
two hyperdeterminants and eliminate as before : a cubic equation having
precisely the same properties, and in fact virtually identical with the former,
will result. When m and consequently M are found, there is no difficulty
whatever, calling the given function F and its Hessian H (F), in forming
linear functions of the two, as
<t>(m)F+ ^(m)H(F)}
which shall be equal to, that is, identical with, (u* + v 2 ) 2 and u-v 2 , whence
u and v are completely determined.
192 Sketch of a Memoir on Elimination, [32
Another and interesting mode of solution is to take, besides the given
function F and its Hessian, either the second Hessian or the post-Hessian
of the given function, by the post-Hessian understanding the determinant in
respect of and 77 of
any three of the four functions will be linearly related, and it may be shown
that, calling either the second Hessian (that is, the Hessian of the Hessian)
or the post-Hessian H , we shall have
where a and b will be rational and integer functions of the coefficients of
F, and numerical multiples of two quantities R and 8, such that the
determinant of F will be equal R 3 + S 2 ; and this, be it observed, without
any previous knowledge of the existence of these hyperdeterminants R
and 8.
If now we go to Hesse s form for a cubic function of three letters, we
shall find that precisely similar modes of investigation apply step for step.
Calling the function F and its Hessian H(F), and the post- Hessian or second
Hessian at choice H (F), we shall find
H (F) + mSH(F) + n&F = 0,
where m and n are numerical quantities and R 3 + S 2 equal the determinant
of F. It is interesting to contrast this equation with the one previously
mentioned as applicable to the 4 C functions of two letters, namely,
H (F) + mRH (F) + nSF = 0.
In both instances there is no difficulty in assigning the relations between
the original R and 8, and the R and 8 of any adjunctive form. All
Aronhold s results may be thus obtained and further extended without
the slightest difficulty. As regards the equation for finding the parameter
in Hesse s canonical form for the cubic of three letters, this will be of the
4th degree in respect to the cube of the parameter, and the roots will be
functionally representable as
as; d(x); 0(ar); ty(x),
where 6- (x) = <p (x} = ^(x) = x;
(a?) =
owing to which property the equation is soluble under the peculiar form
observed by Aronhold.
32] Transformation, and Canonical Forms. 193
I pass on now to a brief account of the method, or rather of a method
(for I doubt not of being able to discover others more practical), of reducing
a function of the 5th degree of two letters (say of x and y) to its canonical
form u s + v s + w 5 , subject to the linear relation au + bv + cw = 0, where the
ratios a : b : c, and the linear relations between u, v, w and the two given
variables are the objects of research. Here I have found great aid from the
method of Relative Determinants; and I may notice that the successful
application of more compendious methods to the question would be greatly
facilitated were there in existence a theory of Relative Hyperdeterminants,
which is still all to form, but which I little doubt, with the blessing of God,
to be able to accomplish. It may some little facilitate the comprehension of
what follows, if c be considered as representing unity.
Calling as before the given quintic function F, the modulus of transforma
tion M, the Hessian and post- Hessian of F, H and H , and its ordinary or
constant determinant D, we shall find
a?v 3 w 3 + bWu 3 + c*u*v 3 = M *
and PPPPM H
where P 1 = a^vw + b%wu + c^uv,
P 3 = a?
wu + c* uv ;
also D = M w multiplied by the product of the sixteen values of
From the above equations it may be shown that H (a known function of
the 8th degree of the given variables x, y) must be capable of being thrown
under the form
L {(x - a,y) (x - a 2 y) x (x - a 3 y) (x - a t y)
x (x - a s y) (x - a 6 y) x (x - a 7 y] (x - a 8 y)},
where (a, - 2 ) 2 x (a 3 - a,) 2 x (a 5 - a 6 ) 2 x (a, - a 8 ) 2
so that K is a known quantity*. Accordingly the said equation of the 8th
degree, considered as an algebraical equation in - , may by known methods be
7
* Or in other words, the post-Hessian determinant of a given function in two letters of the
second degree, may be divided into four quadratic factors in such a way that the product of the
determinants of these several factors shall be equal to the determinant of the given function.
s- 13
194 Sketch of a Memoir on Elimination, [32
found by means of equations not exceeding the 4th or even the 3rd degree :
in fact, to do this it is only necessary to form the equation to the squares of
OC
the differences of the roots of - in the equation H -f- y s = 0, which new equa-
i7
tion will be of the 28th degree. If we then form two other equations of the
378th degree, one having its roots equal to *JK multiplied by the binary
products of the twenty-eight roots of the equation last named, the other to
*JK multiplied by the reciprocal of such binary products, the left-hand
members of these two equations expressed under the usual form will have
a factor in common, which may be found by the process of common measure
and will be of the 6th degree, whose roots consisting of three pairs of
reciprocals may be found by the solution of cubics only.
In this way, by means of cubics and quadratics,
(a.-a^Y, (a s -a,) 2 , (a 5 -a 6 )-, (a 7 -a 8 ) 2 ,
can be found, which being known,
a-iCiz, a 3 a 4 , a 5 a 6 , a 7 a s ,
can be determined in pairs by means of quadratics from the equation
H -r- y 8 = 0. This being supposed to be done, we have
Pi -A,
where L l} L 2 , L s , L 4 , are known quadratic functions of x arid y. To
determine the ratios of/, g, h, k, we have three equations* obtained from
the identity
P 4 ) = ;
f:g:h:k being known, fLj, : gL n _ : hL s : kL 4 are known ratios.
But PI + P a
P, + P 3 =
Hence a*vw = \P,
= \Q,
where P, Q, R are known quadratic functions of x, y.
* For we must have the coefficients of x 2 , xy and y- in
f
of all them zero.
32] Transformation, and Canonical Forms. 195
Hence a : b : c may be found by means of the identical equation
a?w s v 3 + b*u 3 u? + c?tfu* = H (F),
whereby the ratios a~% :b~% : c~% can be obtained without any further
extraction of roots, showing that there is but one single true system of
ratios a 5 : 6 5 : c 5 applicable to the problem; a:b:c being thus found, A, is
easily determined, and thus finally u, v, iv are found in terms of x and y*.
I have little doubt that a more expeditious mode of solution than the
foregoing -f- will be afforded by an examination of the properties and relations
of the quadratic and cubic forms, adjunctive to the general quintic functions,
and indeed to every (4tn+ l) c function of two letters hereinbefore adverted to.
Sufficient space does not remain for detailing the steps whereby the
general cubic function of four letters may, by aid of equations not trans
cending the fifth degree, be reduced to its canonical form u 3 + v 3 + w 3 + p 3 + q 3 ,
wherein u, v, w, p, q are connected by a linear equation
au + bv + cw + dp + eq = ;
the four ratios of whose coefficients a:b:c:d:e give the necessary number
456
4 2 parameters furnished by the general rule. Suffice it for the
JL . Z . o
present to say, that the analytical mode of solution depends upon a cir
cumstance capable of the following geometrical statement : Every surface
of the 4th degree represented by a function which is the Hessian to any
given cubic function whatever of four letters, has lying upon it ten straight
lines meeting three and three in ten points, and these ten points are the only
points which enjoy the following property in respect to the surface of the 3rd
degree denoted by equating to zero the cubical function in question, to wit,
that the cone drawn from any one of them as vertex to envelop the surface,
will meet it not in a continuous double curve of the 6th degree, but in two
curves each of the 3rd degree, lying in planes which intersect in the ten lines
respectively above named ; so that to each of the ten points corresponds one
of the ten lines : these ten points and lines are the intersections taken
respectively three with three, and two with two, of a single and unique
system of five principal planes appurtenant to every surface of the 3rd
degree, and these planes are no other than those denoted by
u = 0, v = 0, w = 0, p = 0, q = 0.
* The problem thus solved may be stated as consisting in reducing the general function
ax 5 + b x 4 y + cx 3 y 2 + dx-y 3 + exy* +fy 5 to the form
(Ix + my) 5 + (I x + m y) 5 + (l"x -f m"t/) 5 .
t The coefficients in the reducing recurrent equation of the 6th degree in the process above
detailed may rise to be of 541632 dimensions in respect to the original coefficients in .F.
132
196 Sketch of a Memoir on Elimination, [32
I have found also by the theory of Sub-resultants, that the analogy
between lines and surfaces of the third degree, in regard to the existence
of double and conical points, is preserved in this wise : that in the same
way as a double point on a curve of the 3rd degree commands the existence
of a double point on its Hessian, so does a conical point in a surface of the
3rd degree command over and above the 10 necessary, and so to speak
natural conical points, at least one extra, that is to say an llth conical
point on its Hessian. And here for the present I must quit my brief and
imperfect notice of this subject, composed amidst the interruptions and
distractions of an official and professional life.
Observation. It may be somewhat interesting and instructive to my
readers, to have a table of the successive scalar* determinants of a quintic
function of two letters presented to them at a single glance. Preserving the
notation above [page 193], we have the following expressions:
The given function = u* + v s + w 5 ,
its Hessian = M 2 (aVw 3 + bWu 3 + c^W),
its post-Hessian = M 6 x the product of the four forms of
a?vw + 6* (\}^wu + c*-
its pra3ter-post-Hessian = M 12 x the product of the nine forms of
and the final determinant = M x the product of the sixteen forms of
d* +(!)*&*
The success of the method applied depends (as above shown) upon the
fact of a certain function of the roots of the post-Hessian (which is an octavic
function of the variables) being known, which fact hinges upon the circum
stance that
(MJ x (M 2 ) 4 = i/ 20 .
P.S. I have much pleasure in subjoining the cubical hyperdeterminant
of the 12th degree function of two letters, worked out upon the principle of
Compound Permutation hinted at in the foregoing pages, for which 1 am
indebted to the kindness and skill of my friend Mr Spottiswoode.
* By which I mean the determinants in respect to , rj of
d
32] Transformation, and Canonical Forms. 197
The function being called
12 11
atf* + l%bx ll y + ^ c# 1( y + &c. . . . + ly l \
Z
the following is* its cubical hyperdeterminant :
agm 6ahl + I5aik + lOcy 2 Qbfm,
- 246M- + Mgl + ZObij - 24>cfl + lUcgk,
145ct 2 + 5Qchj + locem + 2Qcgi +
- 4iQOdgj + 280dhi + 20del + oOdfe
Mr Spottiswoode will I hope publish the work itself in the next number
of the Journal, in which I shall also show how the hyperdeterminants of the
cubical function of three letters, Aron hold s S and T, may be similarly
obtained.
[* See below, p. 202.]
33.
ON THE GENERAL THEORY OF ASSOCIATED
ALGEBRAICAL FORMS.
[Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 289 293.]
THE following brief exposition of the general theory of Associated Forms,
as far as it has been as yet developed by the labours or genius of mathema
ticians, is intended as elucidatory and, to a certain extent, emendative of
some of the statements in my paper* on Linear Transformations, in the
preceding number of the Journal.
In the first place, let a linear equivalent of any given homogeneous
function be understood to mean what the function becomes when linear
functions of the variables are substituted in place of the variables them
selves, subject to the condition of the modulus of transformation (that is,
the value of the determinant formed by the coefficients of transformation)
being unity.
Secondly, let two square arrays of terms (the determinants corresponding
to each of which are unity) be said to be complementary when each term in
the one square is equal to the value of what the determinant represented by
the other square becomes when the corresponding term itself is taken unity,
but all the other terms in the same line and column with it are taken zero.
This relation between the two squares is well known to be reciprocal. Thus,
for instance,
i b c
a
b c
b" c
and
7
7
7"
will be said to be reciprocally complementary to one another when the two
determinants which they represent are each unity, and when we have
[* p. 184, above.]
33] General Theory of Associated Algebraical Forms. 199
a =
1
a =
100
ff 7
V c
" 7"
6" c"
010
=
010
a 7
a c
a" 7"
a" c"
a 7
=
a c
010
010
a" 7"
a" c"
b =
&c. &c.
Accordingly, two transformations, say of F (ar, y, 2) and 6r (w, v, w) respect
ively, may be said to be concurrent when in F for x, y, z, we write
ax +by +cz,
a x +b y +c z,
a"x + b"y + c"z ;
and in for u, v, w, we write
au +bv + cw,
a u + b v +c w,
a"u + b"v -t- c"w ;
but complementary when for u, v, w, we write
OLU
a u + (3 v + yw,
au
a, b, c, &c., a, /3, 7, &c. being related in the manner antecedently explained.
Two forms, each of the same number of variables, may be said to be
associate forms when the coefficients of the one are functions of those of the
other; and when it happens that the coefficients of the first are all explicit
functions of those of the second, the latter may be termed the originant and
the former the derivant.
If now all the linear equivalents of one or of two associated forms are
similarly related to corresponding linear equivalents of the other, so that
each may be derived from each by the same law, the forms so associated will
be said to be concomitant each to the other. This concomitance may be of
two kinds, and very probably, in the nature of things, only of the two kinds
about to be described.
200 On the General Theory of [33
The first species of concomitance is defined by the corresponding
equivalents of the two associated forms being deduced by precisely similar,
or, as we have expressed it, concurrent transformations or substitutions, each
from its given primitive. The second species of concomitance is defined by
the corresponding equivalents being deduced not by similar but by contrary,
that is, reciprocal or complementary substitutions. Concomitants of the first
kind may be called covariants ; concomitants of the second kind may be
called contra variants. When of the two associated forms one is a constant,
the distinction between co- and contra- variants disappears, and the constant
may be termed an invariant of the form with which it is associated*. It
follows readily from these definitions that a covariant of a covariant and a
contravariant of a contravariant are each of them covariants; but a covariant
of a contravariant and a contravariant of a covariant are each of them
contravariants ; and also that an invariant, whether of a covariant or of
a contravariant, is an invariant of the original function f.
It will also readily be seen that as regards functions of two letters
a contravariant becomes a covariant by the simple interchange of x, y
with y, x, respectively. Covariants are Mr Cayley s hyperdeterminants ;
contravariants include, but are not coincident with, M. Hermite s formes-
adjointes, if we understand by the last-named term such forms as may be
derived by the process described by M. Hermite in the third of his letters to
M. Jacobi, " Sur differents objets de la The orie des Nombres," (which process
is an extension of that employed for determining the polar reciprocal of an
algebraical locus*). M. Hermite appears, however, elsewhere to have used
* Accordingly an invariant to a given form may be defined to be such a function of the
coefficients of the form, as remains absolutely unaltered when instead of the given form any
linear equivalent thereto is substituted. Of course if the determinant of the coefficients of the
transformations correspondent to the respective equivalents be not taken unity as supposed in
this definition, the effect will be merely to introduce as a multiplier some power of the deter
minant formed by the coefficients of transformation.
t It may likewise be shown that linear equivalents of covariants and coutravariants are
themselves related to one another as covariants and contravariants respectively, the transforma
tions by which the equivalents are obtained being taken concurrent in the one case and contrary
or reciprocal in the other ; and of course any algebraic function of any number of covariants is
a covariant and of contravariants a contravariant.
This has been further generalized by me in the theorem given in the last number of this
Journal, where I have shown in effect that any invariant in respect to , 77 ... 6 of
/(, 77 ... 6) + (xt + yn + ... + te + p) P n -\
(f being supposed to be of the degree n) is a contravariant of f(x, y ... t). When this invariant
is the determinant of /, it may be shown that we obtain M. Hermite s theorem. It is somewhat
remarkable that contravariants should have been in use among mathematicians as well in
geometry as the theory of numbers (although their character as such was not recognized) before
covariants had ever made their appearance. Invariants of course first came up with the theory
of the equation to the squares of the differences of the roots of equations, the last term in such
equation being an invariant. I believe that I am correct in saying that covariants first made
their appearance in one of Mr Boole s papers, in this Journal ; but Hesse s brilliant application
[ p. 186 above.]
33] Associated Algebraical Forms. 201
the term forme-adjointe in a sense as wide as that in which I employ
contravariants. For instance, he has given a most remarkable theorem,
which admits of being stated as follows:
If we have a function of any number of letters, say of x, y, z, as
ax m + mbx m -^y + mcx- 1 z + rr ^L 2 dx-^ + &c.,
and if / be any invariant of this function, then will
d d d d , V T
i<i i_ ^.m i .. i_ ~i i ~ i_ ,-m 2 7/ 2 fan } /
w -f ^^ t// U -. j \^ vU & -m |^ w If -* i VX/vy. I -*.
da do dc dd J
be a "forme-adjointe" of the given function. It is perfectly true and admits
of being very easily proved, as I shall show in your next number, that this is
a contravariant of the given function*; but it is not (as far as I can see) a
forme-adjointe in the sense in which the use of that word is restricted in the
letter alluded to. If, however, we adopt as the definition of formes-adjointes
generally, that property in regard to their transformers which M. Hermite
has demonstrated of the particular class treated of by him in the letter
alluded to, then his formes-adjointes become coincident with my contra
variants. It will thus be seen that covariants and contravariants form two
distinct and coextensive species of associated forms, which divide between
them the wide and fertile empire of linear transformations so far as its
provinces have been as yet laid open by the researches of analysts. In
your next number I propose to enter much more largely into the subject
generally. More particularly I shall describe the new method of Permutants,
including the theory of Intermutants and Commutants (which latter are
a species of the former, but embrace Determinants as a particular case), and
their application to the theory of Invariants. I shall also exhibit the con
nexion between the theory of Invariants and that of Symmetrical Functions,
and some remarkable theorems on Relative In variants -f*.
Some of your readers may like to be informed that a Supplement to my
last paper, under the title of " An Essay on Canonical Forms," has been since
published*; and that I have there given a much simpler method of solution
of the problem of the reduction of quintic functions to their canonical form
than in the original memoir, arid extended the method successfully to the
of one from among the infinite variety of these forms to the discovery of the points of inflexion
in a curve of the third order, in other words, to the Canonical Reduction of the Cubic Function
of Three Letters, appears to have been the first occasion of their being turned to practical
account.
* This is also true if J be taken any covariant instead of an invariant of the function.
t It will be readily apprehended that the definitions and conceptions above stated, respecting
covariants and contravariants of two single functions, may be extended so as to comprehend
systems of functions covariantive or contravariantive to one another.
J By Mr George Bell, University Bookseller, Fleet Street, [p. 203 below.]
202 General Theory of Associated Algebraical Forms. [33
reduction of all odd-degreed functions to their canonical form. I may take
this occasion to state that the Lemma given in Note (B) of the Supplement,
upon which this method of reduction is based, is an immediate deduction
from the well-known theorem for the multiplication of Determinants.
There is a numerical error in " The Cubical Hyperdeterminant of the
Twelfth Degree," worked out after the method of commutants by Mr Spottis-
woode, given at the end of my paper in the May Number. The correct result
will be stated in the next number of the Journal, where I hope also to be
able to fix the number of distinct solutions of the problem of reducing a
Sextic Function to its canonical form
For odd-degreed functions there is never more than one solution possible, as
shown in the Supplement referred to.
P.S. Since the above was sent to press, I have discovered an uniform
mode of solution for the canonical reduction of functions, whether of odd or
even degrees. The canonical form however, except for the fourth and eighth
degrees, requires to be varied from that assumed in my previous paper. Thus,
for the sixth degree the canonical form will be
au? + bv K + cw 6 4- muvw (v w)(w it) (u v),
where u, v, w are supposed to be connected by the identical equation
u + v + w = 0. And there will be only two solutions a remarkable arid
most unexpected discovery. For functions of the eighth degree there are
five distinct solutions, and in general there is the strongest reason for be
lieving (indeed it may be positively affirmed) that when the canonical form
has been rightly assumed for a function of the even degree n, the, number of
solutions will be (n + 2) when %n is even, but (n + 2) when \n is odd. It
turns out therefore that the theory for functions of the sixth degree is in
some respects simpler than for those of the fourth. The investigation into
canonical forms here referred to has led me to the discovery of a most unex
pected theorem for finding all the invariants of a certain class, belonging to
functions of two letters of an even degree.
34.
AN ESSAY ON CANONICAL FORMS, SUPPLEMENT TO A
SKETCH OF A MEMOIR* ON ELIMINATION, TRANSFOR
MATION AND CANONICAL FORMS.
SINCE the above paper was in print I have succeeded in obtaining a
canonical representation of the quadratic and cubic functions adjunctive to
the general quintic (5th degreed) functions of two letters.
Let F the quintic function of x, y,
= U 5 + V 5 + W*,
and
au + bv + cw = 0,
M being the modulus of the transformation, whereby transition is made from
x, y to u, v. Then the quadratic adjunctive is
M*
- f a 4 vw + b*wu + c*uv} ;
c 4 l
and the cubic adjunctive is simply
-M 6 (cLbcfuvw\.
Cr
Hence we can, in accordance with what I ventured to predict in the preceding
sketch, find u, v, w, by means of a simple and practical co-process. To
wit, call
I0px"y 3 + 5qxy 4 + ry\
[* p. 184 above. See p. 201, note J.]
t The knowledge of the existence of these lower adjunctive forms is mainly a consequence
of Mr Cayley s splendid discovery of hyperdeterminant constants. In fact, they are respectively
the quadratic and cuhic hyperdeterminants in respect to and 77 of i ^ 3 4, 5 \^clx +ri dij) ^
x and y being treated as constants.
The fortunate proclaimer of a new outlying planet has been justly rewarded by the offer of
a baronetcy and a national pension, which the writer of this wishes him long life and health
to enjoy. In the meanwhile, what has been done in honour of the discoverer of a new and
inexhaustible region of exquisite analysis ?
204 On Canonical Forms. [34
Form the determinant
Ix + my, mx + ny, nx + py
mx + ny, nx + py, px + qy
nx + py, px + qy, qx + ry
Let this cubic function, by solving it as a cubic equation, be made
equal to
L ( x +fy) ( x + oy) ( x + %).
then
u = k(x +fy), v = I (x + gy), w = m (x + hy).
By means of the identity, F = u 5 + v 5 + w\ I 5 , m 5 , n 5 , are known by the
solution of linear equations, and thus u, v, w, are determined by solving
a cubic equation instead of one of the eighth degree, as in the method first
given, and the process of canonising a quintic function is rendered practically
possible.
For brevity sake let c represent unity. The constant determinant of the
cubic adjunctive will be found to be
3M 30 (a&c) 10 .
Calling, then, the cubic adjunctive of F, C (F), we have the remarkable
equation
C(F)
uvw =
It may also be shown that if we call the Hessian of F, H (F), we shall have
the following equally remarkable equation :
Again, calling the quadratic adjunctive of F, Q (F), we shall easily find
or, if we please,
-JfioK + ^ + o 10
}_ 2a 8 6 5 - 2a 5 c 5 - 26 5 c 5 ]
When u, v, w are known, a, b, c, which are the resultants of v, w ; w, u ; u, v
respectively are known. But their ratios, or, if we please to say so, the ratios
of a 5 : b 5 : c 8 , may be found independently and very elegantly as follows :
Let M 10 x product of the 4 forms of a* + 1^6^ + 1^ = A,
M w x product of the 16 forms of c^ + 1*6^ + l*c* = B,
M 30 x a 10 . 6 10 . c 10 = C.
34] On Canonical Forms. 205
A, B, C are known quantities, being respectively what we have called
It may easily be shown that
B - A- = 128M 20 a 5 b*c 5 (a 5 + 6 5 + c 5 ).
Hence M s a s , M 5 b 5 , M s c 5 are the roots of p in the cubic equation
B-A n - .
A, B, C, it will be observed, are independent and, as they may be termed,
prime or radical adjunctive constants. Hitherto much mystery and un
certainty have attached to the theory of hyperdeterminants, from its having
been tacitly assumed that they were always either of lower dimensions than
the ordinary determinant, or else algebraical functions of such, and of the
determinant. Whereas we now see that, whilst the determinant of a function
in two letters of the fifth degree is of eight dimensions, one of its radical or
primitive hyperdeterminants is of four, but the other of twelve dimensions.
This is a most valuable consequence, and would seem to indicate that the
number of radical hyperdeterminants to a function, over and above the
common determinant, is always equal to the number of parameters entering
into its canonical form. The importance of this ascertainment of an un
suspected third radical constant, adjunctive to a quintic function of two
letters, in making to march the theory of hyperdeterminants, can hardly
be over-estimated.
From the equation last given we are enabled to assign the conditions in
order that two functions of the fifth degree may be capable of being linearly
transformed either into the other. For if we call F and F two such linearly
equivalent quintic functions, they must be capable each of being thrown
under the same form u 6 + v 5 + (lu + mv} 5 , where I and in shall be the same for
each. Consequently we must have the roots of p in the same ratio for F
and F , which conditions may be expressed by means of the two equations
B - A* B - A *
(B- A*f -
* More strictly speaking (and this correction should be supplied throughout in the "Sketch"),
B is the negative determinant of F. After finding, by the method of characteristics, or any
special artifices, the algebraic part of the value of a resultant or determinant, a process frequently
of some complexity remains over in assigning its numerical multiplier; this part of the operation
being analogous to that which occurs in the Integral Calculus, of determining the constant to be
added after the general form of an integral has been determined. In the " Sketch," a correction
for the numerical multiplier remains also to be applied to the expressions given for the successive
Hessian determinants.
206 On Canonical Forms. [34
A , B , C , of course representing the same functions of the coefficients of
F as A , B, C, respectively of F.
The two conditions required in their simplest form are accordingly
or A*:B*:C::A *:B *:C f ,
that is to say, all quintic functions of two letters of which the determinant
is to the subduplicate power of the radical hyper determinant of the twelfth
order and to the sesquiduplicate power of the radical hyperdeterminant of the
fourth order in given ratios, are mutually convertible.
So for the quartic (that is, biquadratic) function of two letters, calling R
and 8 the radical adjunctive constants of the second and third orders, the
condition of convertibility between different forms of the same is, that
R 3 : $ 2 shall be a given ratio. And, in general, we may infer that the
condition of convertibility between different functions of any degree is,
that the several radical adjunctive constants of each raised respectively
to such powers as will make them of like dimensions, shall be to one another
in given ratios. Of course all cubic functions of two letters, according to this
rule, are mutually convertible without any condition, they having but one
radical adjunctive constant ; and in fact all such functions, being represent-
able as the sum of two cubes of new variables linearly related to those given,
are necessarily convertible.
I have further succeeded in obtaining the canonical form of the quadratic
adjunctive to any odd degreed function of two letters, which presents a
wonderful analogy to the theory of relative determinants of quadratic
functions of any number of letters, and constitutes an important step towards
the construction of the theory of relative byperdeterminants.
Let a function of two letters of the odd degree m (= In- 1) be thrown
under its canonical form,
M! + u 2 m -t . . . + u n m ,
and let there exist the n 2 equations,
!%! +a 2 ii 2 + + aiW n = 0, (1)
&! MJ + & 2 M 2 + + b n Un =0, (2)
= 0. (n - 2)
Then, if M be the modulus of the transformation which converts u l} u 2 into
34]
Chi Canonical Forms.
207
x, y, and if, on making #,, d z ...6 n disjunctively equal to 1, 2...n we
use (0 n -i, 0n) to denote in general the determinant
the quadratic adiunctive of . F will be
m(m- 1) ... 2
*" -* ( / s\ /\ \ , -i / \ i
N.B. By means of this formula, and of the theorem for finding relative
determinants of quadratic functions, we can obtain the general canonical
form for one set of the biquadratic adjunctive constants (hyperdeterminant&
of the fourth order in Mr Cayley s language) of any odd degreed function
of two letters f.
Thus, for the fifth degree, preserving the notation of the " Sketch," we
have the biquadratic adjunctive constant
0, c 4 , b\ a
c 4 , 0, a 4 , b
6 4 , a 4 , 0, c
a, b, c,
For the seventh degree, if we suppose the function to be equal to
u~ + v 7 + w 1 + B 7 ,
x "
and
au + bv + cw + dO = 0,
a u + b v + c w + d = ;
J//14
(cd -c d)
r^r- multiplied by the
the biquadratic adjunctive constant will be
determinant
0,
(ba - b a) 6 ,
(ca f - c a),
(da - d a) 6 ,
a,
a,
* The condition m = 2n-l is only necessary in order that S n (u m ) may be a canonical,
because a possible and determinate, form for any given function of the mth degree. But the
theorem in the text, so far as it serves to obtain the quadratic adjunctive of S n (it "), is true for
all odd values of m, whether greater or less than 2n- 1.
t See Note (A) of Appendix.
(ab - a 6) 6 ,
(ac a cf,
(ad - a d) 6 ,
a,
a
0,
(be - Uc}\
(bd - b d) 6 ,
b,
b
(cV - c b),
o,
(cd - c df,
c,
c
(db -d b?,
(dc - d c)\
0,
d,
d
b,
c,
d,
o,
b ,
c,
d ,
o,
208 On Canonical Forms. [34
The determinants of the Hessian, the post-Hessian, and the praeter-post-
Hessian of F will be found (in the case of the quintic function) to be always
multiples of powers of the determinant of the given function, and of its cubic
adjunctive ; and I believe that in general for a function of two letters of any
degree the determinants of all the derived forms in the Hessian scale*, will
be necessarily algebraical functions of any two of them.
I hope very shortly to accomplish the reduction of functions, as high as
the seventh degree of two letters, to their canonical form, and also to present
a complete theory of the failing or singular cases of canonical forms.
Since the above was in print I have discovered the following
GENERAL THEOREM
for reducing a function of two letters of any odd degree to its canonical
form.
Let the degree of the function be (2rc 1) ; then its canonical form is
with (n 2) linear relations between u lt u 2 , ... u n .
To find u lt u 2 , ... u n , proceed as follows. Let the given function of the
(2w l)th degree be supposed to be
2n 2 j
Form the determinant
i /y [ ft tt n nr \ tt i/ n 0^ i n ?/ ft nr ^^ ft 11
L-j tX/ ^^ LL 2 U , IXgw |^ W/3 V, LcgiA/ ^^ CC/^ (/ t*y^t^ (^ fl^l J
a/f* \ f~l f)l
2/1 i 1 * T ^zny
This determinant is a function of # and y of the nth degree, and by resolving
an equation of the wth degree, may be decomposed into n factors, say
(1-iX + miy) (I 2 x + m 2 y) ... (l n i
* I use the term Hessian (more properly speaking the Boolian) Scale, to denote the deter
minants in respect of and 9iof( + 17 + &c. ) F.
V dx dy J
Neither Hesse, however, nor any other writer up to the present time, had thought of con
structing, and still less of turning to account, the functions (the first only excepted) which figure
in this scale.
34] On Canonical Forms. 209
we shall then have
where the I s and ms are known, and the (2?i - l)th powers of the p s may
be found linearly, by means of the identical equation 2u 2n ~ l = F (x, y). Thus
for example a function of the seventh degree of two letters may be reduced
to its canonical form
(Ix + my) 7 + (I x + my) 7 + (l"x + m"yj + (l "x + m "y)\
by the resolution of a biquadratic equation. My demonstration of this
extraordinary and unexpected consequence rests upon the following lemma*,
itself a very beautiful and striking theorem (no doubt capable of much
generalisation) in the theory of determinants. Form the rectangular matrix
consisting of n rows and (n + 1) columns
J-iy -LZ> *l J-n+it
J-z> J-3> -M J-n+zi
J-3) J- 4> *l * n+3>
J-n> J-n+iy
where
Then all the n + I determinants that can be formed by rejecting any one
column at pleasure out of this matrix are identically zero.
In order the better to realise the proof, suppose
n = 4t, so that 2n 1 = 7.
Let
F(x, y) = a^ + 7a 2 afy + 21a 3 a,y + 35a 4 #y + 35a 5 ay
+ 21a 6 #y + 7a 7 xy 6 + a^y 1 .
Suppose
P + u 1 + v 7 + w 1 = F(x, y)=G(u, v),
aft + b u = w.
Then, if M is the modulus of transition from x, y to u, v the hyper-
* See Note (B) of Appendix.
14
210
On Canonical Forms.
[34
determinant, or, to adopt my new expression, the permutant P 4 (meaning
thereby)
d/\CC ~\~ f^o V) CLnQC |~ W"3 ^/l (JjQvG ~T~ Cv^U y Cu*X/ \~ d/fil/
a 2 x + a 3 y, a 3 x + a 4 y, a 4 x + a 5 y, a s x + a 6 y
a 3 x + a 4 y, a t x + a 5 y> a s x + a 6 y, a G x + a^y
a 4 x + a 5 y, a 5 x + a 6 y, a 6 x + a 7 y, a 7 x + a s y
/ d d\ K
which is a constant adjunctive in respect to and rj of ( %-j- + TI -^ } F, will,
\ CM? ciyj
according to the principles laid down in the preceding " Sketch," be the
product of a power of M multiplied by the corresponding adjunctive constant
/ d d\ 6
of I -!- + v -7- 1 G(u,v\ and is therefore a multiple of the determinant
\ du dvj
i A it A 4- i A it A ~f~ i /j 11
~t" ^Igt- j -^-2^ I **$*j -^-3^ I 1 ^*4**1 J
A 3 t+A 4 u, A 4 t+A 5 u,
where
A t = a? + a 7 , A 2 = a 6 6 + a /G 6 ,
In this determinant the coefficient of w 4 is
111 A 3 , A 4> A;
A =
A=V + b 7 .
A 3>
A 6 , A 7
A 7 , l+A,
which is numerically equal to
AS, A 4 , A 5
A,, A, A 5
A 5 A 4 , A K , A,
-A 6
^ l ;; . - 1 .)i **> <
A 5 , A G , A 7
A A A
-0.4, ^ig, ^i 7
A%, -0.3, ^5
A A A
-tt-Z, -"-3) "<
+ A 7
AS, A 4 , A 6
- (1 + A B )
AS, A 4 , A 5
A A A
"4) -^*-5) "6
= 0, because the second factors of the products are all zero by the lemma.
Hence the permutant P 4 vanishes when t = 0, and consequently it contains
t as a factor, and in like manner it may be proved to contain u, v, w.
Hence t, u, v, w are the algebraical factors of P 4 , and precisely the same
proof applies to show in the case of a function in x and y, say F 2n ^, of any
34] On Canonical Forms. 211
odd degree (2n 1) whatever, that the corresponding permutant P n will
contain the factors u lt u 2 ... u n linear functions of x, y, such that
as was to be shown.
Whenever P n has equal roots, this will denote either (which is the more
general case) that the usual canonical form fails and gives place to a singular
form, (owing to some of the coefficients of transformation becoming infinite),
or, which is the more special supposition, that the canonical form becomes
catalectic by one or more of the linear roots* disappearing. Thus in the
cubic function, if P 2 has equal roots, and consequently its determinant
(which is coincident with that of the function itself) vanish, then the canoni
cal form in general fails ; so that, for example, aa? + bx 2 y cannot in general
be exhibited as the sum of two cubes : if, however, certain further relations
obtain between the coefficients of F, the canonical form reappears catalectically,
the function becoming in fact representable as a single cube. So, again, for
the quintic function (referring back to the notation above [page 205]),
if P 3 have equal roots, that is if C = 0, the canonical form fails, unless at
the same time B A 2 = 0, in which case the function becomes the sum of
two fifth powers; but if furthermore .4 = 0, then this catalectic form again
gives place to a singular form, which, on the satisfaction of a further condition
between the coefficients, again in its turn gives way before a (bicatalectic,
that is) doubly catalectic form, namely, a single fifth power.
It is remarkable, that the form to which Mr Jerrard s method reduces the
function of the fifth degree, expressed homogeneously as ax 5 + bxy 4 + cy 5 , is a
singular form, being incapable of being exhibited as the sum of three cubes ;
such, however, is not the case with the form aaf + ba?y i + cy 5 . It may further
be remarked, that although the singly catalectic form of the quintic function is
expressible by two conditions only, namely, (7=0, B- A 2 =0, it will be indicated
by P 3 (which being a cubic function of x and y contains four terms) completely
disappearing, so that apparently four conditions would appear to be required
or implied. But of course these must be capable of being shown to be
non-independent, and to be merely tantamount to the two independent ones,
(7=0, B A 2 = Q. The theory of the catalectic forms of functions of the
higher degrees of two variables presents many strong points of resemblance
and of contrast to that of the catalectic forms of quadratic functions of
several variables.
One important and immediate corollary from the General Theorem is,
that the constants which enter into the linear functions appurtenant to the
canonical form of any function of an odd degree form a single and unique
system ; or, in other words, the canonical forms for such functions are void of
* MJ, u 2 ... u n may be termed the linear roots of the form -F 2 n-i-
142
212
On Canonical Forms.
[34
multiplicity, a result contrary to what might have been anticipated, and
to what we know is the case for the canonical forms of functions of an even
degree.
It may further be shown that if we have the (n - 2) equations
a^ + a 2 u 2 + . . . + a n u n = 0,
61^1 + &2 M 2 + + b n u n = 0,
and call M the modulus of transformation in respect to w 1( u z , and if we
make
P n = Ku^Uz ... U n ,
then
n n
*> U-4
Jn(n i)
is equal to the product of the
factors of the form
0i, #2 #n-2 being any (n - 2) numbers out of the n numbers 1, 2, 3 ... n.
It may hence be shown that
*a
io
m being a number which is a function of n, and which may be shown to be
equal to a (x^ 1 y + xy n ~ l ) -f- product of the squared differences of the roots
x,y
of l n ~ 2 = 1, that is
m = f-^-VP = (" w)n " 2
and thus
a P 1
***!
* D means the determinant in respect to x and y.
x, y
34]
On Canonical Forms.
As an example of the mode of finding u l} u 2 ... u n , let
F =
then
, 2 2/>
Hence
u =fx, v = g(x + y\ w = h(x-y).
To find / g, h, we have u 5 + if + w 5 = F, hence
f*+g s +h* = 3; g 5 + h 5 = 2: g 6 -h*=l;
whence we have
F = of + (x + y) 5 + (x - yj.
Again, we find
D (4^ _ 4^) = _ 44 x 12,
and accordingly
(a; + y) ( - y) =
according to the general formula above given.
As a second example let
F = 3
then
213
= 4 (o?y - xf) = 4>xy (x -y)(x + y),
and accordingly we shall find
off + y 7 + (x-y) 7 + (x + y) 7 = F.
Moreover
D (4iX 3 y 4txy 3 ) = 4 9 ,
and V-
Thus
P^_
tuvw
agreeable to the general formula.
44-2
214 On Canonical Forms. [34
As a corollary to our general proposition, it may be remarked, that if
Fzn-i be a symmetrical function of oc, y of the (2n l)th degree, P n (F m _ 1 )
will be also a symmetrical function of a; and y, and may therefore be resolved
into its factors by solving a recurring equation of the nth degree, which may,
by well-known methods, be made to depend on the solution of an equation
of the fyiih or \ (n l)th degree, according as n is even or odd.
Hence the reduction of a function of two letters of the degree 4m + 1 to
its canonical form as the sum of powers may be made to depend on the
solution of an equation of the rath degree ; so that, for example, a symmetrical
function of x, y, as high as the fifteenth or seventeenth degree, may be
reduced by means of a biquadratic equation only.
In a short time I hope to present to the public a complete solution
of the canonical forms of functions of two letters of even degrees, and possibly
to exhibit some important applications of the principles of the method to the
theory of numbers.
APPENDIX.
NOTE (A).
The permutants (meaning, in Mr Cayley s language, the hyperdeter-
minants) of F m+l (x, y) of the fourth dimension in respect to the coefficients
of F, may be all obtained by taking the quadratic permutant in respect to x
and y of the quadratic permutant in respect of and rj of
d d
I having any integer value from 1 to n.
In extension of a theorem in the foregoing Supplement, which applies
only to the case of l = n, I am able to state the following more general
theorem, in which the same notation is preserved as above [page 207].
The quadratic permutant in respect to and 77 of
1 / d d
. .
is equal to
MzL
"*
If now we proceed to form the quadratic permutant of the above sum
in respect to x and y, we know A priori, by reason of Mr Cayley s invaluable
researches, that we shall not get radically distinct results for all values, but
only for certain periodically changing values of I.
34]
On Canonical Forms.
215
I have not yet had leisure to seek for an explicit demonstration of this
remarkable law, founded upon the above given canonical representation.
NOTE (B).
The lemma, upon which the general method for reducing odd degreed
functions to their canonical form is founded, may be stated rather more simply
and more generally as follows :
The determinant
T
J-r
at
!+*!
where T e denotes A 1 a 1 e + A 2 a 2 9 + ... + A m a m 6 provided that m is less than n,
is identically zero. In the theorem, as thus stated, there is no substantial
loss of generality arising from the omission of the 6 s.
Thus stated the theorem and its extensions evidently repose upon the
same or the like basis as the theory of partial fractions.
NOTE (C), referring to the original " Sketch."
The Boolo-Hessian scale of determinants furnishes a very pretty general
theorem of geometrical reciprocity in connexion with the doctrine of suc
cessive polars. Let F (x, y, z}, a cubic homogeneous function of x, y, z
equated to zero, express in general a curve of the third degree ; then
( a _|_ & _|_ c ) f 1 w ill express its first polar in respect to the point a, b, c,
\ dx dy dz)
that is, the conic which passes through the six points in which the tangents
drawn from a, b, c to touch the given curve meet the same.
Again, if we take I, m, n the coordinates of any new point,
d d d\ ( d -. d d\
-7- + m -f- + n -7- a-3^ + 0j- + e-j-)jp
dx dy dz) \ dx dy dz)
will express the polar, that is the chord of contact of the above conic, in
respect to the last named point. If now we eliminate I, m, n between the
three equations
d , d d^
dx dy J dz
V c
dx dy dz
d d d
d
m d +
n d ]
dx
U dy
dz)
d
d
^ d\
dx
m -j- +
dy
dz)
d
d
n d }
dx
dy
H dz)
(ilL A
\ dx dy
(i d A
\ dx dy
T-
dx
-j-
dy
c -^-
dz
216 On Canonical Forms. [34
it is easily seen that the resultant of the elimination is the square of the
determinant
a, b, c
a , b , c
a", b", c
multiplied by the Hessian of the given function. And, moreover, that if we
eliminate so, y, z we shall obtain precisely the same result with the letters
I, m, n substituted for a, y, z. Hence it follows, that if we take the doubly
infinite system of first polars to a given curve of the third degree, in respect
to all the points lying in its plane, and then from any point in the Hessian
to the given curve, draw pairs of tangents to each conic of the system so
generated, then all the chords of contact will meet in one and the same point,
which will itself be also a point situated upon the Hessian and conjugate to
the former.
So, in general, for a function of any degree of any number of letters,
viewed with relation to the doctrine of successive polars, the determinants
of the Boolo-Hessian scale take one another up in pairs ; namely the first
takes up the last but one, the second the last but two, and so on ; and
consequently, if the degree of .the function be odd, that function which
(making abstraction of the constant determinant at the end) lies in the
middle of the scale pairs with itself, and, in a sense analogous to that above
exhibited for a function of the third degree, may be said to be always
its own reciprocal.
P.S. I have just discovered the method of reducing functions of two
letters of even degrees to their canonical form, which will shortly be published
in a second Supplement.
At present I offer the annexed theorem (which strikingly contrasts with
the law of uniqueness demonstrated of functions of an odd degree) as a
foretaste of the enchanting developments with which I hope shortly to present
my readers :
If a given homogeneous function of x and y of the degree 2n be supposed to
be thrown under its canonical form,
then will K n have n 2 1 in general distinct values, to each of which will
correspond a single distinct system of the linear functions of x and y,
1 i
n i/ ~\ n n
. 6*2, ... J- Uf|<
35.
EXPLANATION OF THE COINCIDENCE OF A THEOREM GIVEN
BY MR SYLVESTER IN THE DECEMBER NUMBER OF THIS
JOURNAL, WITH ONE STATED BY PROFESSOR DONKIN
IN THE JUNE NUMBER OF THE SAME.
[Philosophical Magazine, (Fourth Series) I. (1851), pp. 4446.]
I WISH to state, without loss of time, that in the theorem given by me* for
the composition of two successive rotations about different axes, I have been
anticipated by Prof. Donkin in the June Number of your Journal.
To my shame I must confess, that, although an occasional contributor to,
I am not invariably a constant reader of your valuable miscellany, otherwise
I should not have introduced the theorem in question without due acknow
ledgment of Professor Donkin s claims to whatever merit may attach to the
priority of publication. The fact is, that I made out the theorem for myself
nine years ago, and had some communication on the subject with Professor
De Morgan, who was then writing the seventeenth chapter of his Differential
Calculus. A recent conversation with this gentleman has brought back to
my mind a vivid recollection of the course of that communication. I brought
under Professor De Morgan s notice the analytical memoir of Sr Gabrio Pola
on the subject in the Memoirs of the Italian Society of Modena, and satisfied
myself of the existence of the single axis of displacement by compounding
the two rotations in the manner given in my paper, which, for the case of two
axes fixed in space, is the same as Professor Donkin s, and for two axes fixed
in the rotating body is materially, although not formally the same.
It then occurred to me that a more simple demonstration ought to be
deducible from the possibility of always finding the point on a sphere, by
revolution about which, as a pole, one equal arc could actually be shown to
be transportable into the place of another. But in proceeding to work out
this idea I fell into a remarkable blunder, in which I have since been followed
by more than one able friend to whom I have proposed the question. The
[* p. 158 above.]
218 Explanation, etc. [35
blunder was of this kind : Two arcs have to be drawn, bisecting at right
angles the arcs joining the extremities of two equal arcs; the point of inter
section of the two bisecting arcs must in all cases fall outside the quadrilateral
formed by the equal and joining arcs. I supposed it to fall inside. There
appears to be a fatal tendency to do so in all who take the subject in hand.
In consequence of this error, the cause of which I did not at the moment
perceive, I was driven to deny and admit in one breath the same proposition.
Mr De Morgan sent me the correct proof after this method (the same as that
given by him at page 489 of his Calculus), I am inclined to think after I had
myself detected my error ; but of this I cannot feel certain.
This is the method alluded to by me in the words "it is right to bear in
mind, &c.," at the time of writing which all recollection of the same thing
having been published by Mr De Morgan had vanished from my memory.
The proof of the triangle of rotations is so simple, that, as Professor
Donkin states (in a letter which he has done me the favour of addressing me
on the subject) was the case with himself, I thought it incredible that it
should not have appeared in some elementary work, and I was therefore at
no pains to publish it as my own ; nor should I have written at all on the
subject, had it not been for the surprise occasioned to my mind by falling in
with Professor Stokes s article in the Cambridge and Dublin Mathematical
Journal, to demonstrate the existence of an instantaneous axis, which
proceeds in apparent unconsciousness of the so simply demonstrable law,
that any number of rotations of any kind (and therefore those that take
place in an instant of time) are representable by a single rotation about
a single axis. I shall feel obliged by the early insertion of this explanation,
more in justice to myself than to Professor Donkin, whose high and worthily
earned reputation, not to speak of the disinterested love of truth for its own
sake, apart from personal considerations, which animates the labours of the
genuine votary of science, must make him indifferent to whatever credit
might be supposed to result from the first authorship or publication of the
very simple (however important) theorem in question.
36.
AN ENUMERATION OF THE CONTACTS OF LINES AND
SURFACES OF THE SECOND ORDER.
[Philosophical Magazine, I. (1851), pp. 119 140.]
IT is well known that in general any two homogeneous quadratic
functions of the same system of variables may be simultaneously trans
formed, so as to be expressed each of them as pure quadratic functions of
a new system of variables equal in number and linearly connected with the
original ones ; a pure quadratic function meaning one in which only the
squares of the variables are retained.
Every homogeneous quadratic function may be treated as the character
istic* of a locus of the second degree : if the function be of two letters, the
locus is a binary system of points in a line wherein the distances of two
fixed points from either point of the given system or given multiples of such
distances correspond to the variables ; if of three letters, the locus is a conic,
the distances or given multiples of the distances of every point in which from
three given lines in the plane of the conic are represented by the variables ;
if of four letters, the locus is a surface of the second order, the coordinates
being the distances or multiples of the distances of any point therein from
four planes drawn in the space in which the surface is contained, and so on
for loci of four and higher dimensions.
I propose, however, in the present paper to restrict myself to the theory
of the contacts of loci not transcending the limits of vulgar space, by which
I mean the space cognizable through the senses f, and shall accordingly be
* According to the definition stated by me in a previous paper, the characteristic of a locus is
the function which, equated to zero, constitutes the equation thereto.
t If the impressions of outward objects came only through the sight, and there were no sense
of touch or resistance, would not space of three dimensions have been physically inconceivable ?
The geometry of three dimensions in ordinary parlance would then have been called trans
cendental. But in very truth the distinction is vain and futile. Geometry, to be properly
understood, must be studied under a universal point of view; every (even the most elementary)
proposition must be regarded as a fact, and but as a single specimen of an infinite series of
homologous facts.
In this way only (discarding as but the transient outward form of a limited portion of an
infinite system of ideas, all notion of extension as essential to the conception of geometry,
however useful as a suggestive element) we may hope to see accomplished an organic and vital
development of the science.
220 An Enumeration of the Contacts of Lines [36
almost exclusively concerned in determining the singular cases of conjugate
systems of quadratic forms of two, three, and four letters respectively.
In order that the reduction of any such system, say U and V, to a pure
quadratic form may be possible (as it generally is), it is necessary that none
of the roots of the complete determinant of U+\V shall be equal; if any
relation of equality exist between these roots, the general reduction is
generally no longer possible ; under peculiar conditions, however, as will
hereafter appear, in spite of the equality of certain of the roots, the
irreducibility in its turn will cease, and the ordinary reduction be capable
of being effected. It is easily seen, that to every relation of equality between
the roots of the determinant of U + \V must correspond a particular species
of contact between the loci which U and V characterize. But we should
make a great mistake were we to suppose that every such relation of equality
corresponded with but one species of contact ; for instance, the characteristics
of U and V of two conies are functions of three letters, and o(U + \V) will
be a cubic function of X. Such a function may have two roots, or all its roots
equal : this would seem to give but two species of contact, whereas we well
know that there are no less than four species of contact possible between two
conies. Accordingly we shall find, that, in order to determine the distinctive
characters of each species of contact, we must look beyond the complete
determinant, and examine into the relations (in themselves and to one
another) of the several systems of minor determinants that can be formed
from U + \V.
By pursuing this method, we may assign a priori all the possible species
of contact between any two loci of the second degree. How important this
method is will be apparent from the fact, that not only have the distinctive
characters of the various contacts possible between surfaces of 1^he second
order never been determined, but their number and the nature of certain of
them have remained until this hour unknown and unsuspected.
The method which we shall pursue is an exhaustive one, and will conduct
us by a natural order to a systematic arrangement of all the different modes
and gradations of such contacts.
In a paper* in this Magazine for November 1850, 1 explained the decline
of minor determinants, and stated a law, called the homaloidal law, con
cerning them.
If U and V be characteristics of the two loci whose contacts are to be
considered, U +\V will be the function, the properties of whose complete
determinant, and of the minor systems of determinants belonging to it, will
serve to specify the nature of the contact.
It will be remembered, that, whatever be the number of variable letters
in any quadratic function U, three of its first minor determinants being zero,
[* p. 150 above.]
36] and Surfaces of the Second Order. 221
makes all the first minors zero ; six of its second minors being zero, makes all
the second minors zero ; and so on for the third, fourth, &c. minor systems
according to the progression of the triangular numbers.
It is well known that whatever linear transformations be applied to
a quadratic function W, the complete determinant thereof will remain un
altered, except by a multiplier depending upon the coefficients introduced
into the equations of transformation; consequently the roots of X in the
equation obtained by making the determinant of U + XF zero remain
unaffected by such transformation ; and any relation or relations of equality
among the roots of the equation n(E7"+\F) = is an immutable property
of the system U, V, which is unaffected by linear transformations. Another
and more general kind of immutable property (comprehending the above as
a particular case), to which I shall have occasion to refer, is the following.
Suppose all the minors of any order of U + \V have a factor X-fe in
common ; this factor will continue common to the same system of minors
when U and F are simultaneously transformed. This is a very important
proposition, and easily demonstrated ; for if X + e be a common factor to all
the rth minors of U+\V,(U-eV) will have its rth minors zero, and there
fore, as explained by me in the paper above referred to, U - eF will be
degraded r orders below U or F. This is clearly a property independent
of linear transformation, consequently X + e will remain a factor of the
transformed rth minors.
In like manner it is demonstrable that any number of distinct factors
X + e 1( X4-e 2 ... common to the rth minors of one form of U+\V, will
remain common factors of any other linearly derived form of the same.
It is consequently necessary that each rth minor of one form of any
quadratic function W shall be a syzygetic* function of all the rth minors
of any other form of the same ; and consequently a function of X of any
degree, whether all its factors be or be not distinct, which is common to
the rth minors of one form of U + \V, will remain so to the rth minors of
any other form of the same.
The law exhibiting the connexion of each rth minor of one form of W
(any homogeneous quadratic function) with all the rth minors of any other
form of W, will form the subject of a distinct communication.
Finally, to fully comprehend the annexed discussion, the following
principle must be apprehended.
* If A = pL + qM + rN + &G., where p, q, r ... do not any of them become infinite when
L, M, N ... or any of them become zero, A may be termed a syzygetic function of L, H, N....
In the theorem above alluded to, it will be shown (as might be expected) that the syzygy in the
case concerned is of the simplest kind, that is, that each rth minor of a quadratic function of any
number of letters is a homogeneous linear function of all the rth minors of the same quadratic
function linearly transformed.
222 An Enumeration of the Contacts of Lines [36
If any factor K e enter into all the rth minors of W, and if K* be the
highest power of K common to all the (r + l)th minors, then K ze ~ { will be
a common factor to all the (r l)th minors.
Let r be taken unity; it is easily proved* that the complete determinant
of any square matrix may be expressed by the difference between two pro
ducts f, each of two first minor determinants divided by a certain second
minor determinant. The proposition is therefore demonstrated for this case,
and thereby in fact implicitly for every case, inasmuch as the first minors of
every rth minor are (r + l)th minors of the original matrix. Hence it
follows, that if any system of rth minor determinants have a common factor
e l , the complete determinant must contain at lowest the factor e (r+1)i , and any
system of (r-s)th minor determinants thereunto will contain at lowest the
factor e (s+1 > 1 .
I now proceed to apply these principles to the determination of the
relative forms of conjugate quadratic functions representing geometrical
loci of the second order. I shall begin with two binary systems of points
in a right line.
The general characteristics V and F of two such systems may be thrown
under the form
F = ax 2 + by
When D(F+A,/") = has its two roots equal, these systems have a point
in common. The above forms cease to be applicable, and convert into
U = xy
v=
where a; represents the common point.
* This will appear in my promised paper on Determinants and Quadratic Functions.
t When the matrix is symmetrical about one of its diagonals (as it is in the case which we are
concerned with), one of these products becomes a square. I may take this occasion of hinting,
that the theory of quadratic functions merges in a larger theory of binary functions, consisting of
the sum of the multiples of binary products formed by combining each of one set of quantities,
x, y, z ... with each of the same number of quantities of another set, as x , y , z .... For
instance,
axx + bxy + cxz
+ a yx + b yy + c yz
+ a"zx + b"zy + c"zz
would be a binary function, and its determinant (no longer, as in a quadratic function,
symmetrical about either diagonal) would correspond to the square matrix
a b c
a b c
a" b" c".
Almost all the properties of quadratic apply, with slight modifications, to binary functions.
36] and Surfaces of the Second Order. 223
Let U and V now represent two conies. When there is no contact, we
have as the types of their characteristics
U = x* + if- + z 2 ,
V=ax- + by 2 + cz*.
The three roots of D ( V + \ U) = are
\ = a, \ = b, \ = c,
showing that there are three distinct pairs of lines in which the intersections
of U and V are contained, the equations to three pairs being respectively
(b-a)f + (c-a)z 2 = 0,
(c - b) z> + (a - b) x 2 = 0,
(a - c) x* + (b - c) y 2 = ;
the four points of the intersection being defined by the equations corre
sponding to the proportions
x:y:z\: \/(b c) : \/(c a) : \/(a b).
Now let a (U + \V) have two equal roots ; the characteristics assume the
form
U = x 1 + y 2 + ccz,
V = ax 2 + by- + cxz *.
Two of the pairs of lines become identical, that is, two of the four points of
intersection coincide.
* We may if we please make a = b; for it may be shown that the equations, in their present
forms, contain an arbitrariness of 10 degrees; namely, 9 on account of x, y, z being arbitrary
linears of f, 17, 6 \ 2 on account of the ratios a : b : c ; together 11 reduced by one degree on
account of x, y, z, changed into Ix, ly, Iz, leaving U=0, F=0 unaffected. Now the degrees of
arbitrariness in two conies, subject to satisfy only one condition, is 2 x 5 - 1 or 9. Hence there
is one degree of arbitrariness to spare. In fact, on making a = b, the axis z becomes the line
joining the two points of intersection distinct from the point of contact ; x remaining the tangent
at the point of contact, aud y, strange to say, still arbitrary, subject only to passing through the
point of contact ; if, however, y be made to pass through the point of contact, and either one of
the distinct intersections, this form,
U=x 2 + y 2 + xz,
V=ax 2 + ay 2 + cxz,
becomes no longer tenable, but gives place to
V= ay 2 + ayx + cxz,
where x is the tangent at the point of contact, z the line joining the two intersections with one
another, and x, x + y respectively the lines joining either of them with the point of contact ; if the
multiplier of yx in F in the above be made b instead of a, x re mains the tangent as before, y
becomes any line through the point of contact, and z any line through one of the distinct inter
sections. A systematic view of similar modulations of form and the study of the laws of
arbitrariness connected with them, as applicable to the general subject-matter of this paper,
must be deferred to a subsequent occasion.
224 An Enumeration of the Contacts of Lines [36
This may be termed " Simple Contact." The tangent at the point of
contact is # = 0; this equation making U and V each become of only one
order.
The intersections are
oc = 0, y = 0, (1)
* = 0, 7/ = 0, (2)
V(a - c) x + V(6 - c) y = 0, * = 0, (3)
V(a - c) x - V(6 - c) y = 0, s = 0. (4)
These are obtained by making V aU= 0, which gives x or z = 0.
# = gives y 2 - = 0, that is, y = twice over, and z = gives
(a - c) x* + (b - c) f = 0.
The number of conditions to be satisfied in this case is one only.
Next let o(U + \V) have all its roots equal. This condition will be
satisfied (still leaving U and F as general as they can remain consistent
with these conditions) by making
U = y? + yz + yx,
V = ax* + ayz + byx.
Here only one distinct pair of lines can be drawn to contain the inter
sections, showing that three out of the four points come together.
This may be termed " Proximal Contact." The number of affirmative
conditions to be satisfied is two, and the contact is therefore entitled of the
second degree.
The tangent at the point of contact is y = 0, and the four intersections
become
x = 0, y = 0,
x = 0, y = 0,
x = 0, y = 0,
x = 0, z = 0.
These may be obtained from the equation V all = 0, which gives y =
or = 0; the former implying concurrently with itself x 2 = 0, and the latter
yz=0.
Thus we obtain three systems,
x = 0, y = 0,
and one
x = 0, z=0,
corresponding to three consecutive points and the single distinct one.
36] and Surfaces of the Second Order. 225
The determinant of U+\V being only of the third degree in X, we
have exhausted the singularities of the system U, V dependent on the form
of the complete determinant of U+\V.
Let now the first minors of U + \V have a factor in common; this will
indicate that U+\V may be made to lose two orders by rightly assigning X,
in other words, that the intersections of U and V are contained upon a pair
of coincident lines. Here it is remarkable that the original forms of U and V
reappear, but with a special relation of equality between the coefficients : we
shall have, in fact,
U = a? f y- + z-,
V = ax 2 + ay 1 + bz 2 .
This gives the law for double, or, as I prefer to call it, diploidal contact*.
By virtue of the Homaloidal law, we kuow that if three first minors of
U+\V be zero, all are zero; we have therefore to express that three
quadratic functions of X have a root in common. This implies the exist
ence of two affirmative conditions ; the contact of the two conies taken
collectively may therefore be still entitled of the second degree, although
the contact at each of the two points where it takes place is simple, or
of the first degree.
These points are evidently defined by the equation
and the ordinary algebraical solution of the equations 7=0, F=0 would
naturally lead to the four systems
= 0,
the two tangents at the point of contact are x + V( 1) y 0, x \/(\)y = 0,
and the coincident pair of lines containing the intersections is z 2 = 0.
* See my remarks t cm the conditions which express double contact in the Cambridge Journal,
Nov. 1850. If 7i functions, being all zero, be the condition of a fact, but r independent syzygetic
equations admit of being formed between these functions, the number of affirmative conditions
required is not n, but (n - r) ; because the fact may be expressed by affirming (n - r) equations and
denying certain others. Thus if P=0, Q = 0, R = 0, S=0 express a fact, and
the fact is expressible by affirming P = 0, Q-0, and denying R S"-R"S =0, for then P = 0, Q =
will imply R = 0, S = 0; or, in like manner, by affirming any other two out of the four necessary
equations, and denying the other equations. Observe, however, that all the required equations
may coexist in the absence of such right of denial.
[t p. 129 above.]
S. 15
226 An Enumeration of the Contacts of Lines [36
It may at first view appear strange, that whilst no condition is required
in order that U and V may be simultaneously metamorphosed into the forms
of # 2 + 2/ 2 + 2 2 , ax* + by 2 + cz 2 , a, b and c being all unequal, for this metamor
phosis to be possible when any two become equal, not one but two conditions
must be satisfied. The reason of this is, that the coefficients of transform
ation, which, as well as a, b, c, are functions of the coefficients of the given
quadratic functions, become infinite ou constituting between the said
coefficients such relations as are necessary for satisfying the equation a = 6,
or a = c, or 6 = c, except upon the assumption of some further particular
relations between them over and above that implied in such equality.
In the ordinary case of diploidal contact, the first minors having a factor
in common, this factor will enter twice into the complete determinant of
U + \V, but it may enter three times: this will indicate, that not only
do the four intersections lie on a coincident pair of lines, but furthermore,
that there is but one pair of lines of any kind on which they lie.
In the ordinary case of diploidal contact, it will be observed that this
latter condition does not obtain ; the four intersections lie on a coincident
pair of lines ; but they lie also on a crossing pair, namely, in the two tangents
at the points of contact. In this higher species of diploidal contact, it is
clear that the two points of contact, which are ordinarily distinct, come
together, and that all four intersections coincide.
This I call confluent contact ; the forms of U and V corresponding thereto
will be
U = a? + 2/ 2 + xz,
V = ay 1 + axz ;
the common tangent at the point of contact being x = Q, and the four
coincident points x* = 0, y* = 0.
The number of affirmative conditions to be satisfied being three, the
contact is to be entitled of the third degree.
Observe, that it is of no use to descend below the first minors in this
case ; because the second minors, being linear functions of X, could not have
a factor in common, unless V: U becomes a numerical ratio, which would
imply that the conies coincided*.
Fortified by the successful application of our general principles to the
preceding more familiar cases of contact, we are now in a condition to apply
with greater confidence the same d priori method to the exhaustion and
characterization of all the varied species of contact possible between surfaces
* No-contact and complete coincidence may be conceived as the two extreme cases in the scale
of relative conjugate forms.
36] and Surfaces of the Second Order. 227
of the second order; a portion of the subject comparatively unexplored, and
never before thought susceptible of reduction to a systematic arrangement.
When there is no contact, we may write
U = x * + f + z* + t 2 ,
V=ax 2 + by 2 + cz 2 + dtf,
and the intersection of the surfaces will lie in each of the four cones,
(a - d) a? + (b - d) f + (c -d).2 2 = 0,
(a - b) a? + ( c -b)z* + (d- b) t 2 = 0,
(a - c) x* + (b - c) f + (d -c)t 2 = 0,
(b -o)if + (c- a) z 2 + (d- a) P = 0.
Whenever the surfaces are in contact, certain of these cones will coincide
with certain others, so that their number will be always less than four. Also,
as we shall find in such event, they may degenerate into pairs of intersecting
or coincident planes.
Let us begin with considering the cases of contact for which the first
minors (and consequently a fortiori the minors inferior to the first) have
no factor in common.
Here a (V+\U) is a biquadratic function.
If X have all its roots unequal, we have U and V as above given.
If two roots are equal, the characteristics assume the form
U = x 2 + y 2 + z* + act -I
V = ax 2 + by*- + cz" + dxt]
The touching plane is x = ; the point of contact is x = 0, y = 0, z = ; the
curve of intersection is one of the fourth degree, with a double point at the
point of contact.
There is but one condition to be satisfied, and the contact may be entitled
" simple " and of the first degree.
Next let A, have three equal values, the equations become
U = X* + yz + 1 2 + xy,
V=x 2 + yz + at 2 + bxy.
The tangent plane at the point of contact y = Q, and the point itself x = 0,
y = 0, t = 0. The curve of intersection is a curve of the fourth order, with a
cusp at the point of contact. The number of affirmative conditions to be
satisfied is two ; the contact is of the second degree, and may be termed
" proximal " or cuspidal.
152
228 An Enumeration of the Contacts of Lines [36
Next let a (U + XF) have two pairs of equal roots, we shall find
U = a? + xy + zt,
V = ayz + bxy + czt.
The line x Q, z = Q will be common to both surfaces. The curve of
intersection will therefore break up into a right line and a line of the
third order.
The former will meet the latter in two points, which will be each of them
points of contact. The contact is therefore diploidal ; but as there is another
species of diploidal contact to which we shall presently come, it will be
expedient to characterize each of them by the nature of the intersections
of the two surfaces ; accordingly this may be termed unilinear-intersection
contact, or more briefly, unilinear contact.
The number of affirmative conditions to be satisfied being two, it may
be said to be collectively of the second degree, but (obviously ?) the contact
at each of the two points is of the nature of simple contact.
Lastly, let us suppose that all four roots of U + \V are equal; we shall
find, as the most simple expressions of the most general forms of the two
surfaces,
U = x 2 + xy + yz + zt,
V = axy + bz z + azt.
In this case the two points of intersection of the curve of the third
degree, and the right line on which the surfaces intersect, come together, so
that the right line becomes a tangent to the curve. The number of conditions
to be satisfied is three : there is but one point of contact which may be con
sidered as the union of two which have coalesced, and the species may be
defined as confluent-unilinear contact.
If we throw the equations to the conoids having an unilinear contact into
the form
x(x + y} + zt = 0,
xy + z(y + ct) = 0,
we obtain
which last equation is no longer satisfied by x = 0, 2 = 0, these systems of
roots having been made to disappear by the process of elimination.
The curve of the third degree, in which the two given conoids intersect,
may thus be defined as their common intersection with the new conical
surface defined by the third of the above equations.
36] and Surfaces of the Second Order. 229
More generally, it is apparent that the three conoids,
xu yt =
yv zu
zt xv =
in which x, y, z, t, u, v may any of them be considered as a homogeneous
linear function of four others, intersect in the same line of the third degree.
Besides which, the first and second intersect in the right line y, u ; the second
and third in z, v ; the third and first in x, t ; each of which lines it is evident
is a chord of the common curve of intersection. For instance, y = 0, it =
may be satisfied concurrently with all the above three equations by satisfying
the equation zt xv = 0, which, as two linear relations exist originally be
tween the six letters, and two more have been thrown in, becomes a quadratic
equation between any two of the letters.
The only case of exception to this reasoning is, when y = 0, u = can be
satisfied concurrently with z 0, v = 0, and with x = 0, t = ; but in this case
the surfaces all become cones ; and as there is no longer a curve of the third
degree, " Cadit qusestio." Even here, however, the intersection of any two of
the surfaces becomes a conic, and two coincident generating lines on the two
cones ; so that if we take one of these and the conic to represent a degenerate
form of a line of the third degree, the remaining straight line passes through
a double point of such degenerate form, and the case passes into that of
confluent-unilinear contact.
The two double points in the intersection of the two conoids
U = x (x + y) + zt = 0,
V = xy + z (y + ct) = 0,
by which I mean the points of intersection of the conic with the right line
common to them, are found by making x = 0, z 0, and substituting in the
derived equation
(x + y} (y + ct) - ty = 0,
which gives y = 0, or y + (c 1) t = ; so that the two points required are
x = 0, y = 0, z = 0,
It appears also that the entire intersection is contained in each of the two
cones,
U-V, that is, x 2 + z{(l-c)t-y]
and
cU V, that is, cx 2 + y {(c l)x z},
the respective vertices of which are at the points above determined.
230 An Enumeration of the Contacts of Lines [36
The equations for confluent-unilinear contact,
xy + z (cz + t) = 0,
give
which, on making # = 0, 2 = 0, is satisfied by 2/ 2 = 0; showing that the
confluence takes place at the point
as = 0, y = 0, 2 = 0.
The number of terms in the two equations for ordinary unilinear contact
being six, and in those given for confluent unilinears seven, and the empirical
rule in all other cases being that the terms tend to dimmish and never
increase in number as the degree of the contact (expressed by the number
of conditions to be satisfied) rises, I am led to suspect that the conjugate
system for the latter species of contact may admit of being reduced to some
more simple form.
I must state here once for all, that all the distinct systems of (at least
consecutive) conjugate forms that have been, and will be given, are mutually
untransformable. This it is which distinguishes singular from particular
forms.
A particular form is included in its primitive ; but a singular form is one,
which, while it responds to the same conditions as some other more general
form, is incapable of being expressed as a particular case of the latter, on
account of the additional condition or conditions which attach to it.
I pass now to the singularities which arise from the first minor deter
minants of U+\V having a factor in common, the second minors being
supposed to be still without a common factor.
When this common factor is linear in respect to X, let it be supposed
to enter not more than twice (twice, we know, by the general principle
enunciated at the commencement of this paper, it must enter) into the
complete determinant.
Two of the cones containing the intersection of U and F then become
coincident, and degenerate each into the same pair of crossing planes. This
may be termed biplanar-contact. The characteristics of such contact are
U^x^ + yt + z^ + V,
V = ax 2 + ay 2 + bz 2 + ct 2 ;
the points of contact are two in number, being at the intersection of the two
plane conies into which the curve of intersection breaks up. The two planes
36] and Surfaces of the Second Order. 231
in which these lie are given by the equation (b a) z- + (c a) t 2 = ; these
intersect in the right line z = 0, t = 0, which meets both surfaces in the same
two points,
z=Q, t = 0, #-V(
the two common tangent planes at these points being
* + V(- 1) y = 0, a - V(- 1) y =
respectively.
This, then, is another species of double contact between two conoids, and,
as far as I know, the only kind hitherto recognized as such. The number of
conditions to be satisfied remains two, as in the former species.
Next suppose that the common factor of the first minor enters three
times into the complete determinant instead of twice only, as in the last
case.
The corresponding characteristics will be found to be
U = x - + zt + y 2 + z 2 ,
V = ax 2 + azt + by 2 + cz 2 .
The intersection of U, V still lies in two planes,
(b - a) if + (c - a] z 2 = ;
but the intersection of these two planes,
y = o, * = o,
meets the surfaces in the two coincident points,
y = 0, z = 0, x 2 = 0.
This, therefore, I call confluent-biplanar contact ; the two conies con
stituting the complete intersection, instead of cutting, touch and at their
point of contact the two conoids have a contact of a superior order. The
conditions to be satisfied for this case are three in number.
Next suppose that the common factor of the first minors enters only
twice into the complete determinant, but that the remaining two factors
become equal.
Here the analytical characters of unilinear and biplanar contact are
blended ; in fact, the intersection consists of a conic and a pair of right
lines meeting one another and the conic. The characteristics are
V = ax* + ay 2 + bz 2 + czt.
232 An Enumeration of the Contacts of Lines [36
The intersection is contained in the two planes
z = Q, (b-a)z + (c-a)t = 0,
and consists of the two lines 2 = 0, x* + f = 0, lying in the common tangent
plane z = 0, and the conic
(b-a)z+(c-a)t = }
(a - c)a? + (a - c)2/ 2 + (b - c)z* = of
There are three points of contact, namely, the point x = 0, y = 0, z = 0,
where the two right lines cut, and a? + y* = 0, t = 0, 2 = 0, where these lines
meet the conic. This, then, is a case of triple contact. I distinguish it by
the name of bilinear-contact. The number of conditions is still three.
Now all else remaining as before, let the two pairs of equal roots in the
complete determinant become identical, or, in other words, let the common
factor of the first minors be contained four times in the complete deter
minant. The characteristics become
V = axz + beat + by- + bz 2 .
The intersection becomes the two right lines
x = 0, y 2 + z- = 0,
and the conic
2 = 0, x 2 + y- = 0.
All these meet in the same point,
a> = 0, y = 0, z = 0;
so that instead of contact in three points, the contact takes place about one
only, in which the three may be conceived as merging. This I call confluent-
bilinear contact. It requires the satisfaction of four conditions.
Next let us suppose that the two distinct factors are common to each
of the first minors. This will imply the existence of four affirmative
conditions.
The complete determinant will of necessity contain each of these factors
twice, so that no additional singularity can enter through this determinant.
The characteristics assume the form
U = & + f 4- 2 + 1\
V = ax 2 + ay 2 + bz- 4- bt\
The two surfaces will meet in four straight lines, forming a wry quadrilateral,
whose equations are
36] and Surfaces of the Second Order. 233
These intersect each other in the four points
x = 0, y = Q, z" + t 2 = 0,
z = Q, t = (\ tf + if = 0,
each of which will be a distinct point. This I term quadrilinear contact.
Now let the two factors common to each of the first minors become
identical; so that a squared function, instead of an ordinary quadratic
function of X, is now their common measure.
The factor which enters twice into each of the first minors will enter
four times into the complete determinant ; the number of conditions to be
satisfied is one more than in the preceding case, namely five, and the
characteristics become
V = ax* + ~by + cxz + cyt.
Here arises a singularity of form in the intersections utterly unlike
anything which has been remarked in the preceding cases. For it will
not fail to have been observed, that the intersection in the nine preceding
cases was always a line or system of lines of the fourth degree, so as to be cut
by any plane in four points.
But in this case, the fact of the first minors having a factor in common,
shows that the intersection is contained in two planes (which is of course to
be viewed as a degenerate species of cone) ; and the fact of the complete
determinant having all its roots equal, shows that there is but one system of
a pair of planes in which the intersection is contained, and no more.
So that the two pairs of planes, into which the wry quadrilateral was
divisible in the case immediately preceding, now become a single pair. This
can only be explained by two of the opposite sides of the quadrilateral
becoming indefinitely near to one another, but still not coinciding in the
same planes; so that the actual visible or quasi- visible* intersection will
be in three right lines, of which the middle one meets each of the
two others.
This will further appear by proceeding regularly to solve the equations
y_ c jj = o gives y = kx, where k = 4^E| > and tberefore xz + kxi =
or xz - kxt = ; whence we see that the complete intersection is represented
by the lines
(x = 0, y = 0) ; (z + kt = 0, y-kx = 0),
(as = 0, y = 0) ; (z - kt = 0, y + kx = 0),
* I use the term quasi-visible, because the intersection may become in part or whole
imaginary.
234 An Enumeration of the Contacts of Lines [36
showing that there are but three physically distinct lines, as already
premised.
This, then, may be considered as derived from the preceding case of
a wry quadrilateral intersection, by conceiving two opposite sides of the
quadrilateral to come indefinitely near, but without coinciding.
Let these two lines be called P and P ; take any point in P and any two
points in P indefinitely near to one another arid the point first taken, then
this indefinitely small plane will be common to both surfaces, and consequently
they ought to touch along every point in the line P. This is again confirmed
by the forms given to U and V. For at any point where the coordinates are
0, 0, f, 6 the equations to the tangent planes to the two surfaces respectively
are
&+0y = 0,
cx + cOy = 0,
that is to say, are identical.
Whilst, therefore, certain grounds of geometrical, and still stronger
grounds of analytical analogy, might seem to justify this species of contact
taking the name of confluent quadrilinear, yet as, in fact, the intersection is
trilinear, and as, moreover, the two indefinitely proximate lines must be con
sidered, not as coincident, but as turned away from one another through an
indefinitely small angle and out of the same plane, I prefer to take advantage
of this striking property of contact at every point along a line (a property
entirely distinct from any that we have yet considered), and confer upon the
species of contact we have been considering the designation of unilinear-
indefinite contact.
Where the line of indefinite contact meets the two other lines of the
intersection, the contact is of course of a higher order ; thus offering a
parallel to what takes place in ordinary unilinear contact, in which there
is no contact, except only at two points of the right line forming part of the
complete intersection.
I believe that this kind of contact, which forms a natural family with two
others about to be described, and which will close the list, has never before
been imagined, and would at first sight have been rejected as impossible.
Having now exhausted the cases of the first class, in which the minors
have no factor in common, and the two sections of the second class, in which
the second minors have no common factor, but the first minors of U + \V a
linear or quadratic function of A, in common, I descend to the third class, in
which the second minors, which are quadratic functions of X, are supposed to
have a common factor.
This common factor must enter twice into each of the first minors by
virtue of the law previously indicated, and cannot enter more than twice, as
36] and Surfaces of the Second Order. 235
otherwise the first minors of U + \V could only differ from one another by a
numerical multiplier, which is obviously impossible, except when U+\V is
of the form (k + X) U, that is, when the two surfaces coincide.
Again, the common factor of the first minor must enter three times into
the complete determinant ; but there is no reason why it may not enter four
times, and thus two cases arise. In the first, the characteristics take the
form
U = x* + f + z 2 + t 2 ,
V = ax- + ay* + az 2 + U 2 .
The second determinant having a factor in common, shows that the inter
section U, V is contained in a pair of coincident planes; but the complete
determinant, having two distinct factors, evidences that these plane inter
sections, viewed as indefinitely near but still distinct, lie in the same cone,
which will be a cone enveloping both the surfaces U and V all along their
mutual intersections. This is also seen easily from the forms of U and V;
for we have V a U = (b a) t 2 , which proves that the intersection lies in
the coincident, or, to speak more strictly, consecutive planes t 2 = ; and at
any point x = , y = rj, z=, the tangent plane to each surface becomes
gx + rjy + z = 0.
As there are six independent, that is, non-necessarily co-evanescent second
minors, that the second minor systems shall all have a common factor, implies
the satisfaction of five conditions. This species of contact I call curvilineo-
indefinite ; it is, I believe, the only kind of indefinite contact between two
surfaces of the second order hitherto taken account of.
There is still, however, a higher species of contact, videlicet, when all the
four roots of the complete determinant of U + \V are identical with the root
common to each of its second minors. In this case the common enveloping
cone becomes identical with the plane (considered as a coincident pair of
planes) in which the surfaces intersect.
The characteristics take the form
U x- + xy -r zt,
V = xy + zt.
The intersection is contained completely in the common tangent plane
x 0, and consists of the two right lines,
(x = 0, z = 0),
(x = 0, t = 0).
This, the highest and crowning species of contact, I call bilineo-indefinite.
It is defined by six conditions.
At each point of the two lines of intersection of U and V there is contact,
and a very peculiar species of contact at the intersection of these two lines
themselves.
236 An Enumeration of the Contacts of Lines [36
To form a distinct idea of this, let the physical visible or quasi-visible
intersection of f7, V take place along the two lines L, M ; the rational inter
section must be conceived as made up of the wry quadrilateral, L, M; L , M ,
in which L is indefinitely near to L , and M to M . It follows, therefore, that
there is contact at the four angles of the quadrilateral ; but as there is
nothing to fix the relative directions of the diagonal joining the intersection
of L and M to that of L and M , because there is nothing to restrict the
position of the latter point, except that it shall lie upon either surface*, it
appears that not only is there contact at the junction of the two lines
constituting the complete intersection of the two surfaces, but that these
surfaces continue to touch at consecutive points taken all round this first,
and indefinitely near to it in any direction f.
Bilineo-iu definite (the highest) contact for two conoids is strictly
analogous to confluence, the highest species of contact between conies.
For this latter may be conceived as an intersection made up of two co
incident pairs of coincident points ; and the former, as an intersection made
up of two coincident pairs of crossing right lines ; and a pair of crossing
lines is to a plane locus of the second degree what a coincident pair of points
is to a rectilinear locus of the same degree.
In the subjoined table I have brought under one point of view the
characters and algebraic forms which I call the condensed forms corre
sponding to each species of contact above detailed.
A. Quadratic loci in a right line.
Simple contact. | xy
One condition. J x" + xy
B. Quadratic loci in a plane.
1st Class.
Simple contact. ) x 2 + y 2 + xz
One condition. j ax z + bif + cxz
Proximal contact. ) x 2 + yx + yz
Two conditions. j ax* + byx + ayz
2nd Class.
Diploidal contact. \ x- + y~ + z*
Two conditions. j ax* + af + bz*
Confluent contact. I a? + if + xz
Three conditions. j y z + xz
* This will be better seen by reference to the analogy presented by the case when the two
conoids touch all along a curve. The rational intersection is made up of this curve and another
indefinitely near it. The two curves, whatever be the position of their node, will lie in the same
enveloping cone, so that the position of the node is indeterminate.
t As the two surfaces jut one close into the other at this point, it would perhaps be not
improper to designate the contact at such point as umbilical.
36]
and Surfaces of the Second Order.
237
C. Quadratic loci in space.
1st Class.
Simple contact.
One condition.
Proximal contact.
Two conditions.
Unilinear contact.
1st species of diploidal.
Two conditions.
Confluent-unilinear, or
triple contact.
Three conditions.
2nd Class, 1st Section.
Biplanar contact.
2nd species of diploidal.
Two conditions.
X 2 + y 2 + 2* + Xt
ax* + by* + cz- + dxt
x- + y 2 + xt + zt ] .
ax 2 + by 2 + cxt + azt }
X* + xy + zt I
ayz + bxy + czt j
x 2 + yz + xy + zt
az- + bxy + bzt
a? + y 2 + z 2 + t 2
ax 2 + ay 2 + bz 2 + ct 2
Confluent-biplanar con- ] x 2 + zt + y 2 + z 2
tact. Three conditions, j ax 2 + azt + by* + cz-
Bilinear contact. ] x z + y 1 + z- + zt
ax 2 + ay 2 + bz 2 + czt ) \ axt + byz
Three conditions.
Confluent-bilinear con- ) xz + xt + y 2 + z"
tact. Four conditions, j axz + bxt + by- + bz*
2nd Class, 2nd Section.
or
f xz + yt
Quadrilinear, or quad
ruple contact.
Four conditions.
Unilineo-indefinite con
tact. Five conditions.
3rd Class.
Curvilineo-indefinite
contact.
Five conditions.
Bilineo-indefinite con
tact. Six conditions.
x 2 + if + z 2 + t"
or
f xy + zt
ax* + ay- + bz 2 + bt 2 ) \ axy + bzt
x 2 + y* 4- xz + yt
ax 2 + by 2 + cxz + cyt
ax 2 + ay 2 + az 2 + bt 2
x 2 + xy + zt |
xy + zt }
238 An Enumeration of the Contacts of Lines [36
Another (and, in a physical sense, more) natural mode of grouping the
twelve species of conoidal contact, which, without observing the same lines
of demarcation, leaves intact the sequence of the species, is into the three
families. The first, or definite-continuous, for which the surfaces touch in
a single point, and intersect in an unbroken curve, comprises simple and
cuspidal contact.
The second definite-discontinuous, for which the surfaces touch in one,
two, three or four points, but intersect in a curve more or less broken up into
distinct parts, comprises all the species from the third to the ninth inclusive.
The third natural family is that of indefinite contact, and comprises the three
last species. It will of course be observed that there are five species of single
contact, that is, contact at one point, namely, simple, cuspidal, and the three
confluent species, two of double, one of treble, one of quadruple, and three of
indefinite contact ; the last being distinguishable inter se lineo-indeh nite as
being special at two points, curvilineo-indefinite as having no speciality, and
bilineo-indefinite as being special at one point only.
I might now proceed to discuss more particularly the nature of the
contact taken, not collectively, but with reference to each single point where
it exists. This, however, must be reserved for a future communication ; as
also, among other important and curious matter, the ascertainment of the
singular forms of quadratic conjugate functions of five or more letters. At
present I shall content myself with stating the following general proposition,
which naturally suggests itself from a consideration of the cases already
considered.
In a conjugate quadratic system of any number of letters, the lowest and
also the highest degree of singularity will be always unique ; the conditions
to be satisfied in the former case being only one in number, and, in the latter
\r (r 1), where r denotes the number of the letters. The first part of this
proposition is self-apparent, the latter part may be inferred from the homa-
loidal law; for the (r 2)nd minors will be quadratic functions, and the
highest degree of contact will correspond to those having a factor in common,
which would involve the satisfaction of |r(r-l)-l conditions only; but
over and above this, that the complete determinant, instead of containing
this common factor, as it needs must, (r-l) times, shall contain it r times :
this gives one condition more, making up the entire number to \r (r 1).
The total number of different species of singularity for conjugate func
tions of a given number of letters ; can only be expressed by aid of formulae
containing expressions for the number of various ways in which numbers
admit of being broken up into a given number of parts.
The computation of this number in particular cases, upon the principle of
the foregoing method, is attended with no difficulty.
36] and Surfaces of the Second Order. 239
We have seen that this number for two, three and four letters, is
respectively one, four, twelve.
I have found that for five letters the number is twenty-four, for six
letters fifty, for seven letters a hundred, and (subject to further examination)
for eight letters one hundred and ninety-three. The series, therefore, as far
as I have yet traced it, is 1, 4, 12, 24, 50, 100, 193. The last number must
not be relied upon at present.
It will be observed, that the foregoing table for the contacts of surfaces
of the second order contains no form corresponding to a complete intersection
in two non-intersecting lines and an undegenerated conic. In fact, if two
such lines form part of the intersection, at least one other right line inter
secting them both, must go to make up the remaining part. This is easily
verified ; for it is readily seen that the most general representation of two
conoids intersecting in two non-meeting lines will be
U = xy + zt,
V= axy + bzt + cxt + eyz,
where the two lines in question are
(x = 0, 2 = 0),
Now it will be found that the first minors of V+\U formed from the
above equation will all contain the common factor (a + X) (b + X,) ce, showing
that the contact is quadrilinear or linear-indefinite, that is bilinear, according
as the roots of
X 2 + (a + b) \ + ab - ce =
distinct or equal ; which explains how it is that only one species of
bilinear contact (that is to say, the case corresponding to the two conoids
agreeing in the two right lines in which each is cut by a common tangent
plane) comes to find a place in the preceding enumeration.
It may not be uninteresting, under an euristic point of view, to state that
the above theory, which, as well in what it accomplishes as in what it
suggests (the author cannot but feel conscious), constitutes a substantial
accession to analytical science, arose out of a theorem which occurred to
him as likely to be true, in the act of reviewing for the press his paper
"On Certain Additions" in the last November Number* of this Magazine,
and which he had only then time to throw into a foot-note as a probable
conjecture.
Wishing to subject it to an analytical test, he found it necessary to obtain
the condensed forms which serve to characterize the confluent contact of
[* p. 148 above.]
240 Contacts of Lines and Surfaces of the Second Order. [36
conies. In this way he became aware of the great utility of these condensed
forms, and of the desideratum to be supplied in obtaining a complete list of
them applicable to all varieties of contact. The happy thought then occurred
to him of inverting the process which he had applied in the treatment of
the contacts of conies, in the November Number* of the Cambridge and
Dublin Mathematical Journal; for whereas the nature of the contacts was
there assumed and translated into the language of determinants, he soon
discovered that it was the more easy and secure course to assume the relations
of every possible immutable kind that could exist between the complete and
minor determinants corresponding to the characteristics, by aid of these
relations to construct the characteristics, and from the characteristics so
obtained, determine the geometrical character of each resulting species of
contact. Thus he has been able to effect the very results stated by himself
as desiderata at the close of the paper in this Magazine above referred to.
Note. It is proper to remark, that all the condensed forms given in this
paper have actually been obtained by the author in the way above pointed
out. The limits imposed by the objects to which the Magazine is devoted
have restricted him from exhibiting the method at full ; but any of his
readers will be able without difficulty to make it out for himself.
The process consists in finding U + Xl^by means of solving for each case
a problem of position (a kind of chess-board problem) on a square table,
containing three places in length and breadth for conies, four places by four
for surfaces, and so on (if need be) according to the number of variable letters
involved. L r +\V being thus determined in form, U and V become readily
cognizable. It is right also to add, that some of the condensed forms here
set forth have been incidentally noticed and employed by previous authors,
as Pliicker and Mr Cayley.
The conditions in each case to which the position-problem is subject
are immediately deducible from the laws which the complete determinant,
and the successive minor systems of determinants of U + \V, are required
to satisfy.
[* p. 119 above.]
37.
ON THE RELATION BETWEEN THE MINOR DETERMINANTS
OF LINEARLY EQUIVALENT QUADRATIC FUNCTIONS.
[Philosophical Magazine, I. (1851), pp. 295 305.]
I SHOWED in the preliminary part of my paper on Contacts in the February
Number of this Magazine*, by a priori reasoning, that if a quadratic function
(7) be linearly converted into another (V), any minor determinant of any
order of F~must be a syzygetic function of all the minor determinants of U of
the same order.
The object of my present communication is to exhibit the syzygy in
question, which, as I indicated, is linear ; by which I mean that a determinant
of the one function is equal to the sum of the pari-ordinal determinants
of the other affected respectively with multipliers formed exclusively out of
the coefficients of the equations of transformation. In order that a clear
enunciation of the theorem in view may be possible, it is necessary to premise
a new but simple, and, as experience has proved to me, a most powerful,
because natural, method of notation applicable to all questions concerning
determinants.
Every determinant is obtained by operating upon a square array of
quantities, which, according to the ordinary method, might be denoted
as follows :
i,i> Ll i,2 "a.M*
2, i > &2, 2 ... Gk>, n >
3, 1 > ^3, 2 ^3, H >
My method consists in expressing the same quantities biliterally as
below :
a.i, a n 2 ... a n a n ,
[* p. 221 above.]
16
242 The Relation between the Minor Determinants of [37
where of course, whenever desirable, instead of c^, a 2 ... a n , and a a , a 2 ... a n ,
we may write simply a, b ... I, arid a, {3 ... X respectively. Each quantity is
now represented by two letters; the letters themselves, taken separately,
being symbols neither of quantity nor of operation, but mere umbrae or ideal
elements of quantitative symbols. We have now a means of representing
the determinant above given in a compact form ; for this purpose we need
but to write one set of umbrsB over the other as follows: ( 1J 2 " n } . If
\*1, Oa ... CtJ
we now wish to obtain the algebraic value of this determinant, it is only
necessary to take a lt a 2 ... a w in all its 1, 2, 3 ... n different positions, and we
shall have
{ N
n \ = S |o 1 a fll x a,a e . 2 x ... x a n a e j,
a i a 2 . . . a n j
in which expression # n 6 2 ...6 n represents some order of the numbers
1, 2... n, and the positive or negative sign is to be taken according to the
well-known dichotomous law. Thus, for example,
[abc] .,, la
\ _ } will represent act x bp x cy \
(oipy)
+ a/3 x by x ca.
+ ay x ba. x c/3
a/3 x ba x cy
aa. x by x c/3
ay x b/3 x ca t
Although not necessary for our immediate object, it may not be inop
portune to observe how readily this notation lends itself to a further natural
extension of its application.
ab cd\ will natura ii y denote
a/3 78)
that is
ab cd ab cd
\x ^_ y
a/8 78 78 a/3
(aa x &)]_ ( (cy x d8)\ _( (ay x 6S){ f (ca x c?/3)|
- (a/S x 6a)j X {- (cS x dy)J j- (aS x 67)) X {- (c/3 X <fe)j
And in general the compound determinant
1 di , 61 .
tab I
,. /,
K, A.
. . Xj, Oa, /S 2 ... X 2
will denote
2 +
\ n
^1 1 Of o j t/9 ^2
( X ) /9
lx...x|
K, A t ...
XeJ (a e4 , p e2 ...X ei
J I
37] Linearly Equivalent Quadratic Functions. 243
where, as before, we have the disjunctive equation
6 lt 2 ... O r = I, 2...r.
As an example of the power of this notation, I will content myself with
stating the following remarkable theorem in compound determinants, one of
the most prolific in results of any with which I am acquainted, but which
is derived from a more particular case of another vastly more general. The
theorem is contained in the annexed equation
1, U 2 ... r > "r+l> u l> tt>...l*y, r+2 1*1 > 1*2 "> u r+g
1, ft... ... ft,., 0r+lj #1, 2 ... Ct r , ftr+2 ^1) a 2 a r> ^r+s
, (7 2 . . . tt r , ftr+1 > ^r+2 flj
[, ft 2 ... Of?-) (^ftj , ft 2 ... CLf, 2j--f.j, ftr+2 ^j
It is obvious, that, without the aid of my system of umbral or biliteral
notation, this important theorem could not be made the subject of statement
without an enormous periphrasis, and could never have been made the object
of distinct contemplation or proof.
To return to the more immediate object of this communication, suppose
that we have any binary function of two sets of quantities, x ll x 2 ...x n ;
i> %2-- t;n, of which the general term will be of the form c rjS x# r s ;
according to the principles of notation above laid down, nothing can be
more natural than to represent c ftS by the biliteral group a r g ; the function
in question will then take the form
the a; s and s denoting quantities, but the a s and a s mere umbrae. The
function may then be thrown under the convenient symbolical form
So if we confine ourselves to quadratic functions, for which x lt x 2 ...x n ;
fn2---n become respectively identical, the general symbolical represen
tation of any such will be
The complete determinant will be denoted by
a 2 ... a n ]
r ,
Lu
and any minor determinant of the rth order by
! fl 2 <
162
244 The Relation between the Minor Determinants of [37
where 1} 6^ ... 6 r are some certain r distinct numbers taken out of the series
1, 2, 3 ... r. Suppose now that we have
U = (d-^x-i + a 2 # 2 + . . . + a n ac n ) 2
linearly transformable into
P=(& 1 y, + 6 2 y a + ...+b n y n )\
by means of the n equations
#1 = aA 2/i + A - y 2 + + a^n . y n
x* = a 2 b, . y l + a a 6 2 . y a + . . . + aj> n .
#n = n6i 3/i + a A . y 2 + . . . + a n b n . y n
in which equations, be it observed, each coefficient a r b s is a single quantity,
perfectly independent of the quantities denoted generally by a r a g , b r b s which
enter into U and V. Our object is to be able to express the minor
determinant
(b kl , b k2 ...b kr
in which the one group of distinct numbers, k l ,k n _...k r may either differ
wholly from, or agree wholly or in part with the other group of distinct
numbers l l} 1 2 ... l r , under the form of
( I n n rt \ ^
/B B B \
The particular value of Q corresponding to each double group, f lj 2 r j,
/ B B B \
may be denoted by Q ( 1( 2 r ) ; so that our problem consists in deter-
V9i , 92 .- 9r/
(B B B \
mining the value of Q ( * 2 " . r } in the equation
\01) 92 9r/
|6 fcl , 6 t , ... b kr \ = ^ | ^!, a ... ^ x ( - a fll , % 2 ... a 0t .
k, 6^ ... 6j "
Accordingly I enunciate that
subject to one sole exception in the case of #j, 2 &r being identical with
^j, </> 2 , ... 9 r ; namely, that for the terms (for such case) of the form
37] Linearly Equivalent Quadratic Functions. 245
/ft 6 9 \
Q ( a a " a I tne va l ue to be taken is not that which the general formula
\0i, 02 . . . 6 r /
would give, namely,
K,
but the half of this, that is simply the square of
/Q 8 9 \
The value of Q ( ls / */), it is obvious, contains only quantities
of the form a r .& g , which are coefficients in the equations of transformation,
but none of the form a. r . a s or b r . b s ; showing that the syzygetic connexion
between the minor determinants of U and V of the same order is linear,
as has been already anticipatively announced.
The problem which I have treated above is only a particular case of
a more general one, which may be stated as follows : given
and supposing m linear equations to be instituted between x ly x^..,x n ,
so that U may be made a function of (n m) letters only, to express any
minor determinant of the reduced form of U without performing the process
of elimination between the given equations. Let the given equations be
written under the form
>
. . . + a n a n+l x n = 0,
... + a n a n+2 x n = 0,
m^z T T ^n^n-^m^n ">
and let it be convened (which takes nothing away from the generality of
these equations) that a n+r a n+g shall signify zero for all values of r and s
concurrently greater than zero. Suppose that x l , x. 2 ...x m , being eliminated,
U becomes of the form
(^m+i^TO+i "I" ^m+a^wi-t-a + + O n OC n ) ,
and suppose that we wish to determine the value of the complete determinant
of this last function ; it will be found to be
the squared divisor being, as is obvious, a function only of the coefficients
of the transforming equations, and depending for its value upon the particular
246 The Relation between the Minor Determinants of [37
m quantities selected for elimination. The dividend, on the contrary, is
independent of this selection, but involves the coefficients of the function
combined with the coefficients of transformation. This is the symbolical
representation of the theorem given by me in the postscript to my paper in
the Cambridge and Dublin Mathematical Journal for November 1850*.
Suppose, now, more generally that we wish to find any minor determinant.
The solution is given f by the equation
(wherein the two groups 6 m+l , m+2 , ...0 m +s , <f>m+i, $m+z - 4>m+s are each
of them s differing, or wholly or in part agreeing individuals arbitrarily
selected out of the (n m) numbers m + 1, m + 2, ... ri)
,. %2 . %n+l> a m-M Cf m+. 4 . ( 1, 2 m
If we make w = 2y and 7/1 = y, and a y+r a y+s = for all positive values
of either r or s, and o y _ i a n+(( = for all values of i and e differing from one
another, and for equal values a y _ e a y+c = 1, it will readily be seen that this
last theorem reduces to the one first considered ; and on careful inspection
it will be found, that the solution given of the general question includes
within it that presented for the particular case in question. Such inclusion,
however, I ought in fairness to state is far from being obvious; and to
demonstrate it exactly, and in general terms, requires the aid of methods
which my readers would probably find to exceed their existing degree of
knowledge or familiarity with the subject.
The theorem above enunciated was in part suggested in the course of a
conversation with Mr Cayley (to whom I am indebted for my restoration
to the enjoyment of mathematical life) on the subject of one of the pre
liminary theorems in my paper on Contacts in this Magazine.
It is wonderful that a theory so purely analytical should originate in
a geometrical speculation. My friend M. Hermite has pointed out to me,
that some faint indications of the same theory may be found in the Recherches
.Arithmetiques of Gauss. The notation which I have employed for deter
minants is very similar to that of Vandermonde, with which I have become
acquainted since writing the above, in Mr Spottiswoode s valuable treatise
On the Elementary Theorems of Determinants. Vandermonde was evidently
on the right road. I do not hesitate to affirm, that the superiority of his
and my notation over that in use in the ordinary methods is as great and
almost as important to the progress of analysis, as the superiority of the
notation of the differential calculus over that of the fiuxional system. For
what is the theory of determinants ? It is an algebra upon algebra ; a
[* p. 136 above.] [t see p. 251 below.]
37] Linearly Equivalent Quadratic Functions. 247
calculus which enables us to combine and foretell the results of algebraical
operations, in the same way as algebra itself enables us to dispense with
the performance of the special operations of arithmetic. All analysis must
ultimately clothe itself under this form*.
I have in previous papers denned a " Matrix " as a rectangular array of
terms, out of which different systems of determinants may be engendered,
as from the womb of a common parent ; these cognate determinants being
by no means isolated in their relations to one another, but subject to certain
simple laws of mutual dependence and simultaneous deperition. The con
densed representation of any such Matrix, according to my improved Vander-
mondian notation, will be
To return to the theorems of the text. Theorem (2) admits of being
presented in a more convenient form for the purposes of analytical operation,
so as to become relieved from all cases of exception appertaining to particular
terms.
The limitation to the generality of the expression for Q arises from our
treating
Ja 01 , a<, 2 ... a dr \
as identical with its equal,
a 9l , a e , ...a 0r }
If, however, we now convene to treat these two forms as distinct, so that
in theorem (2)
17} (n _ 1 ^ ( n r -L-] \) 2
-^ - -iip -- >[ terms, then we may write simply
1.2...r J
(a kl , a ki ... a k ] fr h , a h ... a lr
* Perhaps the most remarkable indirect question to which the method of determinants has
been hitherto applied is Hesse s problem of reducing a cubic function of 3 letters to another
consisting only of 4 terms by linear substitutions a problem which appears to set at defiance
all the processes and artifices of common algebra. I have succeeded in applying a method
founded upon this calculus to the linear reduction of a biquadratic function of two letters to
Cayley s form x 4 + mx^y 2 + y*, and of a 5 C function of two letters to the new form x& + y s + (ax + by) 5 .
This last reduction is effected by means of the properties of a certain other function of the
8th degree connected with the given function of the 5th degree. See a paper on this subject in
the forthcoming May Number of the Cambridge and Dublin Mathematical Journal, [p. 191 above.]
248 The Relation between the Minor Determinants of [37
which equation is subject to no exception for the case of the s and < s
becoming identical. As regards this theorem, it will not fail to strike the
reader that it ought to admit of verification ; for that U may be derived
from V in the same manner as V from U if we express y lt y^...y n in terms
of x lt x^ ... x n , by solving the system of equations (2), which there is no
difficulty in doing. In fact, if we write
7/2 = Oa
we shall obtain
a a _ s
\b lt b 2 .
Accordingly we shall find
a Pl>
r _x, a r+1 ,
=2 n
a 3 ... a
and
a m \
substituting for the a s and /3 s their symbolical equivalents given above,
and applying the theorem given below, we shall easily obtain
If, now, in the expression
!, a 2 ..;a n
we resubstitute for j 9l> fia " 6r [ its value in the form of
we shall obtain
under the form of
r,
37] Linearly Equivalent Quadratic Functions. 249
and R ( , 1J , 2 . r ) must =0, except for the case of w l , &&gt; 2 ...eo r ; >Ir lt TK...iIr. r
VflKi Tfci-W
being respectively identical with &j, k. 2 ...k r ; 1 1} I 2 ...l r , for which case
(k k k \
i> a- r\ mug | je un ity. I have gone through this calculation and
fcj , t 2 r/
verified the result ; in order to effect which, however, the following important
generalization of theorem (1) must be apprehended.
Suppose two sets of umbrae,
Oj , d^ ... O. m+n >
1 , 2 Om+n >
and let r be any number less than m, and let any r-ary combination of
the m numbers 1, 2, 3 ...m be expressed by 9# 1; q d 2 ... q O m , where q goes
through all the values intermediate between 1 and /u, p being
m (m 1 ) ... (m r + 1)
1.2 ...r~
then I say that the compound determinant,
^ii i^ 2 " l e m Vm+i,
is equal to the following product,
ft -i ft r fl ft. a * "
, "" (4)
where
,,_(m- l)(w-2) ... (m-r + 1)
and
f ^??Z ~~ J. j \lfYL ^~ y ... \JIW, ^~ ITj
when r = 1, we have the case already given in theorem (2), and of course
// is to be taken unity.
This very general theorem is itself several degrees removed from my still
unpublished Fundamental Theorem which is a theorem for the expansion
of the products of determinants.
250 On Linearly Equivalent Quadratic Functions. [37
Obs. The analogy upon which the extension of the Vandermondian
notation from simple to compound determinants is grounded, would be better
apprehended if the biliteral symbols of simple quantities were written with
the umbral elements disposed vertically, as , , instead of horizontally, as ab ;
which latter is the method for the purposes of typographical uniformity
adopted in the text above. The other mode is, however, much to be pre
ferred, and is what I propose hereafter to adhere to. For my two general
umbrae, a, b, Vandermonde uses two numbers, one set a-cock upon the other,
as 5 4 . The objection to the use of numbers is apparent as soon as it becomes
necessary to treat of the mutual relations of diverse systems of determinants,
and his mode of writing the umbras militates against the perception of the
most valuable algebraical analogies. The one important point in which
Vandermonde has anticipated me, consists in expressing a simple determinant
by two horizontal rows of umbra; one over the other. But the idea upon
which this depends is so simple and natural, that it was sure to reappear
in any well-constructed system of notation.
38.
NOTE ON QUADRATIC FUNCTIONS AND HYPER-
DETERMINANTS.
[Philosophical Magazine, I. (1851), p. 415.]
PERMIT me to correct an error of transcription in the MS. of my paper
" On Linearly Equivalent Quadratic Functions " in the last number of the
Magazine. The theorem [p. 246 above] marked (3), should read as follows :
a 6 m+a > a n+D &M+2 a n+m \
tt <t> m+s , ft n+i > <3/i+2 ^n+m)
I may take this opportunity of mentioning, that by extending to
algebraical functions generally a multiliteral system of umbral notation,
analogous to the biliteral system explained in the paper above referred
to as applicable to quadratic functions, I have succeeded in reducing to
a mechanical method of compound permutation the process for the discovery
of those memorable forms invented by Mr Cayley, arid named by him hyper-
determinants, which have attracted the notice and just admiration of analysts
all over Europe, and which will remain a perpetual memorial, as long as
the name of algebra survives, of the penetration and sagacity of their author.
39.
ON A CERTAIN FUNDAMENTAL THEOREM OF
DETERMINANTS.
[Philosophical Magazine, II. (1851), pp. 142 145.]
THE subjoined theorem, which is one susceptible of great extension and
generalization, appears to me, and indeed from use and acquaintance (it
having been long in my possession) I know to be so important ami funda
mental, as to induce me to extract it from a mass of memoranda on the same
subject; and as an act of duty to my fellow-labourers in the theory of
determinants, more or less forestall time (the sure discoverer of truth) by
placing it without further delay on record in the pages of this Magazine. Its
developments and applications must be reserved for a more convenient
occasion, when the interest in the New Algebra (for such, truly, it is the
office of the theory of determinants to establish), and the number of its
disciples in this country, shall have received their destined augmentation. In
a recent letter to me, M. Hermite well alludes to the theory of determinants
as "That vast theory, transcendental in point of difficulty, elementary in
regard to its being the basis of researches in the higher arithmetic and in
analytical geometry."
The theorem is as follows : Suppose that there are two determinants of
the ordinary kind, each expressed by a square array of terms made up
of n lines and n columns, so that in each square there are n- terms. Now
let n be broken up in any given manner into two parts p and q, so that
p 4. q = n. Let, firstly, one of the two given squares be divided in a given
definite manner into two parts, one containing p of the n given lines, and the
other part q of the same; and secondly, let the other of the two given squares
be divided in every possible way into two parts, consisting of q and p lines
respectively, so that on tacking on the part containing q lines of the second
square to the part containing p lines of the first square, and the part con
taining p lines of the second square to the part containing q of the first, we
39] On a Fundamental Theorem of Determinants. 253
get back a new couple of squares, each denoting a determinant different
from the two given determinants ; the number of such new couples will
evidently be
and my theorem is, that the product of the given couple of determinants
is equal to the sum of the products (affected with the proper algebraical sign}
of each of the new couples formed as above described. Analytically the theorem
may be stated as follows.
fa 1( a 2 ...a
according to the notation heretofore* employed by me in the preceding
numbers of this Magazine, denote any two common determinants, each of
the nth order, and let the numbers 6 l , # 2 @n be disjunctively equal to the
numbers 1, 2 ... n and p + q = n; then will
I (i, cr 2 ...a n ) ("!, ot 2 ... a n
\b 1} b 2 ... &J X tft, &..
i, 2 a n | x | of!, o
The general term under the sign of summation may be represented by aid
of the disjunctive equations
under the form of
x ... x
1st. When ^> 15 < 2 ^ = ^i, ^ 2 ^p , it will readily be seen, that for
given values of </> 1( <f> . . . (f> p , the product of the third and fourth factors
becomes substantially identical with the general term of the determinant
and consequently, making the system^, </> 2 ...^ (or, which is the same
thing, its equivalent fa, fa... fa) go through all its values, we get back for
the sum of the terms corresponding to the equation
0! , </>o . . . (j) p = fa, fa... fa,
[* p. 242 above.]
254 On a certain Fundamental [39
the product of the determinants
fttj, a 2 ...a n ) , (!, a,... OB]
, V and < } .
2nd. When \ve have not the equality above supposed between the < s
and the i/r s, let
<j> P -h = ^+fc and </> p+r) = i/rp_ f ;
the corresponding term included under the S will contain the factor
a* /3 x cu . & .
Pf+T) ^ B p+>) V-f P-
Now leaving ^>j, (j) 2 ...<j> p , and A/T I} -^ ., . . . -vj^ unaltered, we may take a
system of values 0/, #./ . . . n , such that
j a 1 _ a
and " p-$ a p+ri .
and for all other values of q except p + rj, or p 6 q = 6 q . The correspond
ing new value of the general term so formed by the substitution of the
6 for the 6 series, will be identical with that of the term first spoken of, but
will have the contrary algebraical sign, because the 6 arrangement of the
figures 1, 2, 3 ...p is deducible by a single interchange from the arrange
ment of the same, the rule for the imposition of the algebraical sign plus
or minus being understood to be, that the term in which
enter into the symbolical forms of the respective derived couples of deter
minants, has the same sign as, or the contrary sign to, that in which
so enter, according as an odd or an even number of interchanges is required
to transform the arrangement
into the arrangement
/!/ /,/ /)/ . a Q a
" p+ii Vp +z ...Uni & ii V z ...v p .
I have therefore shown that all the terms arising from the expansion of
the products included under the sign of summation, for which the disjunctive
identity <i, < 2 ... </>? = ^i, ^2 typ d es not exist, enter into the final sum in
pairs, equal in quantity and differing in sign, which consequently mutually
destroy, and that the terms for which the said identity does exist together
make up the sum
( a lt a,... aj _
\&. &-..0J
39] Theorem of Determinants. 255
which proves, upon first principles drawn direct from that notion of polar
dichotomy of permutation systems which rests at the bottom of the whole
theory of the subject, the fundamental, and, as I believe, perfectly new
theorem, which it is the object of this communication to establish.
In applying the theorem thus analytically formulized, it is of course to be
understood that, under the sign 2, permutations within the separate parts of
a given arrangement,
zi /j /i . /j a /j
Vp+i, V p+ <>...V n , t>i, V 2 ...V p ,
are inadmissible, the total number of terms so included being restricted to
n(n l) ... (np + 1)
l. 2...p
The theorem may be extended so as to become a theorem for the ex
pansion of the product of any number of determinants, and adapted so as to
take in that far more general class of functions known to Mr Cayley and
myself under the new name of commutants, of which determinants present
only a particular, and that the most limited instance.
40.
ON EXTENSIONS OF THE DIALYTIC METHOD OF
ELIMINATION.
[Philosophical Magazine, II. (1851), pp. 221230.]
THE theory about to be described is a natural extension of the method of
elimination presented by me ten years ago (in June, 1841) in the pages of
this Magazine, which I have been induced to review in consequence of the
nattering interest recently expressed in the subject by my friend M. Terquem,
and some other continental mathematicians, and because of the importance
of the geometrical and other applications of which it admits, and of the
inquiries to which it indirectly gives rise. We shall be concerned in the
following discussion with systems of homogeneous rational integral functions
of a peculiar form, to which for present purposes I propose to give the name
of aggregative functions, consisting of ordinary homogeneous functions of the
same variables but of different degrees, brought together into one sum made
homogeneous by means of powers of new variables entering factorially.
Thus if F, G, H . . . L be any number of functions of any number of letters
x, y ... t of the degrees m, m - i, in i ... m - (t) respectively,
will be an aggregative function of the variables entering into F, G, &c. and of
\, li...d. I shall further call such a function binary, ternary, quaternary,
and so forth, according to the number of variables contained in the functions
F, G, H, &c. thus brought into coalition.
It will be convenient to recall the attention of the reader to the meaning
of some of the terms, employed by me in the paper above referred to.
If F be any homogeneous function of a;,y t z...t, the term augmentative
of F denotes any function obtained from F of the form
x a ? zi . . . $ x F.
Again, if we have any number of such functions F, G, H ... K of as many
40] Extensions of the Dialytic Method of Elimination. 257
variables x, y, z ... t, and we decompose F, G, H . . . K in any manner so as to
obtain the equations
F = a? PL + /P 2 + z c P 3 + &c. . . . + t d (P),
H =
+ &c. . . . + t d (R\
and then form the determinant
\, P 2 , P,...(P) ,
, ft, ft... (flf)
this determinant, expressed as a function of a, y, z ...t, is what, in the paper
referred to, I called a secondary derivee, but which for the future I shall cite
by the more concise and expressive name of a connective of the system of
functions F, G,H ... K from which it is obtained. One prevailing principle
regulates all the cases treated of in this and the antecedent memoir, namely
that of forming linearly independent systems of augmentatives or connectives,
or both, of the given system whose resultant is to be found, of the same
degree one with the other, and equal in number (when this admits of being
done) to the number of distinct terms in the functions thus formed. The
resultant of these functions, treated as linear functions of the several
combinations of powers of the variables in each term, will then be the
resultant of the given system clear of all irrelevant factors. If the number
of terms to be eliminated exceed the number of the functions, the elimination
of course cannot be executed. If the contrary be the case, but the equality
is restored by the rejection of a certain number of the equations, the resultant
so obtained will vary according to the choice of the equations retained for
the purpose of the elimination. The true resultant will not then coincide
with any of the resultants so obtained, but will enter as a common factor into
them all.
The following simple arithmetical principles will be found applicable and
useful for quotation in the sequel :
(a) The number of terms in a homogeneous function of p letters of the
rath degree is
w(m + l)... (m + p 1)
17
1.2...JJ
258 On Extensions of [40
(6) The number of augmentatives of the (m + ?i)th degree belonging to
a function of p letters of the mth degree is
1.2...J9
(c) The number of solutions in integers (excluding zeros) of the equation
i + 2 + . . . + a p = k is
To begin with the case of binary aggregatives. Let
F m (x, y) + F m _ t (x, y) V + F rn _< (x, y) p + &c. . . . + F m ^ M (x, y} 0<<>
G n (x, y) + Gn_, (x, y} V + G n -s (x, y) ^ + &c. . . . + G n - (i) (x, y} # (t)
, A s
K p (x, y) + K p _i (x, y) V + K p _< (x, y} p + &c. . .. + K P _ M (x, y) 6
be a system of functions (whose Resultant it is proposed to determine) equal
in number to the variables x, y, X, p . . . 6, and similarly aggregative, that is
having only the same powers of X, p, &c. entering into them, but of any
degrees equal or unequal m, n ... p. Let the number of the functions be r.
Raise each of the given functions by augmentation to the degree s, where
the number of augmentatives of the several functions will be
(s + l)-m,
(+!)-*>,
and the total number will therefore be
r (s + 1) (m + n + . . . +p),
which =(r-l)(m + n+ ...+p)-r(i + i + ... + (t)).
Again, the number of terms to be eliminated will be the sum of the
numbers of terms in functions respectively of the sth, (s - t)th, (s t )th, . . .
(s (t))th degrees, which are respectively
* + l,
S+l-L,
s + 1 - i,
s + 1 - (i),
40] the Dialytic Method of Elimination. 259
and the number of these partial functions is r\. Hence the number of
terms to be eliminated is
(r - 1) {m + n + &c. + p-(i + if + &c. + (t))} -(t+i+ &c. + (*))
= (r- 1) (TO + W + &C. + p) r(i + i + ... + (t))
which is exactly equal to the number of the augmentative functions. Hence
the Resultant* of the given functions can be found dialytically by linear
elimination, and the exponent of its dimensions in respect to the coefficients
of the given functions will be the number
(r l)2ra rSt,
as above found.
The method above given may be replaced by another more compendious,
and analogous to that known by the name of Bezout s abridged method for
ordinary functions of two letters. As the method is precisely the same
whatever the number of the functions employed may be, I shall for the sake
of greater simplicity restrict the demonstration to the case of three functions,
U, V, W, whotee degrees (if unequal, written in ascending order of magnitude)
are m, n, p respectively. Let
ET -JUfey) +**-> fey)**
V=G n (x, y) + G^ (x, y) z 1 -,
W = H p (x,y} + H p _ l (x,y}*.
Let 6, a) be taken any two numbers which satisfy in integers greater than
zero the equation 6 + a = m + 1, and let
F m (x, y} = c m _ e . x* + <j> m - w . y",
G n (x, y} = 7n _e . x + 7 W _ . y",
Hp ( x > y} = Vp-e v 6 + IP-* y"
where the < s, 7*8, rj s may be always considered rational integer functions of
x and y\ for every term in each of the functions F m , G n , H p must either
contain x e or # tt , since, if not, its dimensions in x and y would not exceed
(tf-l)+(o>-l),
that is m 1, whereas each term is of m conjoined dimensions, at least, in x
and y. Hence from the equations
[7 = 0,
W=0,
* The Resultant of a system of functions means in general the same thing as the left-hand
side of the final equation (clear of extraneous factors) resulting from the elimination of the
variables between the equations formed by equating the said functions severally to zero.
172
260 On Extensions of [40
by eliminating af, y and z l we obtain the connective determinant
4>m-6> $m-o F m _
7n IV (T
n e> jn co) *-* n
^j? 8) ^7p <>> > -"p
which will be of the degree
m4-w4^-(0 + &&gt;4*)>
that is of the degree (n+p- i - 1) in x and y; and the number of such
connectives by principle (c) is p.
Again, by augmentation we can raise each of the functions U, V, W to
the same degree as the connectives, and by principle (b) the number of such
will be
n 4 p m t,
p-i,
n i,
from U, V, W respectively, together making up the number
2?z + 2jo - m 3 4.
Hence in all we have 2n+2p 3t equations; and the number of terms
to be eliminated will be, n+p i arising from F m , G n , H p , and n+p 2t
from F m _ l} Gn- t , Hp_i ; together making up the proper number 2^4- 2p 3t.
Each connective contains ternary combinations of the coefficients, namely
one of the coefficients belonging to that part of U, V, W which contains z 1 ,
and two coefficients from the other part : the dimensions of the resultant in
respect of the coefficients of the former will hence be readily seen to be equal
to the number of connectives 4- the number of terms in the augmentatives
into which z l enters, that is, will equal m + n+p 2t; the total dimensions
of the resultant in respect to all the coefficients of U, V, W will be
3m 4 (2n. 4 2p m 3t),
that is, 2m 4- 2n + 2p 3t ;
and consequently, in respect to the coefficients of F m ; G n ; H p , will be of
(2m + 2w 4 2p 3t) (m + n + p 2t),
that is, of m + n+p i dimensions. This result, which is of considerable
importance, may be generalized as follows.
Returning to the general system (A), for which we have proved that the
total dimensions of the resultant are
(r - 1) (m + n + . . . + p) - r (i 4 if + . . . + (i))>
40] the Dialytic Method of Elimination. 261
let the coefficients of the column of partial functions
F m ,
On,
Kft
be called the first set ; the coefficients of the column
the second set, and so forth ; then the dimensions in respect of the 1st,
2nd ... (r l)th sets respectively are s, s i, s t, ... s (i), where
s = m + n + &c. + p (i + i + &c. + (*)).
The important observation remains to be made, that all the above results
remain good although any one or more of the indices of dimension of the
partial functions in the system (A), as m i, m i, n i, &c., should become
negative, provided that the terms in which such negative indices occur be
taken zero, as will be apparent on reviewing the processes already indicated
upon this supposition. If we take
m = n = ...=p, and i = L =&c. = (i) = m e,
the exponent of the total dimensions of the resultant becomes
(i 1 ) rm r (i 2) (m e}
= rm +r (r 2)e,
when e = 0, this becomes mr, which is made up of 2m units of dimension
belonging to the coefficients of the first column, and of m belonging to each of
the (i 2) remaining columns. Consequently, if we have
or any other number of equations similarly formed, the result of the
elimination is always of m dimensions only in respect of , 77, , 0, or of
, 77 , ", 6 , and of 2m in respect of the coefficients in F, 0, H, K.
I now proceed to state and to explain some seeming paradoxes connected
with the degree of the resultant of such systems of defective functions as
have been previously treated of in this memoir, as compared with the degree
262 On Extensions of [40
of the general resultant of a corresponding system of complete functions of the
same number of variables.
In order to fix our ideas, let us take a system of only three equations of
the form
F m (x, y} + F m _, (x, y) * = j
G n (x, y}+G n _ i (x, y)*-o|. (B)
H p (x, y) + H p _, (x, y) * = OJ
The resultant of this system found by the preceding method is in all of
2m + 2n + 2p - 3i dimensions. But in general, the resultant of three
equations of the degrees m. n, p is of mn + mp + np dimensions.
Now in order to reason firmly and validly upon the doctrine of elimination,
nothing is so necessary as to have a clear and precise notion, never to be let
go from the mind s grasp, of the proposition that every system of n homo
geneous functions of n variables has a single and invariable Resultant.
The meaning of this proposition is, that a function of the coefficients of the
given functions can be found, such that, whenever it becomes zero, and
O
never except when it becomes zero, the functions may be simultaneously
made zero for some certain system of ratios between the variables. The
function so found, which is sufficient and necessary to condition the possibility
of the coexistence of the equality to zero of each of the given functions, is
their resultant, and by analogy they may be termed its components. It
follows that if jR be a resultant of a given system of functions, any numerical
multiple of any power of R or of any root of R when (upon certain relations
being supposed to be instituted between the coefficients of its components)
R breaks up into equal factors, will also be a resultant. This is just what
happens in system (B) when m = n = p = i; the resultant found by the
method in the text is of the degree 3m ; the general resultant of the system
of three equations to which it belongs is of the degree 3m 2 ; the fact being,
that the latter resultant becomes a perfect mth power for the particular
values of the coefficients which cause its components to take the form of the
functions in system (B).
Suppose, however, that we have still m = n=p, but t less than m,
Qm - 3t will express the degree of the resultant of system (B) ; but this is
no longer in general an aliquot part of 3m 2 , and consequently the resultant
of system (B) that we have found is no longer capable in general of being
a root of the general resultant. The truth is, that on this supposition the
general resultant is zero ; as it evidently should be, because the values
- = 0, - = satisfy the equations in system (B), except for the case of m = i ;
z %
consequently the resultant furnished in the text, although found by the same
process, is something of a different nature from an ordinary resultant ; it
40] the Dialytic Method of Elimination. 263
expresses, not that the system of equations (B) may be capable of coexisting,
M /jj
but that they may be capable of coexisting for values of - , - other than
z z
and 0. This is what I have elsewhere termed a sub-resultant. But there
is yet a further case, to which neither of the above considerations will apply.
This is when m, n, p are not equal, but p i = 0.
On this supposition the degree of the resultant of (B) becomes 2m + 2w p,
which in general will not be a factor of mn + mp + np ; and in this case it
/v> 77
will no longer be true that the values - = 0, - = will satisfy the system (B),
Z 2
inasmuch as the last equation therein cannot so be satisfied. Now, calling
the general resultant R and the particular resultant R < if R should
break up into factors so as to become equal to (r ) a x (s ) b ... (t ), it might be
the case that R should equal (r }- . (s Y ... (t )*, and there would be nothing in
this fact which would be inconsistent with the theory of the resultant as
above set forth ; but suppose that R is indecomposable into factors, then
it is evident that we must have R = R . R", and consequently that the
existence of such a particular resultant as R will argue the necessity of
the existence of another resultant R" ; in other words, the resultant so
found cannot be in a strict sense the true and complete resultant for the
particular case assumed, and yet the process employed appears to give the
complete resultant, or at least it is difficult to see how the wanting factor
escapes detection. To make this matter more clear, take a particular and
a very simple case, where m = 2, n= 2, p = i = 1, so as to form the system of
equations
Aoi? + Bay + Gy- + (Dx + Ey) z=Q\
A a?+ B xy + C y* + (D x + E y) z = i . (C)
Ix + my +nz = 0;
By virtue of my theorem, the degree of the resultant R is
2(2 + 2 + l)-8.1~7,
but the resultant R of the system
Ax- + Bxy + Cy n - +(Dx+Ey)z +Fz 2 = Ch
A *+ B xy+ C"y a + (D oc+ E y) z+F z- = i , (D)
Ix + my +nz = 0;
which becomes identical with the former when F=Q, F = is of
2x2 + 2 x 1 + 2 x 1,
that is, of 8 dimensions. Hence it is evident that when F = 0, F = 0, R must
become R x R".
264 Extensions of the Dialytic Method of Elimination. [40
It will be found in fact*, that on the supposition of F=0, F =Q, R becomes
equal to N x R ; and accordingly, besides the portion R of the resultant
of system (C), found by the method in the text, there is another portion
N which has dropped through ; but it may be asked, is N truly a relevant
factor? were it not so, the theory of the resultant would be completely
invalidated ; but in truth it is ; for N = will make the equations in
system (C), considered as a particular case of system (D), capable of co
existing; the peculiarity, which at first sight prevents this from being
IT 77
obvious, consisting in the fact that the values of - , - which satisfy the
Z Z
three equations when N = become infinite.
Thus, finally, we have arrived at a clear and complete view of the relation
of the particular to the general resultant.
The general resultant may be zero, in which case the particular resultant
is something altogether different from an ordinary resultant ; or the particular
resultant may be a root of the general resultant, or it may be more generally
the product of powers of the simple factors, which enter into the composition
of the general resultant ; or lastly, it may be an incomplete resultant, the
factors wanting to make it complete being such as when equated to zero, will
enable the components of the resultant to coexist, but not for other than
infinite values of certain of the ratios existing between the variables.
Without for the present further enlarging on the hitherto unexplored and
highly interesting theory of Particular Resultants, I will content myself
with stating one beautiful and general theorem relating to them ; to wit,
"if F=0, G=0, &c. be a given system of equations with the coefficients left
general, and R be the resultant of F, G, &c., and if now the coefficients in
F, G be so taken that R comes to contain as a factor or be coincident with
R m , then will R = indicate that (when the coefficients are so taken as
above supposed) F=0, 6r= 0, &c. will be capable of being satisfied, not, as in
general, by one only, but by m distinct systems of values of the variables
in F, G, &c., subject of course to the possibility, in special cases, of certain of
the systems becoming multiple coincident systems."
I pass on now* to the more recondite and interesting theory of the
resultant of Ternary Aggregative Functions, that is to say, functions of
the form
F m (x, y, z)+F m _ l (x, y, z)t +&c. ... + F m _ M (x, y, z}t ( ^,
which will be seen to admit of some remarkable applications to the theory
of reciprocal polars.
[* See the Author s remarks below, p. 283.]
ON A REMARKABLE DISCOVERY IN THE THEORY OF
CANONICAL FORMS AND OF HYPERDETERMINANTS.
[Philosophical Magazine, II. (1851), pp. 391 410.]
IN a recently printed continuation* of a paper which appeared in the
Cambridge and Dublin Mathematical Journal, I published a complete
solution of the following problem. A homogeneous function of x, y of
the degree 2n + 1 being given, required to represent it as the sum of n + 1
powers of linear functions of x, y. I shall prepare the way for the more
remarkable investigations which form the proper object of this paper, by
giving a new and more simple solution of this linear transformation.
Let the given function be
a ac 2n+1 + (2n + 1) a,x m y + (2?i + 1) (2n) a^^y- +... + a m+l f n+ \
and suppose that this is identical with
m+l + *x + , M + &c. + nl x + ] an+1 .
The problem is evidently possible and definite, there being 2n + 2
equations to be satisfied, and (2n + 2) quantities p l} q l} &c. for satisfying
the same.
In order to effect the solution, let
&c. = &c.
[* p. 203 above.]
266 On a remarkable Discovery in the Theory of [41
we have then
271+1
= o ,
= a 2 ,
Eliminate p lt p t ...p n +i between the 1st, 2nd, 3rd ... (n + 2)th equations,
and it is easily seen that we obtain
Again, eliminating in like manner > 1 2n+1 X 1 , p a m+l \i ...p^X+i between
the 2nd, 3rd ... (n+ 3)th equations, we obtain
a n+2 a n+1 zA,! + ... + a^xX^ ... \ n+ i = 0,
and proceeding in the same way until we come to the combination of the
(n + l)th ... (2?i + 2)th equations, and writing
we find
_L (\
Ctfi-\-\ ~~" Cvn "i i * / -n i "2 * * i ^o^w-4-i """ *
/^
.^ /- o _J_ n o -4- /^f Q i ~g? I)
"n+s u n+2*i T W n+i *a X *ilH-l
/\ j
Hence it is obvious that
(x + \y) (x + XgT/) . . . (x + \n+i y)
is a constant multiple of the determinant
,v,7l+l rr n >>l rfU \rffi J- -iyW+1
w , iX/W, tv u.. i ^y
* These equations in their simplified form arise from the ordinary result of elimination, in
this case containing as a factor the product of the differences of the quantities \, X 2 , ... X n +i-
41] Canonical Forms and of Hyper determinants. 267
Hence X 1( X 2 ... X n+1 are known, and consequently
Pi, P*. Pn+i, ft, ?a
are known, by the solution of an equation of the (n + l)th degree.
Thus suppose the given function to be
F = ax 5 + 5bx*y + IQctfy 2 + IQdtfy 3 + oexy* +
= (pix + fry) 5 + (px + q,y) 5 + (p s x + q z y)\
we shall have, by an easy inference from what has preceded,
equal to a numerical multiple of the determinant
x 3 , x 2 y, xy 2
-y 3
d, c, b,
a
e, d, c,
b
/, *> d,
c
The solution of the problem given by me in the paper before alluded to
presents itself under an apparently different and rather less simple form.
Thus, in the case in question, we shall find according to that solution,
( p,x + q,y)( p,x + q,y) ( p s x + q s y)
equal to a numerical multiple of the determinant
ax + by, bx + cy, ex + dy
bx + cy, ex + dy, dx + ey
ex + dy, dx+ ey, ex + fy
The two determinants, however, are in fact identical, as is easily verified,
for the coefficients of x 3 and y 3 are manifestly alike ; and the coefficient of x z y
in the second form will be made up of the three determinants,
a, b, d ,
b, c, e .
c, d, f
a, c, c
b, d, d
c, e, e
, \ b, b, c
c, c, d
d, d, e
of which the latter two vanish, and the first is identical with the coefficient
of a?y in the first solution. The same thing is obviously true in regard of the
coefficients of xf in the two forms, and a like method may be applied to
show that in all cases the determinant above given is identical with the
determinant of my former paper, namely
a 3 y
... a n x + a n+l y
Q n+i E ~r fln+22/
a n x + a n+1 y, a n+1 x
a. M x + a 2n+1 y
268 On a remarkable Discovery in the Theory of [41
Thus, then, we see that for odd-degreed functions, the reduction to their
canonical form of the sum of (n + 1) powers depends upon the solution of one
single equation of the (n + l)th degree, and can never be effected in more
than one way.
This new form of the resolving determinant affords a beautiful criterion
for a function of x, y of the degree 2n + 1 being composed of n instead of,
as in general, (n +1) powers. In order that this may be the case, it is obvious
that two conditions must be satisfied; but I pointed out in my supple
mental paper on canonical forms, that all the coefficients of the resolving
determinant must vanish, which appears to give far too many conditions.
Thus, suppose we have
ax 7 +
+ hy 7 .
The conditions of catalecticism, that is, of its being expressible under the
form of the sum of three (instead of, as in general, four) seventh powers,
requires that all the coefficients of the different powers of x and y must
vanish in the determinant
y > y x > y x > y^ > * >
a, 6, c, c?, e
b, c, d, e, f
c, d, e, f, g
d, e, f, g, h
in other words, we must have five determinants,
a, b, c, d ,
a, c, d, e
, a, b, c, e
b, c, d, e
b, d, e, f
b, c, d, f
c, d, e, f
o, e, f, g
c, d, e, g
d, e, f, g
d, f, g, h
d, e, f, h
a, b, d, e ,
b, c, d, e
,
b, c, e, f
c, d, e, f
c, d, f, g
d, e, / g
d, e, g, h
^ f, g, h
all separately zero. But by my homaloidal law*, all these five equations
amount only to (5 4)(5 3), that is, to 2. I may notice here, that a theorem
substantially identical with this law, and another absolutely identical with
the theorem of compound determinants given by me in this Magazine, and
afterwards generalized in a paper also published f in this Magazine, entitled
[* p. 150 above.]
[t p. 241 above.]
41] Canonical Forms and of Hyper determinants. 269
" On the Relations between the Minor Determinants of Linearly Equivalent
Quadratic Forms," have been subsequently published as original in a recent
number of M. Liouville s journal.
The general condition of mere singularity, as distinguished from cata-
lecticism, that is, of the function of the degree 2w + 1, being incapable of
being expressed as the sum of n + 1 powers, is that the resolving resultant
shall have two equal roots; in other words, that its determinant shall be
zero.
Mr Cayley has pointed out to me a very elegant mode of identifying the
two forms of the resolving resultant, which I have much pleasure in sub
joining. Take as the example a function of the fifth degree, we have by
the multiplication of determinants,
y 3 ,
-y*x
, yx*
, -y?
1,
0,
o,
a,
b,
c,
d
X,
9>
0,
x
b,
c,
d,
e
0,
X,
y.
c,
d,
e,
f
0,
o,
X,
y
if
a,
b,
0, ax + by, bx + cy, ex + dy
0, bx + cy, cx + dy, dx + ey
0, ex + dy, dx+ ey, ex +fy
which dividing out each side of the equation by y 3 , immediately gives the
identity required, and the method is obviously general.
Turn we now to consider the mode of reducing a biquadratic function of
two letters to its canonical form, videlicet
( hx +
Let the given function be written
ax* + 4,bx 3 y
Let # =/*!> k = h\z, m/ 2 A 2
then we have
/ 4 + h* + Qfi = a,
(fa
= 46,
+ 2s 2 ) = 6c,
2 = e.
270 On a remarkable Discovery in the Theory of [41
Eliminating / and h between the first, second and third; the second, third
and fourth; and the third, fourth and fifth equations successively, we obtain
as 2 bsi+c p (8s 2 2sj 2 ) = 0,
6s 2 ci + d p^SiSs 6 1 ! 3 ) =0,
cs 2 -ds^e-fj, (8s 2 2 - 2s 1 2 s. 2 ) = 0.
(2s, 2 - 8s 2 ) fi = v,
as 2 bs l + (c + v) = 0,
bs 2 - ( c ~ j Sj + d = 0,
(c -(- y) s 2 d.?! + e = 0.
Hence ^ will be found from the cubic equation
Let now
and we shall have
a, o, c + j
26, 2c - v, 2d
V,
d,
= 0,
that is,
v*-v (ae - 4bd + 3c 2 )
a, 6, c
6, c, c
c, cZ, e
= 0,
in which equation it will not fail to be noticed that the coefficient of v* is
zero, and the remaining coefficients are the two well-known hyperdeter-
minants, or, as I propose henceforth to call them, the two Invariants of
the form
ax 1 + kba?y + 6cx z y 2 + ^dxy 3 + ey* ;
be it also further remarked that
in which equation the coefficient of S/A is the Determinant or Invariant of
x- + s^xy + s 2 y\
When v is thus found, s 1} s 2 , and /A, being given by the equations in terms of v,
are known, and by the solution of a quadratic X^ X 2 become known in terms
of Sj, s. 2) and /, h in terms of X 1} X 2 , JJL, and the problem is completely deter
mined. The most symmetrical mode of stating this method of solution is
to suppose the given function thrown under the form
(fa + gy}* + (/i* + <M) 4 + 6e (fa + ffyy (/> + )*
Then writing
(fa + gy} (/i* + ffiy) = La? + Mx y + N y n ~>
41] Canonical Forms and of Hyper determinants. 271
v, the quantity to be found by the solution of the cubic last given, becomes
86 (LN-~ r .
V 4 J
I shall now proceed to apply the same method to the reduction of the
function
-f 2
under the form of
+ 70e ( Pl x + q,yY (p z x + q 2 y) 2 (p s x + q 3 y) 2 (p,x + q,y}\
It will be convenient to begin, as in the last case, by taking
qi=pi\, q*=p*\t, qs=p3^ 3 , q t = p^,
epfpfpfpf = m,
and
(x + \,y) (as + \ 2 y) (as + \ 3 y) (x + \ t y) = x 4 + s^y + s^y" + s 3 xy 3 + s 4 y* = U,
we shall then have nine equations for determining the nine unknown
quantities of the general form
XV +KV +P** V + K V + M * m = a "
where t has all values from to 8 inclusive, and where
1.2...,.!. 2... (8-Q
1.2. ..8
multiplied into the coefficient of y -x*~ t in U-.
Taking these nine equations in consecutive fives, beginning with the first,
second, third, fourth, fifth, and ending with the fifth, sixth, seventh, eighth.,
ninth, we obtain the five equations following:
a s 4 tt^s + a 2 s 2 a 3 S! + a 4 s ?/zJV\ = 0,
a^St a 2 s 3 + a 3 s 2 G^SJ + a 5 s mN z 0,
a 2 s 4 a 3 s 3 + a t s 2 a^ -F a e s - mN 3 = 0,
a 3 s 4 - a 4 s 3 + a 5 s 2 a^ + a 7 5 mN^ = 0,
a 4 s 4 a s s 3 + a 6 s 2 a 7 Sj + a s s mN 5 = 0,
where
N l = M s 4 JfjSs -f M z s 2 - M 3 s-i + M t ,
N 2 = M& - M 2 s 3 + M 3 s, M& + M 6 ,
N 3 = i 2 s 4 - M. A s A + 3/ 4 5 2 - M& + M G ,
N 4 = M 3 Si - M,s 3 + M 5 s 2 - MSS, + M 7 ,
N & = l/ 4 s 4 - M 5 s 3 + M s s 2 - M 7 s, + M 9 .
272 On a remarkable Discovery in the Theory of [41
Developing now U 2 , we have
35
5
5* + |k
55
55
, M 6 = 5
~
Hence
-i3 ==
54 ~ SjSj S
*3
Hence we have
,
4
where it will be observed that 7 is the quadratic invariant of U".
Making now
we shall have the five following equations :
+(a 4 -*>) = 0,
*+4j* + . =0.
2 S 4 tt 3 5 3 + I (Z> 4 ft/ 2 ^5*1 + &6 ~"i
3*4~ (-4 + 7 1*3 + ^5*2- 6*1 + 7 = >
so that the problem reduces itself to finding v, which is found from the
equation of the fifth degree:
^o, Oi> &2> a s> a t v
v
Oi,
o,
a 4 -y,
a s , tt 4 + 7, a :
4~ 7T, a o, a e
a
a 8
= 0,
41] Canonical Forms and of Hyper determinants. 273
v, it will be observed, being 72 times the quadratic invariant of
the function being supposed to be thrown under the form of
2 ( p 1 as + q,y) 8 + t IOe(p 1 as + q.y}* (p 2 x + q z y^(p z x + q 3 y)* ( p,x + q 4 y)*.
It is obvious that in 1 the equation for finding v, all the coefficients beino-
functions of the invariable quantities p 1} q lt &c., and e, must be themselves
invariants of the given function ; so that the determinant last given will
present under one point of view four out of the six invariants belonging
to a function of the eighth degree, and these four will be of the degrees
2, 3, 4, 5 respectively*.
I shall now proceed to generalize this remarkable law, and to demonstrate
the existence and mode of finding 2?i consecutively-degreed independent
invariants of any homogeneous function of the degree 4ft, and of n + 1 con-
secutively-even-degreed independent invariants of any homogeneous function
of the degree 4w + 2 ; a result, whether we look to the fact of such invariants
existing, or to the simplicity of the formula for obtaining them, equally
unexpected and important, and tending to clear up some of the most obscure,
and at the same time interesting points in this great theory of algebraical
transformations.
In the first place, let me recall to my readers in the simplest form what is
meant by an invariant f of a homogeneous function, say of two variables
x and y. If the coefficients of the function f(x, y) be called a, b, c ... I, and
if when for x we put Ix + my, and for y, nx+py, where Ip- mn=l, the
coefficients of the corresponding terms become a , b ... I ; and if
I(a, b...l) = I(a , V...I ),
then / is defined to be an invariant of y!
Let now f(x, y) be a homogeneous function in as, y of the 2tth degree,
and write
dx + r) ~dy) f^ X + my> nx
where and 77 are independent of as, y, and Ip mn = 1.
Let x = Ix + my,
then - +77 =t + fc + .
6 cfo; dy ? dx dx ^* dx dy ^ dy dx r V dy dy"
The reasoning in this paragraph seems of doubtful conclusiveness. It may be accepted,
however, as a fact of observation confirmed and generalized by the subsequent theorem, that the
coefficients are invariants.
t Olim, Hyperdeterminant, Constant derivative.
s - 18
274 On a remarkable Discovery in the Theory of [41
and if we now write
l + 1*1**?,
ng + pr) =i),
ni f. d d , d , d
we find Z^r+y j- = f J-T + 17 -T-, .
ax dy b dx dy
Again, from the equations between x , y , x, y, we find
px - my ,
= - - *- =
pi mn
x = - - *- = px my
ly nx
y=j - =ly nx ;
pl-mn
therefore rjx - %y = (prj + ng) x - (mrj + Ig) y = 77 V - gy .
Hence P = (f ^ + V jV) V (* , y ) + X (, V - fy ) .
d , d d
-J- = ^TT, - -T-, .
drj d? f dr)
Hence
\ r> f i f d\ L ~ l d n . fdYn
fc P +^ l ~ 1 [jp T^P + fea + n 1 U-, P ,
|/ V^/ ^ V^/
d v D/ / c? V D , , / d V- 1 d D L
J- -P = m f JF/ -P + * J7T, T-, P + &C. + ^>M j->
drjj \dgj \ag) dt] \drj
But P being of L dimensions in and 77 , and also in oc and y, each
of the equations above written will be of t dimensions in x and y, and of no
dimensions in % , rf ; in fact, the successive terms of the right-hand members
of the above i + 1 equations will be multiples of the (i + 1) quantities
(xj, (x }^y , (xj--y ^.
Consequently a linear resultant may be taken of
treating a; 1 , x l ~ l y ... y 1 as independent, and as quantities to be eliminated;
and this, according to a well-known principle of elimination, will prove
41] Canonical Forms and of Hyper determinants. 275
the linear resultant of the foregoing equations to be equal to the lineal-
resultant of
multiplied by the determinant
I 1 , inl^ 1 ,
1-1 TO, l l ~ l p + (i 1) mnl^ 2 ,
Yfti im l ~ 1 f) J>i(i 1) m "p i ~ 2 . v
This last written determinant may be shown from the method of
its formation to be equal to (lp mn} 2 , that is, to unity, because
Ip mn = 1. Again, since
x 1 = l l x l + tl L ~ l mx -~ 1 y + &c. + mJ-y-,
x^ = l^nx 1 + l l - l n + i-l l c - 2 mri x l ~ l + &c. + m L ~ ll ,
+ p l y l ,
d \ / d \
-7^1 P ... ( -j- ] P , obtained by treating x l , x l ~ l y ... y l as the
eliminables, will be equal to the resultant of the same functions when
x L , x ^y ... y 1 are taken as the eliminables* multiplied by a power of the
determinant
which determinant, like the last, is unity. Thus, then, we have succeeded
in showing that the resultant obtained by eliminating # , x l ~ 1 y ... y L
between
?/
is equal to the resultant obtained by eliminating (x ) 1 , x r< -~ l y ...y 1 between
/ d V" 1 d
* For the statement of the general principle of the change of the variables of elimination,
see my paper in the March Number, 1851, of the Camb. and Dub. Math. Jour. [p. 186 above].
182
276 On a remarkable Discovery in the Theory of [41
or, which is evidently the same thing, the resultant obtained by eliminating
x\ x l ~ l y ... y l between
iVp fly- 1 IP (d_\ P .
?/ Ui; * "W
that is to say, this last resultant remains absolutely unaltered in value when
for x, y we write respectively
Ix + my,
nx+py,
provided that Ip mn = 1.
Hence by definition this resultant is an invariant f(x, y), and A being
arbitrary, all the separate coefficients of the powers of A, in this resultant
must also be invariants. I proceed to express this resultant in terms of
A, and the coefficients of (x, y). Let r = 1 . 2 . 3 . . . t and
dy
-fjfc) (;rM = (;r) (^}f+\(-y) i - 2 x 2 = E 3)
nf \dt;/ \drjj \dx/ \dyl j
and
/(*, y) = o 2t + Zia^-^y + | (2i) (2t - 1) a^ 2 - 2 ,?/ 2 + &c. + a 2t2 / 2t .
We find, writing o-A. for X, where a = 2t (2t- 1) ... (i + 1),
1 .
- &! =
l _ l xy -~ 1 + a L y L + A (
1
K n T 1 - -\-
*l IMC T^
a
t (i - 1) a^tfy 1 - 2 + ta.xy 1 - 1 + a t+1 y l + A (-
- 1) a^y- 2 + ia i+l xy~ l + a l+2 y + A (-
- ^ l+1 = a t ^ + &c.
41] Canonical Forms and of Hyper determinants.
accordingly, by eliminating
we obtain as the required resultant*,
a t + X, a t _!, a t _ 2 ,
_X
277
tto t _! ,
a t + X
Inasmuch as all the coefficients of X in this expression are invariants of
f(x, y), and there are no invariants of the first order, it is clear that the
coefficient of X 1 must be always zero, which is easily verified.
Again, if t is odd, the determinant remains unaltered if we write X for
X ; hence when f(x, y) is of the degree 4e + 2, all the coefficients of the odd
powers of X disappear. Thus, then, our theorem at once demonstrates that a
function of a, y of the degree 4e has 2e invariants of all degrees from
2 up to 2e + 1 inclusive, and that a function of x, y of the degree 4e + 2
has e + 1 invariants whose degrees correspond to all the even numbers in the
series from 2 to 2e + 2.
But in order that the proposition, as above stated, may be understood in
its full import and value, it is necessary to show that these invariants are
independent of one another, which is usually a most troublesome and difficult
task in inquiries of this description, but which the peculiar form of our
grand determinant enables us to accomplish with extraordinary facility. In
order to make the spirit of the demonstration more apparent, take the case
of a function of the twelfth degree, whose coefficients, divided by the
12 11
successive binomial numbers 1, 12, ^ , &c. may be called
Zt
a, b, c, d, e, f, g, h, i, j, k, I, m.
* Mr Cayley has made the valuable observation, that X (given by equating to zero the above
determinant) may be denned by means of the equation
s-0l/fc >**. >-*>.>. .
<(> being itself a certain rational integral form of a function of the ith degree, the ratio of whose
coefficients would be given by virtue of the above equations as functions of X and the coefficients
of f(x, y).
278 On a remarkable Discovery in the Theory of [41
Our grand determinant then takes the form
f, e, d, c, b, a
/, e, d, c, b
i, h, g + , f, e, d, c
A
15
I, k > j> i> h, ff --, f
m, I, k, j, i, h, g + \
Here it will be observed that
a and m appear only 1 time.
b and I
c and k
d and j
e and i
f and h
9
Let now the coefficients be called
2 times.
3 ...
4 ...
5 ...
6 ...
7
H 2 and H 3 manifestly are independent.
Again, if possible, let H i =pH 2 z , then a and in would appear twice in
contrary to the rule.
Hence H 4 is independent of H 2) H 3 .
For a similar reason H 5 cannot depend on H 2 , H 3 .
Again, if possible, let
H 2 3 will contain b 6 l 6 , which by the rule cannot appear in J7 a JJ 4 or in H 3 2 .
Hence p = 0.
Also HI will contain bH 2 x the coefficient of A, 3 in
41] Canonical Forms and of Hyper determinants. 279
which is not zero. And H 2 also contains bl; hence H Z H will contain b 3 l 3 .
But H z will evidently not contain b 3 or I 3 , or tfl or bl 2 , nor can H 6 contain b s l 3 ;
hence q = 0. Finally, H/ will contain c 6 and & 6 , but H 6 can only contain as
to these letters the combination c 3 k s ; hence r = 0.
Consequently H 6 does not depend on H 2 , H i} H 3 . As regards H 2 , H 3 ,
H 4 , H 5 , H 6 not vanishing, this may be made at once apparent by making
all the letters but g vanish ; the H s then become identical with the
coefficients of
none of which are zero except that of \ 6 . The same or a similar demonstra
tion may be extended to H 7 and easily generalized ; hence, then, this most
unexpected and surprising law is fully made out*.
To return to the subject of canonical forms, I have not found the method
so signally successful in its application to the 4th and 8th degrees, conduct to
the solution of other degrees, such as the 6th, 12th, or IGth, of all of which
I have made trial ; possibly another canonical form must be substituted to
meet the exigency of these cases f ; and it may be remarked in general, that
if we have a function of the (2w)th degree, the canonical form assumed
may be taken,
where V, in lieu of being the squared product of
* This demonstration, however, does not extend to show that the coefficients of the powers of
A may not possibly be dependents, that is, explicit functions of one another combined with other
invariants not included among their number, or of these latter alone. For example, in the case
of the 12th degree, we know by Mr Cayley s law that there must be two invariants of the
4th order. Our determinant gives only one of these. Call the other one JT 4 ; by the above
reasoning it is not disproved but that we may have
I believe, however, that the H s may be demonstrated without much difficulty to be primitive
or fundamental invariants. The law of Mr Cayley here adverted to admits of being stated in the
following terms : The number of independent invariants of the 4th order belonging to a
function of x, y of the ?ith degree is equal to the number of solutions in integers (not less than
zero) of the equation 2x + 3y = n- 3. Vide his memorable paper (in which several numerical
errors occur against which the reader should be cautioned) "On Linear Transformations," vol. i.
Cambridge and Dublin Mathematical Journal, new series. There is no great difficulty in showing,
by aid of the doctrine of symmetrical functions, that there can never be more than one quadratic or
one cubic invariant, and in what cases there is one or the other, or each, to any given function
of two variables. The general law, however, for the number of invariants of any order other
than 2, 3, 4 remains to be made out, and is a great desideratum in the theory of linear trans
formations.
t See the Postscript [p. 283] for a verification of this conjecture.
280 On a remarkable Discovery in the Theory of [41
may be any hyperdeterminant, or (as I shall in future call such functions)
covariant of this product, understanding P (x, y) to be a covariant of
f(x, y) when P (Ix + my, nx + py) stands in precisely the same relation to
f(lx + my, nx + py) as P(x, y) to f(x, y}, provided only that lp mn=\.
For the relation and distinction between covariants and contravariants, see
a short article of mine* in the Cambridge and Dublin Mathematical Journal
for this month. In endeavouring to apply the method of the text to the
Sextic Function
ax 6 + Qbafy + loca-y + SOctafy 3 + loexY + Gfxy 5 + gy 6 ,
thrown under the form
where
U = (p 1 x + 0i y) (p 2 x + q 2 y)(p 3 x + q 3 y) = s.x 3 + s^y + s 2 xy n - + s 3 y 3 ,
I obtain the following equations :
as 3 bs 2 + c^ ds = e (162s 2 s 3 oAs^s^ + 12s/),
bs 3 cs 2 + ds! es = e (54s s 1 s 3 + Gsfa 36s s 2 2 ),
cs s ds 2 + esj, fs = e( 54s s 2 s 3 63^ + 3Qs 3 s^),
ds 3 - es z +/Si - #$ = e (- 162s s 3 2 + Ste&Si + 12s 2 3 ).
In these equations, if we call the quantities multiplied by e respectively
L, M, N, P, we shall find
s P = 0,
and 5 3 I-s 2 Jl/-
where / denotes the determinant, or, as I shall in future call such function
(in order to avoid the obscurity and confusion arising from employing the
same word in two different senses), the Discriminant f, which is the biquadratic
(and of course sole) invariant of the cubic function
s o? + s^y + s. 2 xy 2 + s 3 f.
The reduction of the function of the fourth degree to its canonical form
may be effected very easily by means of the properties of the invariants of
[* p. 200 above.]
t "Discriminant," because it affords the discrimen or test for ascertaining whether or not
equal factors enter into a function of two variables, or more generally of the existence or other
wise of multiple points in the locus represented or characterized by any algebraical function, the
most obvious and first observed species of singularity in such function or locus. Progress
in these researches is impossible without the aid of clear expression ; and the first condition of a
good nomenclature is that different things shall be called by different names. The innovations
in mathematical language here and elsewhere (not without high sanction) introduced by the
author, have been never adopted except under actual experience of the embarrassment arising
from the want of them, and will require no vindication to those who have reached that point
where the necessity of some such additions becomes felt.
41] Canonical Forms and of Hyper determinants. 281
the canonical form, as I have shown in the Cambridge and Dublin Mathe
matical Journal. Accordingly I have endeavoured to ascertain whether
the reduction of the sixth degree might not be effected by a similar
method.
If we start with the form ax s +by 6 +cz 6 +9Qmx*y 2 z v - > where oc + y+z = 0,
which is only another mode of representing the canonical form previously
given, we shall find that there are four independent invariants, of the second,
fourth, sixth and tenth degrees. Calling these H 2 , H t , H 6 , H w , and writing
*i, s 2 , s 3 for a+b+c, ab + ac + bc, abc it will be found, after performing
some extremely elaborate computations, that
H 2 = s 2 - 270m 2 ,
HI = Gnis 3 + 45m 2 s 2 + 21 6m 8 *! + 891m 4 ,
H 6 = 4s 3 2 + 120s,s 3 m - {684s,/ + 432s 1 s 3 | m 2
+ (13 . 27 . 64s 3 - 64 . Sls^) m 3 + 8 . 81 . 169s 2 m 4
+ 7 . 128 . 729s 1 m 5 + 16 . 729 . 239m 6 .
H w is too enormously long to attempt to compute ; but we can easily
prove its independent existence by making m = 0, in which case the (deter
minant, or, to use the new term proposed, the) discriminant of aa? + by 6 + cz 6
becomes the product of the twenty-five forms of the expression
Now in general the value of such a product for a*+/3*. 1* + 7*. 1* is obviously
of the form
(a+ /3 + 7) 5 + a/?7 (/(a + /3 + 7)* + g (a/3 + a 7 + 7)} ;
for when a = or yS = or 7 = 0, the product must become respectively
(fi + ryY, (7 + a) 5 and (a + /3)" . Moreover, without caring to calculate/, <jrf, it is
enough for our present purpose to satisfy ourselves that g cannot be zero, as
then the product would have a factor (a + /3 + 7) 2 . Hence, then, on putting
* Such a product in the language of the most modern continental analysis is, I believe,
termed a Norm. If we suppose the general function of x, y of the 4th degree thrown under the
form Au* + Bv*+Cw*, where u + v + w = 0, and the general function of a;, y, z of the 3rd degree
thrown under the form Au i + Bv 3 + Civ 3 + Dd 3 , where u + v + w + = Q, the theory of norms will
afford an instantaneous and, so to speak, intuitive demonstration of the respective related
theorems, and the discriminant (aliter determinant) of each such function is decomposable into
the sum of a square and a cube. Each of these forms is indeterminate, in either case there
being but two relations fixed between the coefficients A, B, C ; A, B, C, D ; and we may easily
establish the following singular species of algebraical porism. In the first case
(ABC) 2 : (AB + AC + BC) 3 ,
and in the second case
(ABCD) 3 :
are invariable ratios.
t/=-625, ,7 = 3125.
282 On a remarkable Discovery in the Theory of [41
a = 6c, /3 = ac, y = ab, we see that the discriminant, when m is 0, will be of
the form
But when m is 0, H 4 vanishes, and there is no term s l or s 3 in H 2 . Hence
evidently the discriminant H w just found cannot be dependent on H 2 , H it
or H 6 ; nor is it possible to make
that is, (p + 1) sf +fsjs,? + gs 3 3 s l
a perfect square on account of g not vanishing ; so there is no H 5 upon which
H lo can depend. Hence, admitting, as there seems every reason to do, that
the number of invariants of a function of x, y of the degree m is in 2,
we find that the four invariants in the case of the first degree are respectively
of the second, fourth, sixth, and tenth dimensions, a determination in
itself, as a step to the completion of the theory of invariants, of no minor
importance.
But it seems hopeless by means of these forms to arrive at the desired
canonical reduction. The forms, however, of H 2 , H i} H 6 are very remarkable
as not rising above the first, first and second degrees respectively in s lt s z , s a .
Also H 4 vanishes when m = and H 4 has been obtained by putting
ax 6 + by 6 + C2 6 + 90w# 2 ?/ 2 .s 2
under the form of
Ax 6 + SBafy + loCxY + 20Ztofy 3 + l5Ex-y* + QFxif + Gy 6 ,
and taking the determinant
A B C D !.
B C D E
C D E F
D E F G
Consequently in general the vanishing of the above-written determinant will
express the condition that a function of the sixth degree maybe decomposable
into three sixth powers. This also is true more generally. If F(x, y) be
a function of 2i dimensions, the vanishing of the resultant in respect to
x 1 , x i ~ l y...y i (taken dialytically) of
/ d \* / d V 1 d f d \^
\dx) \dx) dy " \dy)
will indicate that F admits of being decomposed into i powers of linear
functions of x, y*.
In consequence of the greater interest, at least to the author, of the
preceding investigations, I have delayed the insertion of the promised
continuation of my paper on extensions of the dialytic method, which will
* Such a function so decomposable may be termed meio-catalectic. Meio-catalecticism for
even-degreed functions is the analogue of singularity for odd-degreed functions.
41] Canonical Forms and of Hyper determinants. 283
appear in a subsequent Number. I take this opportunity of correcting a
trifling slip of the pen which occurs towards the end* of the paper alluded to.
CC 7/
The values of - and - become zero, and not infinite, when JV=0; and the
z z
antepenultimate paragraph should end with the words " an incomplete
resultant." The theorem also, in the last paragraph but one, should be
stated more distinctly as subject to an important exception as follows.
Whenever the resultant of a system of equations F = 0, G = 0, &c.
contains a factor R m , this will indicate that, on making R = 0, the given
system of equations will admit of being satisfied by m algebraically distinct
systems of values of the variables, except in those cases where there is a
singularity in the forms of F, G, &c., taken either separately, or in partial
combination with one another. An example will serve to make the meaning
of the exception apparent. Let F, G, H denote three quadratic equations
in x and y, so that F = 0, G = 0, H = may be conceived as representing
three conic sections. Let R be the resultant of F, G, H, and suppose the
relations of the coefficients in F, G, H to be such that R = R 2 ; then R =
will imply the existence of one or the other of the three following conditions :
namely, either that the three conies have a chord in common, which is the
most general inference; or, which is less general, that two of the conies
touch one another; or, which is the most special case of all, that one of the
conies is a pair of right lines.
So, again, if we have two equations in x, and their resultant contains F 2 ,
this may arise either from one of the functions containing a square factor,
or from their being susceptible, on instituting one further condition, namely
of F= 0, of having a quadratic factor in common between them.
P.S. The conjecture made in the preceding pages has been since con
firmed by the discovery of a modification in the canonical form applicable
to functions of the sixth degree, which simplifies the theory in a remarkable
manner. Assume f(x, ?/), a function of the sixth degree, as equal to
au 6 + bv 6 + civ 6 muvw (u v)(v- w) (w u\
where u, v, w, linear functions of x and y, satisfy the equation
u + v + w = ;
then will the product of uvw be capable of being determined by means of the
solution of a quadratic equation, of the square root of whose roots the
coefficients of uvw will be known linear functions. Thus by an affected
quadratic, a pure quadratic, and a cubic equation, the values of u, v, w
may be completely ascertained. The discussion of this theory, and of a
general inverse method for assigning the true (in the sense of the most
manageable) Canonical Form for functions of any even degree, will form
the subject of a subsequent communication.
[* p. 264 above.]
42.
ON THE PRINCIPLES OF THE CALCULUS OF FORMS.
[Cambridge and Dublin Mathematical Journal, vii. (1852), pp. 52 97.]
PART I. GENERATION OF FORMS*.
SECTION I. On Simple Concomitance.
THE primary object of the Calculus of Forms is the determination of
the properties of Rational Integral Homogeneous Functions or systems of
functions : this is effected by means of transformation ; but to effect such
transformation experience has shown that forms or form-systems must be
contemplated not merely as they are in themselves, but with reference to
the ensemble of forms capable of being derived from them, and which
constitute as it were an unseen atmosphere around them. The first part of
this essay will therefore be devoted to the theory of the external relations
of forms or form-systems ; the second part to the analysis of forms : that is to
say, the first part will treat of the Generation and affinities, and the second
part of the Reduction and equivalences of forms.
In its most crude and absolute, or, so to speak, archetypal condition a
Rational Integral Homogeneous Function may be regarded as a linear
function of several distinct and perfectly independent classes of variables.
* It may be well at the outset to give notice to my readers of the exact meaning to be
attached to the following terms :
1. The linear-transformations are supposed to be always taken such that the modulus,
that is, the determinant of the coefficients of transformation, is unity ; or, as it may be phrased,
the transformations are uni-modular.
2. The word Determinant is restricted in all cases to signify the alternate function formed in
the usual manner from a group of quantities arranged in square order.
3. The word Discriminant (typified by the prefix-symbol D ) is used to denote the deter
minant (usually but most perplexingly so called) of a homogeneous function of variables.
4. The resultant of two or more homogeneous functions of as many variables is the left-
hand side of the final equation (in its complete form and free from extraneous factors) which results
from eliminating the variables between the equations obtained by making each of the functions
42] On the Principles of the Calculus of Forms. 285
The first step towards the limitation of this very general but necessary
conception consists in imagining the total number of classes to become
segregated into groups, and certain correspondences to obtain between
the variables of a class in any group with some the variables in each other
class of the same group. The investigations in this and the subsequent
section will be confined exclusively to the theory of functions where the
several classes of variables, if more than one, all belong to a single group, so
that the variables in one class have each their respective correspondents
in the remaining classes. Such a group may again be conceived to become
subdivided into sets each of the same number of variables, and the corre
sponding variables in the different sets to become absolutely identical. This
leads to the conception of a homogeneous function of related classes of
variables of various degrees of exponency in respect to the several classes.
The relation of the different classes, if containing the same number of
variables (in which case the relation may be termed Simple) will be under
stood to be defined by their being simultaneously subject to similar or
contrary operations of linear substitution; so that, for example, if x, y, z:
, 77, are two such classes, when x, y, z are replaced by ax+by + cz,
a x + b y + c z, a"x + b"y + c"z, respectively, , 77, will be, according to the
species of the relation, subject to be at the same time replaced either by
ttf + brj + c, a g + b r) + c , a" + b"r) + c", or otherwise by a + fir) + y%,
+ $ i) + 7 , a"| + /3"77 + 7 " where
a =
b" c"
&c.
=
a" c
&c.
7 =
&
a" b"
&c*
On the former supposition the related classes x, y, z, , 77, will be said to
be cogredient, and on the latter supposition contragredient f. If now we
have one or more functions of classes of variables so related J, such function
or system of functions may have associated with it a concomitant, also made
up of distinct but related classes of variables, such classes being capable
of being either greater or fewer in number than the classes of the given
function or system of functions.
In the primitive function or system, as also in the concomitant, the
related classes may be all of the same species, or some of one and the others
of the contrary species. Even if we limit ourselves to the conception of a
* See rny paper in the previous number of this Journal [p. 199 above.]
t The germ of the notion of contragredience will be found in the immortal Arithmetic of the
great and venerable Gauss.
The relation here spoken of will be observed to be of a dynamical character, not referring
to the systems as they are in themselves, but to the movements to which they are simultaneously
subject.
286 On the Principles of the Calculus of Forms. [42
primitive function or system of functions with only one class of variables, its
concomitant may be composed of various classes of variables, in respect to
some of which it will be covariant with, and in respect to the others contra-
variant to, the primitive function or system*. This is an immense and most
important extension of the conception of a concomitant given in my preceding
paper in this Journal, and will be shown to have the effect of reducing the
whole existing theory under subjection to certain simple abstract and
universal laws of operation.
The relation of concomitance is purely of form. A being a given form,
B is its concomitant, when A being derived from A by simultaneous substi
tutions impressed upon the class of variables or upon each of the classes
(if there be more than one) in A, and B from B by corresponding (coincident
or contrary) substitutions impressed upon the class or classes of variables in
B, B is capable of being derived from A after the same law as B from A ;
or, as it may be otherwise expressed, " functions are concomitant when their
correlated linear derivatives are homogeneous in point of form }*."
This definition implies that one at least of the forms must be the most
general possible of its kind : in a secondary but very important sense, however,
functions obtained by impressing particular values or relations upon the
quantities entering into the primitive and its associate form, will still be
called concomitant. Thus x 3 y 3 will be termed a concomitant to a? + y 3 ,
not that we can affirm that (ax + by) 3 (ex + dy) 3 :
that is (a 3 - c 3 ) x 3 + 3 (a-b - c 2 d) x 2 y + 3 (ab 2 - cd 2 ) xy 2 + (b 3 - d 3 ) y 3 ,
treated as a function of x and y, can be derived from (ax + by) 3 + (ex + dy) 3 ,
that is (a 3 + c 3 ) a? + 3 (a-b + c 2 d) x 2 y + 3 (ab 2 + cd 2 ) xy 2 + (b 3 + d 3 ) y 3 ,
when ad bc= 1 by the same laAv as (x? y 3 ) from (x 3 + y 3 ), for the elements
for forming such comparison are wanting, but because x 3 + y 3 and a? y 3 are
the correspondent particular values respectively assumed by
ax 3 + 3bx 2 y + Sexy- + dy 3 ,
arid its concomitant
(ad- + 2c 3 - 3bcd) x 3 - (Qb 2 d - 3c 2 6 - 3acd) x 2 y
+ (Qac- - 3c6 2 - 3c6a) xy 2 - (a-d + 26 3 - Sbca) y 3 ,
when a = 1, 6 = 0, c = 0, d=l.
With the aid of this extended signification of the term concomitant (whether
it be a covariant or contravariant) we can in all cases speak (as otherwise we
in general could not) of the concomitant of a concomitant. The relation
* And of course the concomitant may be an invariant to its originant in respect of one or
more systems of variables entering into the former.
t Or, more generally, it may be said that concomitance consists in the persistence of morpho
logical affinity.
42] On the Principles of the Calculus of Forms. 287
between systems of variables has been stated to be Simple (whether they be
cogredient or contragredient) when each variable in one system corresponds
with some one in each other. Compound relation arises as follows : Suppose
x > V 5 > V two independent systems of two variables each, and that the
system of four variables u, v, w, t is subject to linear variations imitating,
in the way of cogredience or contragredience, those to which x%, xrj, y, yrj
are subject ; then u, v, iv, t may be said to be cogredient or contragredient
to the continued systems x,y; , vj. If as, y ; ,17 be themselves cogredient,
then a system of only three variables u, v, w, may be cogredient or contra
gredient in respect to x%, art) + y%, yrj, and ifas.y; f , 77 be coincident, u, v, w
may be similarly related to a; 2 , xy, y-. The illustration may easily be
generalized, and it will be seen in the sequel that its conception of compound-
relation between systems of a differing number of variables will greatly
extend the power and application of the methods about to be developed.
Without having recourse to a formal definition, it is obvious that the notion
of a concomitant conveyed in my former paper in this Journal lends itself
without difficulty to the most general supposition which can be made of
functions between which any number of systems of related variables are
distributed, whatever such relation be, whether simple or compound, and
whether of cogredience or of contragredience. The proposition stated in my
last paper relative to a concomitant of the concomitant of a function being
a concomitant of the original still applies to concomitants in the wider sense
in which we now understand that term, and the species of each system of
variables in the second concomitant with respect to the species or either
species (if there be systems of both kinds in the primitive) will be determined
upon the general principle which determines the effect of concurrence and
contrariety being made to operate each upon itself or one in either order
upon the other.
The highest law and the most powerful in its applications which I have
yet discovered in the theory of concomitants may be expressed by affirming
that when several related classes of variables are present in any concomitant,
a new concomitant, derived from the former by treating one or any number of
these classes as independent of the remaining classes, will still be a concomitant
of the primitive. I shall quote this hereafter as the Law of Succession.
This law, to which I have been led up inductively, requires an extended
examination and a rigorous proof. It is the keystone of the subject, and any
one who should suppose that it is a self-evident proposition (as from the
simplicity of the enunciation it might be supposed to be) will commit no
slight error.
If </>(#, y ... z) be any homogeneous form of function of x, y, ... z, every
homogeneous sum in the expansion by Taylor s theorem of
<j>(u->ru, v + v ... w + w ),
288 On the Principles of the Calculus of Forms. [42
which in fact, on making u x^ v = y ... w = z, becomes identical (to a
numerical factor pres) with ( u -y- + v -= \- w -y- ) d>, is what I have elsewhere
\ dx dy dz) 7
termed an Emanant, and by a partial method I had demonstrated that every
invariant of such an emanant in respect to u, v ...w, in which x, y ... z are
treated as constants, or vice versa, would give a covariant of </>. The reason
of this is now apparent. For it may easily be shown* that every emanant
is in fact itself a covariant of the function to which it belongs with respect
to each of the related classes of variables which enter into it, or is as it may
be termed a double covariant. The law of Succession shows therefore that
a concomitant to an emanant from which one of the classes has disappeared
will be a covariant of the primitive in respect to the remaining class.
In applying the law of Succession, great use can be made of a function
of two classes of letters which may be termed a Universal Mixed Concomitant;
this is x% + yrj + ... + z, which has the property of remaining unaltered when
any linear substitution (for which the modulus is unity) is impressed upon
x, y ... z, and the contrary one upon , 77 ... -f-.
If f(x, y) be any function of x, y, of the degree m, f+ A, (x% + yr)) m will
* To demonstrate this it is only necessary to observe that if u, v, ... w, u , v , ... w be
cogredient with themselves and with x, y, ... z,
<f>(u + \u , v + \v , ... io+\w )
will evidently be a concomitant of <j>(x,y, ... z); and, X being arbitrary, the coefficients of the
different powers of X must be separately concomitants of <f> (x, y, ... z), but these coefficients are
the emanants of <f>. Q. E. D.
t Thus, if
x = ax + by + cz , = (gn-hm)g + (hl-fn)-r)
y =fx + gy + hz , -r)=(-nb + mc) + ( - lc + na)r) + (-ma
z lx + my + Jiz , =(bh-cg) +(cf-ah)r) + (ag-bf) ,
a b c\
/ g h
I m n
When the coefficients of transformation correspond to the direction-cosines between one system
of rectangular axes and another, the reciprocal system is identical with the direct system ; so
that x, y, z ; f , 77, f, on this particular supposition, may be regarded indifferently as contragredient
or as cogredient; accordingly they may be made identical, and then x 2 + y 2 + z 2 remains invariable,
which is the well-known characteristic of orthogonal transformation. It may be observed here
that there exists a special theory of concomitance limited to such species of linear transform
ations, which may be termed Conditional Concomitance, and I have found in several cases that
the invariants of conditional concomitants turn out to be absolute invariants of the primitive.
Much more important is the remark that there exists a theory of universal concomitants for
an indefinite number instead of merely two systems of variables, as used in the text. In
the sequel it will be seen that the application of this universal concomitant (like the touch of
an enchanter s wand) serves to transmute covariants into contravariants, and back again, and
causes single invariants to germinate and fructify into complete connected systems of forms.
42] On the Principles of the Calculus of Forms. 289
be a mixed concomitant of /, it being evident that every function of con
comitants of a function is itself a concomitant of the same.
Suppose now
/= ax m + mbx m ~ l y + \m (m-1) cx m ~ 2 y z + &c.,
the concomitant becomes
(a + \% m ) x m + m (b + \% m - 1 7 7 ) x m ~^y + \m (m - 1) (c + Xf ">-y) + &c.
Consequently if P be any concomitant of/, P obtained from P by writing
a + \% m , 6 + X m - 1 ?7, &c. for a, b, &c., will still be a concomitant of/; and by
Taylor s theorem P evidently equals
+ p t - i ?4+
db
+ m - l 7, + &P Y P
+ + J
+ &c.
If we take P an invariant of/ we have M. Hermite s theorem* for
f( x > V\ ar *d precisely the same demonstration applies to the general case
of f(x, y ... z\ P is, by virtue of the general rule, a contravariant of/ in
respect to , rj ... : if P be taken a function containing one single system,
and is also a contravariant to / in respect to that system, P will be a double
contravariant ; and if we make the two systems in P identical, we have the
extension of M. Hermite s theorem alluded to by me in one of the notes f
to my last paper, wherein I have stated that " / may be taken any covariant
of the function" : as regards the purpose of that statement, the word covariant
was used in error for contravariant.
The preceding method may be viewed as a particular application of the
general principle, that if U lt U 2 ... U m be any m functions (whether con
comitants any of them of the others or not), then any concomitant of
XiC/i + X 2 [/2 + ... + \ m U m being expressed as a function of X 1; X 2 ... \ m ,
every coefficient in such expression will be a concomitant of the system
U l , U^... U m . Thus, for example, if U and V be two quadratic functions
of n variables x, y ... z, the discriminant n (X U + //, F) will contain n + 1 terms,
of which the coefficients of the first and last will be D U and D F; and every
one of the (n + 1) coefficients will be a concomitant (of course an invariant)
of U and F. These (n + 1) invariants will in fact constitute the fundamental
scale of invariants to the system U and F, and every other invariant of U
* This theorem was first stated to me by Mr Cayley, who, I understand, derived it from
M. Eisenstein, under the form of a theorem of covariants, which of course it becomes on inter
changing x, y with - y, x. But as a theorem of covariants it could not be extended to functions
of more than two variables. M. Hermite appears to have discovered this theorem, under its
more eligible form, subsequently to, but independently of, M. Eisenstein.
[t p. 201 above, note *.]
s - 19
290 On the Principles of the Calculus of Forms. [42
and F will be an explicit rational function of the (n + l) terms of the scale.
In connexion with this principle may be stated another relative to any system
of homogeneous functions of a greater number of variables of the same class,
namely, that if any set of the variables one less in number than the number
of the functions be selected at will, and any invariant of a given kind be
taken of the resultant of the functions in respect to the variables selected,
all such invariants so formed will have an integral factor in common, and
this common factor will be an invariant of the given system of functions.
It will be convenient to speak hereafter of systems for which the march
of the linear substitutions is coincident as cogredient, and those for which
the march is contrary as contragredient systems.
Suppose m cogredient classes of ra variables, the determinant formed by
writing the m x m quantities in square order will evidently be a universal
covariant. Thus, take the two systems x, y ; f, 77. x^-yt; is a universal
covariant, and evidently therefore F, which I use to denote
will be a covariant to c/> (x, y). Let c (x, y) be of m dimensions ; any invariant
of F will be an invariant of <j>; thus, let the two systems a;, y; , 77 be treated
as perfectly independent, and take the discriminant of F (viewed as a function
dF dF dF dF
of x, y\ |, 77), that is the resultant of the four functions ^, ^-, -^, ^- ;
this resultant will be an invariant of <J> ; and X being arbitrary, all the
coefficients of its different powers will be invariants of <. We thus fall upon
,
l
<b(x, y) x $ (, 77)
another theorem of M. Hermite, namely that if X-
coefficients of the equation which will give the minimum values of X are
invariants of </>. So more generally, any invariant of /<>, y, %, 77) - X (x% - yn) m ,
f being of the degree m in x, y and in , 77, will be an invariant of /; and
among other invariants may be taken the discriminant obtained by treating
x, %, y, 77 as absolutely unrelated.
If / be a function of various classes each containing n covariables, and if
not less than n of these classes be covariable classes, and after selecting at
will any n of such systems, as x l} y l . . . ^ ; x. 2> y^ . . . z 2 ; ...... x n , y n z n , the
symbolical determinant
d
d
d
dx l
dyi"
d Zl
d
d
d
d
A
d
dx n
dy n "
dz n
42] Oti the Principles of the Calculus of Forms. 291
be expanded and written equal to D, then D L f will be a concomitant of/; and,
more generally, by selecting different combinations of the covariable systems
n and n together in every way possible, and forming corresponding symbols
of operation E, F ... H, we shall have D .E 1 ... H^ ./ for all values of
L, i ... (L\ a covariant of/ in respect to the classes so combined. This explains
and contains the whole pith and marrow of Mr Cayley s simple but admirable
method of obtaining covariants and invariants (or, as termed by their author,
hyperdeterminarits) to a function fa of a single system x lt y^...z^\ he forms
similar functions fa ... fa of a? a , y 2 ... z. 2 \ ... x^, y^ ... z^, and uses the product
fax fax ... x fa as a function/ of //, systems: the multiple covariant obtained
by operating thereupon becomes a simple covariant on identifying the
different classes of covariables introduced in the procedure.
SECTION II. On Complex Concomitance.
We have hitherto been engaged in considering only a particular case
of concomitance, the true idea of which relates not to an individual associated
form (as such), but to a complex of forms capable of degenerating into an
individual form. Such a complex may be called a Plexus. A plexus of forms
is concomitant to a given form or combination of forms under the following
circumstances.
If (0) be the originant, meaning thereby the primitive form or system of
forms, and P the concomitant plexus made up of the p forms P 1} P., ... P^,
and if, when by duly related linear substitutions, becomes , the plexus
P becomes P , made up of the forms P 1( P 2 ...P M , and if the plexus P
formed from after the same law as P from be made up of the forms
-Pi, P^ Pn, then will each form in either of the plexuses P, P be a linear
function of all the forms in the other plexus, and the connecting constants in
every such linear function will be functions of the coefficients of the substitu
tion whereby and P have become transformed into and P .
A function forming part of a concomitant plexus may be termed a
concomitantive. Concomitantives therefore usually have a joint relation
to a common plexus and a concomitant is only another name for an unique
concomitantive. Every plexus contains a definite number of concomitantives;
in place of any one of these may be substituted an arbitrary linear function
of all the rest, but the total number of independent forms sufficient and
necessary to make the complete plexus respond to the requirements of the
definition will remain constant.
If now we combine together the whole number of functions contained
in one or more plexuses concomitant to any given originant, all of the same
degree relative to any given selected system or systems of variables, and if
the number of the concomitantives so combined be exactly equal to the
192
292 On the Principles of the Calculus of Forms. [42
number of terms in each, arranged as a function of the selected class or classes
of variables, then the dialytic resultant (obtained by treating each combination
of the selected variables as an independent variable, and forming a deter
minant in the usual manner), will be a concomitant to the given originant.
This, which is only the partial expansion of some much higher law, may
be termed the " Law of Synthesis."
Let /be any function of a single class of variables x 1} x 2 ...x n . Let ^
represent any product of these variables or of their several powers of any
given degree r ; the number of different values of ^ will be p, where
_ n (n + 1) . . . (n + r 1)
/i= 1.2. ..r "
and x,if> X*f /6*/ w iH f rrn a covariantive plexus to/.
Again, let S- represent any product of the degree r of the symbols
d d d
&J/, Sr 2 / . . . ^/ will also form a covariant plexus to /.
The coefficients of connexion between the forms of either plexus depend
in an analogous manner upon the coefficients of the substitution supposed
to be impressed upon the variables, with the sole difference that every
coefficient taken from the line r and column s of the determinant of sub
stitution which appears in any coefficient of connexion of the one plexus
is replaced by the coefficient taken from the line s and the column r in the
corresponding coefficient of connexion for the other plexus.
Let/(#, y) be any function of x, y of the degree 2m ; then
(d_\ m (d\ m ~ l d_ fdV
\dx) (dx) dy \dy)
- m
will form a covariantive plexus ; thus, suppose
f(x, y} = a^ m + 2maox- m ~ l y+... + a 2m+l y-
omitting numerical factors, the plexus will be composed of the (m + 1) lines
following :
a^x + ma^x m - l y + . . . + a m+l y m ,
a m+l x
and consequently, by the law of synthesis, the determinant
is an invariant of/
42] On the Principles of the Calculus of Forms. 293
When this determinant is zero, I have proved in my paper* on Canonical
Forms, in the Philosophical Magazine for November last, that / is resoluble
into the sum of m powers of linear functions of x and y. I shall here
after refer to a determinant formed in this manner from the coefficients
of / as its catalecticant. Mr Cayley was, I believe, the first to observe that
all catalecticants-f* are invariants.
Again, more generally, let /(a?, y, %, ??) be a function of the mth degree
of x, y, and of a like degree in respect of , 77, which are supposed to be
cogredient with x and y ; then
(say F) will be a concomitant of/; and therefore if we take the system
( Y 1 F (\ n ~ 1 Jf .. . . ( \ m F,
\dx) \dxj dy " \dy)
which will be functions of and 77 alone, and take their resultant, this
resultant will be an invariant of/ As a particular case of this theorem, let
/ =
dx T dy)
where <f> is supposed to be a function of x and y only and of 2m dimensions,
/ is a concomitant of <f>, and therefore the invariant of/ obtained in the
manner just explained, will be an invariant of <j>. Thus then we have an
instantaneous demonstration of the theorem given J by me in the paper of
the Philosophical Magazine before named, namely, if
say, in order to fix the ideas, = ax 6 + Qbofy + IScafy 3 + ... + gy 6 ; then the
determinant
a, 6, c, d + ~)
b, c, d $\, e
c, d + ^\, e, f
d-\, e, / g
(and the analogously formed determinant for the general case) will be an
invariant of <. The general determinant so formed is peculiarly interesting,
because it furnishes when equated to zero the one sole equation necessary
to be solved in order to be able to effect the reduction of < (x, y) to its
canonical form, and gives the means, irrespective of any other view of the
theory of invariants, of determining completely and absolutely the condition
[* Bee p. 282 above.]
+ But the catalecticant of the biquadratic function of x, y was first brought into notice
as an invariant by Mr Boole ; and the discriminant of the quadratic function of x, y is identical
with its catalecticant, as also with its Hessian. Meicatalecticizant would more completely express
the meaning of that which, for the sake of brevity, I denominate the catalecticant.
[ p. 277 above.]
294 On the Principles of the Calculus of Forms. [42
of the possibility of two given functions of the same degree of x, y being
linearly transformable one into the other. This theorem will be obtained
in a more general manner in the following section. I only pause now to
make the very important observation, that not only is the determinant an
invariant, but every minor system* of determinants that can be formed from
it (there are of course m such systems) is an invariantive plexus to the given
function <.
The form under which this theorem presents itself suggests a theorem
vastly more general and of peculiar interest, as showing a connexion between
the theory of functions of a certain degree and of a certain number of
variables with other functions of a lower degree but of a greater number
of variables. Here again, under a different aspect, is reproduced the great
principle of dialysis, which, originally discovered in the theory of elimination,
in one shape or another pervades the whole theory of concomitance and
invariants.
Let <f> represent any function of the degree pq (of any number, or, to
fix the ideas, say of three variables c, y, z) ; let the general term of </> be
represented by
where a 4- /3+y=pq, and (a, /?, 7) represents a portion of the coefficient of
aPifzir.
Let
where r + s + 1 =p, so that there are as many # s as there are modes of
* These minor systems mean as follows : the system of rth minors comprises all the distinct
determinants that can be got by striking out from the square array (which I call the Matrix)
from which the complete determinant is formed, any r lines and any r columns selected at will.
The last, or mth minor, is of course a system consisting of the coefficients of <f> (x, y), and it is
evident that if <f> (x, y ... z) be any function of any number of variables x, y ... z, the coefficients
will form an invariantive plexus to <p.
The following remark as to the changes undergone by the coefficients of <f> when the variables
undergo any substitution, is not without interest and importance for the theory.
Let x become fx+f y+... + (/) z,
z hx + h y+ ... + (h)z.
Then the coefficient of the highest power of x becomes
*(Af...*)i
and the coefficient of the term containing y r ... z s becomes
42] On the Principles of the Calculus of Forms. 295
subdividing p into three integral parts (zeros being admissible); that is
^ ^p 4. i) (p 4. 2) (p + 3). Then any product such as x"-y^zi may be divided
in a variety of ways into the product of q of these # s, and it may be shown
that the entire quantity
pq (pq -!)...!
(1.2...o)(1.2...)(1.2... 7 )
where m l + m 2 + . .. + m r = q. Consequently < may be represented under the
form of a function of the degree q of % (p + 1) (p + 2) (p + 3) (say i) variables
#!, ^ 2 ... t , and its general term will be of the form
12 a
. _ ^ _ ( a Q ry) \&*10*I . Q m r \
(1.2... Wl )(1.2...m 2 )...(1.2...m r ) Ve
where a, /3, 7 are the indices respectively of x, y, z, when the last factor
is expressed as a function of these variables*. Now if ^ be used to denote
this new representation of < when 1 , 6 2 ...0, are treated as absolutely
independent variables, and if we attach to it any universal concomitant, as
(# + yn + Z %Y admitting of being written under the form w(0 l> 0.J...0J,
wherein the coefficients will be functions of f, 77, ; then any invariant to
^ and CD, treated as two systems of i variables, will be a concomitant to $,
the original function in x, y, z\. ^ and <o may be termed respectively, for
facility of reference, the Particular and Absolute functions. Thus, for example,
we take (/> a function of x, y of the degree 4w, say
a&M + 4n a^ n ~ l y + &c. + a m+l ^\
and make p = 2n, q 2, so that S- becomes a quadratic function of (2w + 1)
variables obtained by making # 2n = # 1 , x* l ~ l y = # 2 ... y" n = #on+it, an d the
concomitant o>, formed from (jx + nyY 11 , becomes
then if we take R the quadratic invariant of CD, that is
It = 0A,m - *n0J m &c. ^
* See Note (1) in Appendix, [p. 322 below.]
t Iu fact S- is a concomitant to <j>, and w to a power of the universal concomitant ; the s
forming a system of variables cogredient with the compound system x r \y t \z t \, C*tjr*tX*t, &c. :
and it must be well observed that the same substitutions which render S- and w respectively
identical with <f> and a power of the universal concomitant, would render an infinite number
of other functions also coincident with the same ; but none of these other functions would be
concomitants. Herein we see the importance of the definition and conception of compound
relation ; the 6 system being compound by relation with the x, y, z system, after the manner
of cogredience.
t A slight variation upon the method as above explained for the general case has been here
introduced inadvertently by writing x Zn ~ l y = 9 2 , &c., in lieu of 2;ia: !:n ~ 1 y = 2 > &c., which, as it
does not in any degree affect the reasoning, I have not deemed it worth while to alter.
296 On the Principles of the Calculus of Forms. [42
it will readily be seen that the determinant of S- + \R, treated as a quadratic
function of (2n + 1) variables, will give an invariant of $, and this will be the
same as that obtained by the particular method above given. Thus, suppose
</> (x, y} = ax* + 4>ba?y + Gca^y* + 4<dxy 3 + ey*.
Let a? = e^ 2xy = 2 , y> = 3 ,
** = 0,0!* + 260 1 2 + c(9 2 2 + 2C0J0J + 2d0 a 3
w = (x% + yrff = o?e
0*
R = 6^3 -
Then A the discriminant of S- 4- 2\R in respect to 6 l} 2 , 3
a, b, c + \
b, c ^\, d
c+\, d, e
and I may remark that the relations between the several transformees of the
invariantive plexuses formed by the minor determinant systems of A (in this,
and in general for the case of an evenly-even index) may be found by treating
^ + 2\R as a quadratic function of the variables (in this case lt # 2 , 3 ) and
applying the rule given by me in the Philosophical Magazine in my* paper
" On the relation between the Minor Determinants of linearly-equivalent
Quadratic Forms."*f- This second method, however, is not immediately
applicable to the case of indices oddly even, that is of the form 4n + 2, to
which the first method applies, equally as to the case 4>n ; for if we make
2n + 1 =p and q = 2, o> being of an odd degree, has no quadratic invariant ;
it has however a quadratic covariant, which will be of the second degree
in respect to lt 2 ... P+1 as well as in respect to , 77 ; and if we call this R
and take the discriminant of S- + \R in respect to the variables 1} # 2 ... P+1 ,
we shall obtain, as I am indebted to a remark of my valued friend M. Hermite
for bringing under my notice, a very beautiful and interesting function of X,
of which all the coefficients will be contravariants of <f>. Thus, let
<f> = ax 6 + 6bx 5 y + locx*y 2 + 20da?y* + loex^y* + ftfxy* +gy 6 ,
[* p. 241 above.]
+ Moreover, upon the supposition made in the text, the particular and absolute functions
S- and w may be treated in all respects as if they were functions characterizing quadratic loci,
and any singularity in their relation will correspond to and denote a singularity in the given
function <f> to which S- refers. Thus, for instance, if <p be a function of x, y of the eighth degree,
S- and w will be quadratic functions of five letters each. Quadratic loci have no other singularity
of relation than what corresponds to different species of contact. The number of contacts between
loci, characterized by 5 letters, is 24 (see my paper J in the Philosophical Magazine, "On the con
tacts of lines and surfaces of the second order"). Consequently this mode of representing ^ and u
will give rise to the discovery and specification of 24 different kinds of singularity in <p, and the
analytical characteristics of each of them. But there of course may, and in fact will, exist
other singularities in besides those which have their correspondencies in the relations of these
quadratic concomitants. [ p. 237 above.]
42]
make
so that
On the Principles of the Calculus of Forms.
a?=6 lt 3x 2 y = 2 , 3xy 2 =0 3 , y 3 = 0^,
297
2e0 4 2 ,
3 r)
-R =
Consequently the discriminant in respect to lt 6 Z , 3 , # 4 of ^ 2\R becomes
a, b, c-3X 2 , d-
b, c + 2Xp, d + X 77, e
d-
d + \%r], e + 2X?7 2 ,
e - 3X77 2 , /
f
9
If this determinant be expanded as a function of X, all the coefficients of the
various powers of X will be contravariants to the given function </>. The term
involving X 4 is zero. Let become y and ?? become #, then the remaining
terms (abstraction made of the powers of X) become covariants of <. The
first term (the coefficient of X 3 ) becomes <f> itself; the last term is the
catalecticant, and thus we see, in general, that for functions of # and y
of an oddly-even degree, a whole series of covariants may be interpolated
between the function and its catalecticant, the dimensions in respect of the
coefficients of <f> in arriving at each step increasing by 1 unit and the degree
in respect of the variables diminishing by 2 units. This is consequently
a much simpler and more available scale than one with which I have been
long previously acquainted, and which applies alike to functions of any even
degree.
Thus, let (f> (x, y) be of 2k dimensions ; form all the even emanants of <f>
/ d d\ Zi
which will be all of the form I -^- +17 -y- ) <, and take their respective cata-
\ doc y f
lecticants in respect to and 77. We shall in this way obtain a regular scale
of covariants interpolated between the Hessian of <f> (corresponding to i = 1)
and the catalecticant of < (corresponding to i = k). If cf> be of the degree
2& + 1, we shall have an analogous scale interpolated between the Hessian
of (f> and its canonizant ; the latter term denoting the function which is the
product of the k + 1 linear functions of x and y, the sum of whose (2& + l)th
powers is identically equal to (/>*.
By means of the Theory of the Plexus we may obtain various representa-
See Note (2) in Appendix, [p. 322 below.]
298 On the Principles of the Calculus of Forms. [42
tions of the same invariant ; thus, for example, if we take F a function
of x, y of the fifth degree and form its Hessian H, that is
da? dxdy
dydx dy 2
this will be a function of the sixth degree in x, y, and of the two orders
in the coefficients. If we combine the two plexuses
dF dF d*H d?H_ d*H
doe dy } dx* dxdy dy z
we shall have five equations between which x 4 , x 3 y, x 2 y 2 , xy 3 , y 4 may be
eliminated dialytically ; the resultant will be of the 2 + 3.2, that is the
eighth order in the coefficients, and of the form oF I/, where n F and 7 4
are respectively the determinant and quintic invariant of F, each affected
with a proper numerical multiplier (the "B A 2 " of my supplemental* essay
on canonical forms) which, as Mr Cayley has remarked, may also be repre
sented by the resultant of P; --; - where P and Q are respectively the
ctoc ^Jl
quadratic and cubic invariants in respect to f and
It will be well at this point to recapitulate in brief a method of elimination
applicable to certain systems of functions published by me many years since
in the Philosophical Magazine, and to compare this method with that afforded
by the theory of the plexus for finding an invariant for each of the very same
systems, possessing all the external characters, formed in a precisely similar
manner to, and not impossibly identical with, the resultant of every such
system. I shall devote my first moments of leisure to the ascertainment
of this last most important point, as to the identity or otherwise of the
plexus-invariant with the resultant. Take the case of three functions of
x, y, z (say $, ty, w) each of the same degree n; to fix the ideas, suppose
n = 3 : there are two purely algebraical processes (modifications of the same
method and leading to identical results) by which the resultant of </>, ty, &&gt;
may be found. I shall call these processes the first and second respectively.
First process : Write
<f> = x*P +yQ +zR,
^ = x*P +yQ +zR,
<o = x"-P" + yQ" + zR",
decompositions which may be effected in an infinite variety of manners,
so that P, Q, R shall be integer functions of x, y, z ; take the linear resultant
of (j), ty, <w, in respect to x 2 , y, z, which call H^ 1; : ; this will evidently be
[* p. 205 above.]
42] On the Principles of the Calculus of Forms. 299
of 9 4, that is, of 5 dimensions. Form analogously the functions #1,2,1) #1,1,2;
#2,1,1. #1,2,1) #1,1,2 constitute an auxiliary system of functions which vanish
when <f), ty, &) vanish together; combine this auxiliary system with the
augmentative system
xyw, yzw, zxco,
We shall thus have in all 3 + 3x6, that is, 21 functions into which the
21 terms x 5 , x*y, x*z, &c. enter linearly : the linear resultant of these 21
functions is the resultant of <, -fy, a>, clear of all extraneousness.
Second process : Write
<f>=a*P + yQ + zR,
yQ + zR,
and, as before, take the linear resultant # 3 ,i,i, which will however be of 9 5,
that is, of only 4 dimensions.
Again, take
<f> = x-L + y-M + zN,
^ = x-L + y-M + zN ,
a>=x*L" + y*M" + zN",
and form the determinant # L ., 2 ,i ; we shall thus have the auxiliary system
-"3,1,1) -"1,3,1) -"1,1,3) -"2,2,1) -"2,1,2) -"1,2,2-
Let this be combined with the augmentative system
xco, ya>, zo) ; x<f>, ?/</>, z(j> ; x\jr, y^r, z-^r.
Between these G + 9, that is, 15 functions, the 15 terms x 4 , a?y, a?z, &c. may
be linearly eliminated, and the resultant thus obtained will be precisely
the same as that got by the preceding process.
Here we have 6 auxiliaries and 6 augmentatives ; the auxiliaries are
of three dimensions in respect to the coefficients of </>, -v|r, &&gt; ; the augmenta
tives of one dimension only; in the former process there were 3 auxiliaries
and 18 augmentatives, 6x3 + 9 = 27 =3x3-1-1 8.
Now let this method be compared with the following :
First process: Take the 18 augmentatives x-<f>, x-w, x 2 ^r, &c. as in the first
process of the algebraical method above explained ; but in place of the
3 auxiliaries therein given, take another system of 9 as follows :
300 On the Principles of the Calculus of Forms.
Write the determinant
[42
dR dR
dx dy
dR
dcf> d<f> d<f>
dx dy dz
d-fr d-fr d-^r
dx dy dz
dm dm dco
dx dy dz
= R-
-, form a concomitantive plexus ; the 18 augmentatives form
another ; the linear resultant of these two plexuses will be an invariant
of <f), -vjr, w, and of precisely the same dimensions as the resultant last found ;
if they are not identical it will be indeed a matter of exceeding wonder,
and even more interesting than if they should be proved so to be.
Second process : Combine the augmentative plexus
xw, yw, zw x(p, yfa z(j> ; x^r, y^, z^,
with the differential plexus
<m d d?R d 2 R d?R
dx 2 dxdy dy 2 dydz dz 2 dzdx
we thus obtain a linear resultant in a manner precisely similar to that
afforded by the second process of our algebraical method.
In general, if <, -fy, &&gt; be of the degrees n, n, n, as there are two algebraical
varieties of the linear method for finding the resultant, so are there two
varieties of the concomitantive method for finding the resembling invariant.
In both methods the augmentatives are identical ; the only difference being
in the auxiliary system.
In the first process the augmentative system will be got by operating
upon each of the functions <, -ty, &), with the multipliers x n ~ l , y n ~ l , z n ~ l ,
and the other homogeneous products of x, y, z ; the auxiliary system by
/ d \ n ~ 2 / d \ n ~ 2 i d \ n ~ 2
operating upon R with the symbolical multipliers [-5- 1 .1-5-) , IT")
\aaej \dy) \dz)
and the other homogeneous products of - , , -j- , -T of the degree n 2.
ctcc ^y GLZ
In the second process the augmentative system is formed by the aid
of the multipliers x n ~ 2 , y n ~ 2 , z n ~ 2 , &c., and the auxiliary system by aid of
-
dx)
dy
_
dz.
,&c.
For the particular case of n=2 the first process of the concomitantive
method is merely an application under its most symmetrical form of the first
42] On the Principles of the Calculus of Forms. 301
process of the general algebraical method. The second process of the con-
comitantive method for this same case (at least when 0, -v/r, w are the partial
differential coefficients of the same function of the third degree) has been
shown by Dr Hesse to give the resultant, so that for this case, at all events,
we know that each concomitantive auxiliary must be a linear function of the
augmentatives and the algebraical auxiliaries.
Again, if we go to the system where </>, -^r, w are of the respective degrees
n, n, Ti+1. In the algebraical method (for applying which there are no
longer two, but one only process), the augmentative system is obtained
by multiplying <j> by the homogeneous products of x n ~ l , x n ~*y, x n ~ l z, &c.,
^Jr by the like products, and &&gt; by the homogeneous products x n ~ 2 , x n ~-y, &c.
The auxiliary system is made up of functions of the general form
H pi g ir where p + q + r^n+Z,
Hp,q,r being the determinant obtained by writing
<f> = LxP + Myi + Nz r ,
M yi + N z r ,
And in like manner for the case of <, i|r, <y, being of the respective degrees
n, n, n1, the augmentative system is obtained by affecting </>, ty each with
multipliers x n ~*, x n ~*y, &c., and o> with the multipliers x n ~ 1 , x n ~ l y, &c.
The number of functions (for either case) in the augmentative and
auxiliary plexuses thus obtained will be found to be exactly equal to the
number of terms in each such function, as shown by me in the paper alluded
to. Let this be compared with the transcendental method (I use this word
at this point in preference to concomitantive, because in fact the algebraical
and differential auxiliary systems are both alike concomitantive plexuses
to </>). For the case of n, n, n + 1, the Jacobian determinant R of <, ty, <w
(^ \ n 1 / ^ \ n 2 / d\
-5-1 R, ( -j- } I -p- } R, &c.
ax/ \dxj \dyj
combined with the augmentative systems
yi 2 /vtlli 3 ,T/-
will give an invariant resembling (at least in generation and form) if not
identical with the resultant of <, -v/r, &&gt;. For the case of <, ty, w being of the
degrees n, n y n 1, the Jacobian R is of the degree 3n 4 and
d \ n 2 / d \ n * d
\ Z? I \ 7? $(*
302 On the Principles of the Calculus of Forms. [42
is the system which, combined with the augmentative systems
0} n ~ 2 <f), x n ~ 3 y^>, &c.
# n ~ 2 i|r, x n ~ s y\jr, &c.
x n ~ l w, x n ~-yw, &c.
will produce the resembling invariant.
Finally, for the last and more special case which the algebraical method
applies to, namely of <, -\|r, w, 0, four quadratic functions of x, y, z, t, there
can be here little doubt (upon the first impression) that in place of the
algebraically obtained plexus
-"2,1,1,1; -"1, 2,1,1) -"1,1, 2,1) -"1,1,1,2)
may be substituted the differential plexus
dR dR dR dR
dx dy dz dt
which, combined with the augmentatives
xcf), x^, ca), x6 ; yty, y-^r, yw, y6 ; zfy, zty, zw, z6; t$, tty, tw, td,
will render possible the dialytic elimination of the 20 homogeneous products
a? t x-y, x*z, a?t, xyz, y 3 , &c. &c.*
Upon precisely the same principles may be verified instantaneously
the method given by Hesse (without demonstration) for finding the polar
reciprocal of lines of the third and fourth orders, at least to the extent of
seeing that the functions obtained by his methods are contravariants (of the
right degree and order) of the function from which they are derived. The
polar reciprocal to a surface of the third degree may be obtained in the same
manner.
Let $ (x, y, z, t) be the characteristic of such a surface. If we form
a differential plexus of the first emanant of </> taken together with the
concomitant w = x% + yrj + z% + tO, by operating with
d d d d ,., d , d .,, d ,, d
d d d d ( ,., d , d .,,
T . j- T- i j: u P on S-+V:j-+ T-
dx dy dz dt V dx dy dz
and combining this plexus with x% + yn + z + tO , the resultant taken in
respect to , ?/, ^ , 6 (say R) will (according to the law of synthesis) be a
* Subsequent reflection induces me to reject as very improbable the (at first view likely)
conjecture of the identity of the resultant with the invariant which simulates its form, except in
the proved cases of three quadratic functions and the strongly resembling case of four quadratic
functions last adverted to in the text above. Did this identity obtain, analogy would indicate
that the catalecticant of the Hessian of two homogeneous functions of the same degree in x,
should be identical with their resultant, which is easily demonstrated to be false, except when
the functions are of the third degree.
42] On the Principles of the Calculus of Forms. 303
contravariant to the system < + Aw; and w, and therefore to </>, because w is
itself a concomitant to <. R is of the third degree in x, y, z, t, as also in the
coefficients of <. If we form a differential plexus of .R + pw analogous
to that formed above with (/> + \w, and combine these two plexuses with the
augmentative system xw, yw, zw, tw, there will be 4+4 + 4, that is, 12
functions containing the 12 terms x 2 , y*, z*, t 2 , xy, xz, xt, yz, yt, zt, \, p, and
the dialytic resultant, which will be found to be a contravariant of the twelfth
degree in , 77, ", 6, and of the twelfth order in respect of the coefficients of </>,
will be (there can be little doubt) the polar reciprocal to the characteristic </>.
A few remarks upon the analytical character of a polar reciprocal may be
not out of place here. If </> be any homogeneous function of the degree m
of any number (n) of variables (x, y ... z}, the object of the theory of polar
reciprocals is to discover what is the relation between , 77 . . . expressed in the
simplest terms such that, when this equation is satisfied, %x + rjy + . . . + z =
will be tangential to $ = 0. In order for this to take effect it is necessary
that when any one of the variables z is expressed in terms of the others ...y,x,
and this value established in </>, the discriminant of </>, so transformed, should
be zero. Consequently the characteristic of the polar reciprocal to < is
that rational integral function which is common to all the discriminants
obtained by expressing < (by aid of the equation %x + rjy + . . . + %z) as a
function of any (n 1) of the variables. Let I x be any invariant whatever of
the order r of <j) x (meaning by this last symbol what </> becomes when x is
eliminated), and I y ... I z the corresponding invariants when y ... z respectively
pi
are eliminated ; I x will evidently be of the form . ..^ *j- , the numerator being
an integer of r dimensions in the coefficients of $ and of mr dimensions in
respect of , T; ... ; and by the fundamental definition of invariants it may
easily be shown that
and therefore
-x -Ly * _mr mr * ?nr
E x E y E z m(n-2)r
- - - W h PTP 11 - 5 -
p-qp -p> Qp n-l
Consequently all these quotients must be essentially integer, and any one
of them will be of the order r in respect of the coefficients of < and of the
* We see indirectly from this, that for a function of (n-l), say 7, variables of the degree m,
WIT
an invariant of the order r must be subject to the condition that =an integer. This is easily
shown upon independent grounds; when 7 = 2, must be not merely an integer but an even
integer, and doubtless some analogous law applies to the general case.
304 On the Principles of the Calculus of Forms. [42
rar
degree - in respect of f , 77 ... . Consequently the polar characteristic
of (f>, which is the common factor of the discriminants of I x , I y ... I 2 (for which
species of invariant r evidently is equal to (n 1) (m l) w ~ 2 , the function being
in fact the discriminant of a function of the rath degree of (n 1) variables),
will be of the order (n l)(m l) n ~ 2 in respect of the coefficients of </>
and of the degree m (m l) n ~ 2 in respect of the contragredients , 77 ... f.
As to what relates to the reciprocity which exists between </> and its polar
reciprocal i|r, this is included in a much higher theory of elimination, one
proposition of which may be enunciated somewhat to the effect following,
namely that if < be a homogeneous function of x, y ... z, and o> of x, y . . . z,
u, v ... w, and if, by aid of the equations
f-O.
^ +x s=-
tci*/ \JLJu
d<f> da> _
~j r A, -j- \j,
dy dy
d<j> dta
"~T~ " ~T~ ">
dz dz
x,y ... z be eliminated and the resultant be called i/r, then the effect of
performing a similar operation upon -fy, CD, with respect to u, v ... w, as that
just above indicated for the system <f>, co, with respect to a, y ... z, will be to
give a resultant, one factor of which will be the primitive function <f> over
again. There is some reason for supposing that polar reciprocals, which are
scarcely ever (if ever, except indeed for quadratic functions) the simplest
contravariants to a given function, may be expressed algebraically by means
of the simpler contravariants, in the same way as discriminants admit (in
many, if not in all cases, with the same exception as above) of being repre
sented as algebraical functions of invariants of a lower order or simpler
form.
I close this section with the remark that every complete and unambiguous
system of functions of the constants in a given form or set of forms charac
teristic* of any singularity absolute or relative in such form or forms must
* I repeat here that a function or system of functions which severally equated to zero express
unequivocally and completely the existence of any position or negation, is termed the character
istic of such position or negation. Thus for example the resultant of a group of equations
is the characteristic of the possibility of their coexistence. The discriminant of a function of two
variables is the characteristic of its possession of two equal factors; the catalecticant is the
characteristic of its decomposability into the sum of a denned number of powers of linear
functions of the variables, &c.
42] On the Principles of the Calculus of Forms. 305
constitute an invariantive plexus or set of invariantive plexuses. The system
unambiguously characteristic of a singularity of an order n will (except when
n = 1) almost universally consist of far more than n functions, subject of course
to the existence of syzygetic* relations between any (n + 1) of such functions.
The existence of multiple roots of a function of two variables is a specific, but
by no means a peculiar case of singularity, and requires, for its complete and
systematic elucidation, to be treated in connexion with the general theory
of the subject.
SECTION III. On Commutants.
The simplest species of commutant is the well-known common deter
minant.
If we combine each of the n letters a, b ... I with each of the other n,
a., /3 . . . X, we obtain n 2 combinations which may be used to denote the
terms of a determinant of n lines and columns, as thus :
a, a/3 ... a\,
bet, b@ ... b\,
lot, l t
It must be well understood that the single letters of either set are mere
umbrse, or shadows of quantities, and only acquire a real signification when
one letter of one set is combined with one of the other set. Instead of
the inconvenient form above written, we may denote the determinant more
simply by the matrix
a, b, c ...I,
and to find the expanded value of such a matrix the rule is evidently to take
one of the lines in all its 1, 2, 3...n different forms, arising from the
permutations of the letters (or umbrae) which it contains ; and then form the
product of the n quantities formed by the combination of the respective pairs
of letters in the same vertical column, affecting such product with the sign
of + or according to the rule, that all products corresponding to arrange
ments of the terms subject to the permutation derivable from one another
by an even number of interchanges are of the same, and by an odd number
of interchanges of a contrary sign. If both lines are permuted and a similar
rule applied, with the additional circumstance that the sign of the products
* Rational integer functions which admit of being multiplied severally by other rational
integer functions such that the sum of the products is identically zero, are said to be " syzygeti-
cally related."
s. 20
306 On the Principles of the Calculus of Forms. [42
is made to depend on the product of the algebraical signs due to the respective
arrangements in the two lines of umbrse, it is evident that the result will be
the same as when only one line is put into motion, save and except that
a numerical factor 1 . 2 . 3 . . . n will affect each term. If the two sets of umbras
a, b, c ... 1] a, & 7... X be taken identical, and if it be convened that the
order of the combination of any two letters shall not affect the value of the
quantity thereby denoted, a> " will denote a symmetrical determinant.
Ctj 0) C v
If instead of two lines of umbras, three or more be taken, the same
principle of solution will continue to be applicable. Thus, if there be a
matrix of any even number r of lines each of n umbras,
a r , b r ... l r ,
the first may be supposed to remain stationary, and the remaining (r - 1)
lines each be taken in 1, 2 ...n different orders; every order in each line
will be accompanied by its appropriate sign + or - ; and each different
grouping in each line will give rise to a particular grouping of the letters
read off in columns. The value of the commutant expressed by the above
matrix will therefore consist of the sum of (1 .Z...nf^ terms, each term
being the product of n quantities respectively symbolized by a group of
r letters and affected with the sign + or - according as the number of
negative signs in the total of the arrangements of the lines (from the columnar
reading off of which each such term is derived) is even or odd.
For example, the value of
a, b,
c, d,
e, f,
g, h >
will be found by taking the (1 . 2) 3 arrangements, as below,
a, b, a, b, a, b, a, b, a, b, a, b, a, b, a, b,
c, d, d, c, c, d, d, c, c, d, d, c, c, d, d, c,
e, f, e, f, f, e, /, e, e, /, e, f, f, e, f, e,
g, h, g, h, g, h, g, h, h, g, h, g, h, g, h, g.
The signs of c, d ; e,f; g, h being supposed + , those of d, c ; /, e and h, g
will be each - . Consequently the sum of the terms will be expressed by
aceg x bdfh - adeg x bcfh - acfg x bdeh + adfg x bceh
- aceh x bdfg + adeh x bcfg + acfh x bdeg - adfh x bceg.
42 J On the Principles of the Calculus of Forms. 307
Commutants thus formed may be termed total commutants, because the
entire of each line is made to pass through all its possible forms of arrange
ment. In total commutants it is necessary that the number of lines r be
even ; for if taken odd, on making all the r lines to change, instead of
obtaining 1 . 2 . . . n lines, the result obtained when all but one are made
to change, it will be found that the latter will be repeated ^(1.2...n)
times with the sign +, and (1.2...w) times with the sign, so that the
algebraical sum of the terms will be zero. Moreover the commutants of
the species above described, besides being total, are simple, inasmuch as all
the umbra3 to be termed consist of single letters.
My first proposition in the application of the theory of commutants to
that of forms is as follows :
Let (f> be a function homogeneous and linear in respect to an even number
r of any systems whatever of variables, as
#i y\-"t\\ &2> yv- -t?) r> jfr *r
Form the commutant
d_ d d
dx l dy " dt-L*
d d d
d d d
dx r dy r dt r
Let the general term of this commutant, expanded, be called
F dl xF 02 x ... xF 6r ,
then is "F 6l . <f> x F 6i . < x . . . x F 6r . <f>
a covariant or invariant*, as the case may be, of <.
Be it observed that the march of the substitution for the different sets of
variables in the above proposition is supposed to be perfectly independent.
All the systems but one may undergo linear transformation, or they may all
undergo distinct and disconnected transformations at the same time, and
the proposition still continue applicable. It will however evidently be no
less applicable should the march of substitution for any of the systems
become cogredient or contragredient to that of any other systems.
If we suppose $ to be a function of an even degree r of a single system
of n variables x, y ... t, so that the r systems x lt y l} &c., # 2 , y 2 , &c. ... x r , y rt &c.
become identical, we can at once infer from the above scheme the existence
and mode of forming an invariant to <J> of the order n. This last appears
[* See below, p. 324.]
202
308 On the Principles of the Calculus of Forms. [42
for the case n= 2, and ought, for all other values of n, to have been known*
to the author of the immortal discovery of invariants, termed by him
hyperdeterminants, in the sense which, according to the nomenclature here
adopted, would be conveyed by the term hyperdiscriminants.
Before proceeding to discuss the theory of compound total commutants,
or enlarging upon that of partial commutants, I shall make an interesting
application of the preceding general proposition to the discovery of Aronhold s
S and T, the two invariants respectively of the fourth and sixth orders
appertaining to a homogeneous cubic function (say F) of three variables
x, y, z. These may be termed respectively H 4 and H 6 . As to H 6 a
theoretically possible but eminently prolix and ungraceful method im
mediately presents itself, namely to take F 2 = G, and after forming the
commutant with six lines,
d d d
dx dy dz
d
d
d
dx
dy
dz
d
d
d
dx
dy
dz
d
d
d
dx
dy
dz
d
d
d
dx
dy
dz
d
d
d
dx
dy
dz
to operate with the 6 5 ternary products of which this is made up upon G: the
result being an invariant of G, will be so to F, and being of the third degree
in respect to the coefficients of G, will be of the sixth in respect to those of F.
It will evidently therefore be H 6 , or at least a numerical multiple of H 6 , the
form of which, inasmuch as the only other invariant is H it we know in form
to be unique. But the general theorem affords another and probably the
* That this was not known explicitly to and should have escaped the penetration of the
sagacious author of the theory, and those who had studied his papers, must be attributed to the
imperfection of the notation heretofore employed for denoting the coefficients of a homogeneous
polynomial function. The umbral method of denoting such a function <j> of the degree r under
the form of (ax + by + . . . + cz) r , which is equivalent to, but a more compendious and independent
mode of mentally conceiving and handling the representation
/ d d d\ ^
x ,-+y -y-+ + z ~r ) #>
\ dx y dy dz) r<
exhibits the true internal constitution of such functions, and necessarily leads to the discovery
of their essential properties and attributes.
42] On the Principles of the Calculus of Forms. 309
most practically compendious* solution as regards H s , of which the question
admits.
Let Gf represent the mixed concomitant to F formed by the bordered
determinant
d 2 JF
O?f
d 2 F
t
dx 2
dxdy
dxdz
s
dydx
dy 2
dydz
*?
d 2 F
d 2 F
d*F
dz*
5
dzdx
dzdy
fc
V,
r,
G is a function of the second order as to x, y, z, and of the like order in respect
to ff, ?;, which two systems will be respectively cogredient and contra-
gredient in respect to the x, y, z system in F. In other words, which is all
we need to look to, G is a concomitant of F, and so also will be
which may be termed H. Form now the commutant
d d d
dx dy dz
d d d
dx dy dz
A
dt
d
d_
dr)
d
drj
d
this being applied to H will give an invariant (the fact that the march
of the substitutions for the systems x, y, z; , 77, f is contrary, being com
pletely immaterial to the applicability of the general theorem above given) ;
* Having since this was printed been favoured with a view of some of the proof-sheets of
Mr Salmon s most valuable Second Part of his System of Analytical Geometry (about to appear,
and which is calculated, in my opinion, to awaken a higher idea of and excite a new taste for
geometrical researches in this country), I find that I am mistaken in this point ; the less sym
metrical method operated with by Mr Salmon being decidedly the shortest for practically
obtaining S and T in the general case. Symmetry, like the grace of an eastern robe, has not
unfrequently to be purchased at the expense of some sacrifice of freedom and rapidity of action.
t G is the mixed concomitant to the given cubic function, which is halfway (so to speak)
between it and its polar reciprocal. In fact, when the operation is repeated upon G, which was
executed upon the given function to obtain G (that is, when we border the Hessian of G in
respect to x, y, z, vertically and horizontally with the column and line , 17, f ) the determinant
thereby represented becomes the polar reciprocal to the given function.
310 On the Principles of the Calculus of Forms. [42
the commutant so formed will be a cubic function of X, in which the coefficient
of X* is a numerical quantity, that of X 2 is zero, that of X is H^ and the
constant term is H 6 .
Thus for example let F = x 3 + y 3 + z 3 + Qmxyz, then
x, mz, my, ?
mz, y, mx, T?
my, mx, z, ?
?, V, ?.
and therefore
H = 2 {(X - m 2 ) a 2 ? 2 + (X + m 2 ) 2yzrj + yz!; 2 -
the 2 implying the sum of similar terms with reference to the interchanges
between x, ; y, 77 ; z, ?.
In developing the commutant above, the first line may be kept in a
fixed position; for the sake of brevity, (x), (y), (z}\ (), (77), (?) may be
written in the place of
d d d d d d
dx dy dz" 1 5? drj d?
and it will readily be seen that the only effective arrangements will be
as underwritten :
(x)(y)(z} (x)(y)(z) (*) (y) (*)
0*0 (y) ( z } ( x ) (y) (^ 0*0 (y) (^)
(?)(>?)(?) O?) (?)(?) (?)(?) to)
(*0(y)(*) (*)(y)() ()(y)(*) ()(y)(^) 0*)(y)(*) (^(y)^)
(a?) (2) (y) (x) (z) (y) (?) (r/) (?) (z) (y) (*) (y) () (^) (y) (*) ()
(?)(>?)(?) (?)(?) (^) (?)W(?) (?)(^)(?) (?)(^)(?) to) (?)(?)
(?)(?) (17) (?)(^)(?) (?)0?)(?) (?)(^)(?) (^ (?)(?) (?)W(?)
(aO(y)0) (*)(y)() ()(y)(^) ()(y)(*) (^(y)^) (^(y)^)
(a?) () (y) () (a?) (y) (*) (y) (^ (y) (*) (*) (y) () (*)
(?)(;)(?) (?)(?) (^) (?)(?) (^) (^) (?)(?) (?)(?) (^)
(17) (?)(?) (?)(?) (^) (?)(?) 0?) (?)(?) ( r /) Ort (?)(?)
(*) (y) (^) (*) (y) (^) (^) ^y) (^ (a; ) (^) & ^ ^ ^
(*)() (y) (y) (*)(*) (y)(^)(^) (^(^(y) (*)(*) (y)
(17) (?)(?) (?)(^)(?) (^) (?)(?) (?)W(?) (?)(?) 0*)
(y)
(*) (y)
(y) (*)
(?) (?)
(?)
(17) (?)(?) (^)
(?) 07)
(?)
42] On the Principles of the Calculus of Forms. 311
The signs of the four lines in each of these arrangements are two alike,
and two contrary to the signs of the correspondent lines in the first arrange
ment ; hence the effective sign is the same for all, and the result, after
rejecting from each term the common factor 16, is seen, from inspection,
to be
4 (X - m 2 ) 3 - 8??i 3 + 6 (X - m 2 ) (X + m 2 ) 2 - 12i (X -H m 2 ) + 2 (X + ra 2 ) 3 + 1,
which is equal to
12X 3 + . X 2 - 12 (m - m 4 ) \ + 1 - 20i 8 - 8m 8 ;
here the coefficients m m 4 and 1 20m 3 8/?i 6 are the two invariants
(Aronhold s S and T) for the canonical form operated upon ; and it will
be observed that
(1 - 20m 3 - 8m 6 ) 2 + 64 (m - m 4 ) 3 = (1 + 8m 3 ) 3 ,
which is easily proved to be the discriminant of
a? + y 3 + z 3 + Qmxyz.
It may however be observed, that this is not the discriminant of the
function in X just found, as reasons of analogy* might have suggested it
probably would be : in order that this might be the case, the coefficient
of X 3 should be 4 instead of 12, and of X, m m 4 instead of m 4 m. There is
ground for supposing that another function of X may be found by a different
method, in which this relation will take effect.
The theorem above given for simple total commutants admits of an
interesting application to the general case of a function F of the nth degree,
in respect to each of two independent systems of two variables x, y ; , 77.
Let F be symbolically represented by (ax + by) n (aj; + /3rj) n , so that a n a. n
represents the coefficient of # n f n , na n ~ 1 ba n of x n ~ l y^, &c. &c. ; then the
commutant
a, b, (1)
a, b, (2)
a, b,
, A
, &
a, /9, (n)
will represent a quadratic invariant of F, which will contain (n + I) 2 coefficients.
By expanding this commutant we obtain a general expression for the invariant
under a very interesting form.
* The biquadratic function of x, y having only one parameter, and therefore two invariants,
its theory possesses striking analogies to the theory of the cubic function of three letters. The
function in X which gives these invariants for the first-named function, according to the method
given in the first section, has the same discriminant as the function itself.
312 On the Principles of the Calculus of Forms. [42
I now proceed to give the general theorem for compound total commutants
as applicable to the discovery of invariants.
Let there be a function of in disconnected classes of systems of variables ;
let the systems in the same class be supposed all distinct but cogredient
with one another. The function is supposed to be linear in respect to each
system in each class, and the number of systems is the same for all the
classes, and the number of variables the same in each system. This function
may then be represented symbolically under the form
x ( 2 a 1 . a ar 1 + a 6 1 . a y 1 + ... +
X &C.
In this expression the <K, y . . . t s are all real, but the a, b ... I s all umbral ;
in fact, fa ff , ^b g , &c. may be understood to denote -^j , -^ , &c.
a j x g a/yg
The n systems of variables in each of the sets above written are supposed
to be cogredient inter se.
Take the symbolical product of the first set, first making for the moment
l x l = l x z = . . . l x n = x, &c. &c., HI = %= ... l t n = t ;
and let the coefficients of the several terms
x 11 , x n ~ l y ... &c.,
be called 1 A 1} 1 A 2 ... ^
where /JL is the number of terms contained in a homogeneous function of the
nth degree of the m variables x,y...t. In like manner proceed with each
of the lines, and then write down the commutant
lA 1 A lA
ft-D * "Ml
2 A 2 A 2 A
**ii -0-2-" ^-m
This commutant is an invariant of F : it will of course be remembered that,
unless p is even, the commutant vanishes.
42] On the Principles of the Calculus of Forms. 313
Thus, for example, take two sets of two systems of two variables : in all
four systems,
each couple of systems on either side of the colon (:) being cogredient
inter se: and let F be symbolically represented by
(ax + by) (af + 77) (Ip + mq) (\<j> +
then the invariant given by the theorem will be the commutant
act ; a/3 + ab ; 6/3,
l\ ; IfM + Xm ; m/u.
The six positions of this are as below written (the first three being positive
and the second three negative)
act ; a/3 -f ab ; 6/3, aa ; a/3 + a6 ; 6/3, aa ; a/3 + a6 ; 6/3,
ZX; lp+\m; m/j,, l/j, + \m; ra/u,; ZX, w/i; X; l/j,+ \m,
aa ; a/3 + a6 ; 6/3, aa ; a/3 + a6 ; 6/3, aa ; a/3 + ab; 6/3,
/u + Xm ; ZX ; m/i, l\ ; ??i/u, ; ^/a + Xw, m/A ; Z/u + Xra ; ^X.
If we write F under its explicit form,
+ Bxgpty + Cxgqcf) + D
+ B xrjp-fr + C xrjq<f) + D
we have identically the relations following,
aaZX = A, aal/j, = B, aawX = C, aam/i = D,
a/3ZX = A , afilfji = B , a/3mX = C , ajSmp = D ,
bal\ = A", bal/ji = B", bctm\ = C", 6am/a = D",
and the commutant expanded becomes
A (B + C" + C + B") D" + (B+C) (D + D") A" + D(A + A") ( " + C ")
-(B + C) (A + A") D " -A(D + D") (B" f + C" ) -D(B + C" + C + B") A ".
In the foregoing the x s in the several lines were for the moment taken
identical, in order the more easily to explain the law of formation of the
314 On the Principles of the Calculus of Forms. [42
quantities A. But suppose that they become actually identical for the same
line. F then becomes a function of the ?ith degree in respect to each of
p systems of variables, and may be represented symbolically under the form
( l a l x + l b l y + ...+ l H) n x ( 2 a 2 x + *b 2 y + ...+ HH} n
... x (PaPx + PbPy + ...+ nnj\
We may still further limit the generality of the theorem by supposing
^t = H = ...Pt=t;
F then becomes (ax +by+ ...+ lt) np .
Accordingly, as many different factors as can be found contained an even
number of times in the exponent of the function, so many invariants can be
formed immediately from a function of any number of variables m by the
method of total commutation.
If one of these factors be called n, the commutant corresponding thereto
will be of the order
1.2...O-1)
in respect to the coefficients. Thus take m = 2, so that
F = (ax + by) n P.
The general form of such a commutant will be found by taking
A 1} A 2 ...A n+1 the coefficients of the several combinations of x, y in
(ax + by) n , from which the numerical coefficients n, ^n (n 1), &c. may be
rejected, as only introducing a numerical factor into the result; the com
mutant will therefore be expressed by means of the form
a n ; a n ~ l b] a n ~ 2 6 2 ...; b n , (1)
a n ; a n ~ l b; a n ~ 2 & 2 ...; b n , (2)
a n ; a"- 1 6; a n ~ 2 6 2 ... ; 6". (p)
= 2, the compound commutant
a n - a n ~^b] ...; b n ,
a n ; a n ~ l b; ...; b n ,
42] On the Principles of the Calculus of Forms. 315
will easily be seen to be only another form for the catalecticant of (ax + by)" 1 .
Thus, let n = 2,
(ax + by) 4 = Ax 4
so that 4 = A, a s b = B, ft 2 6 2 = C, ab 3 = D, 6 4 = E.
The commutant (which is of the form of the matrix to an ordinary
determinant, with the exception that the umbraa enter compoundly instead
of simply into the several terms separated by the marks of punctuation),
will be
ft 2 ; ab ; 6 2 ,
ft 2 ; ab; 6 2 :
this, written in the six forms
ft 2 ; ab; 6 2 ] ft 2 ; ab; b" | ft 2 ; ab; b
a-; ab; 6 2 J ft 2 ; 6 2 ; ab } ab ; ft 2 ; b
ft 2 ; ab; 6 2 ] ft 2 ; ab; b- } ft 2 ; ab; b-
b-; ab; ft 2 / ab; b 2 ; ft 2 j 6 2 ; a 2 ; ab
gives the expression
a 4 x a-b 2 x 6 4 - ft 4 x (ft& 3 ) 2 - b 4 x (a^) 2 - (ft 2 6 2 ) 3 + 2 3 6 x
that is ACE- AD 2 - EB* - C s + "1BCD.
One important observation may here be made of a fact which otherwise
might easily escape attention, which is, that commutants, where the same
terms simple or compound are found in all or several of the lines, in general
give rise to products, some of them equal and with the same sign, and others
equal but with the contrary sign.
This last phenomenon does not manifest itself in commutants appertaining
to functions of two variables of the two particular and different species which
first and most naturally present themselves, namely where there are only two
lines or only two columns* I believe that it displays itself in every other
case of commutantives to functions of two variables. Thus it is that
algebraical expressions derived from given functions disguise their symmetry ;
to make which come to light it becomes necessary to add terms of contrary
sign to such expressions. As an example, the reader is invited to develope
the cubic invariant of a function of x and y, symbolically expressed by
(ax 4- by) s , where
* These commutants give respectively the quadrinvariant and the catalecticant, the former
of which alone was formerly recognised by Mr Cayley as a commutant.
316 On the Principles of the Calculus of Forms. [42
by means of the commutant
a 2 , ab, b 2 ,
a 2 , ab, b\
a 2 , ab, b",
a 2 , ab, b-.*
Suppose F to be the general even-degreed function of two variables
of the degree
Let H=(i;-j 77 -jr-
V dy doe/
and express H umbrally under the form
(ax
* [See p. 346 below.] The number of terms resulting from the independent permutation
of each of the 3 linear lines is 6 3 , that is 216 ; but the actual result is (using small letters instead
of large) P - Q, where
P= aei + 3fif 2 + I2beh + 3<?i + 24c/ 2 + 24d 2 + loe 3 ,
Q = 4a/7i + ibid + 8bgf+ 22ceg + 8chd + 36def,
so that the effective number of permutations is only 164. The difference between this and
216 divided by 216 may be termed the Index of Demolition, which we see in this case is / T \ or if ;
that is, somewhat less than J. For the cubic invariant of the function of the fourth degree this
index is zero, all the permutations being effective. If we take the cubic invariant of the function
ax l2 + 12bx ll y + S6cx w y 2 + &c. + my 12 under the form P- Q, we shall find
P= Qahl + lOajj + 6bfm + 54bhk + 54c/7 + ISocii + IQddm + 430<%
+ I55eek + 520ehh + 520ffl + 280ggg,
Q = agm + 15aik + 30bgl + 50bij + 15cem + icgk + ISQchj + BOdel + 210dfk
+ 250dhi + 2SOefj + 555egl + 6 6 0/0 /i.
The number of terms in P and Q, is of course the same, and will be found to be 2200 for each ;
so that out of the 6 5 , that is 7776 permutations of the 5 lower rows, only 4400 are effective, and
the index of demolition becomes fff $, that is $|f , or rather greater than T \. The Index of Demo
lition thus goes on constantly increasing as the degree of the function rises ; probably (?) it
converges either towards ^ or else towards unity. In arranging the terms it will be found most
convenient to adopt, as I have done above, the dictionary method of sequence. The computations
are greatly facilitated by the circumstance of the effect of any arrangement of each of the 5 lower
lines not being altered when these lines are permuted with one another ; this gives rise to the
subdivision of the 7776 permutations into groups as follows : 6 of 120 identical terms, 60 of 60,
36 of 20, 60 of 30, 24 of 20, 30 of 10, 30 of 5, and 6 of 1. So that the total number of permuta-
tional arrangements to be constructed is only 252. Other methods of abridging the labour will
readily suggest themselves to the practical computer. The total number of the groups of terms
is of course always known a priori, and, for instance, in the case before us, must be equal to the
number of ways in which (12 x 3), that is the number 18, can be divided into 3 parts, none of
which is to exceed the number 12, that is 25 ; for the cubic invariant of the function of the
eighth degree of two variables it is the number of ways in which 12 can be divided into 3 parts,
of which none shall exceed 8, and so forth, zeros being always understood to be admissible ;
and of course in general for an invariant of the order r to a function of the degree n of i variables,
717*
the number of distinct terms is in general the number of ways in which can be divided into
r parts, of which none shall exceed n, subject however always to the possibility in particular
cases of a diminution in consequence of some of the groups assuming zero for their coefficient.
42] On the Principles of the Calculus of Forms. 317
The commutaut
a 11 , a n ~ l b...b n , (1)
a n , a"^ 1 &...&", (2)
a 1 , a n ~ l b...b n , (p)
a a 1 - 1 /?..-/^, (1)
a", a"-V3 .../3 n , (2)
a n , a"-* &...&", (p)
will be a function of \, and all the several coefficients will be invariants of F*.
When p = 1 we obtain the A given in the preceding section, and origin
ally published by me in the Philosophical Magazine for the month of
November, 1851. The A obtained on this supposition has for its coefficients
a series of independent invariants, commencing with the catalecticant and
closing with the quadratic invariant. When p has any other value, we
observe a similar series commencing with a commutantive invariant of a
lower order than the catalecticant, but always closing with the quadratic
invariant. Thus, for example, when 2np = 8, we may obtain by the preceding
theorem three different quadratic functions ; one giving the invariants of the
orders 5, 4, 3, 2, the second those of the orders 3, 2, the third the invariant
of the order 2.
In this case the invariants of the same order given by the different A s
are the same to numerical factors pres. Whether this is always necessarily
the case is a point reserved for further examination.
The commutants applied in the preceding theorems have been called
by me total commutants, because the total of each line of umbrae is permuted
in every possible manner. If the lines be divided into segments, and the
permutation be local for each segment instead of extending itself over the
whole line, we then arrive at the notion of partial commutants, to which
I have also (in concert with Mr Cayley) given the distinctive name of
Intermutants. In order to find the invariants of functions of odd degrees,
the theory of total commutants requires the process of commutation to be
applied, not immediately to the coefficients of the proposed function, but to
some derived concomitant form. I became early sensible of this imper
fection, and stated to the friend above named, to whom I had previously
* By substituting the symbols y- , -j- > &c. in place of the umbra a, b, &c., the theorem
is easily stated for covariants generally. But in applying the commutantive method to obtain
co variants, or rather in the statement of the results flowing from each application, it is never
necessary to go beyond the case of invariants, because the commutantive covariants of any given
homogeneous function are always identical with commutantive invariants of emanants of the
same function.
318 On the Principles of the Calculus of Forms. [42
imparted my general method of total commutation, my conviction of the
existence of a qualified or restricted method of permutation, whereby the
invariants of the cubic function, for instance, of two and of three letters would
admit, without the aid of a derived form, of being represented. Many months
ago, when I was engaged in this important research, and had made some
considerable steps towards the representation of the invariant, that is, the
discriminant of the cubic function of x and y, under the form of a single
permutant, I was surprised by a note from the friend above alluded to,
announcing that he had succeeded in fixing the form of the permutant of
which I was at that moment in search. It is with no intention of complain
ing of this interference on the part of one to whose example and conversation
I feel so deeply indebted, (and the undisputed author of the theory of
Invariants,) that I may be permitted to say that, independent of the inter
vention of this communication, I must inevitably have succeeded in shaping
my method so as to furnish the form in question ; and that with greater
certainty, after my theory of commutants had furnished me with the prece
dent of permutable forms giving rise to terms identical in value but affected
Avith contrary signs. As I have understood that Mr Cayley is likely to
develop this part of the subject in the present number of the Journal, it will
be the less necessary for me to enter at any length into the theory of partial
commutants on the present occasion.
The method of partial commutation is a simple but most important
corollary from that of total commutation hereinbefore explained. To fix
the ideas, conceive a class of p cogredient systems, and that there are qr
such classes perfectly independent. Proceed to divide these qr classes in
any manner whatever into r sets, each containing q classes ; and form the
symbol of the total commutant corresponding to each such set. Now let
these commutants be placed side by side against one another, arid transpose
the terms in each compound line thus formed once for all, but in any
arbitrary manner. Then permute in every possible way all those symbols
in each line, inter se, which belong to the same class, and operate with the
symbols thus produced by reading off the vertical columns and attending to
the rule of the + and signs, as in the case of a total commutant ; the
result will be a commutant of the form operated upon. For instance, let
p=l, <7=3, r = 2, and let the number of variables in each system be 2.
Form the commutant operators
d
d
d
d
dx
dy
dj>
drj
d
d
d
d
dp
dt
d<f>
d0
d
d
d
d
dr
ds
dp
da-
42] On the Principles of the Calculus of Forms. 319
Interchange in any manner but once for all the symbols in each line, as thus:
d d d d
dx dy d drj
d d d d
d<j> dp Tt dd
d d d d_
ds dp dr da-
Now permute, inter se, the variables of each system, as
d^ d_ m d^ d_ &
dx dy dp dt
the total number of the operative forms resulting will be (1 . 2) 8 , and the
sum of the (1 . 2) 6 quantities, half positive and half negative, formed after the
type of
d d d TT d d d
____ // y f /
dx d<j> ds dy dp dp
d d d TT d d d
X -yz. -TTj 7- U X -y- -j- -j
d da dr drj at da
U being supposed to be a function homogeneous in
> y > %> v > p> 1 5 $> @ 5 r i s i p> "
will be a covariant of U.
The proof of the truth of this proposition is contained in what is shown
in the Notes of the Appendix for total commutants, it being only neces
sary to make the systems which are independent vary consecutively, and
then apply the inference to the supposition of their varying simultaneously.
It may be extended to the more general supposition of classes of an
unequal number of cogredient systems of unequal numbers of variables
in each, the only condition apparently required being that the number of
distinct terms shall be the same in each line of the final commutantive
operator. The important remark to be made is, that in applying this
theorem there is nothing to prevent any of the systems being made identical ;
or, in other words, a given function of one system of variables may be
regarded as a function of as many different, although coincident, sets as we
may choose to suppose. Thus, suppose
we may take the partial commutant formed of the two total commutant
operators
A A
dx dy
d_ A
dx dy
320 On the Principles of the Calculus of Forms. [42
combined with itself. If we write them in the same order,
d d d d
dx dy dx dy
d d d d
das dy dx dy
(where I use the dots and dashes to distinguish those in the same line which
are considered as belonging to the same class, and therefore as permutable,
inter se), we shall evidently obtain 4 {AC jB 2 } 2 ; if we commence with a
permutation, so as to have the form of operation
d d d d
dx dy dx dy
d d d d
dx dy dx dy
it will be found that we obtain 2 {AC-B-} 2 .
Again, suppose that we have
U=Ax* + 3Bx 2 y + Wxf + Dy 3 .
If we write
d d d d
dx dy dx dy
d d d d
dx dy dx dy
d d d d
dx dy dx dy
the value of the commutant would come out zero ; but if we make a permu
tation, and write
d d d d
dx dy dx dy
d d d d_
dx dy dx dy
d d d^ d^
dx dy dx dy
the operation indicated by the above performed upon U, will give a multiple
of the discriminant of U.
42]
On the Principles of the Calculus of Forms. 321
In like manner we may represent Aronhold s Sextic Invariant of the form
(x, y, z) 3 by means of the partial commutant
d d d d d d
dx dy dz dx dy dz
d d d d d d
dx dy dz dx dy dz
d d d d d d
dx dy dz dx dy dz
If we make
d , d .,, d \ ( t d
~T* "t" V ~T~ T C ~Ti, G~T~
+ Y
dy dz)
and use H to signify the determinant
I, 17,
r, v,
which is evidently an universal triple covariant, and make
and apply to W the partial commutantive symbol
d d d d d d
dx dy dz dx dy dz
d d d
d d
d d d
d dr, d
d d d
d%" dr," ~d
we shall obtain a function of X of which all the odd powers and the second
power will disappear, and such that the coefficients of X 2 and the constant
term will be Aronhold s S and T, and the discriminant of the entire function
in respect to X 2 (if not for the distribution assigned to the dots and dashes
in the foregoing, at least for some other distribution) may not improbably be
the discriminant of the given function (x, y, z) 3 .
s. 21
322 On the Principles of the Calculus of Forms. [42
NOTES IN APPENDIX.
(1) [p. 295 above.] More generally, in as many ways as the number n
can be divided into parts, in so many ways can a given function of one set of
variables be as it were unravelled so as to furnish concomitant forms.
For instance, the form an 3 + 3bx 2 y + Sexy"* + dy 3 has for a concomitant
aux + buy + bvx + cvy + cwx + dwy,
where u, v, w are cogredient with # 2 , 2xy, y 2 ; and also
auu x + buu y + buv x + bvu x + cvv x + cvu y + cuv y + dvv y,
where u, v; u , v are cogredient with each other and with x and y\ and the
proposition in the text may be best derived from this more general theorem
by dividing the index into equal parts, forming as many systems as there are
such parts, and then identifying the systems so formed.
(2) [p. 297 above.] The following additional example will illustrate the
power of this method.
Let <f> = (x, y, zf be the general function of the fourth degree. Form by
imravelment the concomitant form (u, v, w, p, q, r) 2 (say P) where u, v, w, p, q, r
are cogredient with a?, y z , z 2 , 2zy, Zacz, 2yx.
Again, the universal concomitant (x% + yrj + z^f will have for its con
comitant
u + vrj 2 + w? + r) + q^ + rfy,
where , 77, are contragredient to x, y, z. Now take the reciprocal polar of
this last form with respect to , 77, ; that is,
2 (vw - ip 2 ) ! 2 + 22 (qr - %pu) y& (say <?),
where x l ,y l ,z l , being contragredient to f, rj, , will be cogredient with x, y,z.
P + \0 is a quadratic function of the six variables u, v, w, p, q, r, and its
discriminant will give a function of X of the sixth degree, all of whose even
coefficients will be covariants of 0. If we replace x l ,y- i , z 1 by x, y, z, these
even coefficients will be respectively (understanding that order refers to the
dimensions quoad the coefficients of and degree to the dimensions quoad
x, y, z} as follows :
42] On the Principles of the Calculus of Forms. 323
Of order
6 degree
o,
5
2,
j,
4
4s
3
6,
2
8,
1
10,
>;
12.
The two last coefficients must evidently be identically zero. It is possible
that some of the others may be so too : as regards the one of the third order
and sixth degree, this is of the same form as, and may be identical with, the
Hessian of <f> \ as regards the one of the fourth order and fourth degree, this
may be <f> itself multiplied by the cubic invariant (which the theory of
Section III. proves to exist) of <. But the covariants of the fifth order
and second degree, and of the second order and eighth degree, if they are
not identically zero, and if the latter is not < 2 (which a trial or two of some
very simple cases will easily establish one way or the other) are probably
irreducible forms. The existence of a correlated conic section to a curve
of the fourth order, if established, would be particularly interesting, and its
geometrical meaning would well deserve being elicited.
(3) [p. 303 above.] If any form (/) of the degree n be written sym
bolically,
(a^ + a 2 #2 + + a t aO n ,
where x lt # 2 ... CC L are real but a l9 a 2 ... a t umbral, and if I r be any invariant
of the order r in respect of the real coefficients of (/), it is easily seen by
reason of I r remaining unaltered when x lt sc 2 ...oo L become respectively
/i#],/2# 2 /i^u provided that/!,/ 2 .../ t = l, that each term in I r expressed
by means of the umbrae, must contain an equal number of times a 1} a 2 ... a t ,
77,7*
so that each such term will contain of each of them, of course differently
fW*
subdivided and grouped; hence we have the universal condition that must
be an integer ; but this is less stringent than the actual condition, which
nr
is that must be an integer of a certain form ; for instance, as before
nr
observed, when i = 2, must be an even integer.
I
(4.) [p. 307 above.] To prove the theorem given in the text for total simple
commutants it is only necessary to bear in mind that whenever two columns
in any total commutant become identical, the commutant vanishes. To fix
the ideas, take the commutant formed of lines similar to j- , -5- , j- , written
dx ay dz
212
324 On the Principles of the Calculus of Forms. [42
under one another; let there be r such lines, the total number of terms
will be (1 . 2 . 3) r : the 1.2.3 positions of the line written above will corre
spond to (1.2. 3) r-1 several groupings of the remaining lines. Now when
x, y, z undergo a unimodular linear substitution, j~ T~ > j~ w ^^ undergo
a related substitution not coincident with that of x, y, z, but still unimodular;
let x, y, z change, all the other systems remaining fixed, and suppose
, -i- , -T- to become respectively
dx ay dz
f d d , d
dx + 9 dy + l dz
/// d d , d
~T~ V ~~7 " ~T~ >
dx dy dz
then each of the (1.2. 3) r-1 groups of the terms arising from the permutation
of -T- , -r- , -7- will subdivide into 27 groups, of which we may reject those
dx dy dz
in which any of the terms [-3-, -j-, -3-] occurs twice or three times; accord -
\dx dy dz)
ingly there will be left only the six effective orders of permutations,
f.d ,d , d\ f~d ,, d_ d\. o,
\! dx 9 dy dz) \r dx dz ^ dy)
consequently each of the (1.2. ^) r ~ 1 groups gives rise to 6 times 6 products
f" n"
whose sum will be
f, 9 >
x the sum of the 6 products corresponding
/> ff, h
to the permutations of -r- , -r- , -r- ; and therefore, the transformations being
7
unimodular, the sum of the products corresponding to the entire (1 . 2 . 3) r
permutations remains constant when x, y, z change. In like manner, all the
systems may change one after the other, and consequently all of them at the
same time without affecting the value of the commutant: and in like manner
for the general case. Q.E.D.
(5) [p. 312 above.] The truth of the proposition relative to compound
commutants and the mode of the demonstration will be apparent from the
subjoined example.
Let the function be supposed to be
(ax + by) (a x r + b y ) (of + /fy) (T + /3V),
42] On the Principles of the Calculus of Forms. 325
where x, y; a; , y are cogredient and , 77; % , ?/ cogredient; the a, 6, a, (3, &c.
are of course mere umbrse. Now take the compound commutant
aa , ab + a b, 66 ,
Let as, y; as , y undergo a linear substitution, and, accordingly,
let a become fa + gb,
a fa + gb ,
b ha + kb,
b ha + kb ,
f, g, h, k being of course actual and not umbral ; then the above commutant
will be easily seen to decompose into 6 others, which will be equal to the
original commutant multiplied by the determinant
f; *fff, 9*
fh, fk + gh, gk
h-, 2hk, k-
which is equal to (fk gh) 3 , that is = 1.
And so in general, which shows, as in the preceding note, that all the
classes of cogredient systems may be transformed successively one after
the other, and therefore simultaneously, without altering the value of the
commutant.
(6) In the last May Number* of the Journal, Mr Boole, to whose modest
labours the subject is perhaps at least as much indebted as to any one other
writer, has given a theorem f, (14) p. 94, the excellent idea contained in
which there is no difficulty in shaping so as to render it generalizable by aid
of the theory of contravariants. It may be regarded in some sort a pendant
or reciprocal to the Eisenstein-Hermite theorem, presented by me under a
wider aspect in the First Section of this paper.
[* Camb. and Dub. Math. Journ. Vol. vi. (1851), pp. 87106.]
t Mr Boole applied his theorem to obtain the cubic invariant of (x, y)*, say <f>(x,y), by
operating upon its Hessian with d>(, - -r- \ . More generally, when cf>(x,y) = (x, y)- n , the
\dy dx)
catalecticant of the antepenultimate emanant of <p is also of the degree 2n ; and this, when
operated upon by <f> ( , - ) , will give an invariant of the order n + 1, which is probably
identical with the catalecticant of < itself. There exists a most interesting transformation of the
catalecticant of any emanant of a function of any degree in x, y, whether even or odd, under the
form of a determinant some of the lines of which contain combinations only of x and y, without
any of the coefficients, and all the rest the coefficients only of the given function without x or y.
The Hessian being the catalecticant of the second emanant is of course included within this
statement.
326 On the Principles of the Calculus of Forms. [42
Let < (x, y . . . z) have any contravariant 6 (x, y . . . z) ; then will
d d d
be a contravariant of (/>. For orthogonal transformations the terms contra-
variant and covariant coincide, and the theorem for this case appears to have
been known to Mr Boole, see (15), same page. More generally, if ^ and B
be any two concomitants of <, the algebraical product tyd will also be a
concomitant of <f>, provided that the systems of variables in -^r and 6 have all
distinct names, or that those which bear the same names are cogredient with
one another. If this proviso does not hold good, the product in question will
evidently be no longer a concomitant of <j>. Let however ^ denote what -fr
becomes, and ^ what 6 becomes, when in place of the variables x, y ... z
of every two coritragredient synonymous systems in i|r and 6 we write
then will S-\> and ^6 be each of them concomitants of 6,
, .. ,
da; ay dz
the synonymous systems becoming cogredient with ^ in the one case and
with 6 in the other.
(7) There is one principle of paramount importance which has not been
touched upon in the preceding pages, which I am very far from supposing
to exhaust the fundamental conceptions of the subject, (indeed, not to name
other points of enquiry, I have reason to suppose that the idea of contra-
gredience itself admits of indefinite extension through the medium of the
reciprocal properties of commutants ; the particular kind of contragredience
hereinbefore considered having reference to the reciprocal properties of
ordinary determinants only).
The principle now in question consists in introducing the idea of con
tinuous or infinitesimal variation into the theory. To fix the ideas, suppose
C to be a function of the coefficients of </> (x, y, z}, such that it remains
unaltered when x, y, z become respectively/^, gy, hz, provided that fgh = l.
Next, suppose that C does not alter when x becomes x + ey + ez, when e and
e are indefinitely small : it is easily and obviously demonstrable that if this
be true for e and e indefinitely small, it must be true for all values of e and e.
Again, suppose that C alters neither when x receives such an infinitesimal
increment, y and z remaining constant, nor when y nor z separately receive
corresponding increments, z, x and x, y in the respective cases remaining
constant ; it then follows from what has been stated above that this remains
true for finite increments to x or y or z separately ; and hence it may easily
be shown that C will remain constant for any concurrent linear transforma
tions of x, y, z, when the modulus is unity. This all-important principle
enables us at once to fix the form of the symmetrical functions of the roots
of (j> (-, 1 ) which represent invariants of $ (x, y) when the coefficient of the
42] On the Principles of the Calculus of Forms. 327
highest power of x is made unity. It also instantaneously gives the neces
sary and sufficient conditions to which an invariant of any given order of any
homogeneous function whatever is subject, and thereby reduces the problem
of discovering invariants to a definite form. But as these conditions coincide
with those which have been stated to me as derived from other considerations
by the gentleman whose labours in this department are concomitant with my
own, I feel myself bound to abstain from pressing my conclusions until he
has given his results to the press.
(8) By aid of the general principle enunciated in Note (6) above, we can
easily obtain Aronhold s S and T. Let U be the given cubic function of
x, y, z, and let G(x, y, z; %, TJ, ) be the polar reciprocal in respect to , 77,
O f g + y. then Q ^ ^ ^
V dx dy dzj
will be a concomitant to U, but the homonymous systems of variables in the
two G s will be contragredient ; and, accordingly,
d d d d d d
will be a concomitant to U ; this concomitant is readily seen to be an
invariant of the fourth order; that is, Aronhold s S. Again, from S, by
means of the Eisenstein-Hermite theorem, we may derive a form K (x, y, z)
of the third degree in x, y, z, and whose coefficients will be of three dimen
sions ; and, accordingly, if the Hessian of U be called H ( U),
r , f d d d\ TT- / r7N
-"- T~ T~ T } " W )
\dx dy dzj
will be a Sextic Invariant of U, that is, Aronhold s T.
43.
ON THE PRINCIPLES OF THE CALCULUS OF FORMS.
[Cambridge and Dublin Mathematical Journal, vu. (1852), pp. 179 217.]
PART I. SECTION IV. Reciprocity, also Properties and Analogies
of certain Invariants, <L~c,
IT will hereafter be found extremely convenient to represent all systems
of variables cogredient with the original system in the primitive form by
letters of the Roman, and all contragredient systems by letters of the Greek
alphabet ; the rules for concomitance may then be applied without paying
any regard to the distinction between the direction of the march of the
substitutions, the variables at the close of each operation as it were telling
their own tale in respect of being cogredients or contragredients. This
distinction has not (as it should have) been uniformly observed in the
preceding sections ; as, for instance, in the notation for emauants which have
, 7 7 v o
been derived by the application of the symbol ( -=- + 77 -=- + &c. ) , instead
\ doe dy )
of the more appropriate one \d-j- + y j~+& c -)
\ doc ^y 9
The observations in this section will refer exclusively to points of doctrine
which have been started in the preceding sections in such order as they more
readily happen to present themselves. And, first, as to some important
applications of the reciprocity method referred to in Notes (6) and (8) of the
Appendix [pp. 325, 327 above].
The practical application of this method will be found greatly facilitated
by the rule that oc, y, z, &c. may always in any combination of concomitants
be replaced respectively by ^ , -j- , -77. , &c., and vice versa. I shall apply
this prolific principle of reciprocity to elucidate some of the properties and
relations of Aronhold s S and T, and certain other kindred forms. This
S and T are the quartinvariant and sextinvariant respectively of a cubic
of three variables. I give the names of s and t to the quadrinvariant and
cubinvariant of the quartic function of two variables. Furthermore, whoever
will consider attentively the remarks made in Section II. of the foregoing
relative to reciprocal polars, will apprehend without any difficulty that to
every invariant of a function of any degree of any number of variables will
43] On the Principles of the Calculus of Forms. 329
correspond a contravariant of a function of the same degree of variables
one more in number, and that between such invariants, whatever relations
exist expressed independently of all other quantities, precisely the same
relations must exist between the corresponding contravariants. Thus, then,
to s and t the two invariants of (oc, y) 4 will correspond two contravariants
or and r of (x, y, z) 4 , and to S and T the two invariants of (x, y, z} 3 will
correspond 2 and S- two contravariants of (x, y, z, t) 3 . Calling r the resultant
of (x, y)*, R the resultant of (x, y, zf t p the polar reciprocal, or, more briefly,
the reciprocant of (x, y, z)*, and (R) the reciprocant of (x, y, z, t) 3 , we have
the following equations (presuming that all the quantities are previously
affected with the proper numerical multipliers), namely
r = s 3 + t-, p = <r 3 + r 2 ,
I propose in this First Annotation to point out the remarkable analogies
which exist between the modes of generating the four pairs of quantities
5, t, &c., the functions severally corresponding to which I shall call u, &&gt;, U, O.
The Hessian corresponding to any of these functions will be denoted by
an H prefixed, and when we have to consider, not the pure Hessian, but the
matrix formed from it by adding a vertical and horizontal border of variables,
the same in number but contragredient to the variable of the function
(as, for instance, the Hessian of u bordered with , 77 horizontally and verti
cally, or of U with , 77, ), then I shall denote the result by the ruled
symbol H, and if there be occasion to add two borders, as ,77, ; , 77 , ",
both repeated in the horizontal and vertical directions, the result will be
typified by the doubly ruled H.
Now, in the first place, as observed by me in Note (8) of the Appendix
in the last number; if we call the coefficients of (7(10 in number) a, b, c, d,
&c., we have
a ( d d d d d d f.
s =
also
dS d 3 H dS d*H &c
da da? db d 2 xdy dc d*xdz
I will now add the further important relation
+ +
da da? db d -xdy dc d 2 xdz
* It will be found hereafter convenient to designate contravariants formed in this manner
from invariants as Evects of such invariants or contravariants, and according to the number of
times that such process of derivation is applied, 1st, 2nd, 3rd, &c. evects. Such evects form
a peculiar class, and when considered generally, without reference to the base to which they
refer, they may be termed evectants. Evectants will be again distinguishable according as their
base is an invariant simply or a contravariant. Perhaps the terms pure and affected evectants
may serve to mark this distinction.
330 On the Principles of the Calculus of Forms. [43
so that it will be observed if all the derivatives of S are zero, T is zero,
and vice versa.
Precisely in the same way, using h and h to denote respectively the
Hessian of u and the same bordered with , rj, we have
r i d d d d\ r/
s = h -y- , ; -y- , h (x, y; , 77).
\a drj dx dyj
_ ds d*h ds d*h ds d*h
da dx* db dx 3 dy dc dx-dy*
, dt d*li dt d*h dt d*h
g2 . I i i PL-
da dx* db dx 3 dy dc dx 2 dy 2
Again, taking (H) the second bordered Hessian of H; that is, H bordered
as well horizontally as vertically with the double lines and columns , 77, , 6 ;
dc dx z dz + rf ~ +
^^db&B d& d 3 S
da da? + dbdx*dy +
In like manner again
,(d d d d d d
p
o- 2 =-r- , + &c.,
aa aa i4
<r and r are the same quantities as are calculated by Mr Salmon, in his
inestimable work On Higher Plane Curves, but are there expressed under
the names of S and T, with the sole difference that in place of x, y, z, used
by Mr Salmon, the contragredient variables , r) , are used in the expressions
above. Mr Salmon has also pointed out to me that a- may be obtained
by operating with
da db dc
directly upon / a cubic invariant of the function u, or (x, y, z)*. This
/ is no other than the simple commutant obtained by operating upon u
with the commutantive symbol formed by taking four times over the line
-r- , -j- , -T- , agreeable to the remark made in the third section that
43] On the Principles of the Calculus of Forms. 331
every function of an even degree of n variables possesses an invariant of
the nth order in extension of Mr Cayley s observation that every such function
of two variables possesses a quadrin variant, that is an invariant of the second
order.
I need hardly remark that cr is of 2 dimensions in the coefficients and of
4 in the contragredient variables, T of 3 in the coefficients and of 5 in the
contragredients, 2 of 4 in the constants and 4 in the contragredients, ^ of 6
in the constants and 6 in the contragredients, or that the single- bordered
Hessians of u and U and the double-bordered Hessians of a> and ft are each
of them quadratic in respect of the x &c. as well as of the &c. systems.
If the right numerical factors be attributed to S, T, Aronhold has shown
that
and in my paper in the last May Number*, I gave the equation
h {h (u)} +s.h (u) + tu = 0.
I think it highly probable that it will be found that the analogous equations
obtain, namely
h {h (to)} + a- . h (to) + TO) = 0.
These remarkable equations, if verified (of which I can scarcely doubt), will be
most powerful aids to the dissection of the forms &&gt;, ft, and thereby to the
detection of the fundamental properties of curves of the fourth and surfaces
of the third degree, of which at present so little is known. It will have been
observed that in the preceding developments the contravariants of w and ft
were derived in precisely the same way from to and ft as the corresponding
invariants of u and U from u and U, with the sole difference that the Hessian
used in the two latter cases is replaced by a single-bordered Hessian in the
two former cases, and a single-bordered Hessian in the two latter by a
double-bordered Hessian in the two former. The analogies are not even yet
stated exhaustively ; for it will be remembered (as shown in the third section),
that T and S can be derived directly and concurrently by means of operating
with the commutantive symbol
d d d
dx dy* dz
d d d
dx dy dz
\ upon
d d d
d d d
[* p. 192 above.]
332 On the Principles of the Calculus of Forms. [43
which gives a result of the form m (X 3 + S\ + T), m being a number ; and
I conjecture that if
d d d d
dx dy dz dt
d d d d
dx dy dz dt
d d d d
d d d d
dj dy d <W
be made to operate upon
(<r) what cr becomes when , ?/, are replaced by -y- , -r- , -- , (&) h (o>)
and the result be put under the form
m (X 4 + A\ A + B\ 2 +C\+ D),
that A will be zero, B and C will be respectively 2 and S-, and perhaps
D (a contravariant, if it effectively exist, of 8 dimensions in the coefficients
of fl, and of a like number in the contragredients f, ?/, , 6 ), also zero.
But of the evanescence of D I do not speak with any degree of assurance.
Mr Salmon has made an excellent observation to the effect that if we call
-r-
will represent a covariant to w of 3 + 2, that is, 5 dimensions in the coefficients,
and of 6 - 4, that is, of 2 dimensions in x, y, z, h (<w) being of 3 and 6 dimen
sions in these respectively, and <r of 2 and 4 dimensions respectively in the
same. Now these resulting dimensions 5 and 2 precisely agree with the
form especially noticed by me in Note* (2) of the Appendix, ( where it was
derived as one of a group by the method of unravelment. There can
be little doubt that these two conies each of them indissolubly connected
with every curve of the fourth degree are identical. The form (cr)A(ro)
enables us to prove readily (thanks to Mr Salmon s calculation of a, given
in his Higher Plane Curves, under the name of S) that this is a bond fide
existent conic.
For if we take a particular case of o>, say
we find
h (to) =
a^x 2 , 0,
0, b 2 y 2 + dz 2 , dyz
0, dyz, c 3 z 2 + dy 2
[* p. 323 above.]
43] On the Principles of the Calculus of Forms. 333
and o- becomes
M*?T 2 ,
and consequently (a-) is
,/dwdy
a l d(- r \ 1-5-1 ,
\dy) \dzj
and therefore
(a) h (&&gt;) = 4afc (6 2 c 3 + cZ 2 ) ^c 2 ,
the conic here reducing to a pair of coincident straight lines. This example
demonstrates that the conic is in general actually existent.
As I have said so much upon S and T it may not be irrelevant to state
in this place how I obtained the conditions for V, the characteristic of
the curve of the third degree becoming the characteristic of a conic and
a straight line, that is breaking up into a linear and a quadratic factor,
which Mr Salmon has inserted in the notes to his work above referred
to. When U is of this form it may obviously by linear transformations be
expressed by ax 3 + dxyz, but when starting with the general form,
a^o? + b$ 3 + c 3 z 3 + &c. + QDxyz,
we form two contravariants from 8 and T, to wit
and then make o^ = a, D = d, and all the other coefficients zero, it will easily
be seen on examining the forms of 8 and T, given by Mr Salmon, that (8)
and (T) (the evectants of 8 and T) become respectively
we have therefore (T) + A, ($) = : and (T) and (8), although contravariantive
to their primitive U, are covariantive with one another, so that (T) + \(S) =
is a persistent relation unaffected by linear transformations ; it follows
therefore that when U is of, or reducible to, the form supposed,
dS _ dS m dS . &c . dS
da^ db 2 dc 3 dD
= dT ,dTdT_^ c ,dT
dcii db 2 dc s dD
which is the criterion given in the note referred to*.
I am also able to obtain these equations more directly by another method
founded upon a New View of the Theory of Elimination, an account of which,
* Mr Salmon has remarked that the two evectants (S) and (T) intersect in the nine cuspidal
points of the polar reciprocal to the curve.
334 On the Principles of the Calculus of Forms. [43
however, I must reserve for another occasion, but which, I may mention,
serves to fix not merely the conditions, as in the ordinary restricted theory,
that a given set of equations may be simultaneously satisfiable by some one
system of values of the variables, but the conditions that such set of equations
may be simultaneously satisfiable by any given number of distinct systems
of variables.
Mr Salmon has remarked to me to the effect that if in T we write
-T- , -7- , -r , in place of the contragredients, and call r so altered (T),
then (r)h(a)) will be an invariant of 6 dimensions in the coefficients of <w.
This sextinvariant I have little doubt is identical with that obtained by
operating upon <o with the commutantive symbol
/rfy
d d
/ d\ 2
d d
(d\ 2
d d }
(dx)
dx dy
(dy)
dy dz
(dz)
dz dx
/dy
d d
/dy
d d
fd\-
d d
UW
dx dy
(dy)
dy dz
(dz)
dz dx)
This, like every other commutant of 2 lines only, is of course capable of being
expressed under the form of an ordinary determinant, and the remark is not
without interest, as showing how the proposition known with respect to
quadratic functions of any number of variables, namely of every such having
an invariantive determinant, lends itself to the general case of functions
of any even degree of any number of variables which also have always an
invariantive determinant attached to them, of which the terms are simple
coefficients of such functions. The only peculiarity (if it be one) of quadratic
functions in this respect being that they have each but one invariant of such
form and no other. In the case before us, if we write
the sextinvariant in question becomes representable under the form of
the determinant
2 . /, &i, m, n, I
f, b 1} 6 2 , 6 3 , d, m
I, m, b 3 , d, c 2 , n
6, 71) CL, C% } 3, C\
a 3 , I, m, n, c l , e
* This determinant is identical with the determinant formed by taking the second differential
coefficients of the function and arranging in the usual manner the coefficients of the several
powers and combinations of powers of the variables treated as if they were independent quantities.
43] On the Principles of the Calculus of Forms, 335
Before quitting the subject of 8 and T the two invariants of the cubic
function of 3 variables, or, as it may be termed, of the cubic curve, it may
not be amiss to give the complete table which I have formed corresponding
to all the singular cases which can befall such curve, which will be seen below
to be eight in number ; it is of the highest importance to push forward the
advanced posts of geometry, and for this purpose to obtain the same kind
of absolute power and authority over, and clear and absolute knowledge of,
the properties and affections of cubic forms as have been already attained for
forms of the second degree.
Let U = aa? + 4sbx?y + 4<cz?z + &c.
(1) When U has one double point S 3 + T- = 0.
(2) When U has two double points, that is becomes a conic and
right line
dSdT dSdT
T~ ~M~ - jl T~ = > <* c - ^ c -
da do do da
(3) When U has a cusp 8 = 0, T = Q.
(4) When U has two coincident double points, that is, is a conic
and a tangent line thereto, which comprises the two preceding cases in one,
dT dT
-T- = 0, -jT * 0, (XC.
da db
and also therefore 8 = 0.
(5) When U becomes three right lines forming a triangle
d*S d-T d*T d n -S
= 0, &c.
dadb dcde dadb dcde
where a, b, c, e each represent any of the coefficients arbitrarily chosen,
whether distinct or identical.
Another, and lower in degree system of equations, may be substituted
for the above, obtained by affirming the equality of the ratios between
the coefficients of U and the corresponding coefficients of its Hessian.
(6) When U represents a pencil of three rays meeting in a point
dS dS
-r- = 0, =0, &c.
da db
and also therefore T=0.
Also in place of this system may be substituted the system obtained by
taking all the coefficients of the Hessian zero.
336 On the Principles of the Calculus of Forms. [43
(7) When U becomes a line, and two other coincident lines,
dS dS
-7- = 0, -; = 0, &c.
da do
I have not ascertained whether this second system necessarily implies the
first : I rather think that it does not. In the preceding case also it would
be interesting to show the direct algebraical connexion between the system
formed by the coefficients of the Hessian and the system consisting of the
first derivatives of S.
(8) When U becomes a perfect cube representing three coincident
right lines
& 8 n & 8 _A A
^ =: dadb "
The first of these systems of equations necessarily implies the equations
T/TT jrp
_ = 0, = 0, &c., as is obvious from the equation
dS^H dS_*H +&c>
da da? ^ db dtfdy T
d*T
but not necessarily the second and lower system ^ = 0, &c. above written.
So if we take
u = ax* + 46afy + 6c#y
when 2 roots are equal
s 3 + i 2 = 0,
when 2 pairs of roots are equal
ds dt^ _ds^dt_
da db db da ~
when 3 roots are equal s = 0, t = 0,
and when all 4 roots are equal
at _ A dt _ A o
da db~ *
Before closing this Section I may make a remark, in reference to the sextic
invariant of o>, which admits of being extended to all commutants formed
by operating upon the function with a commutantive symbol obtained
writing over one another lines consisting of powers of ^, ^ , &c. and
43] On the Principles of the Calculus of Forms. 337
their combinations (to which, in the Third Section, I gave the name of
compound commutants, a qualification which, for reasons that will hereafter
be adduced, I think it advisable to withdraw). The remark I have to make
is this, namely that the invariant obtained by operating upon o> with
d\* d d I d y d_ d / d y d d_\
dx) dx dy (dy) dy dz (dz) dz dxl
_d_y d d f d y d d (d_\* d^ d [
dxl dx dy (dy) dy dz (dz) dz dx>
is precisely the same as may be obtained by operating with
d d d d d d
du dv dw dp dq dr\
d d d d d d
du dv dw dp dq dr)
upon the concomitant quadratic function to &&gt; obtained by the method of un-
ravelment, as in Note (2) of the Appendix [p. 322 above] ; and so, in general,
every commutant obtained by operating upon a function of any number
of variables of the degree 2mp with a symbol consisting of %p lines in which
the mth powers of -=- , --.- . &c. and their mth combinations occur, will
dx dy
be identical with the commutant obtained by operating with a symbol
also of 2p lines, in which only the simple powers occur of -j- , -y- , &c.
(where u, v, &c. are cogredient with a?, x p ~ l y, &c.), upon a function of
u, v, &c., formed by the method of unravelment from the given function.
Finally, before quitting the subject of reciprocity, I may state, it follows
from the general statement made at the commencement of this Section, that
inasmuch as
x + yn + z
is a universal concomitant form, so also must
d d d d d d
jfj-+-j--T + ji,j-
a dx drj dy d dz
be a universal concomitant symbol of operation ; accordingly it is certain
that any concomitant in which x, y, z, &c., , 77, , &c. enter, operated
upon with this symbol, will remain a concomitant: in several cases which
I have examined, the effect of this operation will be to produce an evanescent
form, but I see no ground for supposing that this is other than an accidental,
or at all events for supposing that it is a necessary and universal consequence
of the operation. It may also be observed that in the case of as many
cogredient sets of variables as variables in each set, as for instance 3 sets
s. 22
338 On the Principles of the Calculus of Forms. [43
of 3 variables each, the determinant which may be formed by arranging them
in regular order, as
, y, 2
", y", *
is evidently a universal concomitant, and moreover an equivocal concomitant,
possessing the property of remaining a concomitant when the variables
are respectively but simultaneously exchanged for their contragredients
V, ; > "n t " ; "> V. "; which shows also that in place of the variables
may be written the differential operators
d d d d d d d d d
dx dy dz dx" ~dy" dz ] ~dx" ~dy" dz 7
a remark which leads us to see the exact place in the general theory occupied
by Mr Cayley s method of generating covariants given in the concluding
paragraph of the First Section [p. 290 above]. I may likewise add, that
inasmuch as (x % + y t) + z + &c.) 2 is a universal concomitant,
7 . y 7
dx dy
will be so too, by virtue of the general law of interchange, which conducts
immediately to the theory of emanation, showing that this last symbol,
operating upon any function, furnishes covariants thereunto for any integer
value of z.
One additional interesting remark presents itself to be made concerning
U, the cubic function of x, y, z, which is, that calling as before T its sextic
invariant, and a, 36, 3c, d, &c. the coefficients, the formula
(& ^ + 2 4 + p ^ + ft;? $-, + &c.Y T
\ /-7^f * W/i ^ /V^ /Y/Y /
\ 061* Ct-C/ Ct O tc/ti/ /
will give the polar reciprocal, or, as it has been agreed to term it, the
reciprocant of U. I believe the remark of the probability of this being
the case originated with myself, but Mr Cayley first verified it by actual
calculation, using for that purpose the value of T, given by Mr Salmon
in his work On the Higher Plane Curves, already frequently alluded to,
which is an indispensable manual equally for the objects of the higher
special geometry as for the new or universal algebra, being in fact a common
ground where the two sciences meet and render mutual aid.
Mr Salmon also observed, that the first evect of T, namely
43] On the Principles of the Calculus of Forms. 339
was identical in form with what may be termed the first devect of the polar
reciprocal, that is, the result of operating upon the polar reciprocal with
what U becomes when -7=, -5, -7^, are substituted in the stead of x. y, z.
dg dr) d%
And inasmuch as, by Euler s law,
d \ 3 _. / d * d I d d ,
-j- + f rj -jj +
da db
-
da db
it follows that T is the second devect of the polar reciprocal, or at least
identical with it in point of form. But, since the preceding matter was
printed, I have discovered in the course of a most instructive and suggestive
correspondence with Mr Salmon, the principle upon which these and similar
identifications depend, thereby dispensing with the necessity for the exces
sively tedious labour of verification which, even in the simple example before
us, would be found to extend over several pages of work.
The theory in which this principle is involved will be given, along with
other very important matter, in the next number of the Journal.
Supplementary Observations on the Method of Reciprocity.
It has been observed, that , 77, &c. may always be inserted in place of
-j- , -T- , &c., and vice versa, in a concomitant form, without destroying
its concomitance. Accordingly, instead of the evector symbol
we may employ
/ d\ 3 d fdVdd ,
j~ ;r + ( j- j- ji. + &c -
Veto/ aa Vote/ # 6
and operating with this upon any concomitant, the result will be a concomitant.
Hence we see, for example, that if we take the concomitant 8H formed
by the product of the invariant 8 and the covariant H,
_d_\* d_ (d\* _d_ d I
dx) da \dx) dy db J
will be a covariant ; in fact this will be found to be T, the difference
between this and the expression before given for T, namely
d *,dS i d V d 7T dS
j-\ H -j- + [j-] , H .-+&C.,
\dxJ da \dx) dy db
being
* <v *i(iyi*+i4,
db\dx) dy j
22 2
340 On the Principles of the Calculus of Forms. [43
which is zero, there being no invariant to (#, y, zf of the 3rd degree in
a, b, c, &c., as the factor multiplied by 8 would be were it not evanescent.
The same observation may be extended to analogous equations given
previously.
I have chiefly, however, made the above observation with a view to
making more clear the enunciation of the theorem which I am now about
to state, the most important perhaps in its application of any yet brought
to light on the subject, but the consequences of which, as I have but quite
recently discovered it, must be reserved for a future number of the Journal.
Let any function of any number of variables be supposed to have for
its coefficients the letters a, b, &c. affected with the ordinary binomial or
multinomial coefficients; and let another function be taken identical with
the former in all respects, except in the circumstance that all their numerical
multipliers are suppressed. Let this function or form be termed the respondent
to the primitive : furthermore, by the inverse of any form understand what
that form becomes when, in place of x, y, z, &c., , ?;, , &c.,
d^ d_ d d d^ d .
da: dy dz ~d drj ~d
are respectively substituted (and so for all the systems of the variables), and
likewise at the same time similar substitutions are made of -7- , -77 , -r- , &c.,
da do dc
in place of a, b, c, &c. ; then we have this grand and simple law The inverse
of any concomitant to a respondent is a concomitant to its primitive. When
the inverse of any concomitant to the respondent is made to operate upon
the same concomitant of the primitive, it will be found that the result
is a power of the universal concomitant. If the concomitant to the respondent
be an invariant thereof, the rule indicates that on merely replacing in the
respondent a, b, c, &c. by -7- , -77 , -7- , &c., the result operating on any
invariant or other concomitant of the primitive, leaves it still an invariant
or other concomitant. For instance, if we take the function
aa? + 5ba?y + 10cx?y 2 + lOefoy + 5exy* +fy,
which has three invariants L, M, N, of the degrees 4, 8, 12, respectively :
and if we call \, p, v what L, M, N become when, in place of a, b, c, d, e,f
respectively, we write
A i^ i^. -L-l 1.1 *.
da odb 10 dc 10 dd ode df
we shall find that
and
\N = a linear function of M and L 2 .
43] On the Principles of the Calculus of Forms. 341
Again, if in the case of any function of x, y, z, &c., we take, instead of any
other concomitant to the respondent, the respondent itself, its inverse gives
the symbol of operation
d\ f d\ s . d
, M , , , v-H br; +*
da) \dxj do \dxj \dyj
just previously treated of. If again, in the case of a function of x, y, say
ax n + nbx n ~ l y + . . . + nb ay n ~ l + a y n ,
we take the inverse of the polar reciprocal of the respondent, we get the
operator
d ( d V d ( d \ n ~ l d p
T- I j~ - -jT j- -Jl- + &C - 5
and replacing -7- , -^ by y, x, we find that
y n -i y n ~ l x -jr + &c.,
1 da db
operating on any concomitant, leaves it still a concomitant, which is
M. Eisenstein s theorem before adverted to, only generalized by the in
troduction of any concomitant in lieu of the discriminant.
This extraordinary theorem of respondence will be found on reflection
to favour the notion of treating the coefficients of a general function as
themselves a system of variables, in a manner contragredient to the terms
to which they are affixed.
Finally, there is yet another mode of applying the principle of reciprocity,
which must be carefully distinguished from any previously stated in these
pages.
I have said that in place of the quantitative symbols of one alphabet, as
d d d s
x, y, z, &c., we may always substitute the operation symbols -=^, -j- , -TL., &c.
of the opposite alphabet. But now I say, in place of the quantitative symbols
x, y, z, &c. occurring in the concomitant to any form /, may be substituted
. . . dF dF
the quantities (observe, no longer operative symbols but quantities) -r^ , -7- ,
dF
-jj, , &c., F being itself any concomitant to /. Thus, for instance, taking F
dc,
identical with /, we see that /H*i 4- , T, &c.J is concomitant to /:
or again, if / be a function of x, y only, say f(x, y), taking F the polar
reciprocal of /, that is f(-rj, ), we see that/Y-^S ^J will be a
342 On the Principles of the Calculus of Forms, [43
concomitant to /: this concomitant, by the way it may be observed, will
j / 7 /
always contain / as a factor, because when /= 0, # -^ + y -/ = 0. Possibly
ct#/ ^2/
it may be true that, when / is a function of any number of variables
x, y, 2, &c., and F (%, 77, f, &c.) its polar reciprocal,
f (dF(x,y,z,bc.) dF(x, y, z, &cc.) \
f ( dx ~dy~ - ^J
which is a concomitant to / contains y as a factor ; but I have not had time
to see how this is. It is rather singular that Mr Cayley and Professor
Borchardt of Berlin have both independently made to me the observation
that, when f(x, y) is taken a cubic function of x and y, /( 7 , , ) is
\d/y dx /
equal to the product of/ by the first evectant of the discriminant of /
The general consideration of the consequences of this new and important
application of the idea of reciprocity must be reserved for a future section.
SECTION V. Applications and Extension of the Theory of the Plexus.
If $ = ax* + 4tbx 3 y + 6cx*y* + 4>dxy 3 + ey*,
we can obtain, by operating catalectically with x , y upon
, d
dx
,_d
dy
ti
dx
A
1 dy
the two concomitants
a# 2 + Zbxy + cy 2 , ba;
bo? + 2cxy + dy 2 , ex 2 -f 2dxy + ey 2
a, b, c
b, c, d
c, d, e
(1)
(2)
the one in fact being the Hessian, the other the catalecticant of < itself.
Again, if
. . . +fy 5 ,
by operating catalectically with x, y upon the second and fourth emanants,
as in the last case, we obtain the two covariants
aa? + 3bx*y + Sexy* + dy 3 , bx 3 + 3cx 2 y + 3dxy* + ey 5
bx 3 + %cx*y + Sdxy 2 + ey 3 , ex 3 + 3dx 2 y + Sexy* -f fy 3
(1)
43] On the Principles of the Calculus of Forms. 343
(2)
ax + by, bx + cy, ex + dy
bx + cy, ex + dy, dx + ey
ex + dy, dx + ey, ex +fy
which are in fact the Hessian and canon izant respectively of <. So in
general, for a function of x, y of the degree 2t or 2i + 1, we can obtain t
covariantive forms, the first being the Hessian, and the last the catalecticant
on the first supposition and the canonizant on the second : calling the index
of the function for either case n, the forms appearing in this scale will be
of the degree (r + 1) in the constants, and of the degree (r + l)(w-2r) in
x and y.
It has previously* been intimated that all these determinants admit of a
remarkable transformation.
This transformation may be expressed more elegantly by dealing not
directly with the covariant forms as above given, but with their polar recipro-
cants obtained immediately by writing for - y and 77 for x.
(1) Suppose
, = ax 3 + Zbx^y + Sexy* + dy 3 ;
a, 26, c
6, 2c, d
will be found to be the reciprocant of its Hessian.
(2) Let < = ax 4 + 4>bx 3 y 4- ... + (.
the reciprocant of its Hessian will be found to be
a, 36, 3c, d
b, 3c, 3d,
F,
(3) Let </> =
the reciprocant of its Hessian will be
a, 46, 6c,
6, 4c, 6d,
I 2 ,
4rf,
4e,
[* p. 325 above, note t]-
344 On the Principles of the Calculus of Forms. [43
and the reciprocant of its canonizant is
a, 36, 3c, d
b, 3c, 3d, e
c, 3d, 3e, f
The numerical coefficients in this and in the first case are inserted for
the sake of uniformity, but it will of course be readily observed that when
there is but one line of and ij, that the numerical coefficients being
the same for each column may be rejected without affecting the form of
the result.
So again, if
(f) = ax 6 -f Qbafy + . . . + gy K ,
the reciprocant of the Hessian is
a, 56, lOc, Wd, be, f
b, be, lOd, We, bf, g
P,
r,
and the reciprocant of the second form in the scale, which comes between
the Hessian and the catalecticant, is
a,
b,
c, d,
e
b,
c,
d, e,
f
c,
d,
/
J
9
P
Ptf,
^Trf, rf,
P,
2 ?7, rj y ,
if
and so in general. The rule of formation is sufficiently plain not to need
formulating in general terms. It is easy to see that all these forms are con
comitants to the function from which they are formed ; for example, take
then
form a plexus.
<f> =
+ . . . + gy 6 ;
43] On the Principles of the Calculus of Forms. 345
So likewise if we take ^ = (x% + yrj) 4 ,
form a plexus. But -^ and $ are concomitantive, -v/r being a universal con
comitant. Hence we may combine together these two plexuses, that is
ax 4 + 4bx*y + Qcx 2 y* + 4>dxy 3 + ey 4 \
^ifx-y* + r) 3 xy s
and, by the principle of the plexus, x 4 , x s y, x?y 2 , xy 3 , y 4 may be eliminated
dialytically, and the resultant will be the determinant last given, which is
therefore a contravariant to <f>.
The manner in which I was led to notice this singular transformation is
somewhat remarkable.
In the supplemental part of my essay On Canonical Forms [p. 203 above],
my method of solution of the problem of throwing the quintic function of
two variables under the form u 5 + v 5 + w 5 , led me to see that u, v, w are the
three factors of
ax + by, bx + cy, ex + dy
bx + cy, ex + dy, dx + ey
ex + dy, dx + ey, ex +fy
the more simple mode of the solution of the same problem, given by me in
the Philosophical Magazine for the month of November last [p. 266 above],
led to
a,
b,
c,
d
b,
c,
d,
e
c,
d,
e,
f
y 3 ,
-xy\
x*y,
-X 3
as the product of the same three factors ; whence the identity of the two
forms becomes manifest. In the paper last named I gave two proofs, one my
own, the other Mr Cayley s, of a like kind of identity for the canonizant
of any odd-degreed function of x, y in general. The proof of the identity
of the corresponding forms in the much more general proposition above
indicated [p. 325 above, footnote *f*] must be reserved until more pressing and
important matters are disposed of. In the footnote referred to I ought to
have added, in order to make the sense more clear, that the degree of the
catalecticant there referred to in respect of the coefficients would be n.
346 On the Principles of the Calculus of Forms. [43
I regret to think that there are many other typographical errors in the
earlier sections; the most unfortunate of these is in the note at page [316],
in the values of P and Q belonging to the cubic commutant dodecadic
function of x and y, the corrected values of which will be given in my next
communication. I ought also to observe, in correction of the remark made in
the footnote to page [302], that it follows as a consequence of a recent paper
by Dr Hesse in Crelles Journal, that the method given by me in the text
applied (according to what I have there termed the 1st process for obtaining
an invariant resembling the resultant) to a system of three cubic equations
(in which application only the 1st powers of -7- , -y-, -j- enter) produces for
doc d y d/z
that case also, as well as for the cases specified in the note, not a counterfeit
resemblance of, but the actual resultant itself.
Returning to the theory of the plexus of which I am about to enunciate
a most important extension, I beg to refer my readers to the last paragraph,
p. [291], in the last number of the Journal, where I have shown how to
form, under certain conditions, a determinant by combining together various
concomitants and eliminating dialytically one set of the variables, which
determinant will be concomitantive to the concomitants out of which it is
formed, and of course also therefore to their common original.
O
Now the extension of this theorem, to which I wish to call attention,
is this, that not only such determinant as a whole is a concomitant to such
original, but every minor system of determinants that can be formed out of it
will form a concomitantive plexus complete within itself to the same original.
But, much more generally, it should be observed that there is no occasion
to begin with a square determinant ; it is sufficient to have a rectangular
array of terms formed by taking the several terms of one plexus or of several
plexuses combined, provided that they are of the same degree in respect
to the variables (or to the selected system of variables, if there be several
systems), and forming out of such rectangular array any minor system of
determinants at will. Every such system will be a concomitantive plexus.
The simple illustrations which follow will make my meaning clear.
Suppose
<f) aof + tibofy + loca^y 2 + 2ldx s y 3 + I5ex~y* + 6fay 5 + gy 6 .
I have previously remarked, in the foregoing sections, that a, b, c, d, e, f, g,
the coefficients form an invariantive plexus to < ; so also we know that the
catalecticant
a, b, c, d
b, c, d, e
c, d, e, f
d, e, f, g
43] On the Principles of the Calculus of Forms. 347
is an invariant to </>. But we are now able to couple together these facts
and see the law which is contained between them ; for if we take
\~ d A ( d \*
dx) a5r*"<W*
i being any number, as for instance, if we take i = 3, we shall have as a plexus
aa? + 3bx*y + 3cxf + dy 3 ,
~bo? + Zcx^y + 3dxf + ef,
ex 3 + %da?y + Sexy 2 +ff,
dx 3 + 3ex*y
accordingly not only is the determinant
a, b, c,
gy 3 ;
d
b, c, d, e
c, d, e, f
d, e, f, g
an invariant, but also the system obtained by striking out any one line and
one column, being what I term the first minors, will be an invariantive
plexus, so too will the system of second minors
ac 6 2 , bd c 2 , ce d 2 , ad be, ae bd, be cd, &c.
form an invariantive plexus, as well as the last minors, that is, the simple
terms a, b, c, d, e, f, g. Again, we might have taken the plexus
dx dy "
which would give the array
a, b, c, d, e
b, c, d, e, f
> &&gt; &&gt; j > g i
but the minor systems of determinants herein comprised will be found to be
identical with those last considered, with the exception that the highest
system, containing a single determinant only, will now be wanting. So in
general it will easily be seen that a similar method in general, when < is
of 2i dimensions, will lead to t + 1 invariantive plexuses comprising the
given coefficients grouped together at one extremity of the scale, and the
catalecticant alone at the other ; and if < is of 2t + 1 dimensions, there will
still be i + 1 such plexuses, commencing with the coefficients as one group
and ending with a system of combinations of the (t+ l)th degree in regard
to the coefficients, which system accordingly takes the place of the cata
lecticant of the former case, which for this case is non-existent.
348 On the Principles of the Calculus of Forms. [43
As a profitable example of the application of this law of synthesis, in
its present extended form, let it be required to determine the conditions that
a function of x, y of the fifth degree may have three equal roots. In general,
let = ax 5 + 5bx*y + lOcx 3 ^ 2 + 10dx*y 3 + 5exy* +fy 5 , then < has a quadratic
and cubic covariant of which I have written at large in my supplemental
essay above referred to, being in fact the s and t (that is the quadrinvariant
and cubin variant) in respect to x, y (x, y being treated as constants) of
,d , d
Let these co variants respectively be called
Ax + By}
then I
Bx + Cy)
forms a plexus, and
ax 2 + Z/3xy + yy
fix 2 + Zyxy + 8y
will form another.
Now when a = 0, b = 0, c = 0, < will have three equal roots, and
(x -r+y -r\ <f>
V dx y dyj i
becomes
Qdy . x 2 y 2 + 4 (dx + ey) x y 3 + (ex +fy) y *,
of which the quadrinvariant in respect to x , y is easily seen to be
and the cubinvariant d 3 y 3 . Accordingly the grouping
A, B . 0,0)
B, (7 bec mes O, ^|
and the grouping
a, & 7 , 0, 0, )
L becomes , \ .
13, 7, 8 0, 0, d 3 }
Accordingly, we see that the determinant
A, B
and all the first minors of
B, C
, that is ay /3 2 , (38 7 2 , aS ^7, become zero ; but the former
P>
single quantity
A, B
B, G
being an invariant, and this last system being
an invariantive plexus, all the quantities so affirmed to be zero will remain
zero, notwithstanding any linear transformations to which <f> may be
subjected ; thus then we obtain an immediate proof of the theorem that
43] On the Principles of the Calculus of Forms. 349
when a function of x and y of the fifth degree contains three equal roots
the determinant of its quadratic covariant, which in fact is its sole quart-
invariant, and the first minors of its cubinvariant will be all separately zero.
This theorem may be made still more stringent ; for by combining
Ax* + 2Bxy + Cy\
CLX 2 + 2/3xy + yy*,
fix* + fyx + Sy 2 ,
it becomes manifest that in the case supposed all the first minor deter
minants of
A, B, C
will be zero, showing in addition to the theorem last enunciated that also
A : B : C :: a : $ : 7 :: (3 : 7 : B.
It is curious and instructive to remark that this last set of equations,
stringent as they appear, and far more than enough to express a duplex
condition, are not sufficient to imply unequivocally the existence of three
equal roots, unless we have also AC - B"- = ; for suppose < to take the form
ax 5 + fy s (b, c, d, e all vanishing) ; then it will easily be seen that
= 0,
= af, C =
If we take L, M, N a system of fundamental invariants to 0, of which all the other
invariants of
N are
are rational integer functions, then L =
A, B
B, C
and the simplest forms for M and
M=
A, B, C
, /3, 7
B. y, 8
and N=
a, 2j8, 7
a, 2/3,
/3, 27, 8,
P, 2-y,
where L and N are the discriminants of the quadratic and cubic covariants of <j> respectively, and
a linear function of M, L 2 is the discriminant of <f> itself (L, M, N being of 4, 8, and 12 dimen
sions respectively in the coefficients of <f>).
For many purposes of the calculus of forms it is desirable to have the command of cases for
which any two out of these three invariants may be made to vanish without the third vanishing ;
and it will be found that when <f> is of the form y 2 (cx 3 +fy*), L = 0, M =0 ; when <p is of the form
y (bx*+fy*), N=0, L = 0; and when <f> is of the form atf + ey*, M = 0, N=0 ; and of course when
<f> is of the form y 3 (dx 2 +fy 2 ), L = 0, M = 0, A T =0; it being obviously true in general, as remarked
by Mr Cayley, that when not less than half the roots of a function of two variables are equal,
all its invariants must vanish together.
350 On the Principles of the Calculus of Forms. [43
Consequently we shall still have all the first minors of
A, B, C
a, ft, 7
ft, 7. $
zero, although there is not even so much as a pair of equal roots in </> ; AC B 2
however, it will be observed, is not zero in this supposition.
The theory of Hessians, simple or bordered, may be regarded as one
among the infinite diversity of applications of the principle of the plexus.
Let U, V, W, &c. be any number of concomitants having the common system
of variables x, y ... z. Let % represent
, d , d , d
dx dy dz
and take
dS dS dS
dx" dy ~dz
forms a plexus; and this, combined with ^F, &c. ... %W, enables us to
eliminate dialytically x, y , z, \ ... p. The result is a Hessian of U,
bordered with
dV dV dV
dx dy " dz
horizontally and vertically, and also with
dW dW dW
dx dy dz
fec. &c.
similarly dispersed ; which Hessian, so bordered, is thus seen to be a
concomitant to U, V ... W. The Hessian, as ordinarily bordered with
, rj ... , is derived by taking for V the universal concomitant
x % + yn +
and for W (if there be a double border)
and so forth.
If V be taken identical with U, the resulting form, consisting of U
bordered with -... , has been shown* in my paper "On certain
dx dy dz
general Properties of Homogeneous Functions," in this Journal, to be equal
to the product of the simple Hessian of U and of U itself multiplied by a
[* p. 173 above.]
43] On the Principles of the Calculus of Forms. 351
numerical factor. The theory of the bordered Hessian may be profitably
extended by taking
and combining with % r F...% r Tf the plexus obtained by operating upon
S with the rth powers and products of -j- , j~ :/- an( ^ eliminating
dialytically the rth powers and products of x , y ...z . Thus if
U = ax* + 4>bx s y + Qcx 2 y 2 + 4>dxy 3 + ey* and F=
+
we obtain, by taking 8**jfU+\jV, and proceeding as indicated in the
preceding,
S
a, o,
b, c,
c, d,
d,
e,
as a concomitant to U. So again, if
we find
... +fy 5 ,
ax + by, bx + cy, ex + dy, 2
bx + cy, ex + dy, dx + ey,
ex + dy, dx + ey, ex +fy, rf
P, to, *?.
a concomitant to U.
These extensions of the ordinary theory of Hessians will be found to
be of considerable practical importance in the treatment of forms, for which
reason they are here introduced.
SECTION VI. On the Partial Differential Equations to Concomitants,
Orthogonal and Plagiogonal Invariants, &c.
In the 7th note of the Appendix to the three preceding sections* I alluded
to the partial differential equations by which every invariant may be defined.
This method may also be extended to concomitants generally. M. Aron-
hold, as I collect from private information, was the first to think of the
application of this method to the subject ; but it was Mr Cayley who com
municated to me the equations which define the invariants of functions of
[* p. 326 above.]
352 On the Principles of the Calculus of Forms. [43
two variables*. The method by which I obtain these equations and prove
their sufficiency is my own, but I believe has been adopted by Mr Cayley in
a memoir about to appear in Grelle s Journal. I have also recently been
informed of a paper about to appear in Liouville s Journal from the pen of
M. Eisenstein, where it appears the same idea and mode of treatment have
been made use of. Mr Cayley s communication to me was made in the
early part of December last, and my method (the result of a remark made
long before) of obtaining these and the more general equations, and of
demonstrating their sufficiency, imparted a few weeks subsequently
I believe between January and February of the present year.
The method which I employ, in fact, springs from the very conception of
what an invariant means, and does but throw this conception into a concise
analytical form.
Suppose, to fix the ideas,
< = ax n + nbx n ~ l y + %n(n-l) cx n ~ z y z + ...+ ly n ,
and let /(a, b, c ... 1} be any invariant to (f>.
Now suppose x to become x + ey, but y to remain unchanged ; the
modulus of the transformation,
1,
, being unity, / cannot alter in con-
0,1
sequence of this substitution ; but the effect of this substitution is to convert
<) into the form
ax n + npx n ~ l y + n (n -
where a = a, ft = b + ae, y = c + 2be + ae 2 , &c. &c.
A, = I + . . . + nbe n ~ l + ae n .
Consequently, if we make
A6 = ae, Ac = 2be + ae 2 , &c. &c.,
we have by Taylor s theorem, observing that Aa = 0,
i + A 4 + &C ) + 172
* It is extremely desirable to know whether M. Aronhold s equations are the same in form
as those here subjoined. It is difficult to imagine what else they can be in substance. Should
these pages meet the eye of that distinguished mathematician he will confer a great obligation
on the author and be rendering a service to the theory by communicating with him on the
subject: and I take this opportunity of adding that I shall feel grateful for the communication
of any ideas or suggestions relating to this new Calculus from any quarter and in any of the
ordinary mediums of language French, Italian, Latin or German, provided that it be in the
Latin character.
43] On the Principles of the Calculus of Forms. 353
and this being true for all the values of e, every separate coefficient of e
in A/ must be zero : hence we obtain n different equations by equating
to zero the coefficients of e, e" ... e n respectively. The first of these equations
will be
(a^-,+2b~ + 3c~
\ db dc dd
and it is obvious that this will imply all the rest ; for, when e is taken
indefinitely small, I (a, b. c ...) does not alter (when this equation is satisfied)
by changing a, b, c ... into a , b , c ...; consequently / (a, b , c, &c.)
will not alter, when in place of a, b , c we write a", b", c", &c., obtained
from a , b , c , &c., by the same law as a, b , c , &c., from a, b, c, &c.
Thus we may go on giving an indefinite number of increments, ey to x,
without changing the value of /. Consequently, if the equation above
written be satisfied, a priori all the rest must be so too. But there is not
any difficulty in showing the same thing by a direct method*.
For we have
.
db dc dd
an identical equation. Hence
--
db dc dd J \ db dc dd
hence
.--
db dc dd J \ db dc dd
that is
repeating the application of the symbolic operator
-
db dc
The method above given has the advantage however of being immediately applicable to
every species of concomitant, and we learn from it that concomitance, whether absolute or
conditional, is sufficiently determined when affirmed to exist for infinitesimal variations; it
cannot exist for infinitesimal variations without, by necessary implication, existing for finite
variations also ; a most important consideration this in conducing to a true idea of the nature
of invariance and the other kinds of concomitance, and in cutting off all superfluous matter from
the statement of the conditions by which they are defined.
s - 23
. 2 . 3 a j + 4>b + We -. + &c.
( da de df
+ 1.2 4 + 2&4-+&J a + 36- + &c.
-
do dc dc ad
354 On the Principles of the Calculus of Forms. [43
we obtain
V/=o,
v, db dc dd
and so on ; the numerical multipliers of the terms of the several series
within the parentheses forming the regular succession of figurate numbers
1, 2, 3, &c.
1, 3, 6, &c.
1, 4, 10, &c.
It is easy to see that these equations correspond to the results of making
the coefficients of the successive powers of e equal to zero.
I may remark, that the first instance as far as I know on record of this,
(as some may regard it rather bold) but in point of fact perfectly safe and
legitimate method of differentiating conjointly operator and operand, occurs
in a paper by myself in this Journal, Feb. 1851, " On certain General
Properties of Homogeneous Functions" [p. 165 above]; where I have applied
it in operating with
- + &c.i
upon
d
* 7 I \ ^^Q i-vot/ / j
da-i da 2
which, as I have there noticed, gives the result
\ d ,
/Vi _ Q g\ _1_ (VQ
1 ddi
The equation ( a -j-, + 2b -=- + &c] 1=0 is evidently not enough to define
V db dc J
I as an invariant; it merely serves to show that / does not alter when
in place of x we write x + ey, but this is true for any function of the
differences of the roots of the form multiplied by a suitable power of a,
namely that power which is just sufficient to cause the product to become
integer. But if we now, for convenience, write
(j) = ax n + nbx n ~ 1 y + \n (n 1) cx n ~ i y 1 + . . .
+*>,
43] On the Principles of the Calculus of Forms. 355
and form the similar equation from the other side, namely
a A + 26 ~ + 3c J ^ + &c] I = 0,
ao ac aa /
these two equations together will suffice to define any invariant, as I shall
proceed to show these are the two equations alluded to brought under
my notice by Mr Cayley. If they coexist, it follows from the method by
which I have deduced them that x may be changed into x + ey, or y into
y +fx, without / being altered, e and / having any values whatever : and
it is obvious that these substitutions may be performed, not merely alter
natively but successively, because the equations between the coefficients
are identical equations, and depend only on the form of /.
Let now as become 00+ ey, and then y become y +fx; the result of these
substitutions is to convert
x into x + efx + ey,
and y into fx + y.
Finally, let x become x+gy; then x is converted into (1 + ef) (x + gy) + ey,
and y into y +f(x + gy),
that is x becomes (1 + ef) x + (eg + efg) y,
and y becomes fx + (1 +fg) y-
The modulus of substitution it is evident, d priori, always remains unity,
and nothing would be gained by pushing the substitutions any further, as it
is clear that we may satisfy the equations
1 + ef=p, e + g + efg = q,
for all values of p, q, p , q, which satisfy the equation
pq -p q = I,
and for none other except such values ; hence / remains unaltered for any
unit-modular linear transformation of x, y, and is therefore an invariant
by definition.
If (f> be taken a function of three variables, x, y, z, and be thrown under
the form
/-* ^Tl I (ri /v> [_, A i/\ /y?l~~]. I I ft /y>2 i 9A /y-1/ I /* -j/2 \ ^-7l~~2 I /vr/^
f / _/ T^ I W/jt/x 1^ \J J U I & \^ \ Us^As 1^ ^\J(Aj<.l ^^ ^^7 / 1^ VX\^.j
and / be any invariant of $, by supposing x to become x -f ey, and giving
bi, 6 2 , c 2 , &c., the corresponding variations, and taking e indefinitely small,
we obtain
d f d d\ / d d d
d f d _ 7 d\ . p ) rn
h -j h C 2 - + 26 2 j + &c. &c. > 7 = :
aa a \ ao 2 cla 2 / j
232
356 On the Principles of the Calculus of Forms. [43
and in like manner, by arranging < according to the powers of y and of x, we
obtain two other pairs of equations : it is clear, however, that three equations
(it would seem any three out of the six) would suffice and imply the other
three. The method of demonstration will be the same as in the instance of
two variables : First, it can be shown by the method of successive accretions,
that / remaining invariable when x receives an indefinitely small increment
ey, or y an indefinitely small increment ez, or z an indefinitely small increment
ex, it will also remain invariable when these increments are taken of any
finite magnitude. Secondly, by eight successive transformations, admissible
by virtue of the preceding conclusion, x, y, z may be changed into any linear
functions of x, y, z, consistent with the modulus of transformation being unity.
And in general for a function of m variables, in partial differential equations
similarly constructed (but not however arbitrarily selected) will be necessary
and sufficient to determine any invariant : and it is clear that all the general
properties of invariants must be contained in and be capable of being educed
out of such equations.
The same method enables us also to establish the partial differential
equations for any covariant, or indeed any concomitant whatever.
Thus let
<j> = ax n + nbx n ~ l y + n (n - 1) cx n ~Y + ...+ nb xy n ~ l + a y n = 0,
and let K (a, b, c, &c. ; x, y, x , y , &c. ; ff, 17, &c.) represent any concomitant,
x, y ; x , y being cogredient, and , 77, &c. contragredient systems ; when
x, y become x + ey, y, any such system x, y becomes x + ey , y ; and any
such system as , 77 becomes 77 - el; ; and taking e indefinitely small, the
second coefficients a, b, c, &c. become a, b + ae, c + 2be, &c. as before ; hence
the equation to the concomitant becomes
d ., d d , d d | * .
and in like manner, by changing y into y + ex, results the corresponding
equation
[ a ^_ + 2& A + ...-x^-x ~+ ... + TI-J- -&c\K=0.
( db dc dy dy d% }
These two equations define in a perfectly general manner every concomitant
(with any given number of cogredient and contragredient systems) to the
form <; and the due number of pairs of similarly constituted equations
will serve to define the concomitant to a function of any given number
of variables^.
* For we have
K (a, b + ae, c + 2be, &c. ; x, y, &c. ; , 17, &c.)
= K(a, b, c, &c. ; x, x + ey, &c.; , 7?-ef, &c.; &c.).
t Vide Note (10) [p. 361 below].
43] On the Principles of the Calculus of Forms. 357
In like manner we may proceed to form the equations corresponding
to what may be termed conditional concomitants, whether orthogonal or
plagiogonal. The concomitants previously considered may be termed absolute,
the linear transformations admissible being independent of any but the one
general relation, imposed merely for the purpose of convenience, namely of
their modulus being made unity. An orthogonal concomitant is a form
which remains invariable, not for arbitrary unit-modular, but for orthogonal
transformation, that is for linear substitutions of a?, y ...z, which leave
unchanged x 2 + y z + . . . + z" : in like manner, a plagiogonal concomitant may
be defined of a form which remains invariable for all linear substitutions
of x, y . . . z, which leave unaltered any given quadratic function of x, y . . . z.
Thus, let it be required to express the condition of Q (a, b, c ... x, y ; , ?/),
being an orthogonal concomitant to the form
ax n + nbx n ~ l y + . . . + nb xy n ~ l + ay n .
Let x become x + ey, e being indefinitely small, then y must become y - ex,
and the variations of a,b...b , a will be the sum of the variations produced
by taking separately x + ey for x and y - ex for y. Hence the one sole
condition for Q being of the required form becomes
-fa 4+26 jU...
V db dc
or, as it may be written, 0Q - o>Q = 0, where OQ = 0, eoQ = are the two
equations expressing the conditions of Q, being an unconditional or absolute
concomitant ; and so in general if be a function of ra variables, we may
obtain ^ra(ra-l) equations of the form L-M=0 for the concomitant,
of which however (ra- 1) only will be independent.
Supposing, again, the substitutions to which x, y are subject to be
conditioned by la? + 2mxy + nf remaining unalterable, or which is a more
convenient and only in appearance less general supposition by # 2 + 2mxy + y*
remaining unalterable, the general type of an infinitesimal system of substi
tutions will be rendered by supposing x, y to become (1 + me) x + ey,
_ ex + (i _ me j y } respectively, for then x n ~ + 2mxy + f becomes
(1 - ra 2 e 2 ) # 2 + {2m + (2m - 2t a ) e*} xy + (1 - ra 2 e 2 ) y\
which differs from x 2 + 2mxy + y- only by quantities of the second order
of smallness which may be neglected, and \ and 77 will therefore become
(l-me) %-er), -ex + (l + me)y, respectively: then, as to the coefficients
of </>, in addition to the variations which they undergo when ra is zero, there
will be the variations consequent upon x assuming the increment mex, and y
358 On the Principles of the Calculus of Forms. [43
the increment mey : but by making x become a; + mex, a, b, c, &c., b , a
assume respectively the variations
n . mea, (n 1) meb, . . . meb , 0, respectively ;
and by making y become y mey, the corresponding variations become
0, meb, ... (n 1) meb , n . mea , respectively.
Hence the equation becomes
where 6 and G> have the same signification as before, and where \ denotes
and fi denotes
d IN , d ,, d d d
na-r- +(n Ijo-yr 4- ... +6 - Jr , + x-, -jg.,
da v d6 ao dx * at
, d _ d , d d d
6-TJ- + 2c-j-+ +na -, , y-j- + tj -y- .
do dc da J dy arj
If there be several systems of x, y or of , 77, or of both, the only difference
in the equation of condition will consist in putting
d \ -? ( ^_\ W ^.\ v ( d\
dx) \ dy) \ dx/ * v dy)
d\ ( d\ ^ ( <, d\ - / d
instead of the single quantities included within the sign of definite sum
mation.
Fearing to encroach too much on the limited space of the Journal,
I must conclude for the present with showing how to integrate the general
equation to the orthogonal invariant of <f>, the general function of x, y.
Beginning with < = ax 2 + Zbxy + cy 2 , the equation becomes
; d . d ^, d d
2b- r + (a-c)- T r + 2b- r + y- r -x- r
da db dc y dx dy)
Write now
da = 2bdd, dx = ydd,
db = (a c) d&, dy = xdO,
dc = + 2bd0 ;
we have then
\da + pdb + vdc = dd {//.a + 2 (v X) b /AC}.
Let /A = K\, 2(v \) = Kfji, //, = KV ;
then d log (Xa + fjb 4- vc) = icd6 ;
or Xa + /j.b + vc be* e .
43] On the Principles of the Calculus of Forms. 359
To find K we have the determinant
*, -1,
2, *, -2 =0,
0, 1, *
that is, K 3 + 4ttc = 0,
and calling the three roots of this equation x lt K Z , 8 . we have
K-L = 0, K Z = 2t, K 3 = 2i ;
accordingly we may put
or K = 2t, X=l, ft = 2t, v = 1,
or K = 2t, X, = 1, /A = 2t, v = 1 .
Again, pdx + qdy = (py qx) d6 ;
and putting -q = ep, p = eq, so that px + qy = Ee* 9 ,
e? = l, e-i- i, e 2 = i;
and we may put
e = t, p = l, q = -<<,
or e = - i, p = 1, q = + i
Consequently the complete integral of the given partial differential equation
is found by writing
a + c = l, x-iy- Ee i6 ,
a + 2ib -c = l e* e , x + iy = E e~ l \
a-2ib-c = l"e-~ i6 .
By means of these five equations, after eliminating 0, we may obtain four
independent equations between a, b, c ; x, y. Suppose
Q 1 = 0, & = 0, Q 3 = 0, Q 4 = 0;
then Q = F(Q li Q 2 , Q 3 , Q,) is the complete integral required.
Pursuing precisely the same method for the general case, it will be found
that, calling the degree of the given function n when n is even, the equation
in K to be solved will be
and when n is odd (say 2wi + 1), the equation in K to solve will be
360 On the Principles of the Calculus of Forms. [43
and performing the necessary reductions, and calling the roots of the
equation, arranged in order of magnitude, KI I, K S I ... Kn i, respectively, it will
be found that the equations containing the integral become
x ty = Ee l6
x + iy = E e- ld
where l lt 1 2 ... l n+1 - E, E are arbitrary constants, and where L lt L 2 ... L n+l
are the values assumed by the 1st, 2nd ... (n + l)th coefficients of the given
function <, or
ax n -f nbx n ~ 1 y + . . . + nb xy n ~ i + a y n ,
when it is transformed by writing x+iy in place of x, and y+ix in place oft/.
i is of course employed in the foregoing according to the usual notation to
represent V( !) The same method applies to the general theory of plagio-
gonal concomitants, where the linear substitutions are supposed such as to
leave Ix* + 2mxy + ny z unaltered in form, and the equations in 6 which
contain the integral present themselves under a similar aspect. But a more
full discussion of these interesting integrals must be reserved until the
ensuing number of the Journal.
NOTES IN APPENDIX.
(9) The scale of covariants to a function of (x, y} obtained by the
method of unravelment [on p. 297 above], may be otherwise deduced
in a form more closely analogous to that of the corresponding theorems
for the corresponding invariantive scale [on p. 295 above], by a method
which has the advantage of exhibiting the scale equally well for the case
of functions of the degree 4t + 2 or 4u + 4, the only difference being that
in the latter case the coefficients of the odd powers of X will be found all
to vanish, so that the degrees of the covariants will rise by steps of 4 instead
of by steps of 2, just conversely to what happens in the invariantive scale;
whereas in the invariantive scale alluded to the forms containing odd powers
of X vanish when the degree of the function is of the form 4t + 2, but do not
vanish when it is of the form 4t. This method in the form here subjoined
is a slight modification of one suggested to me by my friend Mr Cay ley.
Let F be the given function of x, y of the degree 2n ; take the systems
x , y ; x l} y l cogredient with one another and with x, y. Then form the
concomitant
,d_ ,
dx <
43] On the Principles of the Calculus of Forms. 361
Then (by what may be termed the Divellent method, which has been pre
viously applied by me in the Philosophical Magazine for Nov. 1851)
calling , 6 lt 2 ...0 n , the coefficients of
x n , x n ~ l , y , ... y n in K,
we shall have
A A rrtTl _l_ 7? rf.n1 n, _|_ T. y n
C/o = *l$iw T D^ y ~r . . . T^ LJ oy )
r->y+...+L n y n ,
the coefficients being functions of the coefficients of / and of quadratic
combinations of x l} y 1} affected with the multiplier A ; and the determinant
A,, B ...L
A,, B....L,
A n , -tf n ... L n
will give a function of A in which the coefficients of the several powers of A
will be all zero or covariants of F.
The actual form of this determinant is not here given for want of space
and time, but will be exhibited hereafter. Precisely an analogous method
applies to obtain the scale to (x, y, z)* given in Note (2) [p. 322 above].
Calling F=(x, y, z}\ let the systems x , y , /; x lt y,, ^, be taken cogredient with
one another and with x, y, z. Then, using R to express the determinant
x, y , z ,
x,
2/1
and making
~~ v" doc y dy dzj
and proceeding as above by the divellent method, we obtain the scale required.
(10) [p. 356 above.] It is obvious that these denning equations ought
to give the means of discovering and verifying all the properties of con
comitants ; but it is very difficult to see how in the present state of analysis
many of the general theorems that have been stated, readily admit of being
deduced from them.
The comparatively simple but eminently important theory of the evector
symbol does however admit of a very pretty verification by aid of these
equations. Thus, suppose any concomitant ; suppose a contravariant to a
function F of x, y, say
ax 11 + nl)x n ~ l y + ... + nb xy n ~ l + a y n .
362 On the Principles of the Calculus of Forms. [43
Then d must satisfy the two equations
where - a + 26 + ... + n& ,,
i = a -^, + 2& -^ + ... + n&^-.
d& rfc aa
Now let </> = % (#) where
*=^+^ I 4 + ?"- ! ! s + ---
then L(xff)=Ti(L8)-(x.L)8
Hence z + f%W = X i + f * -X>>-0.
Similarly fi + 1)4) % (0) = 0.
\ <*S
Hence if ^ is an integral of the two conditioning equations, so also is % (0).
In like manner, if 6 be a covariant or any other kind of concomitant of F,
it may be proved that its evectant % (6) is the same.
(11) [p. 331 above.] Very much akin with the supposed equations is the
following most remarkable equation, which can be proved to exist. Let
be a function of x and y of the 5th degree. Let P and Q be the quadratic
and cubic covariants of <. P is of two dimensions in the coefficients and
also in the variables, and Q of three dimensions in both ; they are in fact
the s and t (in respect to x and y) of (x ^+y gA f Then, giving P and
Q proper numerical factors, it will be found that
H 2 <f> + PH<j> + Q<f> = Q.
I believe that a similar equation connects any function of x and y above
the 3rd degree with its first and second Hessians. The proof will be given
in a subsequent Section, where also I shall give a complete proof, which
occurred to me immediately after sending the preceding note to the press,
of the complete Theory of the Respondent by means of the general equations
of concomitance.
43] On the Principles of the Calculus of Forms. 363
P.S. Since the preceding was in type, I have ascertained the existence
and sufficiency of a general method for forming the polar reciprocal and
probably also the discriminant to functions of any degree of three variables
by an explicit process of permutation and differentiation. In particular
I am enabled to give the actual rule for constructing the polar reciprocal
and the discriminant curves of the 4th and oth degrees. So far as regards
the polar reciprocal of curves of the 4th degree M. Hesse has already given
a method of obtaining it, but mine is entirely unlike to this, and rests upon
certain extremely simple and universal principles of the calculus of forms.
The only thing necessary to be done in order to carry on the process to
curves of the 6th or higher degrees, is to ascertain the relation of the
discriminants of functions of two variables of those respective degrees to such
of the fundamental invariants as are of an inferior order to the discriminant.
The theory applies equally well to surfaces and to functions of any
number of variables, and may, I believe, without any serious difficulty be
extended so as to reduce to an explicit process the general problem of
effecting the elimination between functions of any degree and of any number
of variables. The method above adverted to will appear in a subsequent
Section.
[Continued pp. 402 and 411 below.]
44.
SUR UNE PROPRltiTE NOUVELLE DE LIQUATION QUI
SERT A DETERMINER LES INEGALIT^S SEOUL AIRES
DES PLANETES.
[Nouvelles Annales de Mathematiques, XL (1852), pp. 438 440.]
[Extract.]
6. Soit le determinant carre symetrique
(M)
dans lequel on a, d apres la definition,
filevant le determinant a la puissance p, on obtient le determinant
A A A \
"I,!! - a -l,2 **l,*l
-^-2,1> -^2,2 -"-2,n
; .n. Wi i, -"-71,2 **-n,n
et ce determinant est symetrique aussi par rapport a la diagonale A lt i,
-"-2,2 -"-n,n-
Retranchant de chaque terme de la diagonale symetrique de (M) la meme
quantite X, on obtient le determinant
, 1 > 2, 2
44]
Sur line propriete nouvelle.
365
Developpant ce determinant et ordonnant par rapport a X, on obtient une
expression qui, ttant egalee a zero, donne I equation
x -/X 1 - 1 -f g\ n ~ 2 +...(- l) n t = 0, (1)
Equation qui a n racines rdelles (voir t. x. p. 259).
Retranchant de chaque terme de la diagonale syme trique du determinant
(N) la quantite /j,, et operant comme ci-dessus, on parvient a I equation
^n _ FfJI -i + G>n- 2 + ...(-!) T - 0, (2)
equation qui a aussi n racines reelles. Les racines de cette equation sont
les racines de I equation (1), elevees chacune a la puissance p.
Demonstration. Representons par
PI
PS---PP,
les p racines de I equation ^-1=0. Ecrivons le determinant
et faisons q e gal successivement a tons les nombres de la suite 1, 2, 3 ...p,
on aura p determinants ; le produit de tous ces determinants reste evidem-
ment le meme dans quelque ordre qu on prenne ces determinants, et, d apres
les proprie te s connues des racines de 1 unite, tous les termes en p qui ne
seront pas eleves a une puissance p disparaitront, et X accompagnant toujours
p, il ne reste done que des X*, et le de terminant-produit sera
-1,1 \ P , -4l,2> -^-1,3 "If*
A n>1 , A n ,i, A n>n -\P
ou, faisant abstraction de X, on a le determinant (N). Ainsi
C. Q. F. D.
7. Application.
determinant
?i = 2, et 2? = 2;
a, b
b, c
elevant ce determinant au carre\ on a
a 2 + b 2 , ab + be
ab + be, 6 2 + c 2
(M)
(N)
366
determinant
determinant
Faisons
Sur une propriete nouvelle.
a \, b
b, c X
a 2 + 6 2 - 11, ab + be
ab + bc, 6 2 + c 2 -
- (a 2 + c 2 + 2& 2 )//, + (ac - 6 2 ) 2 = 0, ou p = X 2 .
n = 2, p = 3,
[44
(P)
(1)
(2)
(M) ne change pas, et Ton a
a 3 + 2a& 2
6 s
(N)
a-b + abc + b 3 + 6 2 c, ab 2 +
le determinant (P) et 1 equation (1) restent les memes; mais 1 equation
(2) devient
p? - (a 3 + c 3 + 3a6 2 + 3c& 2 ) /A + (ac - 6 2 ) 3 = 0,
ou /A = X 3 ,
car, \! et X 2 etant les deux racines de 1 equation (1), on a
V + X 2 3 = a 3 + c 3 + 3a6 2 + 3c6 2 , V X 2 3 = (ac - 6 2 ) 3 .
8. M. Sylvester fait observer que son theoreme est un cas particulier
d un the oreme plus general, de montre par M. Borchardt, pour des determi
nants quelconques, et qui devient le theoreme demontre ci-dessus, lorsque
le determinant est symetrique (Journal de Mathematiques, t. xn. p. 63, 1847).
45.
ON A REMARKABLE THEOREM IN THE THEORY OF EQUAL
ROOTS AND MULTIPLE POINTS.
[Philosophical Magazine, III. (1852), pp. 375378.]
IN order that the theorem which I propose to state may be the more
easily understood, and with the least ambiguity expressed, I shall commence
with the case of a homogeneous function of two variables only, a; and y.
Let
(f> = ax n + ribx n ~ l y + ^n(n l) cx n ~^y i + ... + nb xy n ~ l + a y n ,
and let the result of operating with the symbol
~ ~
n ~ l
x n -j- + x n ~y TV + . . . + y n ~x ^r, + y n -j-,,
da * do do da
on any function of a, b, c ... b , a be called the Evectant of such function,
and the result of repeating this process r times the rth Evectant.
Understand by the multiplicity of the equation the number of equalities
between the roots that exist ; so that a pair of equal roots will signify a
multiplicity 1, two pairs of equal roots, or three equal roots a multiplicity 2;
a pair of equal roots and a set of three equal roots, a multiplicity 1 + 2 or 3,
and so on. Now suppose the total multiplicity of </> to be ra : the first part
of the proposition consists in the assertion that the 1st, 2nd, 3rd ... (m l)th
Evectants of the discriminant of </>, that is of the result of eliminating x and
y between -=?- , -^ (as well as the discriminant itself), will all vanish in
dx ay
whatever way the multiplicity is distributed ; the second part of the
proposition about to be stated requires that the mode should be taken
into account of the manner in which the multiplicity (m) is made up.
Suppose, then, that there are r groups of roots, for one of which the
368 On a remarkable Theorem in the [45
multiplicity is m ls for the second m 2 , &c., and for the rth m r , so that
fttj -f- ?w 2 + . . . + m r = ??i. Then, I say, that the ??ith evectant of the deter
minant of <f> is of the form
b. 2 y) m " n . . . (arX + b r y) m r n ,
where a l : b 1 , a^:b z ... a r : b r are the ratios of x : y corresponding to the several
sets of equal roots.
This latter part of the theorem for the case of m = 1 was discovered
inductively by Mr Cayley, by considering the cases when < is a cubic,
or a biquadratic function. I extended the theory to functions of any
number of variables, and supplied a demonstration, that is for the case
of one pair of equal roots. Mr Salmon showed that my demonstration could
be applied to the case of two pairs of equal roots, or two double points,
&c., and very nearly at the same time I made the like extension to the case
of three equal roots, cusps, &c., and almost immediately after I obtained
a demonstration for the theorem in its most general form. This demon
stration reposes upon a very refined principle, which I had previously
discovered but have not yet published, in the Theory of Elimination.
I have here anticipated a little in speaking of the theorem as applicable
to curves and other loci.
Suppose (/> (x, y, 2) = to be the equation to a curve expressed homo
geneously.
Let
$ (x, y, z) = ax n + (na x n ~ l y + nb x n ~ l z}
+ n ( w _ 1) a V-y + n (n - 1) b"x n ~ 2 yz + %n (n - 1) c"x n -*z\
+ &c. &c.,
and understand by the evectant of any quantity the result of operating upon
it with the symbol
Clf CL CL o Cv ,o
x n -r- + x n ~^y -T-, + x n ~ l z -JT-, + x n z y -7-77 + &c.
da da do aa
Suppose, now, the curve to have double points, the (r l)th evectant
(and of course all the inferior evectants) of the discriminant of <f> (meaning
thereby the result of eliminating x, y, z between -=- , -5- , -j- ) will
CLOO (-by CL2 /
all vanish, and the rth evectant will be of the form
... x (ar% + b r y + c r z) n ,
where a^-.b^-.c-i, a 2 :b t :c 2 ...a r :b r :c r are the ratios of the coordinates at
the respective double points. If there be cusps the multiplicity of each
45] Theory of Equal Roots and Multiple Points. 369
such will be 2; and calling the total multiplicity m, to every cusp will
correspond a factor of the 2wth power in the with evectant ; and so on in
general for various degrees of multiplicity at the singular points respectively.
The like theorem extends to conical and other singular points of surfaces ;
so that there exists a method, when a locus is given having any degree of
multiplicity, of at once detecting the amount and distribution of this multi
plicity, and the positions of the one or more singular points. In conclusion
I may state, that precisely analogous results (mutatis mutandis) obtain,
when, in place of a single function having multiplicity, we take the more
general supposition of any number of homogeneous functions being subject
to the condition of pluri-simultaneity, that is being capable of being made
to vanish by each of several different systems of values for the ratios between
the variables. Multiplicity in a single function is, in fact, nothing more
nor less than pluri-simultaneity existing between the functions derived from
it by differentiating with respect to each of the given variables successively.
But as I purpose to give these theorems and their demonstration, which
I have already imparted to my mathematical correspondents, in a paper
destined for reading before the Royal Society, I need not further enlarge
upon them on the present occasion.
P.S. In the above statement I have spoken only of cusps of curves which
are the precise and unambiguous analogues of three coincident points in
point-systems, in order to avoid the necessity of entering into any disquisition
as to the species of singularity in curves or other loci corresponding to
higher degrees of multiplicity in point-systems, a subject which has not
hitherto been completely made out. I may here also add a remark, which
gives a still higher interest to the theory, which is (to confine ourselves, for
the sake of brevity, to functions of two variables), that if any root of x : y,
say a: b, occur I + p times, the total multiplicity of the equation being
supposed m, and its degree n, then taking t any integer number not exceed
ing p, the (m + t)th evectant of the discriminant will contain the factor
(oa? + 6y) (**- ) S that, for instance, if there be but a single group of equal
roots, and they be 1 + p. in number, every evectant up to the (//. l)th
inclusive will vanish, and from the /*th to the (2/i - t)th will contain a power
of (CUB + by)".
46.
OBSERVATIONS ON A NEW THEORY OF MULTIPLICITY.
[Philosophical Magazine, ill. (1852), pp. 460467.]
IN the Postscript to my paper in the last number of the Magazine,
I mis-stated, or to speak more correctly, I understated the law of Evection
applicable to functions having any given amount of distributive multiplicity.
The law may be stated more perfectly, and at the same time more concisely,
as follows. Every point represented by the coordinates a ly ft l ...yi, for
which the multiplicity is m^ will give rise in every evectant* of the discrimi
nant of the function to a factor (a& + fry + ... + y^)^, n being supposed
to be the degree of the function. Hence if there be r such points, for which
the several multiplicities are ra^ ra 2 ...ra r , every evectant must contain
(raj + ra 2 + . . . + m r ) n linear factors ; and as the tth evectant is of the degree
in, it follows that all the evectants below the (m 1 + m 2 + ... +m r )th evectant
must vanish completely, and this Evectant itself be contained as a factor
in all above itf. When a function of only two variables is in question, there
is no difficulty in understanding what property of the function it is which
is indicated by the allegation of the existence of multiplicities m 1( w 2 ... m r ;
* Frequent use being made in what follows of the word Evectant, I repeat that the evectant
of any expression connected with the coefficients of a given function (supposed to be expressed
in the more usual manner with letters for the coefficients affected with the proper binomial or
polynomial numerical multipliers) means the result of operating upon such expressions with a
symbol formed from the given function by suppressing all the binomial or polynomial numerical
parts of the coefficients to be suppressed, and writing in place of the literal parts of the coeffi
cients a, b, c, &c. the symbols of differentiation ^, ^, ^, &c.; in all that follows it is the
successive evectants of the discriminant alone which come under consideration. I need hardly
repeat, that the discriminant of a function is the result of the process of elimination (clear from
extraneous factors) performed between the partial differential quotients of the function in respect
to the several variables which it contains, or to speak more accurately, is the characteristic of
their coevanescibility.
t The constitution of the quotients obtained by dividing all the other evectants of the
discriminant by the first non-evanescent one, presents many remarkable features which remain
yet to be fully studied out, and promise a wide extension of the existing theory.
46] On a new Theory of Multiplicity. 371
as already remarked, this simply means that there are r distinct groups
of equal roots, such groups containing 1 + m lt l+m 2 ...l+m r roots re
spectively. So for curves and higher loci, the total distributive multiplicity
is the sum of the multiplicities at the several multiple points. But the true
theory of the higher degrees of multiplicity separately considered at any
point remains yet to be elaborated, and will be found to involve the considera
tion of the theory of elimination from a point of view under which it has
never hitherto been contemplated.
Confining our attention for the present to curves, we have a clear notion
of the multiplicity 1 : this is what exists at an ordinary double point. As
well known, its analytical character may be expressed by saying that the
function of x, y, z, which characterizes the curve, is capable, when proper
linear transformations are made, of being expanded under the form of a series
descending according to the powers of z, such that the constant coefficient
of the highest power of z, and the linear function of x, y, which is the
coefficient of the next descending power of z, may both disappear. Again,
when the multiplicity is 2, the third coefficient, which is a quadratic function
of x and y, will become a perfect square. This is the case of a cusp, which,
as I have said, is the precise analogue to that of three equal roots for a function
of two variables. Before proceeding to consider what it is which constitutes
a multiplicity 3 for a curve, it will be well to pause for a moment to fix the
geometrical characters of the ordinary double point and the cusp.
If we agree to understand by a first polar to a curve the curve of one
degree lower which passes through all the points in which the curve is met
by tangents drawn from an arbitrary point taken anywhere in its own plane,
we readily perceive that at an ordinary double point all the infinite number
of first polars which can be drawn to the curve will intersect one another
at the double point. Again, at a cusp all these polars will not only all
intersect, they will moreover all touch one another at the cusp. Now we
may proceed to inquire as to the meaning of a multiplicity of the third degree,
which, strange to say, I believe has never yet been distinctly assigned by
geometricians.
This is not the case of a so-called triple point, that is a point where three
branches of the curve intersect. Supposing x = 0, y = 0, to represent such a
point, the characteristic of the curve must be reducible to the form
(go? + ha?y + kxf + ly 3 ) z n ~ 3 + &c.,
which, as is well known, involves the existence of four conditions. This,
however, would not in itself be at all conclusive against the multiplicity at a
triple point being only of the third degree ; for it can readily be shown that
there may exist singular points of any degree of singularity (as measured
by the number of conditions necessary to be satisfied in order that such
242
372 On a new Theory of Multiplicity. [46
singularity may come into existence), but for which the multiplicity may be
as low as we please ; as, for instance, if at a double point (which is not a cusp)
there be a point of inflexion on one branch or on both, or a point of undulation,
or any other singularity whatever, still provided there be no cusps, the
multiplicity will stick at the first degree and never exceed it ; for only the
discriminant itself will vanish on these suppositions, but no evectant of the
discriminant. The reason, on the contrary, why a so-called triple point
must be said to have a multiplicity of the degree 4, and not merely of the
degree 3, springs from the fact that the 1st, 2nd and 3rd evectants of the
discriminant all vanish at such a point.
It is clear, then, that there ought to exist a species of multiplicity for
which the 1st and 2nd evectants vanish, but not the 3rd. In fact, as at a
double point the first polars all merely intersect, but at a cusp have all
a contact with one another of the first degree, so we ought to expect that
there should exist a species of multiple point such that all the first polars
should have with each other a contact of the second degree (or if we like so
to say, the same curvature) at that point. When the curve has a triple point,
all its first polars will have that point upon them as a double point ; and it
is not at the first glance, easy a priori to say what is the nature of the
contact between two curves which intersect at a point which is a double
point to each of them : we know upon settled analytical principles, that when
one curve having a double point is crossed there by another curve not having
a double point, that the two must be said to have with one another, a contact
of the 1st degree ; and we now learn from our theory of evection, that if each
have a double point at the meeting-point, the degree of the contact must
from principles of analogy be considered to be of the 3rd degree*. Now, then,
we come to the question of deciding definitely what is a multiple point for
which the degree of multiplicity is 3. It is, adopting either test, whether
of first polar contact or of evection, a cusp situated or having its nidus, so to
say, at a point of inflexion. In other words, x = 0, y = will be a point
whose multiplicity is intermediate between that of the cusp and that of
a so-called triple point, when the characteristic of the curve admits of being
written under the form
y*) + g^-t & C . \
or in other words, when over and above the vanishing of the constant and
linear coefficients, and the quadratic coefficient being a perfect square,
as in the case of an ordinary cusp, this square has a factor in common with
the next (the cubic) coefficient ; or again, in other words, a curve has a point
* This may easily be verified by direct analytical means ; as also the more general pro
position, that two curves meeting at a point where there are m branches of the one and n
branches of the other, must be considered to have mn coincident points in common, that is, if
like so to express it, to have a contact of the degree mn - 1.
46] On a new Theory of Multiplicity. 373
for which the multiplicity is 3 when its characteristic function admits of being
expanded according to the powers of one of the variables, in such a manner
that the first coefficient and the second (the linear) coefficient vanish, and
that the discriminant of the third and the resultant of the third and fourth
are both at the same time zero. This being the case, it may be shown that
the first polars will all have with each other a contact of the second degree ;
and moreover, that all the evectants of the discriminant will have as a
common factor a linear function of the variables, raised to a power whose
index is three times that of the characteristic function. As, then, there is but
one kind of ordinary double point, and but one kind of point with multiplicity
2, so there is one, and only one, kind of point with a multiplicity 3. A cusp
is a peculiar double point ; a flex-cusp (as for the moment I call the point
last above discussed) is a peculiar cusp. This law of unambiguity, however,
appears to stop at the third degree. A so-called triple point (which ought
in fact to be called a quintuple point) is a point for which the multiplicity,
as shown above, is of the fourth degree ; but it is not the only point of that
degree of multiplicity. Without assuming to have exhausted every possible
supposition upon which such a degree of multiplicity may be brought into
existence, it will be sufficient to take as an example a curve whose character
istic is capable of assuming the form
z n-* x * + z n-3 (gtf + fjtfy} _|_ ^n-4 (Jctf + I x 3y + m tfyi + nXlf) + Z^ 5 &C.
It may readily be demonstrated that the first polars of this curve have
all with one another at the point x, y a contact of a degree exceeding the
2nd, that is of at least the 3rd degree (and, I believe, in general not higher).
Now the point x, y is evidently not a triple-branched point, but a cusp with
three additional degrees of singularity ; so that we have evidence of the
existence of a point whose degree of singularity is 5, and whose multiplicity
is at least 4, but which is in no sense a modified triple point. It is probably
true (but to demonstrate this requires a further advance to be made than has
yet been realized in the theory of the constitution of discriminants) that a
cusp may be so modified by the nidus at which it is posited, as, without ever
passing into a triple point, to be capable of furnishing any amount of mul
tiplicity whatever, curiously in this contrasting with an ordinary double point,
no amount whatever of extraordinary singularity imparted to which, or so to
speak, to its nidus, can ever heighten its multiplicity so as to make it surpass
the first degree without first converting it into a cusp. I may illustrate the
nature of a flex-cusp by what happens to a curve of the third degree. When
it breaks up into a conic and a right line, there are two ordinary double points;
for the existence of these double points, as for the existence of a cusp, two
conditions are required. When, however, the right line and conic touch one
another (a casus omissus this in the works of the special geometers), the
characters of the cusp and the point of inflexion are combined at the point
374 On a new Theory of Multiplicity. [46
of contact ; the multiplicity is of the third degree, and the singularity also
of a degree not exceeding this ; three conditions only being necessary to
be satisfied in order that a given cubic may degenerate into such a form ;
and it will be found that the discriminant and the first and second evectants
thereof vanish for this case, and that the third evectant of the discriminant
will be a perfect 9th power; whereas in order that the cubic may have a
so-called triple point, that is may degenerate into a trident of diverging rays,
four conditions must be satisfied, and it will be found that when this is the
case, the first, second, and third evectants of the discriminant will all vanish,
and the fourth will be a perfect 12th power of a linear function of the
variables. I may mention, by the way, at this place, that the law of a
discriminant and the successive evectants up to the rath inclusive, all
vanishing, may be expressed otherwise (not in identical, but in equivalent
or equipollent terms), by saying that the discriminant and all its derivatives
of a degree not exceeding the mth will all vanish understanding by a
derivative of the discriminant any function obtained from the discriminant
by differentiating it any specified* number of times with respect to the
constants of the function to which it belongs, the same constants being
repeated or not indifferently*. And very surprising it must be allowed
to be, stated as a bare analytical fact, that (in + 1) conditions imposed upon
the coefficients of a function of any number of variables and of any degree
should suffice to make the inordinately greater number of functions which
swarm among the derivatives of the mth and inferior degrees of the dis
criminant each and all simultaneously vanish.
Without pushing these observations too far for the patience of the general
reader, it may be remarked by way of setting foot with our new theory upon
the almost unvisited region of the singularities of surfaces, that by the light
of analogy we may proceed with a safe and firm step as far as multiplicity
of the third degree inclusive.
The function characteristic of the surface being supposed to be expressed
in terms of the four variables as, y, z, t, and expanded according to descending
powers of t, then when as, y, z is an ordinary double point of the first degree
of multiplicity, the constant and the linear coefficient disappear ; when the
point has a multiplicity 2, the discriminant of the quadratic coefficient
will be zero, that is this coefficient will be expressible by means of due linear
transformations under the form of x 2 + y-\ arid when the multiplicity is to be
of the degree 3, the cubic coefficient will, at the same time that the quadratic
coefficient is put under the form as 2 + y*, itself (for the same system of x and
y) assume the form of a cubic function of x, y, z, in which the highest power
of z, that is z 3 , will not appear ; or in other words (restoring to x, y, z their
* Or, to speak more simply, the discriminant and its successive differentials up to the mth
exclusive must all vanish simultaneously.
46] On a new Theory of Multiplicity. 375
generality), not only will the first derivatives of the quadratic function be
nullifiable simultaneously with each other, but likewise at the same time
with the cubic function itself. These three cases will be for surfaces, the
analogues so far, but only so far as regards the degree of the multiplicity,
to the double point, cusp, and flex-cusp of curves*. The analogue to the
so-called triple point of the curves will be a point whose degree of singularity,
depending upon the vanishing of the six constants in the third coefficient
(which is a quadratic function of x, y, z) at the same time as the three
constants in the linear factor, would seem to be but 6 more than for a double
point, that is in all 1 + 6 or 7, but whose multiplicity, as inferred from
the nature of the contact of its first polars, which will be of the 7th order,
would appear to be 8 (a seeming incongruity which I am not at present in a
condition to explain)!; so that there will apparently be 4 steps of multiplicity
to interpolate between this case and the case analogous (sub modo) to the
flex-cusp, last considered. Whether these intervening degrees correspond
to singularities of an unambiguous kind, no one is at present in a condition
to offer an opinion. I will conclude with a remark, the result of my experi
ence in this kind of inquiry as far as I have yet gone in it, namely that
it would be most erroneous to regard it as a branch of isolated and merely
curious or fantastic speculation. Every singularity in a locus corresponds
to the imposition of certain conditions upon the form of its characteristic;
by aid of the theory of evection we are able to connect the existence of these
conditions with certain consequences happening to the form of the discrimi
nant, and thereby it becomes possible, upon known principles of analysis,
to infer particulars relating to the constitution of the discriminant itself
in its absolutely general form, very much upon the same principle as when
the values of a function for particular values of its variable or variables are
known, the general form of the function thereby itself, to some corresponding
extent, becomes known. Thus, for instance, I have by the theory of evection
in its most simple application, been led to a representation of the discriminant
* At an ordinary conical point of a surface for which the multiplicity is 1, every section
of the surface is a curve with a double point. When the multiplicity is 2, the cone of contact
becomes a pair of planes, through the intersection of which any other plane that can be drawn
cuts the surface in a section having an ordinary cusp of multiplicity 2, but which themselves
cut the surface in sections, having so-called triple points, so that for these two principal sections
(which is rather surprising) the multiplicity suddenly jumps up from 2 to 4. All other things
remaining unaltered when the multiplicity of the conical point is 3, the cusp belonging to any
section of the surface drawn through any intersection of the two tangent planes passes from an
ordinary cusp to a flex-cusp.
f So, too, at a so-called quadruple point in a curve, the degree of the contact of the 1st polars
is 8, and therefore the multiplicity of the curve at such point is 9 ; but the number of constants
which vanish for this case (namely all those of the cubic coefficient in x, y) over and above what
vanish for the case of a so-called triple point is only 4, which is a unit less than the difference
between the measures of the multiplicities at the respective points ; and this difference continues
to increase as we pass on to so-called quintuple and higher multiple points in the curves.
376 On a new Theory of Multiplicity. [46
of a function of two variables under a form very different and very much
more complete and fecund in consequences than has ever been supposed,
or than I had myself previously imagined, to be possible.
According to the opinion expressed by an analyst of the French school,
of pre-eminent force and sagacity, it is through this theory of multiplicity,
here for the first time indicated, that we may hope to be able to bridge over
for the purposes of the highest transcendental analysis, the immense chasm
which at present separates our knowledge of the intimate constitution of
functions of two from that of three, or any greater number of variables.
It is, as I take pleasure in repeating, to a hint from Mr Cayley*, who
habitually discourses pearls and rubies, that I am indebted for the precious
and pregnant observation on the form assumed by the first discriminantal
evectant of a binary function with a pair of equal roots, out of which,
combined with some antecedent reflections of my own, this new theory of
multiplicity has taken its rise. The idea of the process of evection, and the
discovery of its fundamental property of generating what, in my calculus
of forms (Cambridge and Dublin Mathematical Journal), I have called
contravariants, is due to my friend M. Hermite. The polar reciprocals of
curves and other loci are contravariants and, as I have recently succeeded
in showing, for curves at least, evectants, but of course not discriminantal
evectants; and I am already able to give the actual explicit rule for the
formation of the polar reciprocal of curves as high as the 5th degree, which
with a little labour and consideration can be carried on to the 6th, and in
fact to curves of any degree n when once we are acquainted with any mode
of determining all such independent invariants of a function of two variables
as are of dimensions not exceeding 2 (n 1) in respect of the coefficients.
By the special geometers (by whom I mean those who, unvisited by a
higher inspiration, continue to regard and to cultivate geometry as the
science of mere sensible space) this problem has only been accomplished, and
that but recently, for curves whose degrees do not exceed the 4th. Mr Salmon
has made the happy and brilliant (and by the calculus of forms instantaneously
demonstrable) discovery, communicated to me in the course of a most
instructive and suggestive correspondence, that a certain readily ascertainable
* Mr Cay ley s theorem stood thus : If
ax n + nbx n ~ l y 4 ... +nb xy n ~ l + a y n
have two equal roots, and w be its discriminant, then will
d d d I
j y #-jr(feC. -: ;}- ZZT
da do da]
be a perfect 7ith power. It will easily be seen that this theorem is convertible into a theorem of
evection by interchanging in the result x and y with y and - x.
46] On a new Theory of Multiplicity. 377
evectant of every discriminant of any function whatever is an exact power of
its polar reciprocal *.
I believe that it may be shown, that, with the sole exception of odd-
degreed functions of two variables, the polar reciprocal itself (BS, distinguished
from a power thereof) of every function is an evectant, not (of course) of the
discriminant, but of some determinable inferior invariant.
P.S. The terms pluri-simultaneous and pluri-simultaneity, used or
suggested by me in my last paper in the Magazine, may be advantageously
replaced by the more euphonious and regularly formed words consimul-
taneous, consimultaneity. Multiplicity and all its attributes and consequences
are included as particular cases in the general conception and theory of
consimultaneity, that is of consimultaneous equations, or, which is the same
thing, of consimulevanescent functions.
* Namely, for a function of degree n, and variability (that is, having a number of variables)
p, the (n - l) p ~ l ih evect of the discriminant is the (n - l)th power of the polar reciprocal.
47.
A DEMONSTRATION OF THE THEOREM THAT EVERY HOMO
GENEOUS QUADRATIC POLYNOMIAL IS REDUCIBLE BY
REAL ORTHOGONAL SUBSTITUTIONS TO THE FORM OF
A SUM OF POSITIVE AND NEGATIVE SQUARES.
[Philosophical Magazine, iv. (1852), pp. 138 142.]
IT is well known that the reduction of any quadratic polynomial
to the form a^ + a 2 ij 2 + ... + a n &*, where %,?)... 6 are linear functions of
x,y...t, such that a? + f+ ... +t 2 remains identical with 2 + ?7 2 + ... + 2
(which identity is the characteristic test of orthogonal transformation),
depends upon the solution of the equation
= 0.
(1 1) + X
(1, 2)
(~L n~)
(2, 1),
(2, 2) + X .
. (2, n)
(n. 1),
(n. 2}..
.. (n. n~) + \
The roots of this equation give a 1} a 2 ... a n \ and if they are real, it is easily
shown that the connexions between as, y ... t] %, rj ... 6, are also real.
M. Cauchy has somewhere given a proof of the theorem*, that the roots of X
in the above equation must necessarily always be real ; but the annexed
demonstration is, I believe, new ; and being very simple, and reposing upon
a theorem of interest in itself, and capable no doubt of many other applica
tions, will, I think, be interesting to the mathematical readers of this
Magazine.
* Jacobi and M. Borchardt have also given demonstrations ; that of the latter consists in
showing that Sturm s functions for ascertaining the total number of real roots expressed by my
formulae (many years ago given in this Magazine) are all, in the case of /(\), representable as the
sums of squares, and are therefore essentially positive.
47] On Homogeneous Quadratic Polynomials.
Let
379
/(X)=
(1, 1)
(2,1),
(3,1),
, (1,2)
(2,2) + X ............ (2,n)
(3,2), (3,3) + X...(3,n)
(n, 1), (, 2) (n,w)+X
it is easily proved that/(X) x/(-X)
[lil]-X [1,2] [l,n]
[2,1], [2, 2]- X 2 ... [2, 7t]
[n, 1], [, 2] [n, n]-X 2
where [t, e] = (i, 1) x (1, e) + (t, 2) x (2, )+...+(*, w) x (n, e).
If, now, for all values of r and s, (r, s) = (s, r), that is, if /(O) becomes the
complete determinant to a symmetrical matrix, then every term [r, s] in
the derived matrix becomes a sum of squares, and is essentially positive,
and (- l) n /(X) x/(- X) assumes the form
(X 2 ) n - F (x 2 ) 1 " 1 + G- (X 2 ) n ~ 2 + L,
where F, G, ... L will evidently be all positive ; for it may be shown that F
will be the sum of the squares of the separate terms, that is, of the last
minor determinants of the given matrix, G the sum of the squares of the
last but one minors, and so on, L being the square of the complete deter
minant. For instance, if
a 4- X, 7, /3
7, b + X, a
/3, a, c + X
c /(_ x) = x 6 - F\* + G\ 2 - H,
where F = a 2 + 6 2 + c 2 + 2a 2 + 2/3 2 + 2 7 2 ,
G = (ab - 7 2 ) 2 + (6c - a 2 ) 2 + (ac - /3 2 ) 2
+ 2 (aa - /37) 2 + 2 (6/3 - 7) 2 + 2 (07 - a/3) 2 ,
a, V, /
7, &&gt; <
/3, a,
Hence it follows immediately that /(X) = cannot have imaginary roots ;
for, if possible, let X =p + q V(- 1), and write
i c + p =c
380 On Homogeneous Quadratic Polynomials. [47
y (X) becomes
a! + V, 7, ft
7, b + \ , a.
3, , c +X
or say < (X ), and the equation < (V) x </> (- X ) = will be of the form
where J", Cr , # are all essentially positive. Hence, by Descartes rule, no
value of X 2 can be negative, that is, (X pf cannot be of the form <? 2 ;
that is to say, it is impossible for any of the roots of /(X) = to be imaginary,
or, as was to be demonstrated, all the roots are real.
I may take this occasion to remark, that by whatever linear substitutions,
orthogonal or otherwise, a given polynomial be reduced to the form SJ.^ 2 ,
the number of positive and negative coefficients is invariable : this is easily
proved. If now we proceed to reduce the form (expressed under the umbral
notation) (a^ + a z x 2 + ... + a n x n } 2 to the form
A 1 tf + A& + . . . + A n _, tVi + A n t n \
by first driving out the mixed terms in which ^ enters, then those in which
# 2 enters, and so forth until eventually only x n of the original variables is
left, it may readily be shown that
A I {JU \\ A I "1 *> \ / uv l 1 A
A^ = A 2 =( )*( , A 3 =
\aj \Oi<V \ a i
- f Wl Vt/Q U>1
~ An = >~~ a ,
It follows, therefore, that in whatever order we arrange the umbrae a^ ... a n ,
the number of variations and of continuations of sign in the series
\aj
will be invariable, and in fact will be the same as the number of positive
and negative roots in the generating function in A, above treated of, that is,
since all the roots are real, will be the same as the number of variations
and continuations in the series formed by the coefficients of the several
powers of X, that is
2/aA ^ /ia 2 \
UJ \OiaJ "
The first part of this theorem admits of an easy direct demonstration;
for by my theory of compound determinants, given in this Magazine*, we
know that _ _
a a . . . a r _i a r a x a 2 . . . a r _! a r+ i\
[* Cf. pp. 241, 252 above.]
47] On Homogeneous Quadratic Polynomials. 381
The first member of this equation is equivalent to
r -ia r \ fa^a 2
XI i i
Hence it follows, that if the two factors on the right-hand side of the
equation have the same sign,
[ l 2 " r I and
\a^a^ ... a r i(i r >
have also the same sign inter se, and consequently the two triads
r | r * I -
Cvj Cvj CLf i I I t/-j Ct ij (.*"p i (J/f I I (Zj C&2 Cvy i Ct y(Z^_Lj
I Cvj Cc-j \JLf i I 1 (Ti t/-o Cvy i G/f I I Ctj Ctg Cvy j cZj* tt j_i_j
r ~i r r/ "i r/7 /7 n /r n n
and , J J J~ ! J + a a " a * a + * a
will in all cases present the same number of changes and continuations,
Avhich proves that the contiguous umbrae, a r , a r+l , may be interchanged
without affecting the number of variations and continuations in the entire
series ; but, as is well known, any one order of elements is always convertible
into any other order by means of successive interchanges of contiguous
elements, which demonstrates that, in whatever order the elements a lt a^...a n
be arranged, the number of continuations and variations in
1, ( )i ( a 2 ) j >
\Ct~l \^1 ^ 2/ \b\ ^2 * ^71
is invariable. But that the same thing is true (as we know it to be), for the
relation between any one of these unsymmetrical series and the symmetrical
series (resulting from the method of orthogonal transformation)
\Oio
is by no means so easily demonstrable in the general case by a direct method,
and the attention of algebraists is invited to supply such direct method of
demonstration. My knowledge of the fact of this equivalence is, as I have
stated, deduced from that remarkable but simple law to which I have
adverted, which affirms the invariability of the number of the positive and
negative signs between all linearly equivalent functions of the form S c r x r
(subject, of course, to the condition that the equivalence is expressible by
means of equations into which only real quantities enter) ; a law to which
my view of the physical meaning of quantity of matter inclines me, upon the
ground of analogy, to give the name of the Law of Inertia for Quadratic
Forms, as expressing the fact of the existence of an invariable number
inseparably attached to such forms.
48.
ON STAUDT S THEOREMS CONCERNING THE CONTENTS OF
POLYGONS AND POLYHEDRONS, WITH A NOTE ON A
NEW AND RESEMBLING CLASS OF THEOREMS.
[Philosophical Magazine, iv. (1852), pp. 335 345.]
THE beautiful and important geometrical theorems of Staudt are, I
believe, little, if at all, known to English mathematicians. They originally
appeared in Crelle s Journal for the year 1843, and have been recently
reproduced in M. Terquem s Nouvelles Annales for the August Number of
the present year.
These theorems may be summed up, in a word, as intended to show the
possibility and method of expressing the product of any two polygons or any
two polyhedrons as entire functions of the squares of the distances of the
angular points of the two figures from one another. The well-known expres
sion for the square of the area of a triangle in terms of the sides (in which,
when expanded, only even powers of the lengths of the sides appear), is but
a particular case of Staudt s theorem for polygons, for it may be considered
as the case of two equal and similar triangles whose angular points coincide.
So in like manner, as observed by Staudt, a similar expression in terms of
its sides may be found for the square of a pyramid. This expression had,
however, been previously given (although, by a strangle negligence, not
named for what it was) by Mr Cayley in the Cambridge Mathematical
Journal for the year 1841*, in his paper on the relations between the
mutual distances to one another of four points in a plane and five points in
space ; the singularly ingenious (and as singularly undisclosed) principle of
that paper consisting in obtaining an expression for the volume of a pyramid
in terms of its sides, and equating this, or rather its square, to zero as the
conditions of the four angular points lying in the same plane.
* Query, Is not this expression for the volume of a pyramid in terms of its sides to be found
in some previous writer ? It can hardly have escaped inquiry.
48]
On Staudt s Theorems.
383
The analogous condition for five points in space is virtually deduced by
going out into rational space of four dimensions, and equating to zero the
expression obtained for the volume of a plupyramid ; meaning thereby the
figure which stands in the same relation to space of four as a pyramid to
space of three dimensions. Mr Cayley s method, if it had been pursued
a step further, would have led him to a complete anticipation of the principal
part of Staudt s discovery. The method here given is not substantially
different from Mr Cayley s, but is made to rest upon a more general
principle of transformation than that which he has employed. As to
Staudt s own method, it is as clumsy and circuitous as his results are simple
and beautiful. Geometry, trigonometry and statics, are laid under contri
bution to demonstrate relations which will be seen to flow as immediate and
obvious consequences from the most elementary principles in the algorithm
of determinants. Perhaps, however, M. Staudt s method is as good as could
be found in the absence of the application of the method of determinants,
the powers of which, even so recently as ten years ago, were not so well
understood or so freely applied as at the present day.
The following new but simple theorem, of which I shall have occasion to
make use, will be found to be a very useful addition to the ordinary method
for the multiplication of determinants. " If the determinants represented
by two square matrices are to be multiplied together, any number of columns
may be cut off from the one matrix, and a corresponding number of columns
from the other. Each of the lines in either one of the matrices so reduced
in width as aforesaid being then multiplied by each line of the other, and
the results of the multiplication arranged as a square matrix and bordered
with the two respective sets of columns cut off arranged symmetrically (the
one set parallel to the new columns, the other set parallel to the new lines),
the complete determinant represented by the new matrix so bordered
(abstraction made of the algebraical sign) will be the product of the two
original determinants."
rru (b
Thus ,
\ca
or
may be put under any one of the three following forms :
ay + bS
cy + d&
2, 2,
aa, ay
ca,
A
C7:
a,
act +
ca +
b
d
or
2,
,
7.
2,
a,
c,
0,
0,
* Any quantities might be substituted instead of 2 in the places occupied by the figure in the
above determinant, as such terms do not influence the result ; this figure is probably, however,
the proper quantity arising from the application of the rule, because (as all who have calculated
with determinants are aware) the value of the determinant represented by a matrix of no places
is not zero but unity.
384
On Staudt s Theorems concerning the
[48
And in general for two matrices of /i 2 terms each, this rule of multiplication
will give (n + l) distinct forms representing their products.
Thus, as a further example,
a,
b,
c
a,
ft,
7
a ,
b ,
c
X
,
ft ,
7
a",
b",
c"
",
ft",
7"
besides the first and last forms, will be representable by the two intermediate
forms
aa + 6/3, aa+b/3 , aa" + b8", c
a a + b @, a a +b /3 , a a" + b ft", c
a"a + b"/3, a" a + b"ft , a"a
and
+ b"ft", c"
" C\
7
aa ,
aa",
b,
c
a a ,
a a",
b ,
c
a"a ,
a" a",
b",
c"
ft ,
ft",
0,
y,
7",
0,
aa,
a a,
a a,
ft,
V,
To arrive, for instance, at the latter of these two forms, we have only
to write the two given matrices under the respective forms
a, b, c, 0,
a, b , c , 0,
a", b", c", 0,
0, 0, 0, 1,
0, 0, 0, 0, 1
and then apply the ordinary rule of multiplication. So, again, to arrive
at the first of the above written two forms, we must write the two given
matrices under the respective forms
a, b, c,
a, b , c,
a", b", c",
0, 0, 0,
and proceed as before.
This rule is interesting as exhibiting, as above shown, a complete scale
whereby we may descend from the ordinary mode of representing the product
of two determinants to the form, also known, where the two original deter-
,
,
o,
o,
A
7
a ,
o,
o,
ft ,
7
a",
o,
o,
ft",
II
7
0,
1,
o,
o,
0,
o,
1,
o,
, ft,
0, 7
, ft ,
0, 7
and
"> ft",
0, 7 "
1
0, 0,
1,
48]
contents of Polygons and Polyhedrons.
385
minants are made to occupy opposite quadrants of a square whose places
in one of the remaining quadrants are left vacant, and shows us that under
one aspect at least this latter form may be regarded as a matrix bordered
by the two given matrices.
A second but obvious theorem requiring preliminary notice is the
following, namely that the value of the determinant to the matrix
i, i, ... i, o,
is the same as the value of the determinant to the matrix
-"-1,1, -"-1,2 -"-1,71) 1,
^2,1, --2,2 -"-2,71) 1)
-<4-n,i "",! "-n,ni 1,
i, i, ... i, o,
where in general
**-r,s = ^r,s ft-r " gj
AI, & 2 . . . h n and k^k^... k n being any two perfectly arbitrary series of quantities.
This simple transformation is of course derived by adding to the respective
columns in the first matrix the last column (consisting of units) multiplied
respectively by h l} h 2 ...h n , 0; and to the respective lines, the last line
(consisting of units) multiplied respectively by k lt k 2 ...k n , 0.
Suppose, now, that we have two tetrahedrons whose volumes are repre
sented respectively by one-sixth of the respective determinants
2/2,
2/3,
2/4 ,
.,
%r, y r , z r representing the orthogonal coordinates of the point r in one
tetrahedron, and % r , rj r , % r the same for the point r in the other.
By the first theorem their product may be represented (striking off the
last column only from each matrix) by the matrix
1,
25
386 On Staudfs Theorems concerning the
where, in general, any such term as 2# r s represents
Again, by virtue of the second theorem, adding
[48
to the respective lines, and
to the respective columns, the above matrix becomes (after a change of signs
not affecting the result) the - th of
20*!- ft)*, 2 (*!-
2(* 4 -ft) 1 , 2(*4-
1, 1,
2 OF, -
1,
2 (*,-),
1,
or calling the angular points of the one tetrahedron a, b, c, d, and of the
other p, q, r, s, 8 x 36, that is 288 times, their product is representable by
1 x the determinant
(ap) 2 , (aq) 2 , (ar) 2 , (as) 2 , 1
>
(bp) 2 , (bq) 2 , (6r) 2 , (bs) 2 , 1
(cp) 2 , (cq) 2 , (cr) 2 , (cs) 2 , 1
(dp) 2 , (dq) 2 , (dr) 2 , (ds) 2 , I
1, 1, 1, 1,
and of course if p, q, r, s coincide respectively with a, b, c, d, 576 times the
square of the tetrahedron abed will be represented under Mr Cayley s form,
0, (ab) 2 , (ac) 2 , (ad) 2 , 1
*
i
(6a) 2 , 0, (be) 2 , (bd) 2 , 1
(ca) 2 , (cb) 2 , 0, (cd) 2 , 1
(da) 2 , (db) 2 , (dc) 2 , 0, 1
1, 1, 1, 1,
four out of the sixteen distances vanishing, and the remaining twelve
reducing to six pairs of equal distances. The demonstration of Staudt s
* The corresponding quantity to the above determinant for the case of the triangle (hereafter
given) is identical with the Norm to the sum of the sides. I have succeeded in finding the
Factor (of ten dimensions in respect of the edges), which, multiplied by the above Determinant
itself, expresses the Norm to the sum of the Faces, that is, the superficial area of the Tetrahedron.
48]
387
theorem for triangles is obtained in precisely the same way by throwing the
product of the two determinants
T* 1J
#2, 7/0. 1 and
-5 / * *
3, 2/3> *
under the form of th of
V ( _ fc ^2 "? / r _ fc \2
**\^i iTi/i -^ V^i ?2/ ,
2
2 (^-
2(*2-
2 -
1,
1,
When the two triangles coincide, calling their angular points a, b, c
the above written determinant becomes
0, (a&) 2 , (ac) 2 , 1
(6a) 2 , 0, (6c) 2 , 1
(ca) 2 , (c&) 2 , 0, 1
1, 1, 1,
or
(a&) 4 + (ac) 4 + (6c) 4 - 2 (a&) 2 . (ac) 2 - 2 (a&) 2 . (6c) 2 - 2 (ac) 2 . (6c) 2 ,
the negative of which is the well-known form expressing the square of four
times the area of the triangle a&c.
There is another and more general theorem of Staudt for two triangles
not in the same plane, which may be obtained with equal facility. In fact,
if we start from the determinant
(aa) 2 , (a/3) 2 , (a 7 ) 2 , 1
(6) 2 , (6/3) 2 , (6 7 ) 2 , 1
(ca) 2 , (c/3) 2 , (c 7 ) 2 , 1
1, 1, 1,
and add to each column respectively the last column multiplied by e^ 2 , e 2 2 >
e%3 respectively, we arrive at the form
(c 7 ) 2
1,
and considering , rj 1 ;
1,
, ^2; Is, fJa as the coordinates of a, /3, 7, the
252
388 On Staudt s Theorems concerning the [48
projections upon the plane of abc of a triangle ABC, whose plane intersects
the former plane in the axis of y, and makes with that plane an angle whose
tangent is e, it is easily seen that this determinant is term for term identical
with the determinant
1, 1, 1>
which therefore expresses 16 times the product of the triangles abc and
a/37, tnat i- s a ^ c x ABC x cosine of the angle between the two. A similar
method, if we ascend from sensible to rational geometry, may be given for
expressing in terms of the distances the product of any two pyramids (in
a hyperspace) by the cosine of the angle included between the two infinite
spaces * in which they respectively lie. To pass from the cases which have
been considered of two triangles to two polygons, or of two tetrahedrons to
two polyhedrons, generally presents no difficulty ; and for Professor Staudt s
method of doing so, which is simple and ingenious, and does not admit of
material improvement, the reader is referred to the memoir in Crelle s Journal
or Terquem s Annales already adverted to. It is, however, to be remarked
(and this does not appear to be sufficiently noticed in the memoirs referred
to), that whilst the expression for the product of any two polygons in terms
of the distances given by Staudt s theorem is unique, that for the product
of two polyhedrons given by the same is not so, but will admit of as many
varieties of representation as there are units in the product of the numbers
respectively expressing the number of ways in which each polygonal face of
each polyhedron admits of being mapped out into triangles. I, cannot help
conjecturing (and it is to be wished that Professor Staudt or some other
geometrician would consider this point) that in every case there exists,
linearly derivable from Staudt s optional formulae (but not coincident with
any one of them), some unique and best, because most symmetrical, formula
for expressing the product of two polyhedrons in terms of the distances of
the angular points of the one from those of the other. In conclusion I may
observe, that there is a theorem for distances measured on a given straight
line, which, although not mentioned by Staudt, belongs to precisely the same
class as his theorems for areas in a plane and volumes in space ; namely
a theorem which expresses twice the rectangle of any two such distances
under the form of an aggregate of four squares, two taken positively and two
* In rational or universal geometry, that which is commonly termed infinite space (as if it
were something absolute and unique, and to which, by the conditions of our being, the repre
sentative power of the understanding is limited), is regarded as a single homaloid related to a
plane precisely in the same way as a plane is to a right line. Universal geometry brings home
to the mind with an irresistible force of conviction the truth of the Kantian doctrine of locality.
48]
389
negatively; that is to say, if A, B, C, D be any four points on a right line
2ABx CD = AD Z + B&-A & - BD 2 . I know not whether this theorem
be new, but it is one which evidently must be of considerable utility to the
practical geometer.
Note on the above.
The fundamental theorem in determinants, published by me in the
Philosophical Magazine in the course of last year*, leads immediately to a
class of theorems strongly resembling, and doubtless intimately connected
with, those of Staudt.
Thus for triangles we have by this fundamental theorem
ffi, a?a, *s
SI) S 2 ) S3
2/1 2/2, 2/3
X
*?1, ^, *7s
1, 1, 1
1, 1, 1
t- c-
S3) &2> ^3
2/1. *7l, %
X
^3) 2/2) 2/3
+
1, 1, 1
1, 1, 1
52 > S3
2/l> 7 /2:
1, 1,
^3
^ll 2/2 2/3
1, 1, 1
1, 1,
2/3
1
1, 1,
and consequently, if ABC, DEF be any two triangles,
ABCxDEF=ADExFBC + AEFxDBC + AFDx
This may be considered a theorem relating to two ternary systems of
points in a plane. The analogous and similarly obtainable theorem for two
binary systems of points in the same right line is
As in applying this last theorem to obtain correct numerical results we must
give the same algebraical sign to any two lengths denoted by the two
arrangements XY, ZT, according as the direction from X to Y is the same
as that from Z to T, or contrary to it, so in the theorem for the products
of triangles, the areas denoted by any two ternary arrangements XYZ, TUV
must be taken with the like or the contrary sign, according as the direction
of the rotation XYZ is consentient with or contrary to that of TUV] so
that three of the six possible arrangements of X YZ may be used indifferently
for one another, but the other three would imply a change of sign. If we
[* See pp. 249, 253 above.]
390
On StaudCs Theorems concerning the
[48
analyse what we mean by fixing the direction of the rotation of XYZ, and
reduce this form of speech to its simplest terms, we easily see that it amounts
to ascertaining on which side of B, C lies, that is whether to its right or left,
to a spectator stationed at A on a given side of the plane ABC.
Let us now pass to the corresponding theorems for two tetrahedrons put
respectively under the forms
51, 52, S3, 54
2/i, 2/2, 2/3, 2/4
I, 1, 1,
1, 1, 1,
b4
1
We may represent this product in either of two ways by the application
of our fundamental theorem, namely as
or as
&C.
&C.
Ok,
ft, ft,
ft
ft,
b,
%,
^4
2/i >
f?!, ^2,
7/3
774,
yi,
2/3,
2/4
X
*,
Si, 2,
3
4,
^2,
^3,
^4
1,
1, 1,
1
1,
1,
1,
1
,
*2, ft,
ft
&,
ft,
^3,
^4
2/i
2/2, *7l,
?l
?i,
fc,
2/3,
2/4
X
4i
* &,
C*
j:,,
4,
,
^4
1,
i, i,
1
i,
1,
1,
1
there being four products to be added together in the first expression and
six in the latter; and the rule, if we wish that all the products may be
additive, being that on removing the sign of multiplication the determinant
to the square matrix formed by the Greek letters in situ shall always preserve
the same sign. Hence we derive two geometrical formulae concerning the
products of polyhedrons, namely
(1) ABCD x EFGH = ABCE x FGHD - ABCF x GHED
+ ABCG x H EFD - ABCH x FGED.
(2) ABCD x EFGH = ABEF x GHCD + ABGH x EFCD
+ ABEG x HFCD + ABHF x EGCD
+ABEH* FGCD + ABFG x EHCD.
These formula give rise to an exceedingly interesting observation. In
order that they shall be numerically true, we must have a rule for fixing
the sign to be given to the solid content represented by any reading off of
the four points of a tetrahedron, that is we must have a rule for determining
48] contents of Polygons and Polyhedrons. 391
the sign of solid contents of figures situated anywhere in space analogous
to that which, as applied to linear distances reckoned on a given right line,
is the true foundation of the language of trigonometry, and the condition
precedent for the possibility of any system of analytical geometry such as
exists, and which, not altogether without surprise, I have observed in the
pages of this Magazine one of the learned contributors has thought it necessary
to vindicate the propriety of importing into his theory of quaternions.
Various rules may be given for fixing the sign of a tetrahedron denoted
by a given order of four letters. One is the following : the content of A BCD
is to be taken positive or negative, according as to a spectator at A the
rotation of BCD is positive or negative. Another, again, is to consider AB
and CD as representing, say two electrical currents, and to suppose a spectator
so placed that the current AB shall pass through the longitudinal axis of his
body from the head towards the feet, and looking towards the other current
CD ; the sign of the solid content of the tetrahedron (and, indeed, also the
effect, in a general sense, of the action of the two currents upon one another)
will depend upon the circumstance of this latter current appearing to flow
from the right to the left, or contrariwise in respect of the spectator. Last
and simplest mode of all, the sign of the solid content of ABCD will depend
upon the nature (in respect to its being a right-handed or left-handed-screw)
of any regular screw-line (whether the common helix or one in which the
increase or decrease of the inclination is always in the same direction)
terminating at B and C, and so taken that BA shall be the direction of the
tangent produced at B, and CD the direction of the tangent produced at C.
Inasmuch as of the twenty-four permutations of a quaternary arrangement
a defined twelve have one sign, and the other twelve the contrary sign, these
various definitions of the direction, or, as it may be termed, polarity, of a
tetrahedron corresponding to a given reading, whether as taken each in itself
or compared one with another, give rise to, or rather imply a considerable
number of interesting theorems included in our intuitions of space, and
probably belonging to the, in my belief, inexhaustible class of primary and
indemonstrable truths of the understanding.
49.
ON A SIMPLE GEOMETRICAL PROBLEM ILLUSTRATING A
CONJECTURED PRINCIPLE IN THE THEORY OF GEO
METRICAL METHOD.
[Philosophical Magazine, n 7 . (1852), pp. 366 369.]
THE following theorem deserves attention as illustrating a principle of
geometrical method which will be presently adverted to. It is curious, also,
from the fact of its solution being by no means so obvious and self-evident
as one would expect from the extreme simplicity of its enunciation. It
appeared, and for the first time, it is believed, at the University of Cambridge
about a twelvemonth back, where it excited considerable attention among
some of the mathematicians of the place. The proposition, as originally
presented, was merely to prove that if ABC be a triangle, and if AD and
BE drawn bisecting the angles at A and B and meeting the opposite sides
in D and E be equal, then the triangle must be isosceles. It is particularly
noticeable that all the geometrical demonstrations yet given of this theorem
are indirect. Thus the first and simplest (communicated to me by a promising
young geometrician, Mr B. L. Smith of Jesus College, Cambridge), was the
following : Assume one of the angles at DAB to be greater than the corre
sponding angle EBA ; it can easily be shown that, upon this supposition,
D will be higher up from AB than E; so that if DF and EG be drawn
parallel to AB, DF will be above EG ; it is then easily shown that DF= AF,
EG = BG, and consequently DF and AF are each respectively less than EG
49] On a simple Geometrical Problem, 393
and BG; and also DFA, which is the supplement of twice DAB, will be less
than EGB, which is the supplement of twice FBA ; from which it is readily
inferred, by an easy corollary to a proposition of Euclid, that DA will be less
than FB, whereas it should be equal to it ; so that neither of the half angles
at the base can be greater than the other, and the triangle is proved to be
isosceles. Another and independent demonstration by the writer of this
article is less simple, but has the advantage of lending itself at once to a
considerable generalization of the theorem as proposed. Assuming, as above,
that DAB is greater than EBA, it is easily seen that DE produced will cut
BA at K on the side of it : also if AD and BE intersect in H, it is readily
demonstrable, by a suitably constructed apparatus of similar triangles, that
AH :BH:: CE : CD.
But as HBA is less than HAB, AH is less than BH, and therefore
CE is less than CD, and therefore CED is greater than CDE; that is to say,
CAB less K is greater than CBA plus K, and therefore DAB less K is
greater than EBA, that is ADE is greater than ABE, and therefore the
perpendicular from A upon DE is greater than that from E on AB, which
is easily proved to be absurd. Hence, as before, the triangle is proved to be
isosceles. This proof, it is obvious, remains good for all cases in which EB
and DA, drawn on either side of the base, divide the angles at the base
proportionally, provided that these lines remain equal, and make positive
or negative angles with the base not less than one-half of the respective
corresponding angles which the sides of the triangle are supposed to make
with it. The analytical solution of the question, as might be expected,
extends the result still further. To obtain this, let
BAC = n.BAD, ABC = n. ABE,
n for the present being any numerical quantity, positive or negative;
calling BAC=2na, ABC=2n(3, we readily obtain, by comparison of the
equal dividing lines with the base of the triangle,
sin (2?ia + 2/3) sin (2r?,/3 + 2a)
sin 2na sin 2n/3
sin (2na. + 2/3) sin 2?ia
nr x _ . _ - _
sin (2nj3 + 2a) sin
and by an obvious reduction,
tan (n 1) (a /3) _ tan(w + !)( + /3)
tan n (a yS) tan n (a + /3)
When this equation is put under an integer form, it is of course satisfied
by making a = /3; on any other supposition than a = /3 it evidently cannot
be satisfied by admissible values of the angles for any value of n between
394 On a simple Geometrical Problem.
+ 1 and + oo ; for on that supposition, since (a ft) and (a + ft) are each less
ISO
than , the first side of the equation will be necessarily a proper fraction
and positive ; but the second side, either a positive improper fraction if
(n + 1) (a + ft) be less, and a negative proper or a negative improper fraction
if (n + 1) (a + ft) be greater than a right angle.
If n be negative, let it equal v, then
tan (v + 1) (a - ft) _ tan (v -1) (a + ft)
tan v (a ft) tan v(a+ft)
and for the same reason as before, if v lies between oo and 1, this equation
cannot be satisfied. Hence the theorem is proved to be true for all values
of n, except between + 1 and 1. For these values it ceases to be true ;
in fact, for such values for any given values of (a ft) there will be always,
as it may be easily proved, one or more values of (a + ft); thus if n = |, the
equation becomes
CL+ft
tan -
and if n =
a-ft
tan
z
showing that a + ft = 90 and a ft = 90 in these respective cases will
afford a solution over and above the solution a = ft, which is easily verified
geometrically*. It would be an interesting inquiry (for those who have
leisure for such investigations) to determine for any given value of n between
+ 1 and 1 the superior and inferior limits to the number of admissible values
of a + ft corresponding to any given value of a /3f.
My reader will now be prepared to see why it is that all the geometrical
demonstrations given of this theorem, even in the simplest case of all, namely
when n = 2, are indirect, I believe I may venture to say necessarily indirect.
It is because the truth of the theorem depends on the necessary non-existence
of real roots (between prescribed limits) of the analytical equation expressing
the conditions of the question ; and I believe that it may be safely taken
as an axiom in geometrical method, that whenever this is the case no other
* In the first of these cases, if the base of the triangle is supposed given, the locus of the
vertex is a right line and a circle ; in the second case, a right line and an equilateral hyperbola.
t When n lies between - - and - - (t being any positive integer), it is easily seen that
-:- 1 2ll + 1
the superior limit must be at least as great as t.
49] On a simple Geometrical Problem. 395
form of proof than that of the reductio ad absurdum is possible in the nature
of things. If this principle is erroneous, it must admit of an easy refutation
in particular instances.
As an example, I throw out (not a challenge, but) an invitation to discover
a direct proof, if such exist, of the following geometrical theorem, as simple
a one as it is perhaps possible to imagine : " To prove that if from the
middle of a circular arc two chords be drawn, and the remoter segments
of these chords cut off by the line joining the end of the arc be equal, the
nearer segments will also be equal." The analytical proof depends upon the
fact of the equation x^ + ax 6 2 (where a is the given length of each segment,
and b the length of the chord of half the given arc) having only one admis
sible root ; and if the principle assumed or presumed to be true be valid, no
other form of pure geometrical demonstration than the reductio ad absurdum
should be applicable in this case. For the converse case, where the nearer
segments are given equal, the reducing equation is a (a + x) = b 2 , indicating
nothing to the contrary of the possibility of there being a direct solution,
which accordingly is easily shown to exist. The indirect form of demonstra
tion, it may be mentioned, is sometimes liable to be introduced in a mariner
to escape notice. As, for instance, if it should be taken for granted in the
course of an argument, that one triangle upon the same base and the same
side of it as another triangle, and having the same vertical angle, must have
its vertex lying on the same arc ; this would seem to be immediately true by
virtue of the well-known theorem, that angles in the same circular segment
are equal, but in reality can only be inferred from it indirectly by showing
the impossibility of its lying outside or inside the arc in question. To go one
step further, I believe it to be the case, that granted to be true all those
fundamental propositions in geometry which are presupposed in the principles
upon which the language of analytical geometry is constructed, then that the
reductio ad absurdum not only is of necessity to be employed, but moreover
in propositions of an affirmative character never need be employed, except
when as above explained the analytical demonstration is founded on the
impossibility or inadmissibility of certain roots due to the degree of the
equation implied in the conditions of the question. If this surmise turn out
to be correct, we are furnished with a universal criterion for determining
when the use of the indirect method of geometrical proof should be considered
valid and admissible and when not*.
* If report may be believed, intellects capable of extending the bounds of the planetary
system and lighting up new regions of the universe with the torch of analysis, have been baffled
by the difficulties of the elementary problem stated at the outset of this paper, in consequence,
it is to be presumed, of seeking a form of geometrical demonstration of which the question from
its nature does not admit. If this be so, no better evidence could be desired to evince the
importance of such a criterion as that suggested in the text.
50.
ON THE EXPRESSIONS FOR THE QUOTIENTS WHICH APPEAR
IN THE APPLICATION OF STURM S METHOD TO THE
DISCOVERY OF THE REAL ROOTS OF AN EQUATION.
[Hull British Association Report (1853), Part n., pp. 1 3.]
MANY years ago I published expressions for the residues which appear in
the application of the process of common measure to fx and f x, and which
constitute Sturm s auxiliary functions. These expressions are complete
functions of the factors of fx and of differences of the roots of fx, and are
therefore in effect functions of the factors exclusively, since the difference
between any two roots may be expressed as the difference between two
corresponding factors. Having found that in the practical applications of
Sturm s theorem the quotients may be employed with advantage to replace
the use of the residues, I have been led to consider their constitution ; and
having succeeded in expressing these quotients (which are of course linear
functions of x) under a similar form to that of the residues, that is, as
complete functions of the factors and differences of the roots of fx, I have
pleasure in submitting the result to the notice of the Mathematical Section
of the British Association.
Let Aj, A 2 > ^s h n be the n roots of fa.
Let (a, b, c ... I) in general denote the squared product of the differences
of a, b, c ... I.
Let Zi denote in general 2%(h 0t h et ... h di ), where O l} # 2 ... t indicate any
combination of i out of the n quantities a,b,c,... I, with the convention that
Z = l, Z 1 = n; and let (i) denote {1 + ( l)*j, being zero when i is odd, and
unity when i is even ; then I find that the ith quotient Q t may be written
under the form
Qi = t P* (X - A,) + ;P 2 2 (X - h 2 ) + . . . + iP ?l 2 (X - h n ),
where in general
50] On Sturm s Method of Real Roots of an Equation. 397
f x
If we suppose ^r , by means of the common measure process, to be
f x
expanded under the form of an improper continued fraction, the successive
quotients will be the values of Q,, Q* ... Q n above found, that is
x 1 1
the successive convergents of this fraction will be
The numerators and denominators of these convergents will consequently
also be functions of the factors exclusively. They are the quantities the sum
of the products of which multiplied respectively by fx and f x produce (to
constant factors pres) the residues. The denominators are expressible very
simply in terms of the factors and the differences of the roots ; and their
values under such forms were published by me about the same time as the
values of the residues in the Philosophical Magazine; the expression for
the numerators is much more complicated, but is given in my paper, " The
Syzygetic Relations," &c., in the Philosophical Transactions, [p. 429 below.]
By comparing the expression for any quotient with the expressions for
the two residues from which it may be derived, we obtain the following
remarkable identity : Z^ x Zi, that is
When the roots are all real, we have thus the product of one sum of squares
by the product of another sum of squares (the number in each sum depend
ing upon the arbitrary quantity i), brought under the form of a sum of
a constant number n of squares, which in itself is an interesting theorem.
The expression above given for Q; leads to a remarkable relation between
f x
the quotients and convergents to -7- .
Let it be supposed, as before, that
fx = J __ 1 __ L_ J_
fx Q^x -Q^-Q 3 x- " Q n x
and let the successive convergents to this continued fraction be
A () A(*) D 3 (x} D n (xY
where the numerators and denominators are not supposed to undergo any
reductions, but are retained in their crude forms as deduced from the law
398 On Sturm s Method of Real Roots of an Equation. [50
lYj (x) being I, and AO) being & (x) ; then it may be deduced from the
published results above adverted to that
< " i 2 " (i) +1
Hence 2 { (A 9l ^ . . . /O x (A, - h tl ) (h e - h 8i ) . . . (A e
and we have therefore
P _ ~ t-i ~ t-3 ~ t-5 -" (i) 7) /7 x
^- ^ 4 - ^ 4 - " y? 4 ,-, "* 1 "*/
and consequently
A-!^-3^-5 Ai)
which is the general equation connecting the form of each quotient with
that of the denominator to the immediately preceding unreduced convergent
f x
in the expansion of > under the form of an improper continued fraction.
J x
If instead of the denominator of the unreduced convergents, the denom
inators of the convergents reduced to their simplest forms be employed,
the powers of Z in the constant factor will undergo a diminution. The
essential part of this theorem admits of being stated in general terms as
follows :
" If the quotient of an algebraical function of x by its first differential
coefficient be expressed under the form of a continued fraction whose
successive partial quotients are linear functions of x, any one of these
quotients may be found (to a constant factor pres) by taking the sum of the
products formed by multiplying each factor (x h) of the given function by
the square of what the denominator of the immediately antecedent conver
gent fraction becomes after substituting in it for x the root corresponding to
such factor."
P.S. Since the above was read before the British Association, the
theory has been extended by the author to comprise the general case of
the expansion of any two algebraical functions under the form of a continued
fraction, and has been incorporated into the paper in the Philosophical
Transactions above referred to.
51.
ON A THEOREM CONCERNING THE COMBINATION
OF DETERMINANTS.
[Cambridge and Dublin Mathematical Journal, vm. (1853), pp. 60 62.]
Let 1 A represent the line of terms 1 a l , J a 2 , ... l a m ,
Let 1 A x 1 B represent 2 ( l a r x l b r \ where of course there are m terms
within the symbol of summation.
Again, let *A represent the line Z a 1 , 2 a 2 , ... 2 a m ,
and let
IB
l a r , l a g
2 a r , *a s
represent S
l a r , l a g
l b r ,
denoting the determinant ( J a r . 2 a s x a g . J a r ),
there being of course ra(ra-l) terms comprised within the sign of
summation ; and so, in general, let
, n being less than m,
400 On a Theorem concerning the [51
and where in general T A denotes r a l} r a 2 , ... r a m \
and r B denotes &,, & ... r b j represent
"A,, I a>h 2 , ... ^ n ^ l b h ^ ... J 6 An
s 2 ^, ^ ... a*. x ^, 6^, ...*&,
n ^, n % 2 , ... n a An 6 Ai , ^ 2 , ... "6 An
Now let r be any integer less than m, and let
m (m 1 ) . . . (m r -f 1 )
/* = 1 2 r
and, supposing 0^ (9 2 , ... ^ r to be r numbers of the set 1, 2, ... m, let
#1, 6r 2 , ... G> denote the /u, rectangular matrices of the forms
respectively,
and let H 1} H 2 , ... H^ denote the p, rectangular matrices of the forms
respectively.
Now form the determinant
G 1 x H 1} G l xH 2 , . G! x H
then, if we give r the successive values 1, 2, 3 ... m (in which last case the
determinant in question reduces to a single term), the values of the deter
minant above written will be severally in the proportions of
K, K m ,
that is to say, the logarithms of these several determinants will be as the
coefficients of the binomial expansion (1 +x) m .
When we make r = m, and equate the determinant corresponding to this
value of r with that formed by making r = 1, the theorem becomes identical
with a theorem previously given by M. Cauchy, for the Product of Rect
angular Matrices.
51]
Combination of Determinants.
401
It would be tedious to set forth the demonstration of the general theorem
in detail. Suffice it here to say that it is a direct corollary from the formula
marked (4) in my paper in the Philosophical Magazine for April 1851,
entitled "On the Relations between the Minor Determinants of Linearly
Equivalent Quadratic Functions*," when that formula is particularized by
making
(
{
m+n
represent a determinant all whose terms are zeros except those which lie in
one of the diagonals, these latter being all units, which comes, in fact, to
defining that
m+e
= 1, and
= 0.
The important theorem here referred to is made almost unintelligible
by an unfortunate misprint of q d m , a ^ m , 2 m , ^Om, in place of v0 r , *6 r , *0 r , *Q r .
I may here take notice of another and still more inexplicable blunder in the
same paper, formula (3)f, in the latter part of the equation belonging to
which
a <t>
a <l> m + 8 a n
is written in lieu of
[* p. 249 above.]
[t See pp. 246, 251 above.]
52.
NOTE ON THE CALCULUS OF FORMS.
[See pp. 363 and 411.]
[Cambridge and Dublin Mathematical Journal, vm. (1853), pp. 62 64.]
ACCIDENTAL causes have prevented me from composing the additional
sections on the Calculus of Forms, which 1 had destined for the present
Number of this Journal. In the meanwhile the subject has not remained
stationary. Among the principal recent advances may be mentioned the
following.
1. The discovery of Combinants ; that is to say, of concomitants to
systems of functions remaining invariable, not only when combinations of
the variables are substituted for the variables, but also when combinations
of the functions are substituted for the functions ; and as a remarkable first-
fruit of this new theory of double invariability, the representation of the
Resultant of any three quadratic functions under the form of the square of
a certain combinantive sextic invariant added to another combinant which
is itself a biquadratic function of 10 cubic invariants. When the three
quadratic functions are derived from the same cubic function, this expression
merges in M. Aronhold s for the discriminant of the cubic. The theory
of combinants naturally leads to the theory of invariability for non-linear
substitutions, and I have already made a successful advance in this new
direction.
2. The unexpected and surprising discovery of a quadratic covariant
to any homogeneous function in x, y of the nth degree, containing (n 1)
variables cogredient with x n ~ 2 , x n ~ 3 y . . . y n ~ 2 and possessing the property of
indicating the number of real and imaginary roots in the given function.
This covariant, on substituting for the (n 1) variables the combinations of
the powers of x, y with which they are cogredient, becomes the Hessian
of the given function*.
* This covariant furnishes, if we please, functions symmetrical in respect to the two ends
of an equation for determining the number of its real and imaginary roots. The ordinary
Sturmian functions, it is well known, have not this symmetry. As another example of the
successful application of the new methods to subjects which have been long before the mathe
matical world and supposed to be exhausted, I may notice that I obtain without an effort,
by their aid, a much more simple, practical, and complete solution of the question of the simul
taneous transformation of two quadratic functions, or the orthogonal transformation of one
such function, than any previously given, even by the great masters Cauchy and Jacobi, who
have treated this question.
52] Note on the Calculus of Forms. 403
3. The demonstration due to M. Hermite of a law of reciprocity connect
ing the degree or degrees of any function or system of functions with the
order or orders of the invariants belonging to the system. The theorem
itself was first propounded by me about a twelvemonth back, and com
municated to Messrs Cayley, Polignac, and Hermite, as serving to connect
together certain phenomena which had presented themselves to me in the
theory : unfortunately it appeared to contradict another law too hastily
assumed by myself and others as probably true, and I consequently laid
aside the consideration of this great law of reciprocality. To M. Hermite,
therefore, belongs the honour of reviving and establishing, to myself what
ever lower degree of credit may attach to suggesting and originating,
this theorem of numerical reciprocity, destined probably to become the
corner-stone of the first part of our new calculus ; that part, I mean, which
relates to the generation and affinities of forms *.
4. I may notice that the Calculus of Forms may now with correctness
be termed the Calculus of Invariants, by virtue of the important observation
that every concomitant of a given form or system of forms may be regarded
as an invariant of the given system and of an absolute form or system of
absolute forms combined with the given form or system. As regards that
particular branch of the theory of invariants which relates to resultants, or,
in other words, to the doctrine of elimination, I may here state the theorem
alluded to in a preceding Number of the Journal, to wit that if R be the
resultant of a system of n homogeneous functions of n variables, written
out in their complete and most general form (so that by definition R = Q
is the condition that the equations got by making the n given functions
zero, shall be simultaneously satisfiable by one system of ratios), then the
condition that these equations may be satisfied by i distinct systems of
ratios between the n variables is &R = 0, the variation 8 being taken in
respect to every constant entering into each of the n equations.
* This theorem of numerical reciprocity promises to play as great a part in the Theory of
Forms as Legendre s celebrated theorem of reciprocity in that of Numbers. Another demonstra
tion of it, which leaves nothing to be desired for beauty and simplicity, has been since
discovered by Mr Cayley, which ultimately rests upon that simple law (essentially although not
on the face of it a law of reciprocity) given by Euler, which affirms that the number of modes in
which a number admits of being partitioned is the same whether the condition imposed upon
the mode of partitionment be that no part shall exceed a given number, or that the number of
parts constituting any one partition shall not exceed the same number.
262
53.
ON THE RELATION BETWEEN THE VOLUME OF A TETRA
HEDRON AND THE PRODUCT OF THE SIXTEEN ALGE
BRAICAL VALUES OF ITS SUPERFICIES.
[Cambridge and Dublin Mathematical Journal, vili. (1853), pp. 171 178.]
THE area of a triangle is related (as is well known) in a very simple manner
to the eight algebraical values of its perimeter : If we call the values of the
squared sides of the triangle a, 6, c, there will be nothing to distinguish the
algebraical affections of sign of the simple lengths so as to entitle one to a
preference over the other. The area of the triangle can only vanish by reason
of the three vertices coming into a straight line; hence, according to the
general doctrine of characteristics, we must have the Norm of \/a + \/b + \/c,
containing as a factor some root or power of the expressions for the area of
the triangle. The Norm in question being representable as N 2 where N
is the Norm of o> + 6* c*, which is of four dimensions in the elements a, b, c,
and undecomposable into rational factors, we infer that to a numerical factor
pres the square of the area must be identical with the Norm N, and thus,
by a logical coup-de-main, completely supersede all occasion for the ordinary
geometrical demonstration given of this proposition, which in its turn, with
certain superadded definitions, would admit of being adopted as the basis
of an absolutely pure system of Analytical Trigonometry that should borrow
nothing from the methods and results of sensuous or practical geometry.
But into this speculation it is, not my present purpose to enter: what I
propose to do is to extend a similar mode of reasoning to space of three
dimensions, and to point out a general theorem in determinants which is
involved as a consequence in the generalization of the result of the inquiry
when pushed forward into the regions of what may be termed Absolute or
Universal Rational Space.
Let F, G, H, K be the four squared areas of the faces of a tetrahedron,
and V the volume ; then, since V only becomes zero in the case of the four
vertices coming into the same plane, which is characterised by the equation
53] Relation between the Volume of a Tetrahedron, etc. 405
subsisting, we infer that N the Norm of
must contain a power of V as a rational factor. F 2 is rational and of
three dimensions in the squared edges; the Norm above spoken of is of
eight dimensions in the same. Consequently there is a rational factor,
say Q, remaining, which is of five dimensions in the squared edges, and this
factor I now proceed to determine, the other factor F 2 being, as is well
known, a numerical product of the determinant
o,
bd 2 ,
ca 2 ,
da?,
1,
ab 2 ,
o,
cb 2 ,
db\
1,
ac\
be 2 ,
o,
dc 2 ,
1,
ad 2 ,
bd-,
cd 2 ,
0,
1,
1
1
1
1
a, b, c, d being the four angular points of the tetrahedron. See London
and Edinburgh Philosophical Magazine, 1852. [p. 386 above.]
The quantity Q possesses an interest of a geometrical character ; for if we
call the radii of the eight spheres which can be inscribed in a tetrahedron
TI, r zt r 3 , r t , r s , r 6 , ?>, r 8 , we evidently have nryyvWVe x N= (3F) 8 . Hence
38^-8 38-J/6
(R), the product of the eight radii in question, = =r=- = .
Consequently Q is the quantity which characterises the fact of one or
more of the radii of the inscribed spheres becoming infinite. For the triangle
there exists no corresponding property ; this we know d priori, and can
explain also analytically from the fact that if we call P the product of the
radii of the four inscribable circles, v the Norm of the perimeter, and A
the area, we have
Pv = 2*
and
which contains no denominator capable of becoming zero, so that as long
as the sides remain finite the curvature of the inscribed circles is incapable
of vanishing.
To determine N as a function of the edges, and then to discover by actual
N
division the value of -pr , would be the direct but an excessively tedious
and almost impracticably difficult process. I have ever felt a preference
for the d priori method of discovering forms whose properties are known, and
never yet have met with an instance where analysis has denied to gentle
406
On the Relation between the
[53
solicitation conclusions which she would be loth to grant to the application
of force. The case before us offers no exception to the truth of this remark.
Q is a function of five dimensions in terms of the squared edges : let us
begin by finding the value of that part of Q in which at most a certain
set of four of these edges make their appearance, and to find which con
sequently the other two edges may be supposed zero without affecting the
result. We may make two distinct hypotheses concerning these two edges ;
we may suppose that they are opposite, that is non-intersecting edges, or
that they are contiguous, that is intersecting edges.
To meet the first hypothesis suppose ab = 0, ce = 0.
For convenience sake, use F, G, H, K to denote 16 times the square
of each area, instead of the simple square of the areas. Call
lQ(abc) 2 = K, 16 (a&d) 2 = H, 16(acd) 2 =, 16 (bed) 2 = F.
Then
- K = (a&) 4 + (ac) 4 + (be)* - 2 (ab) 2 (ac) 2 - 2 (a&) 2 (be) 2 - 2 (ac) 2 (be) 2
= ac 4 + &c 4 -2(ac) 2 (6c) 2 .
Similarly,
- H= ad 4 + bd* - 2 (ad) 2 (bd) 2 ,
- G = ca 4 + da 4 - 2ca 2 da 2 ,
- F = c6 4 + d& 4 - 2c6 2 d& 2 .
Hence one value of <JF + ^G + ^H + ^K will be
V(- 1) {(ac 2 - be 2 ) + (bd 2 - ad 2 ) + (da 2 - ac 2 ) + (be 2 - bd 2 )} = 0.
Hence, on this first supposition, the Norm vanishes. But V 2 does not vanish
when ab = 0, cd = 0, for it becomes, saving a numerical factor,
o,
o,
ac 2 ,
ad 2 ,
1
o,
0,
be 2 ,
bd 2 ,
1
ca 2 ,
cb 2 ,
0,
0,
1
da 2 ,
db 2 ,
0,
0,
1
1,
1,
1,
1,
that is
(ac 2 . bd 2 - ad 2 . be 2 ) (cb 2 + ad 2 - ca 2 - bd 2 )
+ (be 2 - ac 2 ) (ca 2 . db 2 - cb 2 . da 2 )
+ (ad 2 - bd 2 ) (ca 2 . db 2 - cb 2 . da 2 )
= 2 (ac 2 . bd 2 - ad 2 . be 2 ) (ad 2 + be 2 - ac 2 - bd 2 ) ;
and consequently, since N vanishes but V 2 does not vanish, Q vanishes,
showing that there is no term in Q but what contains one at least of any
53]
Volume of a Tetrahedron, etc.
407
two opposite edges as a factor; or, in other words, there is no term in Q
of which the product of the square of the product of all three sides of some
one or other of the four faces does not form a constituent part.
Next, let us suppose ab = 0, ac = 0, then
K 2 = 16afec 2 = - fee 4 ,
# 2 = 16afed 2 = -(ad 2 -fed 2 ) 2 ,
G 2 = 16acd 2 = - (ad 2 - cd 2 ) 2 ,
F 2 = IGfecd 2 = - fee 4 - fed 4 - cd 4 + 2fec 2 . fed 2 + 2fec 2 . cd 2 + 2fed 2 . cd 2 .
Four of the factors of N will be therefore
{t, (be 2 + cd 2 - bd 2 ) F}, {i (be 2 - cd 2 + bd 2 ) F],
i denoting V(- 1). an d the product of these four factors will be
{(fee 2 + cd? - bd 2 ) 2 + F 2 } x {(be 2 - cd 2 + bd 2 ) 2 + F 2 },
which is equal to
16fec 4 .fed 2 .cd 2 ;
and similarly, the remaining part of the Norm will be
{(2ad 2 - bd 2 - cd 2 + be 2 ) 2 + F>] x {(2ad 2 - bd 2 - cd 2 - be 2 ) 2 + F 2 },
that is
[4-ad 4 - 4>ad 2 (bd 2 + cd 2 + be 2 ) + 4fec 2 . bd 2 + 4bd 2 . cd 2 + 4cd 2 . fee 2 }
x {4ad 4 - 4ad 2 (fed 2 + cd 2 - fee 2 ) + 4fed 2 . cd 2 }.
Again, since ac 2 = and fee 2 = 0, V 2 becomes
0,
0,
0,
da 2
1,
0, 0, ad 2 , 1
0, fee 2 , fed 2 , 1
cfe 2 , 0, cd 2 , 1
dfe 2 , dc 2 , 0, 1
1,
1,
which is evidently equal to
0, 0, ad 2 , 1
0, cfe 2 , cd 2 , 1
26c 2
da 2 , dfe 2 , 0, 1
1, 1, 1,
= 2fec a {2fec 2 ad 2 + ad 4 - ad 2 fed 2 - cd 2 ad 2 + 6d 2 cd 2 } - 26c 4 ad 2
= 2fec 2 {ad 4 - ad 2 (fed 2 + cd 2 - fee 2 ) + fed 2 . cd 2 }.
0,
ad 2 ,
1
-fee 4
da 2 ,
0,
1
1,
o,
408 On the Relation between the [53
Hence, paying no attention to any mere numerical factor, we have found that
N
when ac = and be = 0, Q or ^ becomes
be 2 . bd* . cd 2 {ac? 4 - ad 2 (bd 2 + cd 2 + be 2 ) + be 2 . bd 2 + bd 2 . cd 2 + cd 2 . be 2 }.
Hence, with the exception of the terms in which five out of the six edges
enter, the complete value of Q will be
2 (be 2 . bd 2 . cd 2 ) [ad* - ad 2 (bd 2 + cd 2 + be 2 ) + be 2 . bd 2 + bd 2 . cd 2 + cd 2 . be 2 },
or more fully expressed, and still abstracting from terms containing five
edges,
= 26c 2 . bd 2 . cd 2 {(ab* + ac 4 + ad 4 ) - (ab 2 + ac 2 + be 2 ) (bd 2 + be 2 + cd 2 )
+ be 2 . bd 2 + bd 2 . cd 2 + cd 2 . fee 2 }.
It remains only to determine the value of the numerical coefficient
affecting each of the six terms of the form
ab* . ac 2 . ad 2 . be 2 . bd*.
To find this, let
ab 2 = ac 2 = ad 2 = be 2 = bd 2 = cd 2 = 1 ;
then evidently, since all the squared areas are equal, several of the factors
of N will become zero, but V 2 evidently does not become zero for a regular
tetrahedron ; hence Q becomes zero : and if we call the numerical factor
sought for X, we must have (observing that the 2 includes four parts cor
responding to each of the four faces)
4 [3 - 9 + 3} + 6X = 0,
therefore 12 + 6X = 0, or A, = 2.
Hence the complete value of Q is
2a6 J . be 2 . ca 2 {(da* + db* + dc*) - (da* + db* + dc*) (ab 2 + be* + ca 2 )
+ ab 2 . be 2 + be* . ca 2 + ca 2 . ab 2 }
+ 22 (ab 2 . be 2 . cd 2 . da 2 . ac 2 ) ;
or, which is the same quantity somewhat differently and more simply
arranged,
Q = 2 (ab 2 . be* . ca 2 ) {(da 4 + db* + dc* + da 2 . db 2 + db 2 . dc 2 + dc* . da 2 )
+ (ab 2 . be* + be 2 . ca 2 + ca 2 . ab 2 ) - (da 2 + db* + dc 2 ) (ab* + be* + ca 2 )},
and this quantity equated to zero expresses the conditions of a radius of an
53] Volume of a Tetrahedron, etc. 409
inscribed sphere becoming infinite. The direct method would have involved,
as the first step, the formation of the Norm of a numerator consisting of
the value of which is
and contains 4 + 6 + 12, that is 22 positive terms, and 12, that is 13 negative
terms, together 35 terms, each of which might be an aggregate of 6 4 or
1296 quantities, and thus involve in all the consideration of 45360 separate
parts, for each of the quantities F, G, H, K being a quadratic function of
three of the squared edges, will contain six terms. It is not uninteresting to
notice that in addition to the case already mentioned of two opposite edges
being each zero, as ab = 0, cd = 0, Q will also vanish for the case of ab = cd,
be = ad ; that is for the case of two intersecting edges being each equal in
length to the edges respectively opposite to them. This is evident from
the fact that on the hypothesis supposed the face acb = acd and the face
N
bdc = bda ; hence N = 0, and therefore, V not vanishing, -^ , that is Q, will
vanish.
We may moreover remark that since ab = and cd = does not make
V vanish, the perpendicular distance of ab from cd, which, multiplied by
ab x cd, gives six times the volumes, must on this supposition become infinite.
When three edges lying in the same plane all vanish simultaneously, Q
vanishes, since one edge at least in every face of the pyramid vanishes,
and V also vanishes, as is evident from the expression for F 2 , when ab = 0,
ac =0,bc = 0, becoming a multiple of
o,
o,
0,
ad 2 ,
1,
o,
0,
o,
bd\
1,
o,
0,
o,
cd 2 ,
1,
ad 3 ,
bd\
cd 2 ,
0,
0,
1
1
1
which is evidently zero.
It appeared to me not unlikely, from the situation and look of Q (the
characteristic of one of the inscribed spheres becoming infinite), that it might
admit of being represented as a determinant, but I have not succeeded in
throwing it under that form. I have a strong suspicion that if we take
Q a function corresponding to a tetrahedron a b c d , in the same way as
Q corresponds to abed, QQ , and not improbably *J(QQ ), will be found to be
410 Relation between the Volume of a Tetrahedron, etc. [53
(as we know from Staudt s Theorem of \/( V 2 V 2 ) ) a rational integral
function of the squares of the distances of the points a, b, c, d from the points
a , b , c , d .
That N should divide out by V z is in itself an analytical theorem relating
to 6 arbitrary quantities ab 2 , ac 2 , ad 2 , bc z , bd 2 , cd 2 , which evidently admits
of extension to any triangular number 10, 15, &c. of arbitrary quantities.
Thus we may affirm, d priori, that the norm of
where (for the sake of symmetry, retaining double letters, as AB, AC, &c.,
to denote simple quantities)
0, AB, AC, AE, 1
AB, 0, BC, BE, 1
AC, BC, 0, CE, 1
AE, BE, CE,
1, 1, 1,
0,
1,
0, AB, AC, AD, I
AB, 0, BC, BD, 1
AC, BC, 0, CD, 1
AD, BD, CD, 0, 1
1, 1, 1, 1,
N = &c., M = &c., L = Scc.,
will contain as a factor the determinant
0, AB, AC, AD, AE, 1
AB, 0, BC, BD, BE, 1
AC, BC, 0, CD, CE, 1
AD, BD, CD, 0, DE, 1
AE, BE, CE, DE, 0, 1
1, 1, 1, 1, 1,
and a similar theorem may evidently be extended to the case of any
arbitrary quantities whatever.
n (n + 1)
54.
ON THE CALCULUS OF FORMS, OTHERWISE THE THEORY
OF INVARIANTS.
[Continued from p. 363 above.]
[Cambridge and Lublin Mathematical Journal, vm. (1853), pp. 256 269.]
SECTION VII. On Combinants.
REASONS of convenience have induced me to depart from the plan to
which I originally intended to adhere in the development of this theory,
and I shall hereafter, from time to time, continue to add sections on such
parts of the subject as may chance to be most present to my mind or most
urgent upon my attention, without waiting for the exact place which they
ought to occupy in a more formal treatise, and without having regard to the
separation of the subject into the two several divisions stated at the outset
of the first section. The present section will be devoted to a brief and
partial exposition of the theory of Combinants*, with a view to the applica
tion of this theory to the solution of the problem of throwing the resultant
of three general homogeneous quadratic functions under its most simple form,
being analogous to that given by Aronhold in the particular case where
the three functions are derived from the same cubic, and becoming identical
therewith when the coefficients are accommodated to this particular supposi
tion f. I shall confine myself for the present to combinants relating to
systems of functions, all of the same degree.
If (/>!, < 2 , $r, be homogeneous functions of any number of variables, any
invariant or other concomitant of the system which remains unchanged, not
only for linear substitutions impressed upon the variables contained within the
functions, but also for linear combinations impressed upon the functions them
selves, is what I term a Combinant. A Combinant is thus an invariant or other
concomitant of a system in its corporate capacity (qua system), being in fact
* Discovered by the Author of this paper in the winter of 1852.
t A similar method will subsequently be applied to the representation of the resultant of two
cubic equations as a function of Combinants bearing relations to the quadratic and cubic
invariants of a quartic function of x and y, precisely analogous to those which the Combinants
that enter into the solution above alluded to bear to the Aronholdian invariants of a cubic
function.
412 On the Calculus of Forms. [54
common to the whole family of forms designated by A^ + A^ + ... + A r $ r ,
where \, A 2 , ... A r , are arbitrary constants. If the coefficients of <^> l , </> 2 , ... </>,.,
be supposed to be written out in r lines (the coefficients of corresponding
terms occupying the same place in each line), so as to form a rectangular
matrix, any combinantive invariant will be a function of the determinants
corresponding to the several squares of r 2 terms each that can be formed out
of such matrix, or, as they may be termed, the full determinants belonging
to such rectangular matrix. If we call any such combinant K, then, over
and above the ordinary partial differential equations which belong to it in its
character of an invariant, it will be necessary and sufficient, in order to
establish its combinantive character, that K shall be subject to satisfy (r - 1)
pairs of equations of the form
, d j, d , d
da db dc"
d . d d
a j~ + & ^T + c ~j~>
da do dc
where a, b, c...; a ,b ,c ..., are respectively lines in the matrix above
referred to.
So any combinantive concomitant will be a function of the full deter
minants of the matrix formed by the coefficients of the given system of forms
and of the variables, and will be subject to satisfy the additional differential
equations just above written.
It will readily be understood furthermore, that an invariant or other
concomitant may be combinantive in respect to a certain number of forms
of a system, and not in respect of other forms therein ; or more generally,
may be combinantive in respect of each, separately considered, of a series of
groups into which a given system may be considered to be subdivided,
without being so in respect of the several groups taken collectively.
In the fourth section of my memoir [p. 429 below] on a "Theory of the
Conjugate Properties of two rational integral Algebraical Functions," recently
presented to the Royal Society of London, the case actually arises of an
invariant of a system of three functions, which is combinantive in respect
only to two of them.
For greater simplicity, let the attention for the present be kept fixed
upon combinants which are such in respect of a single group of functions,
all of the same degree in the -variables. (It will of course have been
perceived that when the system is made up of several groups, there would
be nothing gained by limiting the groups to be all of the same degree
inter se; it is sufficient that all of the same group be of the same degree
per se.)
54] On the Calculus of Forms. 413
All such combinants will admit of an obvious and immediate classification.
Let us suppose that a combinant is proposed which is in its lowest terms,
that is to say, incapable of being expressed as a rational integral algebraical
function of combinants of an inferior order. Such a combinant may, notwith
standing this, admit of being decomposed into non-combinantive invariants
of inferior dimensions to its own, and in such event will be termed a complex
combinant ; or it may be indecomposable after this method, in which event
it will be termed a simple combinant. It will presently be shown, that the
resultant of a system of three quadratic functions is made up of a complex
combinant of twelve dimensions, and of the square of a simple combinant
of six dimensions, expressible as a biquadratic function of ten non-com
binantive invariants, each of three dimensions in the coefficients. There
is an obvious mode of generating complex combinants ; according to which
they admit of being viewed as invariants of invariants. Supposing
<>!, < 2 , ... (f> r , to be the functions of the given system, X 1 < 1 + X 2 </> 2 + ... +\ r <f) r
may conveniently be termed the conjunctive of the system : if now one or
more invariants or other concomitants be taken of this conjunctive, there
results a derivative function or system of functions of the quantities
Xj, X.j, ... X r , in which every term affecting any power or combination of
powers of the X series is necessarily an invariant or concomitant of the
given system. If now an invariant or other concomitant be taken of the
new system in respect to \, X 2 , ... X r , (the original variables (supposing them
to enter) being treated as constants), this secondarily derived invariant will
be itself an Invariant, or at all events a Concomitant in respect of the
original system, and being unaffected by linear substitutions impressed upon
the X system, is by definition a combinant of such system. A similar
method will obviously apply if the original system be made up of various
groups; each group will give rise to a conjunctive, and one or more con
comitants being taken of this system of conjunctives and treated as in the
case first supposed, (the only difference being, that there will on the present
supposition be several unrelated systems instead of a single system of new
variables, that is, several X systems instead of one only) the result, when all
the X systems have been invariantized out (that is, made to disappear by any
process for forming invariants), will be a combinant in respect to each of the
groups, severally considered, of the given system of functions.
Here let it be permitted to me to make a momentary digression, in order
to be enabled to avoid for the future the inconvenience of using the phrase
" invariant or other concomitant," and so to be enabled at one and the same
time to simplify the language and to give a more complete unity to the
matter of the theory, by showing how every concomitant may in fact be
viewed as a simple invariant, so that the calculus of forms may hereafter
admit of being cited, as I propose to cite it, under the name of the Theory
of Invariants.
414 On the Calculus of Forms. [54
Thus, to begin with the case of simple contragredience and cogredience,
if > 7 ?> ? are contragredient to cc, y, z ..., any form containing |:, 17, ...,
which is concomitantive to a given form or system of forms S, which contains
a, y, 2 ..., may be regarded as concomitantive to the system S , made up of
S and the superadded absolute form gx + r)y + z+ ..., say S-; where , ?/, ...
are treated no longer as variables, but as constants. In like manner every
system of variables contragredient to x, y, z . . . , or to any other system of
variables in 8, will give rise to a superadded form analogous to S-, the totality
of which may be termed $ x ; and thus the various systems f , 77, . . . will no
longer exist as variables in the derived form, but purely as constants. Again,
if S contain any system of variables <, ty, ^, &c., contragredient to x, y, z, &c.,
the system of variables u, v, w, &c., cogredient with x, y, z, &c., may be
considered as constants belonging to the superadded form <^u + ^rv + ^w ... ;
but if S do not contain any system contragredient to x, y, z, &c., then
u, v, w, &c. may be treated as constants belonging to the superadded system
of forms xv yu, yw zv, zu xw, &c. ; and so in general any concomitant
containing any sets of variables in simple relation, whether of cogredience
or contragredience, with any of the sets in the given system S, may in all
cases be treated as an invariant of the system S , made up of S and a
certain superadded system S 1} all the forms contained in which are ab
solute, by which I mean, that they contain no literal coefficient. The same
conclusion may be extended to the case of concomitants containing sets of
variables in compound relation with the sets in the given system of forms S.
Thus, suppose u jt u. 2 , ... u n , to be in compound relation of cogredience with
x n ~ l , x n ~ 2 y, x n ~ 3 y 2 , ... y n ~ 1 , u lt u it ... u n , may be regarded as constants
belonging to the superadded form
Uiy n ~ l - (n - 1) u 2 y n ~ 2 x + (n - 1) (n - 2) u 3 y n ~ 3 a} 2 + ... u n x n ~\
say fl. And thus universally we are enabled to affirm, that a concomitant
of whatever nature to a given system of forms, may be reduced to the form
of an invariant of a system made up of the given system and a certain other
superadded system of absolute forms : without, therefore, abandoning the use
of the terms concomitant, cogredience, contragredience, &c., which for many
purposes are highly convenient and save much circumlocution, we may
regard every concomitant as a disguised invariant, and under the name of
the Theory of Invariants comprise the totality of the theory of Concomitance.
I have already had occasion to make use of the superadded form H in
discussing the theory of the Bezoutiant (a quadratic form concomitant to
two functions of the same degree in x, y, which plays a most important part
in the theory of the relations of their real roots), in the memoir for the Royal
Society previously adverted to.
I now return to the question of applying the theory of combinants to
the decomposition of the resultant of three general quadratic functions of
54] On the Calculus of Forms. 415
x, y, z. It will of course be apparent that every resultant of any system of n
functions of the same degree of a single set of n variables is a combinantive
invariant of the system. This is an immediate and simple corollary to the
theorem given by me in this Journal, in May, 1851. Accordingly, in pro
ceeding to analyse the composition of the resultant of three quadratic
functions, I may, besides impressing linear combinations upon the variables,
impress linear combinations upon the functions themselves, in any way most
conducive to simplicity and facility of expression and calculation; and
whatever relations shall be proved to exist between the resultant and other
combinants for such specific representation, must be universal, and hold good
for the functions in their most general form.
(1) The system, by means of linear substitutions impressed upon the
variables which enter into the functions, may be made to assume the form
ax* + 6?/ 2 + cz\
la? + my 2 + nz z + 2pyz + 2qzx + 2rxy.
(2) By means of linear combinations of the functions themselves the
system may evidently be made to take the form
(c - a) x* + (c - 6) y\
(a -b)y* + (a- c) z\
ky - + 2pyz + 2qzx + 2rxy ;
and finally, by taking suitable multipliers of x, y, z in lieu of x, y, z, it may
be made to become
y z + 2fyz + 2gzx
We have thus reduced the number of constants in the system from
eighteen to five ; and as it will readily be seen that in any combinant of the
system in its reduced form p and <r can only enter as factors of the simple
quantity, (pa-)\ for all purposes of comparison of the combinants of the
system of like dimensions with one another, p and a might admit of being
treated as being each unity, and accordingly, practically speaking, we have
only to deal with three in place of eighteen constants, a marvellous simplifi
cation, and which makes it obvious, a priori, or at least affords a presumption
almost amounting to and capable of being reduced to certainty, that the
number of fundamental combinants of the system, of which all the rest must
be explicit rational functions, will be exactly four in number ; which, for the
canonical form hereinbefore written, on making p and <r each unity, will
correspond to
A 2 /, fgh,
416 On the Calculus of Forms. [54
and will be of the 3rd, 6th, 12th, and 9th degrees respectively. The reason
why the squares of /, g, h, instead of the simple terms /, g, h, appear in the
2nd and 3rd of these forms is, because, on changing x into x,y into y,
or z into z, two of the quantities f, g, h will change their sign, but the
forms representing the invariants of even degrees ought to remain absolutely
unaltered for such transformations. I shall in the course of the present
section set forth the methods for obtaining these four combinants, which,
although of the regularly ascending dimensions 3, 6, 9, 12, belong obviously
to two different groups, the one of three dimensions forming a class in itself,
and the natural order of the three others being that denoted by the sequence
6, 12, and 9, and not that which would be denoted by the sequence 6, 9, 12,
the combinant of the ninth degree being properly to be regarded as in some
sort an accidentally rational square root of a combinant of 18 dimensions.
Let now p(x 2 -f)=U,
<r(f-z 2 )=W,
f + Zfyz + Zgzx + Zhxy = V.
The resultant will be found by making
Hence the resultant R
2/+ 2g + 2A) (1 - 2/- 2g + 2A) (1 + 2/- 2g - 2A) (1 -
= (pa)* {(1 + 4^ 2 - 4/ - 4sr 2 ) 2 - (4>h
= (pa)* [1 - 8 (/ + g* + A 2 ) + 16 |(
Let now K = \U + pV + vW,
K being what I term a linear conjunctive of U, V, W. The invariant of K,
in respect to x, y, z, will be the determinant
54]
that is
417
= - pa;
3a,X 2 i/ + S
b. 2 = IZfgh - 6g 2 ,
or, multiplying by 6, we may write
where
the notation being accommodated to that employed by Mr Salmon in The
Higher Plane Curves, X, /n, v in IK being correspondent to ac, y, z in
Mr Salmon s form. If now we employ Mr Salmon s expression for the 8
(the biquadratic Aronholdian of IK), observing that
a 2 = 0, c 2 = 0,
we have the complex combinaut
- 16 0* -
j = 0, c 3 = 0,
-f 16 (/ 2 - >9 2 ) 2
Hence, calling the resultant R, we have
R + 4S^, v I x , y , z K = 1 - 8 (/ 2 + g* + A 2 ) + 16 (
+ 32 (/V + g*K + Kf-) =
Let O be taken the polar reciprocal to the conjunctive
and for greater simplicity, as we know, a priori, from the fundamental
definition of a combinant, which (save as to a factor) must remain unaltered
by any linear modification impressed upon the functions to which it apper
tains, that p and a- can enter factorially only in any combinant, let p and <r
be each taken equal to unity in performing the intermediary operations.
Then
X,
hp,
X + p + v,
fa, 77
V, C,
- (y- + vfi + i>X +f~fJ?)
(- \v + g-fj?)
" (X 2 + X/z + Xi/ + /i 2 /* 2 )
27
418
On the Calculus of Forms.
[54
Upon fl, which is a quadratic function in respect of each of the two
unrelated systems 17, ; X, p, v, and also in respect of the coefficients in
(U, V, W), we may operate with the commutantive symbol
A A d_\
d% drj d
d d d
d% drj dt,
d d d
d\ dp dv
d d d
d\ dp dv
which, for facility of reference, I shall term 8E.
Considering the first line as stationary, we shall obtain, for the value of
8#(fl), 216 commutantives, which may be expressed under the following
forms :
V
d
df
d
drj
d
d
d
d
d?
drj
d?
[i
i z
T~ 2
dp 2
d^\
d
d
d
d%
drj
<K
d
d
d
d
drj
5?
[dx 2
d d
dp dv
d d~\
dp dv]
d
d
d
d
drj
d
d
d
d
d?
drj
d *
[_d\
dv
dp*
d\ dv]
d
d
d
d%
drj
d$
d
d
d
d?
drj
dS
[. A
[d\ dp
d
dp dv 2 ]
54]
On the Calculus of Forms.
419
5/4
In this expression the first lines may be considered stationary, the
second lines are subject to the usual process of commutation, which makes
three of the six permutations positive and three negative ; and the third
or bracketed lines are subject to the simple process which makes all the
permutations of the same sign. In the three middle groups two of the
terms in the final line are always identical ; it will therefore be more
convenient to introduce the multiplier 2, and then to consider each such line
to represent the three distinct permutations, taken singly.
Let now
1 f d 2 d 2 i
1 j d 2 d d d d , o
8 \d% 2 dri d dij r 7 "
1 f d d d d (
8 Idijdf d^df d?
( d d d d d
And let
|_dX 2 d/ji dv d/j, dv\
[d d d 2 d d~\ T/ ,
d\ dv dfj 2 d\ dv\
- T "
Ju >
d d d
d_ d d d d d~\ _ j
d\ d/x dp dv dv d\]
Then, attending to the convention just previously explained, we shall have
= (L - 2L - 2L" - 2L " + 2 A)
- 2 (ny - 2 (ny - 2 (ay + 2 (n^},
272
420 On the Calculus of Forms. [54
a symbolical product, any term in which such as L l" will mean
( [".*_ <L*L A
I |_dX 2 dp dv dp ^
I H">
d d d* d d
and a similar interpretation must be extended to each of the 25 partial
products ; we have then
4>L " (0) = 8/ 2 ,
and, finally, the five terms comprised in
each = 0. All the above equations can be easily verified by direct inspection,
it being observed that 8 (ft) represents
that 8 (H) represents
that 8 (O)" represents
-Xr+^V 2 , # (/AX + /AI/) + (g -fh) v?,
that 8 (O)" represents
X 2 + /uX + vX + /i> 2 , - hfjuv
and that (l\ represents
-f\fi - hgp?, g (fi\ + nv] + (g -fh) p
We have thus
E (fl) = 8f - 4^ 2 - 2 + 8/ 2 + 8/
Hence
(A)
54] On the Calculus of Forms. 421
If we restore to U, V, W their general values, and make
U=ax* + by 2 + cz 2 + 2fyz + 2gzx + 2hxy,
V=a x 2 + b y 2 + c z 2 + 2f yz + 2g zx + 2h xy,
W = a"x 2 + b"y 2 + c"z 2 + 2f"yz + 2g"zx + 2h"xy,
and construct the cubic function
^ = (ax + a y + a"z) (bx + b y + b"z) (ex + c y + c"z)
- (ax + a y + a"z) (fx +fy +f"z)* - (bx + b y + b"z) (gx + g y + g"zj
(ex + c y + c"z) (hx + h y -f h"z)*
+ 2 (fx +f y +f"z) (gx + g y + g" z ) (hx + h y + h"z),
that is
2 (abc - af 2 - bg 2 - ch 2 + 2fgh) x 3
+ 2 {a bc + ab c + abc - (af 2 + 2aff) - (b g 2 + 2bgg ) - (c A 8 + 2chh )
+ {a b"c + a bc" + a"b c + a"bc + ab c" + ab"c - Za ff" - 2aff" - 2a"ff
- Zb gg" - 2bg g" - 2b"gg - 2c hh" - Zch h" - 2c"hh
+ 2f"gh + 2f g"h + 2fg h" + Zf gh + Zf gh + 2fg"h } xyz,
S*,n, v Ix,y,tK in the preceding equation becomes simply the Aronholdian
S to ^, which may be calculated by Mr Salmon s formula previously quoted.
fi may be taken equal to the determinant
, bx + b y + b"z, fx+f y+f"z, 77
+ g y + g"z, fx+f y+f z, cx+c y + c"z,
f, <?> r, o
And the cubic commutant of this, obtained by affecting it with the com-
mutantive operator,
d d d^
dx dy dz
d d d
dx dy dz
d d d
dj~ drj rff
d d d
422 On the Calculus of Forms. [54
will give 482? (H) if each of the four lines of the operator undergoes permuta
tion, or 8E(l), if one of the four lines is kept stationary. Thus it falls
within the limits of practical possibility to calculate explicity, by the formula
(A), the value of the resultant. I give to the S of ^ the appellation of the
Hebrew letter & (shin), and to the commutant of O the appellation of the
Hebrew letter b (teth). These letters are chosen with design; for I shall
presently show that when the three given quadratic functions are the
differential derivatives of the same cubic function ty, the b becomes the
Aronholdian T to the cubic function, or, as we may write it, Tty, and the
fcjj becomes the Aronholdian S of the Hessian thereto, that is SH^jr.
Thus for the first time the true inward constitution of the resultant of
three quadratics is brought to light. The methods anteriorly given by me,
and the one subsequently added by M. Hesse for finding this resultant,
adverted to in Section II., lead, it is true, to the construction of the form,
but throw no light upon the essential mode of its composition.
55.
THtiOREME SUR LES LIMITES DES RACINES REELLES DBS
AQUATIONS ALG^BRIQUES.
[Nouvelles Annales de Mathematiques, xn. (1853), pp. 286 287.]
SOIT /O) =
une equation alge*brique de degre n, et supposons qu en operant sur f(x)
et (#) comme dans le theoreme de M. Sturm, on obtienne les n quotients
il faut remarquer seulemerit qu on obtient le w ifeme quotient, a^x + b n , en
divisant 1 avant-dernier residu par le dernier residu.
Formons la serie de 2n quantites
2-^ +2-6o 2-6 3 2 - b n t
a-i a 2 a 3 a n
il n y a aucune racine de liquation
/WrP
entre la plus grande de ces quantite s et + oo , ni entre la plus petite de ces
quantite s et oo *.
* Prochainement, une demonstration de ce theoreme generalise, [p. 424 below.]
56.
NOUVELLE METHODE POUR TROUVER UNE LIMITE SUPERI-
EURE ET UNE LIMITE INF^RIEURE DES RACINES
D UNE EQUATION ALG^BRIQUE QUELCONQUE.
[Nouvelles Annales de MatMmatiques, xn. (1853), pp. 329 336.]
1. LEMME. Soient
C 1} (7 2 , C7 3 . ..>_!, C r
une suite de quantity s positives, assujetties a cette loi
ou. les p sont des quantites positives quelconques.
Si, dans la fraction continue
III _L 1
qi + & + q 3 + - - + q r -i + q r
(les quantites q lt q^ ... dtant des quantites positives ou negatives), on a les
inegalites
[?J > Q , [? 2 ] > C a , [q t ] >C 3 ... [q r -i] > Cr_! , [q r ] > C r
(les crochets indiquent la racine carree positive du carre de la quantit^ que
ces crochets renferment), le denominateur de la fraction continue aura meme
signe que le produit q^qa ... q r -iq r -
Demonstration. Posons
qi = m 1}
1
2 H -- =m 2 ,
1
q r +
56] Nouvelle Methode, etc. 425
il est aise de verifier que les denominateurs successifs de la fraction continue
sont
m^ a meme signe que q^.
111111 1 r , 1
- = , r -..< , < , k/ 2 ] > , [<7 2 ] > , etc. ;
q 1 m, [q,] /V m, ^ ^ mC
done <? 2 a meme signe que ra 2 , et aussi m l m 2 est de meme signe que q^ :
, 1 ! ! r i !
,>/*, + -, m. 2 >^ 2 , <-, [}.]>-;
/ti m 2 /4 2 fa
done g 3 a meme signe que w s ; ainsi m^i^m,, est de meme signe que qiq 2 q 3 ,
et, en continuant, on parvient a demontrer que r m-jn^ r m z ... m r ^m r , c est-a-dire
le d^nominateur de la fraction continue, est de meme signe que le produit
2. TmfoRfcME. Si /"(#) est une fonction algebrique entiere de degre n,
et si Von prend arbitrairement une autre <j> (x) algebrique et entiere, et d un
degre" moindre que n, et qu on developpe la fraction ^. en fraction continue
j W
ou X lt X 2 ... X r sont des fonctions rationnelles de x, et si I on forme V equation
(0) (Z/ - C7,") (XJ - Cf) ... (X\_, - C M ) (Z r - C r ) = 0,
Za racine reelle superieure de cette equation sera plus grande, et la racine reelle
inferieure de cette equation sera moindre quaucune des racines reelles de
Vequation
/(*) = 0;
et si toutes les racines de liquation (0) sont imaginaires, I dquation
aura aussi toutes ses racines imaginaires.
Demonstration.. Tous les quotients de la fraction continue qui suivent le
premier quotient, savoir: X z , X 3 ... X rt sont en general des fonctions lineaires
de x, et X^ sera aussi lineaire, si </> {x) est de degr^ n 1 ; les cas particuliers
ne changent pas la marche de la demonstration ; mais il faut remarquer
que lorsque f(x) et <f>(x) ont des racines communes, le dernier quotient aura
|"Y"|
la forme L ~ , [%] e tant 1 avant-dernier terme, et alors, dans 1 equation (0),
au lieu de X r * C> 2 , on dent simplement X r *.
426 Nouvelle Methode, etc. [56
Solent L la plus grande racine et A la plus petite racine de Pe quation (6) ;
alors aucuri facteur de (6) ne peut devenir nul pour des valeurs de x comprises
entre + oo et L, et entre A et oo ; done on aura toujours
[X r ] > C r .
Or /(*) est evidemment egal au denominateur de la fraction continue
multiplie par un facteur constant. Done, en vertu du lemme, le denominateur
de la fraction continue est de meme signe que le produit X^^X^ ... X r _^X r
pour les valeurs de x comprises entre + oo et L, et entre A et oo ; mais dans
ces intervalles la fonction generate X{ n e tant pas comprise entre + Ci et C{
ne peut devenir nulle, et, par consequent, ne peut changer de signe ; done le
denominateur de la fraction continue conserve le meme signe pour toute
valeur de x renfermee entre ces intervalles, et de meme f(x) ; L est done une
limite superieure et A une limite inferieure des racines de 1 equation
Le nombre des racines reelles de 1 equation (9} est evidemment pair, zero
compris ; dans ce dernier cas, c est-a-dire (6) n ayant aucune racine reelle,
f(x} ne changera done pas de signe pour des valeurs de x comprises entre
+ 00 et oo ; autrement toutes les racines de f(x} = sont imaginaires. Le
theoreme est done completement demontre.
r
3. Si (f) (x) est de degre n 1, la fraction continue renferme en general
(sauf les cas ou quelques-uns des coefficients deviennent nuls), comme il
a ete dit plus haut, n quotients lineaires de la forme
a^x b-i, a. 2 x b 2 . . . a n ^x 6 n -i , a n x b n ;
done, d apres le the oreme, la plus grande et la plus petite des 2w quantites
b 1 C l 6 2 (7 2 bn-i C nr . l b n C n
&i G/2 &ni "n
sont respectivement une limite superieure et une limite inferieure des
racines de 1 equation
/(*) = 0.
Si Ton prend (r = n)
P\ = 1*2= = /^n-i = 1, A t n = 2,
on vient au theoreme enonce [p. 423].
56] Nouvelle Methode, etc. 427
4. Lors meme que les quotients X ly X 2 , etc., ne sont pas lindaires, on
n aura pourtant jamais a resoudre que des equations du premier degrd. En
effet, soient les 2r equations de degre quelconque
II suffit de trouver une quantite / superieure aux racines de ces equations,
et une quantite \ inferieure a ces memes racines, I et A, seront des limites
pour 1 equation
/(*>-&
Si done une de ces equations est de degre p > 1, on applique a cette equation
le procede ci-dessus, en choisissant une fonction <f> (x) de degre p 1, et, en
agissant ainsi, on arrivera par une sorte de trituration a n avoir a traiter que
des Equations du premier degrd
5. On a
plus la valeur de /u. t - est petite, et plus on aura de chances a resserrer les
limites dans les deux fractions - ; par contre, on aura un desavantage sous
di
ce rapport dans les deux fractions suivantes - ; car C i+1 = /Ai +l H ;
a i+l pi
plus fMi diminue, et plus C i+1 augmente. Get inconvenient n a pas lieu pour
la derniere fraction ; on peut done prendre p, n = et C n = .
P"n i
6. II est a remarquer que tous les raisonnements precedents subsistent
en renversant la suite des A et 1 e crivaut ainsi :
P r 1 /^r 2
7. II y a lieu a des recherches interessantes sur la forme a donner a
</>(#), et sur les valeurs a donner aux quantites p pour obtenir les limites
les plus resserrees, et je crois etre parvenu a demontrer que la forme la plus
avantageuse est f (x), precisement la forme que M. Sturm a adoptee.
(h (oc ]
8. Dans la reduction en fraction continue de TT~\ > nous n avons con-
/(*)
sidere que des quotients binomes ; mais on peut pousser les divisions plus
loin et obtenir des quantite s de la forme
c d I
+ -+- + ...+ -;
n\ ffi* m*
Wt mt *As
428 Nouvelle Methode, etc. [56
le reste correspondant sera de la forme
I
a v +1 + Vx r + cx r ~ l + ...+ .
X T
En operant ainsi, le nombre de termes dans chaque reste ira en diminuant,
comme dans le proce de ordinaire, et le dernier reste sera de la forme CoF,
fi etant un entier positif ou negatif, et le dernier quotient de la forme
p x p _|- QxP- 1 , p etant|un entier positif ou negatif ; nommant les quotients
ainsi obtenus q lt q 2 ... q r , on voit aisement qu on aura
f(x) = Mx^D,
ou M est une constante, i un nombre entier positif ou negatif dont la valeur
depend de la maniere dont on a opere dans les divisions successives, et D est
le denominateur de la fraction continue
Done, si Ton dent, comme ci-dessus,
Z = ( 9l - C/) (q? - Cf) . . . (q* - (7 r 2 ) = 0,
nommant L et A les racines extremes de cette equation, si zdro n est pas
compris entre + oo et L, ni entre A et oo , la demonstration donnee ci-
dessus subsiste encore pour le cas general. Et lors meme que zdro est
compris entre ces limites, i et A restent tout de meme les limites pour les
racines, abstraction faite de la racine zero.
57.
ON A THEORY OF THE SYZYGETIC* RELATIONS OF TWO
RATIONAL INTEGRAL FUNCTIONS, COMPRISING AN
APPLICATION TO THE THEORY OF STURM S FUNCTIONS,
AND THAT OF THE GREATEST ALGEBRAICAL COMMON
MEASURE.
[Philosophical Transactions of the Royal Society of London, CXLIII. (1853),
Part in., pp. 407548.]
INTRODUCTION.
" How charming is divine philosophy !
Not harsh and crabbed as dull fools suppose,
But musical as is Apollo s lute,
And a perpetual feast of nectar d sweets,
Where no crude surfeit reigns ! " COMUS.
IN the first section of the ensuing memoir, which is divided into five
sections, I consider the nature and properties of the residues which result
from the ordinary process of successive division (such as is employed for the
purpose of finding the greatest common measure) applied to f(x) and < (x\
two perfectly independent rational integral functions of x. Every such
residue, as will be evident from considering the mode in which it arises,
is a syzygetic function of the two given functions ; that is to say, each of the
given functions being multiplied by an appropriate other function of a given
degree in as, the sum of the two products will express a corresponding residue.
These multipliers, in fact, are the numerators and denominators to the
successive convergents to ^ expressed under the form of a continued frac-
J x
tion. If now we proceed a priori by means of the given conditions as to
* Conjugate would imply something very different from Syzygetic, namely, a theory of the
Invariantive properties of a system of two algebraical functions.
430 On a Theory of the Syzygetic Relations [57
the degree in x of the multipliers and of any residue, to determine such
residue, we find, as shown in Art. 2, that there are as many homogeneous
equations to be solved as there are constants to be determined ; accordingly,
with the exception of one arbitrary factor which enters into the solution,
the problem is definite ; and if it be further agreed that the quantities
entering into the solution shall be of the lowest possible dimensions in
respect of the coefficients of / and <, and also of the lowest numerical
denomination, then the problem (save as to the algebraical sign of plus or
minus) becomes absolutely determinate, and we can assign the numbers
of the dimensions for the respective residues and syzygetic multipliers.
The residues given by the method of successive division are easily seen not
to be of these lowest dimensions ; accordingly there must enter into each
of them a certain unnecessary factor, which, however, as it cannot be
properly called irrelevant, I distinguish by the name of the Allotrious
Factor. The successive residues, when divested of these allotrious factors,
I term the Simplified Residues, and in Arts. 3 and 4 I express the
allotrious factor of each residue in terms of the leading coefficients of the
preceding simplified residues of / and <. In Art. 5 I proceed to determine
by a direct method these simplified residues in terms of the coefficients
of f and </>. Beginning with the case where f and <j> are of the same
dimensions (m) in x, I observe that we may deduce, from f and <, m linearly
independent functions of x each of the degree (m 1) in x, all of them
syzygetic functions of f and <f> (vanishing when these two simultaneously
vanish), and with coefficients which are made up of terms, each of which
is the product of one coefficient of/ and one coefficient of <. These, in fact,
are the very same m functions as are employed in the method which goes
by the name of Bezout s abridged method to obtain the resultant to (that is,
the result of the elimination of x performed upon) / and <f>. As these derived
functions are of frequent occurrence, I find it necessary to give them a name,
and I term them the m Bezoutics or Bezoutian Primaries ; from these m
primaries m Bezoutian secondaries may be deduced by eliminating linearly
between them in the order in which they are generated, first, the highest
power of x between two, then the two highest powers of x between three,
and finally, all the powers of x between them all : along with the system
thus formed it is necessary to include the first Bezoutian primary, and to
consider it accordingly as being also the first Bezoutian secondary; the last
Bezoutian secondary is a constant identical with the Resultant of f and (f).
When the m times m coefficients of the Bezoutian primaries are conceived
as separated from the powers of x and arranged in a square, I term such
square the Bezoutic square. This square, as shown in Art. 7, is sym
metrical about one of its diagonals, and corresponds therefore (as every
symmetrical matrix must do) to a homogeneous quadratic function of m
variables of which it expresses the determinant. This quadratic function,
57] of two Algebraical Functions. 431
which plays a great part in the last section and in the theory of real roots,
I term the Bezoutiant; it may be regarded as a species of generating
function. Returning to the Bezoutic system, I prove that the Bezoutian
secondaries are identical in form with the successive simplified residues.
In Art. 6 I extend these results to the case of / and < being of different
dimensions in x. In Art. 7 I give a mechanical rule for the construction
of the Bezoutic square. In Art. 8 I show how the theory of f(x) and </>(#),
where the latter is of an inferior degree to /, may be brought under the
operation of the rule applicable to two functions of the same degree at the
expense of the introduction of a known and very simple factor, which in fact
will be a constant power of the leading coefficient in f(x}. In Art. 9 I give
another method of obtaining directly the simplified residues in all cases.
In Art. 10 I present the process of successive division under its most general
aspect. In Arts. 11 and 12 I demonstrate the identity of the algebraical
sign of the Bezoutian secondaries with that of the simplified residues,
(T)1T
generated by a process corresponding to the development of ~ under the
/*
form of an improper continued fraction (where the negative sign takes the
place of the positive sign which connects the several terms of an ordinary
continued fraction). As the simplified residue is obtained by driving out
an allotrious factor, the signs of the former will of course be governed by the
signs accorded by previous convention to the latter ; the convention made is,
that the allotrious factors shall be taken with a sign which renders them
always essentially positive when the coefficients of the given functions are
real. I close the section with remarking the relation of the syzygetic
factors and the residues to the convergents of the continued fraction which
fT\np
expresses *=- , and of the continued fraction which is formed by reversing
J x
the order of the quotients in the first named fraction.
In the second section I proceed to express the residues and syzygetic
multipliers in terms of the roots and factors of the given functions ; the
method becoming as it may be said endoscopic instead of being exoscopic*,
as in the first section. I begin in Arts. 14 and 15 with obtaining in this
* These words admit of an extensive and important application in analysis. Thus the
methods for resolving an equation (or to speak more accurately, for making one equation depend
upon another of a simpler form) furnished by Tschirnhausen and Mr Jerrard (although not so
presented by the latter) are essentially exoscopic ; on the other hand, the methods of Lagrange
and Abel for effecting similar objects are endoscopic. So again, the memoir of Jacobi, " De
Eliminatione," hereinafter referred to, takes the exoscopic, and the valuable " Nota ad Elimina-
tionem pertinens" of Professor Ilichelot in Crelle s Journal, the endoscopic view of the subject.
In the present memoir (in which the two trains of thought arising out of these distinct views are
brought into mutual relation) the subject is treated (chiefly but not exclusively) under its
endoscopic aspect in the second, third and fourth sections, and exoscopically in the first and last
sections.
432 On a Theory of the Syzygetic Relations [57
way, under the form of a sum or double sum of terms involving factors
and roots of / and <, and certain arbitrary functions of the roots in each
term, a general representative, or to speak more precisely, a group of general
representatives for a conjunctive of any given degree in x to/ and <, that is,
a rational integral function of x, which is the sum of the products of f and
</> multiplied respectively by rational integral functions of x, so as to vanish
of necessity when /and <f> simultaneously vanish. This variety of representa
tives refers not merely to the appearance of arbitrary functions, but to an
essential and precedent difference of representation quite irrespective of such
arbitrariness.
In Arts. 16, 17, 18, 19, 20, 21, I show how the arbitrary form of function
entering into the several terms of any one (at pleasure) of the formulas that
represent a conjunctive of any given degree may be assigned, so as to make
such conjunctive identical in form with a simplified residue of the same
degree. The form of arbitrary function so assigned, it may be noticed,
is a fractional function of the roots, so that the expression becomes a sum
or double sum of fractions. I first prove in Arts. 16, 17 that such sum is
essentially integral, and I determine the weight of its leading coefficient in
respect of the roots of / and < (this weight being measured by the number
of roots of / and conjointly, which appear in any term of such coefficient).
Now in the succeeding articles I revert to the Bezoutic system of the first
section, and beginning with the supposition of m and n being equal, I demon
strate that the most general form of a conjunctive of any degree in x will be
a linear function of the Bezoutics, from which it is easy to deduce that the
simplified residues of any given degree in x are the conjunctives whose
weight in respect of the roots is a minimum ; so that all conjunctives having
that weight must be identical (to a numerical factor pres}, and any integral
form of less weight apparently representing a conjunctive must be nugatory,
every term vanishing identically. These results are then extended to the
case of two functions of unlike degrees. The conclusion is, that the weight
of the forms assumed in Arts. 16 and 17 being equal to the minimum weight,
they must (unless they were to vanish, which is easily disproved) represent
the simplified residues, or which is the same thing, the Bezoutian secondaries.
We thus obtain for each simplified residue a number of essentially
distinct forms of representation, but all of which must be identical to a
numerical factor pres, a result which leads to remarkable algebraical
theorems.
The number of these different formulae depends upon the degree of the
residue; there being only one for the last or constant residue, two for the
last but one, three for the last but two, and so on. The formula continue to
have a meaning when their degree in x exceeds that of / or <; but then,
as although always representing conjunctives, they no longer represent
57] of two Algebraical Functions. 433
residues, this identity no longer continues to subsist. In Arts. 22, 23, 24, 25,
I enter into some developments connected with the general formulae in
question ; these, it may be observed, are all expressed by means of fractions
containing in the numerator and denominator products of differences; the
differences in the numerator products being taken between groups of roots
of / and groups of roots of </> ; and in the denominator between roots of /
inter se and roots of (f> inter se. A great enlargement is thus opened out to
the ordinary theory of partial fractions.
In Art. 26 I find the numerical ratios between the different formula}
which represent (to a numerical factor pres} the same simplified residue,
and in Arts. 27 and 28 I determine the relations of algebraical sign of these
formulas to the simplified residues or Bezoutian secondaries. In Art. 2.9
I determine the syzygetic multipliers corresponding to any given residue
in terms of the factors and roots of the given functions ; but the expressions
for these, which are closely analogous to those for the residues, cease to be
polymorphic. They are obtained separately from the syzygetic equation,
and it is worthy of notice, that to obtain the one we use the first of the
polymorphic expressions for the residue, and to obtain the other the opposite
extremity of the polymorphic scale. In the subsequent articles of this
section, by aid of certain general properties of continued fractions, I establish
a theorem of reciprocity between the series of residues and either series of
syzygetic multipliers.
Section III. is devoted to a determination of the values of the preceding
formulae for the residues and multipliers in the case applicable to M. Sturm s
theorem, where <j>x becomes the differential derivative of fx. It becomes
of importance to express the formula? for this case in terms of their roots
and factors of fa alone, without the use of the roots and factors of fx, which
will of course be functions of the former.
By selecting a proper form out of the polymorphic scale, the fractional
terms of the series for each residue in this case become separately integral,
and we obtain my well-known formula? for the simplified residues (Sturm s
reduced auxiliary functions) in terms of the factors and the squared differ
ences of partial groups of roots. This is shown in Art. 35. In Art. 36 the
multiplier of f x in the syzygetic equation is expressed by formulae of equal
simplicity, and in a certain sense complementary to the former. This
method, however, does not apply to obtaining expressions for the multiplier
of fx in the same equation in terms of the roots and factors of fx ; for the
separate fractions whose sum represents any one of these factors, it will
be found, do not admit of being expressed as integral functions of the roots
and factors. To obviate this difficulty I look to the syzygetic equation itself,
which contains five quantities, namely, the given function, its first differential
derivative, the residue of a given degree, and the two multipliers, all of
s - 28
434 On a Theory of the Syzygetic Relations [57
which, except the multiplier of fx, are known, or have been previously deter
mined as rational integral functions of the roots and factors of fx. I use
this equation itself for determining the fifth quantity, the multiplier in
question. To perform the general operations by a direct method required
for this would be impossible ; the difficulty is got over by finding, by means
of the syzygetic equation, the particular form that the result must assume
when certain relations of equality spring up between the roots of/; and
then, by aid of these particular determinations, the general form is demon
stratively inferred.
This investigation extends over Arts. 38, 39, 40, 41, 42, 43. It turns
out that the expressions for the multipliers of fx are of much greater
complexity than for the multipliers of / a; or for the residues. Any such
multiplier consists of a sum of parts, each of which, as in the case of the
residues and the factors of fx, is affected with a factor consisting of the
squared differences of a group of roots ; but the other factor, instead of being
simply (as for the residues and factors before mentioned) a product of certain
factors of fx, consists of the sum of a series of products of sums of powers
by products of combinations of factors of fx, each of which series is affected
with the curious anomaly of its last term becoming augmented in a certain
numerical ratio beyond what it should be in order to be conformable to the
regular flow of the preceding terms in the series*.
The fourth section opens with the establishment of two propositions
concerning quadratic functions which are made use of in the sequel. Art. 44
contains the proof of a law which, although of extreme simplicity, I do not
remember to have seen, and with which I have not found that analysts are
familiar : I mean the law of the constancy of signs (as regards the number
of positive and negative signs) in any sum of positive and negative squares
into which a given quadratic function admits of being transformed by
substituting for the variables linear functions of the variables with real
coefficients. This constant number of positive signs which attaches to
a quadratic function under all its transformations, which is a transcen
dental function of the coefficients invariable for real substitutions, may be
termed conveniently its inertia, until a better word be found. This inertia
it is shown in Art. 45, by aid of a theorem identical with one formerly given
by M. Cauchy, is measured by the number of combinations of sign m the
series of determinants of which the first is the complete determinant of the
function, the second, the determinant when one variable is made zero, the
next, the determinant when another variable as well as the first is made
zero, and so on, until all the variables are exhausted, and the determinant
* The syzygetic multipliers are identical with the numerators and denominators (expressed in
f x
their simplest form) of the successive convergents to the continued fraction which expresses -^- .
57] of two Algebraical Functions. 435
becomes positive unity. In Art. 46 I give some curious and interesting
expressions for the residues and syzygetic multipliers, under the form of
determinants, communicated to me by M. Hermite ; and in Art. 47 I show
how, by the aid of the generating function which M. Hermite employs,
and of the law of inertia stated at the opening of the section, an instan
taneous demonstration may be given of the applicability of my formulae for
M. Sturm s functions for discovering the number of real roots of fx, without
any reference to the rule of common measure ; and moreover, that these
formulae may be indefinitely varied, and give the generating function, out
of which they may be evolved, in its most general form. Had the law of
inertia been familiar to mathematicians, this constructive and instantaneous
method of finding formulae for determining the number of real roots within
prescribed limits would, in all probability, have been discovered long ago,
as an obvious consequence of such law. I then proceed in Arts. 48 and 49,
to inquire as to the nature of the indications afforded by the successive
simplified residues to two general functions / and </> ; and I find that the
succession of signs of these residues serves to determine the number of roots
of / or </> comprised between given limits, after all pairs of roots of either
function contained within the given limits and not separated by roots of the
other function have been removed, and the operation, if necessary, repeated
toties quoties until no two roots of either function are left unseparated by
roots of the other ; or in other words, until every root finally retained in one
function is followed by a root of the other, or else by one of the assigned
limits. The system of roots comprised between given limits thus reduced
I call the effective scale of intercalations ; such a scale may begin with a root
of the numerator or of the denominator of ~- ; and upon this and the
J x
relative magnitudes of the greatest root of <j>x and fx it will depend whether
in the series of residues (among which fx and $x are for this purpose to be
counted) changes will be lost or gained as x passes from positive infinity to
negative infinity. In Art. 50 I observe that the theory of real roots of a
single function given by M. Sturm s theorem is a corollary to this theory
of the intercalations of real roots of two functions, depending upon the well-
known law, that odd groups of the limiting function fx lie between every
two consecutive real roots of fx. In Art. 511 verify the law of reciprocity,
already stated to exist between the residues of fx and <f>x, by an d posteriori
method founded on the theory of intercalations. In Arts. 52, 53, 54, I obtain
a remarkable rule, founded upon the process of common measure, for finding
a superior and inferior limit in an infinite variety of ways to the roots of any
given function. This method stands in a singular relation of contrast to
those previously known. All previous methods (including those derived
through Newton s Rule) proceed upon the idea of treating the function
whose roots are to be limited as made up of the sum of parts, each of which
282
436 On a Theory of the Syzygetic Relations [57
retains a constant sign for all values of the variable external to the quantities
which are to be shown to limit the roots. My method, on the other hand,
proceeds upon the idea of treating the function as the product of factors
retaining a constant sign for such values of the variable. In Art. 55, the
concluding article of the fourth section, I point out a conceivable mode in
which the theory of intercalations may be extended to sj stems of three or
more functions.
In Section V. Arts. 56, 57, I show how the total number of effective
intercalations between the roots of two functions of the same degree is given
by the inertia of that quadratic form which we agreed to term the Bezoutiant
to / and <f> ; and in the following article (58) the result is extended to
embrace the case contemplated in M. Sturm s theorem; that is to say,
I show, that on replacing the function of a; by a homogeneous function of
x and y, the Bezoutiant to the two functions, which are respectively the
differential derivatives of / with respect to x and with respect to y, will
serve to determine by its form or inertia the total number of real roots and
of equal roots in f(x). The subject is pursued in the following Arts. 59, 60.
The concluding portion of this section is devoted to a consideration of the
properties of the Bezoutiant under a purely morphological point of view ;
for this purpose / and < are treated as homogeneous functions of two
variables x, y, instead of being regarded as functions of x alone. In Arts.
61, 62, 63, it is proved that the Bezoutiant is an invariantive function of the
functions from which it is derived ; and in Art. 64 the important remark is
added, that it is an invariant of that particular class to which I have given
the name of Combinants, which have the property of remaining unaltered,
not only for linear transformations of the variables, but also . for linear
combinations of the functions containing the variables, possessing thus a
character of double invariability. In Arts. 65, 66, I consider the relation
of the Bezoutiant to the differential determinant, so called by Jacobi, but
which for greater brevity I call the Jacobian. On proper substitutions
being made in the Bezoutiant for the m variables which it contains (m
being the degree in x, y of / and <), the Bezoutiant becomes identical with
the Jacobian to / and < ; but as it is afterwards shown, this is not a property
peculiar to the Bezoutiant ; in fact there exists a whole family of quadratic
forms of m variables, lineo-linear (like the Bezoutiant) in respect of the
coefficients in / and <f>, all of which enjoy the same property. The number
of individuals of such family must evidently be infinite, because any linear
combination of any two of them must possess a similar property; I have
discovered, however, that the number of independent forms of this kind
is limited, being equal to the number of odd integers not greater than the
degree of the two functions / and </>. In Arts. 67 and 68, I give the means
of constructing the scale of forms, which I term the constituent or funda-
57] of two Algebraical Functions. 437
mental scale, of which all others of the kind are merely numerico-linear
combinations. This scale does not directly include the Bezoutiant within it,
and it becomes an object of interest to determine the numbers which connect
the Bezoutiant with the fundamental forms ; this calculation I have carried
on (in Arts. 69, 70, 71) from m = l to m = 6 inclusive, and added an easy
method of continuing indefinitely. In this method the numbers in the
linear equation corresponding to any value of m are determined successively,
and each made subject to a verification before the next is determined, there
being always pairs of equations which ought to bring out the same result for
each coefficient.
In the next and concluding Art. 72, 1 remark upon the different directions
in which a generalization may be sought of the subject-matter of the ideas
involved in M. Sturm s theorem, and of which the most promising is, in my
opinion, that which leads through the theory of intercalations. Some of the
theorems given by me in this paper have been enunciated by me many
years ago, but the demonstrations have not been published, nor have they
ever before been put together and embodied in that compact and organic
order in which they are arranged in this memoir, the fruit of much thought
and patient toil, which I have now the honour of presenting to the Royal
Society.
P.S. In a supplemental part to the third section I have given expressions
in terms of the roots of $x and fx for the quotients which arise in developing
fr) /
.,- under the form of a continued fraction, and some remarkable properties
f*
concerning these quotients. In a supplemental part to the fourth section
I have given an extended theory of my new method of finding limits to the
real roots of any algebraical equation. This method, so extended, possesses a
marked feature of distinction from all preceding methods used for the same
purpose, inasmuch as it admits in every case of the limits being brought up
into actual coincidence with the extreme roots, whereas in other methods a
wide and arbitrary interval is in general necessarily left between the roots
and the limits.
438 On a Theory of the Syzygetic Relations [57
SECTION I.
On the complete and simplified residues generated in the process of developing
under the form of a continued fraction, an ordinary rational algebraical
fraction.
Art. 1. Let P and Q be two rational integral functions of x, and suppose
that the process of continued successive division leads to the equations
P -
Q -
(i)
so that
i=sr-B^ & - , < 2 >
which is what I propose to call an improper continued fraction, differing from
a proper only in the circumstance of the successive terms being connected
by negative instead of positive signs.
MO, M lt MZ, &c., E lt E 2 , E 3 , &c. are, of course, functions of to: the latter
we may agree to call the 1st, 2nd, 3rd, &c. residues (in order to avoid the use
of the longer term "residues with the signs changed"); and by way of
distinction from what they become when certain factors are rejected, we may
call E 1} E 2 , E 3 , &c. the complete residues. Each such complete residue
No
will in general be of the form -yp , N^ and D t being integral functions of the
coefficients only of P and Q, but p t an integral function of these coefficients,
N
and of x ; p L may then be termed the tth simplified residue, and yr- the tth
allotrious factor. Suppose P to be of m and Q of n dimensions in x, and
m n = e, the process of continued division may be so conducted, that all the
residues may contain only integer powers of x\ and we may upon this
supposition make M of e dimensions, and M lt M a , M 3 , &c. each of one
dimension only in x; so that E l} E 2 , E 3 , ... will be respectively of (n 1),
(n 2), (n 3), &c. dimensions in x.
57] of two Algebraical Functions. 439
P and Q are supposed to be perfectly unrelated, and each the most general
function that can be formed of the same degree. From (1) we obtain
(3 )
R 3 = (M.M.M, + M + M 2 ) Q - (M.M, - 1) P
&c. = &c.
and in general we shall have
where it is evident that Q t will be of e + (c - 1), and P t of (i - 1) dimensions
in x.
Art. 2. Hence it follows that the ratios P t : Q L : R L may be ascertained
by the direct application of the method of indeterminate coefficients, for Q L
will contain e + 1, and P t will contain t disposable constants, making e + 2i
disposable constants in all. Again, Q,Q and P t P will each rise to the degree
n + e + t 1 in x ; but their sum R L is to be only of n i dimensions in x.
Hence we have to make (n + e + i - 1) - (n - i), that is e + 2t - 1 quantities
(which are linear in respect to the given coefficients in P and Q, as well as in
respect to the new disposable constants in P t and Q L ) all vanish, that is to
say, there will be e + 2il linear homogeneous equations to be satisfied by
means of e + 2t disposable quantities ; the ratios of these latter are, therefore,
determinate, so that we may write
(5)
and when (P t ), (Q t ), (R t ) are taken prime to one another, it is obvious that
(R L ) will be in all of e + 2i dimensions in the given coefficients, that is of i in
respect of the coefficients of P, and of e + 1 in respect of those of Q ; \ will
correspond to what I have previously called the allotrious factor; being in
fact foreign to the value of R, as determined by means of the equation (4),
and arising only from the particular method employed to obtain it through
the medium of the system (1): it becomes a matter of some interest and
importance to determine the values of this allotrious factor for different
values of i*.
* These are identical with what I termed quotients of succession in the London and Edinburgh
Philosophical Magazine (December, 1839) [p. 43 above] ; but by an easily explicable error of
inadvertence, the quantities Q lt Q 2 , &c. therein set out are not as they are therein stated to be,
440 On a Theory of the Syzygetic Relations [57
Art. 3. This may be done by the following method, which is extremely
simple, and would admit of a considerable extension in its applications, were
it not beside my immediate purpose to digress from the objects set out in
the title to the memoir, by entering upon an investigation of the special or
singular cases which may arise in the process of forming the continued
fraction, when one or more of the leading coefficients in any of the residues
vanish ; such an inquiry would require a more general character to be
imparted to the values of the quotients and residues than I shall for my
present purposes care to suppose.
Let us begin with supposing e = 1, and write
/= ax n + bx n ~ l + cx n ~ 2 + &c.
f (h rh
Let ilr be the first residue of ^ , and w of -v , and therefore of ~ , so that
<p y a-ty
f
to is the second residue of .
Let to = A, (to), 6) being entirely integer, and X a function of the coefficients
N
in f and </>. If we make \ = -^.,N and D being integer functions, D will
evidently be L-, where L denotes the first coefficient in the simplified residue
a 2 -\|r, and is evidently of two dimensions in a, /3, &c., and of one in a, b, &c. ;
Dw is therefore of 2x2 + 1, that is five dimensions in a, /9, &c., and of two
dimensions in a, b, &c. ; but to (by virtue of what has been observed of the
equations in system (5)) is of three dimensions in a, /3, &c., and of two in
a, b, &c. Hence N is of two dimensions in a, /3, &c., and of none in a, b, &c.
This enables us at once to perceive that N = a".
For ^r is of the form / (px + q)(j), }
and to is of the form (p x + q) ty)
the quotients of succession or allotrious factors themselves, but the ratios of each such to the
one preceding, if in the series ; so that
Q is \
_ . Ao
<?3 IS 3
A 2
&C
This error is corrected by my distinguished friend M. Sturm (Liouville s Journal, t. vm. 1842,
Sur un theoreme d Algebre de M. Sylvester), who appears, however, to have overlooked that I
was obviously well acquainted with the existence and nature of these factors, and their essential
character, of being perfect squares in the case contemplated in his memoir and my own.
MM. Borchardt, Terquem, and other writers, in quoting my formulae for M. Sturm s auxiliary
functions, have thus been led into the error of alluding to them as completed by M. Sturm.
57]
of two Algebraical Functions.
441
but N = makes <u vanish, and therefore, upon this supposition, / and <
would appear to have a common algebraical factor -^, that is to say, N
vanishing would appear to imply that the resultant of/ and < must vanish,
so that N would appear to be contained as a factor in this general resultant,
which latter is, however, clearly indecomposable into factors a seeming
paradox the solution of which must be sought for in the fact, that the
equation N = is incompatible with the existence of the usual equations (7)
connecting/, <j>, ty and w. but this failure of the existence of the equations
(7) (bearing in mind that N has been shown to be a function only of the
set of coefficients a, ft, &c.), can only happen by reason of a vanishing when
ever N vanishes ; a must therefore be a root of N, or which is the same thing,
N a power of a and hence N = a 2 .
The same result may be obtained a posteriori by actually performing the
successive divisions ; if the coefficients of any dividend be a, b, c, d, &c., and
of the divisor a, ft, 7, 8, &c., the first remainder, forming the second divisor,
will be easily seen to have for its coefficients
a, b, c
a, b, d
a, 6, e
1
a?
0, , ft
a 2
0, a, ft
1
a 2
0, a, ft
&c.
, ft, 7
, ft, 8
a, ft e
a, b, c
Hence the coefficients in the next remainder (making
0, a, ft
a, A 7
will be each of the form of the compound determinant,
<*, ft, 7
a, b, c
a, b, d
o,
0, a, ft
,
0, a, 7
a, ft, 7
a, ft, ^
a, b, c
a, b, d
a, b, e
0, a, ft
i
0, a, 7
>
0, a, 8
a. ft, 7
a, ft, 8
a, ft, e
The compound determinant above written will be the first coefficient
in the remainder under consideration ; the subsequent coefficients will be
represented by writing / <f> ; g, 7, &c., respectively in lieu of e, e. Omitting
the common multiplier , the determinant above written is equal to
m-
442 On a Theory of the Syzygetic Relations [57
a, b, c
a, b, e
a, 6, d
a, b, d
0, a, /3
X
0, , $
0, a, 7
X
0, a, 7
a, ft, 7
a, & e
a, ft, B
, &
a, b, c
f
a, b, d
ft, b, c
)
+
0, a, /3
x J ft
0, a, 7
~ 7
0, a,
> & 7
(
a, ft, S
, A 7
The last written pair of terms are together equal to
a, b, c
0, a, ft x {- d/3a 2 + C7a 2 + aa (ftS - 7%
a, /3, 7
which is of the form a? A a 2 /3 2 (/3S 7 2 ) a ; and the sum of the first written
pair is of the form a?B + (a/3 2 a/35 ay/3 ay/3) a. Hence the entire deter
minant is of the form a 2 ( A + B), showing that a 2 will enter as a factor into
this and every subsequent coefficient in the second remainder, as previously
demonstrated above.
It may, moreover, be noticed, that this remainder, when a 2 has been
expelled, will for general values of the coefficients be numerically as well
as literally in its lowest terms, as evinced by the fact that there exist terms
(for example cta 2 ye) having + 1 for their numerical part. The same explicit
method might be applied to show, that if the first divisor were e degrees
instead of being only one degree in a; lower than the first dividend, a e+1
would be contained in every term of the second residue : the difficulty,
however, of the proof by this method augments with the value of e ; but the
same result springs as an immediate consequence from the method first
given, which remains good mutatis mutandis for the general case, as may
easily be verified by the reader. Applying now this result to the functions
P and Q, supposed to be of the respective degrees n and n e in x, and calling
the coefficients of the leading terms in the successive simplified residues
i> a 2> a 3> &c., and denoting by a the leading coefficient in Q, and as before denot
ing the successive allotrious factors by \ 1} \. 2 , &c., it will readily be seen that
1111
that is
and in general
a 2
X 4 =
a,-a a -a s
(8)
57] of two Algebraical Functions. 443
Art. 4. Strictly speaking, we have not yet fully demonstrated that the
complete allotrious factors are represented by the values above given for X,
but only that these latter are contained as factors in the allotrious factors ;
we must further prove that there exist no other such factors. This may be
shown as follows : it is obvious from the nature of the process that the
complete residues will always remain of one dimension in respect of the given
coefficients, that is, first of one dimension in the set a, 6, c, &c., and of zero
dimensions in a, /3, y, &c. ; then conversely, of one dimension in a, /3, y, &c.,
and of zero dimensions in a, b, c, &c., and so on, the residues being evidently
required to conform in their dimensions to those of the first dividend and the
first divisor alternately. These coefficients then are always of unit dimensions
in respect to the given coefficients ; whereas it has been shown (Art. 2) that
the simplified residues in respect to these coefficients are successively of the
dimensions 2 + e, 4> + e, 6 + e, &c.
Let the complete residue corresponding to X 2m be M\^ m a zm , that is
or say ML ; in passing from a 2q to a 2q+1 the dimensions rise 2 units for all
values of q except zero, and when q = the dimensions increase per saltum
from 1 to 2 + e ; hence the total dimensions of L in the joint coefficients
will be
{(e + 1) - 2 (e + 2)} - 4 (m -l)
and therefore M is of zero dimensions, and X 2m is the complete allotrious
factor. In like manner if the complete residue corresponding to X 2m+1 be
M\ 2m+l a. 2m+l , that is
,, 1 C^CCs 2 a 2 2n.-i
M n e+i ri 775 ~(,2 "zrn+i,
a c* 2 4 a 2m
or say ML, the dimensions of L will be
- (e + 1) - 4m +{e + 2 (2m + 1)], that is, 1,
and hence, as in the preceding case, M is of zero dimensions, and X, m+1 is the
complete ailotrious factor.
Art. 5. I proceed to show how the simplified residues may be most
conveniently obtained by a direct process, identical with that which comes
into operation in applying to the two given functions of a; the method
familiarly known under the name of Bezout s abridged method of elimination.
Let us call the two given functions U and F, and commence with the case
where U and F are of equal dimensions (n) in x. The simplified tth residue
will then be a function of n L dimensions in #, and of t dimensions in respect
of each given set of coefficients, and may be taken equal to VJJ + U.V, where
F t and U, are each of (t 1) dimensions in x.
444 On a Theory of the Syzygetic Relations [57
Let
U = a x n + a! a?"" 1 + a 2 x n ~- + ... +a n ,
V = b x n + &X 1 " 1 + k n " 2 + ...+&;
we may write in general, m being taken any positive integer not exceed
ing n,
U = (a x m + ai z m - l + . .. + a m ) x n ~ m + (a m+l x n - m ^ + a m+2 x n ~ m ~ 2 +... + a n \
V = (b.x m + b,x m ~ l +...+ b m ) x n ~ m + (b m+l x n - m ~ l + b m+2 x n - m -* + ... + b n ).
Hence
(b x m + hx- 1 + ... + b m ) U - (a x m + a t x m ~ l +... + a )n ) V
m^i^ ~\- m"-i% ~ ~r m-^-2^ "i" ~r m-"-m \y)
where if we use (r, s) to denote a r b s a s b r for all values of r and s, we have
m K, = (0, m + 1), m K s = (0, m + 2) + (1, m + 1),
m K 3 = (0, m + 3) + (1, m + 2) + (2, m + 1),
and in general m Ki = 2 (r, s), the values of r and s admissible within the sign
of summation being subject to the two conditions, one the equality r+s=m+i,
the other the inequality r less than i. By giving to m all the different values
from to m 1 in succession, and calling
respectively Q m and P m , we have
Q U - P F = K,x^ + K,x n ~* + ... + K n
Q 2 U - P 2 V= &ar-*+ 2 K 2 x-*+...+ 2 K n [. (10)
Q n -,U-P n _ l V= n _ l K l x^ + n _ 1 K 2 x-+ ... + n _ 1 R
The right-hand members of these n equations I shall henceforth term the
Bezoutians to U and F.
The determinant formed by arranging in a square the n sets of coefficients
of the n Bezoutians, and which I shall term the Bezoutian matrix, gives, as
is well known, the Resultant (meaning thereby the Result in its simplest
form of eliminating the variables out) of U and F.
Eliminating dialytically, first x n ~ l between the first and second, then x n ~ l
and x n ~ 2 between the first, second and third, and so on, and finally, all the
powers of x between the first, second, third, ... ?th of these Bezoutians, and
repeating the first of them, we obtain a derived set of n equations, the
right-hand members of which I shall term the secondary Bezoutians to U
and F, this secondary system of equations being
57] of ttvo Algebraical Functions. 445
/"V TT 7~> -IT" jr n -i , T7- ~* n , fr tft O , , JT
I j 1 1 __ i j V H ff" * -I- ft / / t * -1- ft T 1 * " -i- -I- M
V^JQ L/ JL Q r JTX j LV ^ /i g 1 ^ -** 3** * T **H
(^ Qo - AT^O tT - (JA P - K, PO F= Zj^ - 2 + L,x n ~ 3 +...+ Z n _!
&c. = &c.
And we can now already without difficulty establish the important proposition,
that the successive simplified residues to expanded under the form of an
improper continued fraction, abstracting from the algebraical sign (the
correctness of which also will be established subsequently), will be repre
sented by the n successive Secondary Bezoutians to the system U, V.
For if we write the system of equations (11) under the general form
- - 1 + &c.,
the degree of ^ t and H, in x will be that of Q^ and P t _ l5 that is i 1 ; and
the dimensions of A t , B,, &c., in respect of each set of coefficients is evidently
t; consequently, by virtue of Art. 2, AiX n ~ l + B i x n ~ 2 + &c., which is the
tth Bezoutian, will (saving at least a numerical factor of a magnitude and
algebraical sign to be determined, but which, when proper conventions are
made, will be subsequently proved to be +1) represent the tth simplified
residue to p.*, as was to be shown.
Art. 6. More generally, suppose U and V to be respectively of n + e and
n dimensions in x.
Let U = a x n+e + a 1 n+e - 1 + a 2 x n+e ~" + &c.
V = b,x n + b^- 1 + &c.
Making
U = (a x e + m + a 1 af +m - 1 + &c. + a e+m ) x n ~ m + (a e+m+1 x n ~ m - 1 + &c. + a n+e ),
V = (b x m + hx- 1 +...+ b m ) x n ~ m + (b m+l x n ~ m - 1 + &c. + 6 n ),
we obtain the equation
Q m U- P e+m V= mK^+t- 1 + m ^2^ n+e - 2 + &c. + m K n+e , (12)
* V is supposed to be taken as the first divisor, and the term residue is used, as hitherto in
this paper, throughout in the sense appertaining to the expansion conducted, so as to lead to an
improper continued fraction, in that sense, in fact, in which it would, more strictly speaking, be
entitled to the appellation of excess rather than that of residue.
446 On a Theory of the Syzygetic Relations [57
where
Q m = (hx + + b m \ P e+m = (a x e + m + . . . + a e+m ) ;
mJ^i = &t)0m+i > m--2 = a ob m+2 + a i^m+i j mfce ~ a o^m+e + ^i^m+e-i + &C. + d e b m ;
m-^e+i = flo&m+e+i + &C. + a e+1 6 m a e+m+l b ; &C. = &C.
By giving to m every integer value from to (n 1) inclusive, we thus
obtain n equations of the form of (12), each of the degree n + e 1 in #, and
of one dimension in regard to each set of coefficients.
In addition to these equations we have the e equations of the form
x* V = b x n+ * + b^x 11 ^- 1 + &c. + b n ar, (13)
in which fi may be made to assume every value from to (e 1) inclusive,
and the right-hand side of the equation for all such values of /* will remain
of a degree in x not exceeding n + e 1, the degree of the equations of the
system above described. There will thus be e equations in which only the
(b) set of coefficients appear, and n equations containing in every term one
coefficient out of each of the two sets.
The total number of equations is of course n + e. Between the e
equations of the second system (13) and the r occurring first in order of the
first system (12), we may eliminate dialytically the e + r 1 highest powers
of x, and there will thus arise an equation of the form
e r _i U - o) e+r _ 1 F = Lx n ~ r + L x n - r ~ l + &c. + (L), (14)
where #,._! and ft> e+r _i are respectively of the degrees r 1 and e + r 1 in x,
and L, L ... (L) are of r dimensions in the (a) set, and of (e + r) dimensions
in the (b) set of coefficients, and consequently Lx n ~ l + L x n ~ r - 1 + . . . + (L)
must satisfy the conditions necessary and sufficient to prove its being (to a
numerical factor pres) a simplified residue to ( U, F).
Thus suppose
U = a # 4 + a-^x 3 + a^x 2 + a 3 x + a 4 ,
V box 2 + b^ + b 2 .
Then, corresponding to the system of which equation (13) is the type,
we have
F= 6 # 2 + ^x + b. 2 ,
xV=
Again, to form the system of which equation (12) is the type, we write
b U (doX* + a^x + or 2 ) F = b (a z x + a 4 ) (a x 2 + a^x + a 2 ) (b^ + b 2 )
= a b l x 3 (a & 2 + A) x z + (b a 3 a^ a 2 b l )x + (6 a 4 a 2 6 2 ),
U (a a? + a^ + a^x + a 3 ) F= (b x + b,) 4 (a Q x* + a^ + a 2 x + a 3 ) b 2
= ctybzX 3 a^x 2 + (6 a 4 a 2 6 2 ) x + (^a 4 b 2 a 3 ).
57]
of two Algebraical Functions.
447
Combining the two equations of the first system with the first of the second
system, we obtain the first simplified residue Lx + L , where
0,
b ,
L =
and
L =
0,
6 ,
b lt
&,
b a 3
b 2
2 6 a 4
By again combining the two equations of the first system with both of the
second system, we have the determinant
0, b , & 1} b. 2
b , b lf b 2 ,
which is the last simplified residue, or in other terms, the resultant to the
system U, V.
Art. 7. It is most important to observe that the Bezoutian matrix to two
functions of the same degree (n) is a symmetrical matrix, the terms similarly
disposed in respect to one of the diagonals being equal.
Thus retaining the notation of Art. 5, so that
(0, 1) = a/3 - ba, (1, 2) = by - c/3, (2, 3) = cS - dy,
(0, 2) = ay -COL, (1, 3) = b& - dp, &c.
(0, 3) = aS - da, &c.
&c.
when n=l the Bezoutian matrix consists of a single term (0, 1);
when ?i = 2, it becomes
when n = 3, it becomes
(0, 1)
(0, 2)
(0, 2)
(1, 2);
(0, 1)
(0, 2)
(0, 3)
/(O, 3)\
(0, 2)
( + )
(1, 3)
\(i, 2)7
(0, 3)
(1, 3)
(2, 3);
448 On a Theory of the Syzygetic Relations [57
when n = 4, it becomes
(0, 1) (0, 2) (0, 3) (0, 4)
/(O, 3)\ /(O, 4)\
(0, 2) ( + ) ( + ) (1, 4)
(0, 3)
(0, 4)
when n = 5, it becomes
(0, 1) (0, 2) (0, 3) (0, 4) (0, 5)
(0, 3) N
(0. 2) ( + )( + )(+) (1, 5)
(0,3) + (1. 4) + (2,5)
x/ \ * x / i / ^ / \
(0, 4) ( + + + ) (3, 5)
\(1, 4)/ \(2, 4)/ \(3, 5)/
(0, 5) (1, 5) (2, 5) (3, 5) (4, 5),
and so forth. Every such square it is apparent may be conceived as a sort
of sloped pyramid, formed by the successive superposition of square layers,
which layers possess not merely a simple symmetry about a diagonal (such
as is proper to a multiplication table), but the higher symmetry (such as
exists in an addition table), evinced in all the terms in any line of terms
parallel to the diagonal transverse to the axis of symmetry being alike*.
Thus for n = 5, the three layers or stages in question will be seen to be,
the first
(0, 1)
(0, 2)
(0, 3)
(0, 4)
(0, 5)
(0, 2)
(0, 3)
(0, 4)
(0, 5)
(1, 5)
(0, 3)
(0, 4)
(0, 5)
(1, 5)
(2, 5)
(0, 4)
(0, 5)
(1, 5)
(2, 5)
(3, 5)
(0, 5)
(1, 5)
(2, 5)
(3, 5)
(4, 5);
* A square arrangement having this kind of symmetry, namely, such as obtains in the
so-called Pythagorean addition table as distinguished from that which obtains in the multiplica
tion table, may be universally called Persymmetric.
57] of two Algebraical Functions. 449
the second
(1, 2)
(1, 3)
(1, 4)
(1, 3)
(1.- 4)
(2, 4)
(1, 4)
(2, 4)
(3, 4);
and the third
(2, 3).
In general, when n is odd, say 2p + 1, the pyramid will end with a single
term (p, (p + 1)), and when even, as 2p, with a square of four terms,
(o>-2), (p-i)), ((p-np)
Each stage may be considered as consisting of three parts, a diagonal set of
equal terms transverse to the axis of symmetry, and two triangular wings,
one to the left, and the other to the right of this diagonal ; the terms in each
such diagonal for the respective stages will be
(0, n), (1, n - 1), (2, (n - 2)) ... (p, (p + 1)),
-n ft 1
p being - 1 when n is even, and - when n is odd.
If we change the order of the coefficients in each of the two given functions,
it will be seen that the only effect will be to make the left and right triangular
wings to change places, the diagonals in each stage remaining unaltered.
The mode of forming these triangles is an operation of the most simple and
mechanical nature, too obvious to need to be further insisted on here.
Art. 8. When we are dealing with two functions of unequal degrees,
n and n + e, we can still form a square matrix with the coefficients of the
two systems of e and n equations respectively, but this will no longer be
symmetrical about a diagonal ; it is obvious, however, that if we treat the
function of the lower degree, as if it were of the same degree as the other
function, which we may do by filling up the vacant places with terms
affected with zero coefficients, the symmetry will be recovered ; and it is
somewhat important (as will appear hereafter) to compare the values of the
Bezoutian secondaries as obtained, first in their simplest form by treating
each of the two functions as complete in itself, and secondly, as they come out,
when that of the functions which is of the lower degree is looked upon as a
defective form of a function of the same degree as the other. A single
example will suffice to make the nature of the relation between the two sets
of results apparent.
Take
fas = a ac* + b at? + co? + dx + e,
<J3X = Q x* + a? + 7# 2 + Sx + .
s. 29
450 On a Theory of the Syzygetic Relations [57
The general method of Art. 7 then gives for the Bezoutian matrix
0, ay, aB, ae
ay,
ce ey
ae, Oe, ce ey, de eB.
We shall not affect the value either of the complete determinant, or
of any of the minor determinants appertaining to the above matrix, by
subtracting the first line of terms, each increased in the ratio of b : a, from
the second line of terms respectively ; the matrix so modified becomes
0, ay, aB, ae
ay, aB,
aB, ae + 68, I + 1 , ce ey
\c8 - dy/
ae, be, ce - ey, de - eB.
Again, adopting the method of Art. 6, we should obtain the matrix
0, y, ^, e
7, S, e,
/ be \
B, ae-bB, I + j, ce-ey
\c8 - dy/
ae, be, ce - ey, de - eB.
Hence it is apparent that the secondary Bezoutians obtained by the
symmetrizing method will differ from those obtained by the unsymmetrical
method by a constant factor a 2 ; and so in general it may readily be shown
that the secondary Bezoutians, by the use of the symmetrizing method, will
each become affected with a constant irrelevant factor a", where <y is the
difference of the degrees of the two functions, and a the leading coefficient
of the higher one of the two. When a is taken unity, the Bezoutian
secondaries, as obtained by either method, will of course be identical.
Art. 9. There is another method* of obtaining the simplified residues
to any two functions U and V of the degrees n and n+e respectively, which,
* Originally given by myself in the London and Edinburgh Philosophical Magazine, as long
ago as 1839 or 1840 [p. 54 above] ; and some years subsequently in unconsciousness of that
fact reproduced by my friend Mr Cayley, to whom the method is sometimes erroneously
ascribed, and who arrived at the same equations by an entirely different circle of reasoning.
57] of two Algebraical Functions. 451
although less elegant, ought not to be passed over in silence. This method
consists in forming the identical equations (of which for greater brevity the
right-hand members are suppressed)
F=&c.
xV= &c.
=&c.
xll = &c.
= &c.
+ 2 F = & c .
&c. = &c.
If we equate the right-hand members of (e + 2t) of the above equations
to zero, and then eliminate dialytically the several powers of a? from x n+f - +l ~ l
to aj"- +1 (both inclusive), the result of this process will evidently be of (e + t)
dimensions in respect of the coefficients in F, of t dimensions in respect
of the coefficients in U and of the degree fc n ~ l in x\ it will also be of the
form
(A+Bx+ ...+La?- 1 )U + (F+ GX + ... + Qx e +^) V,
and by virtue of Art. 2, must consequently be the tth simplified residue to
the system U, V.
Art. 10. The most general view of the subject of expansion by the
method of continued division, consists in treating the process as having
reference solely to the two systems of coefficients in U and F, which them
selves are to be regarded in the light of generating functions. To carry out
this conception, we ought to write
U=ct + a l y + a 2 f + a.y 3 + &c. ad inf.
V=b + b 1 y + % 2 + % 3 + &c. ad inf.,
and might then suppose the process of successive division applied to U and
F, so as to obtain the successive equations
U -M
V -
E 1 -M 3 R, +
&c. &c.,
292
452 On a Theory of the Syzygetic Relations [57
M lt M 2 , M s , &c. being each severally of any degree whatever in y, and in
general the degree of y in M t being any given arbitrary function </> (t) of i.
The values of the coefficients of the residues R l} R. 2 , R s ... , or of these forms
simplified by the rejection of detachable factors, become then the distinct
object of the inquiry, and will, of course, depend only upon the coefficients
in U and V and the nature of the arbitrary continuous or discontinuous
function < (t), which regulates the number of steps through which each
successive process of division is to be pursued. Following out this idea in a
particular case, if we again reduce our two initial functions to the forms
previously employed, and write
V=b x n + b i x n ~ l + &c.;
and if, instead of making, according to the more usual course of proceeding,
the divisions proceed first through one step and ever after through two steps
at a time, which is tantamount to making </>! = 1, $ (1 + &&gt;) = 2, we push each
division through one step only at a time, and no more (so that in fact </> (i)
is always 1), we shall have
U - m, V + R, = 0,
V m.&R 1 + Ra=Q,
R 1 -m 3 R 2 +R 3 =0,
J? 2 - m&R, + R 4 =0,
&c. &c.,
m lt w 2 , m 3 , &c. being functions of the coefficients only of U and V; and it is
not without interest to observe (which is capable of an easy demonstration)
that the simplified residues contained in R l} R 2 , &c., found according to this
mode of development, will be the successive dialytic resultants obtained
by eliminating the (t - l)th highest powers of as between the t first of the
system of annexed equations (supposed to be expressed in terms of x)
[7-0,
F=0,
=0,
= 0,
&c. = &c
57] of two Algebraical Functions. 453
If we combine together 2i + 1 of the above equations, the highest power of
a; entering on the left-hand side will be x n+i , and we shall be able to eliminate
2i of these factors, leaving x^ 1 the highest power remaining uneliminated.
If we take 2i, that is i pairs of the equations, the highest power of x appear
ing in any of them will be x n+i ~ l , and we shall be able to eliminate between
them so as still to leave #+*-!- fei-i^ that is # M ~ f as before, the highest power
of x remaining uneliminated ; and it will be readily seen that such of the
simplified residues corresponding to this mode of development as occupy the
odd places in the series of such residues, will be identical with the successive
simplified residues resulting from the ordinary mode of developing -^ under
the form of a continued fraction.
Art. 11. It has been shown that the simplified residues of fx and <j>x
resulting from the process of continued division are identical in point of
form with the secondary Bezoutians of these functions, but it remains to
assign the numerical relations between any such residue and the corre
sponding secondary.
To determine this numerical relation, it will of course be sufficient to
compare the magnitude of the coefficient of any one power of x in the one,
with that of the same power in the other ; and for this purpose I shall make
choice of the leading coefficients in each. In what follows, and throughout
this paper, it will always be understood that in calculating the determinant
corresponding to any square the product of the terms situated in the diagonal
descending from left to right will always be taken with the positive sign,
which convention will serve to determine the sign of all the other products
entering into such determinant. Now adopting the umbral notation for
determinants*, we have, by virtue of a much more general theorem for
compound determinants, the following identical equation :
//- ft ft n n \ /ft n n n \
I u^u^i^ (*fn_iUfn ^ / u-ju-2 . . u-j/jiUjH,-)-! \
I I
Vctitto . . . ajrtittjn+i /
&m i &w , ..
and consequently
( l " l ~ 1 I x (
...a,,,-!/ V^a^s ...,_!
See London and Edinburgh Philosophical Magazine, April 1851 [p. 242 above].
454 On a Theory of the Syzygetic Relations
and consequently when
-o,
r 2 ... fl^t-lOaH-!/
[57
will have different algebraical signs, it being of course understood that all the
quantities entering into the determinants thus umbrally represented above
are supposed to be real quantities. This theorem, translated into the ordinary
language of determinants, may be stated as follows : Begin with any square
of terms whether symmetrical or otherwise, say of r lines and r columns: let this
square be bordered laterally and longitudinally by the same number r of new
quantities symmetrically disposed in respect to one of the diagonals, the term
common to the superadded line and column being filled up with any quantity
whatever ; we thus obtain a square of (r + 1) lines and columns ; let this
be again bordered laterally and longitudinally by (r+1) quantities symme
trically disposed above the same diagonal as that last selected, the place
in which this new line and column meet being also filled up with any arbitrary
quantity ; and proceeding in this manner, let the determinants corresponding
to the square matrices thus formed be called D r , D r +i, D r +i--- - this
series of quantities will possess the property, that no term in it can vanish
without the terms on either side of that so vanishing having contrary signs.
Thus if we begin with a square consisting of one single term, we may suppose
that by accretions formed after the above rule it has been developed into
the square (M) below written, and which of course may be indefinitely
extended :
a, I,
I, b,
m,
p, s,
q, t,
m, n, c, r, u,
P, q, r, d, v,
s, t, u, v, e.
(M)
Here D , D 1} IX, D 3 , D 4 , D 5 will represent the progression
1, a,
a, I, m, p, s
a, I, m, p
a, 1
a, I, m
I, b, n, q
I, b, n, q, t
>
I, b, n
}
)
m, n, c, r, u
1, b
m, n, c, r
m, n, c
-t
P, q, r, d, v
p, q, r, d
s, t, u, v, e
57]
of two Algebraical Functions.
455
a, I, m, p
a, I, m
a, I
I , b, n, q
)
I , b, n
.
I , b
m, n, c, r
m, n, c
p, q, r, d
so if we use the matrix
a, I, m, p, s,
I , b, n, q, t,
m, n, c, r, u,
p, q, r, d, v,
S, t, u, v, e,
the determinants D 1} _D 2 , D 3 , D 4 , representing
a,
will possess the property in question ; the line and column I, b; I , b not
being identical, the first determinant D representing unity must not be
included in the progression.
We shall have occasion to use this theorem as applicable to the case of a
matrix symmetrical throughout, and we may term the progression (II), above
written, a progression of the successive principal determinants about the
axis of symmetry of the square matrix (M), and so in general. Now it is
obvious that the leading coefficients of the successive Bezoutian secondaries
are the successive principal determinants about the axis of symmetry of the
Bezoutian squares; they will therefore have the property which has been
demonstrated of such progressions ; to wit, if the first of them vanishes, the
second will have a sign contrary to that of + 1 ; if the second vanishes,
the third will have a sign contrary to that of the first, and so on.
Art. 12. Now let fx and <$>x be any two algebraical functions of x with
the leading coefficients in each, for greater simplicity, supposed positive:
and in the course of developing ^ under the form of an improper continued
fraction by the common process of successive division, let any two consecutive
residues (the word residue being used in the same conventional sense as
employed throughout) be
Ax 1 + Bx l ~ l + C^ l ~ 2 + &c.
B x^- 1 + (7V- 2 + DV~ 3 + &c.
The residue next following, obtained by actually performing the division and
duly changing the sign of the remainder, will be
i^ ^w^- -^ 1
456 On a Theory of the Syzygetic Relations [57
which is of the form
Thus the leading coefficients in the complete unreduced residues will be
A, B , {B M-
and when reduced by the expulsion of the allotrious factor will become
A, B , B M AC 2 , and consequently, when B the leading coefficient of one
of the simplified residues vanishes, the leading coefficients of the residues
immediately preceding and following that one will have contrary signs.
First, let fx and (f>x be of the same degree. As regards the numerical
ratio of each Bezoutian secondary to the corresponding simplified residue,
it has been already observed that there are always unit coefficients in the
latter of these, and the same is obviously true of the former ; hence if we
call the progression of the leading coefficients of the simplified residues
R,, Ro, R 3 , R,, &c,
and that of the leading coefficients of the Bezoutian secondaries
B lt B 2 , B 3 , B 4 , &c.,
we have
B l = R 1 , B 2 =R,, B, = R,, B 4 = R., &c.
It may be proved by actual trial that B 1 = R l and B 2 = R. Moreover,
since the signs are invariable, and do not depend upon the values of the
coefficients, we may suppose B 2 = (which may always be satisfied by real
values of the quantities of which B., is a function); we shall also, therefore,
have .Ro = 0, and consequently B 3 has the opposite sign to that of B l , and R 3
the opposite sign to that of R lt which is equal to B^. hence when B 2 =0,
B 3 and R 3 are equal, and consequently are always equal ; in like manner we
can prove that R 4 and 5 4 have the same sign when R 3 and B 3 vanish, and
consequently are always equal, and so on ad libitum, which proves that the
series B l} B 2 , ... B n is identical with the series R lt R z , ... R n , and con
sequently that the Bezoutian secondaries are identical in form, magnitude
and algebraical sign with the simplified residues.
Secondly, when fx and <f>x are not of the same degree, it has been
shown that the secondaries formed from the non-symmetrical matrix corre
sponding to this case will be the same as those formed from the symmetrical
matrix corresponding to fx and 3?x (where 3>x is <f>x treated by aid of
evanescent terms as of the same degree as fx), with the exception merely
of a constant multiplier (a power of the leading coefficient of fx) being
introduced into each secondary. By aid of this observation, the proposition
57] of two Algebraical Functions. 457
established for the case of two functions of the same degree may be
readily seen to be capable of being extended, from the case of / and <
being of equal dimensions in x, to the general case of their dimensions being
any whatever.
Art. 13. Before closing this section, it may be well to call attention to
the nature of the relation which connects the successive residues of fx and
<f>x with these functions themselves, and with the improper continued
fractional form into which ^- is supposed to be developed in the process of
TOO
obtaining these residues.
If <j)x be of ft degrees, and/a? of n + e degrees in x, we shall have
fa ~ & - & - qs - " q
where Q l may be supposed to be a function of x of the degree e, and
(ft, $3, q n , are all linear functions of x; the total number of the quotients
Q!, <? 2 , qn being of course n when the process of continued division is
supposed to be carried out until the last residue is zero. Upon this supposi
tion the last but one residue is a constant, the preceding one a function of x
of the first degree, the one preceding that a function of x of the second
degree, and so on.
Let us call the residue of the degree t, in x, ^ t ; it will readily be seen
that the successive complete residues arranged in an ascending order will be
S-o , S-o qn , S"o (?n-i qn - 1 ), So (ffn-sffn-i ?n ~ ?n-a - <?n), &C.,
being in the ratios
1, q n , <?n-i . <?n-2 -, & c -
qn qni *n
Again, we shall have in general
AJ-L i <i> = ^, (15)
A t being an integral function of x of the degree n t 1, and L, an integral
function of x of the degree (n + e) i 1 ; and it is easy to see that the
successive convergents to the continued fraction
1 1 1 .
- fee,
have their respective numerators and denominators identical with those of
the fractions
A-n i A w _ 2 * c,
458 On a Theory of the Syzygetic Relations [57
Adopting the language which I have frequently employed elsewhere,
I call S- t a syzygetic function, or more briefly a conjunctive of/ and <, and
A t and L, may be termed the syzygetic factors to S- t so considered. If we
divide each term of the equation (15) by the allotrious factor (M), we have
where R t is the tth simplified residue to (/, <) ; and if we call -W = T t , and
~ = t l; so as to obtain the equation
r l f-t l <f} = R l , (16)
f
we see that the fraction formed by the component factors to any simplified
t(.
residue of (/, <), will be identical in value (although no longer in its separate
terms) with one of the corresponding convergents to ^ , exhibited under the
form of an improper continued fraction. I shall in the next section show
how, not only the successive simplified residues, but also the component
syzygetic factors of each of them, and consequently the successive con
vergents, may be expressed in terms of the roots of the two given functions..
Since the preceding section was composed the valuable memoir of the
lamented Jacobi, entitled "De Eliminatione Variabilis e duabus Equationibus
Algebraicis," Crelle, Vol. XVL, has fallen under my notice. That memoir is
restricted to the consideration of two equations of the same degree, and the
principal results in this section as regards the Bezoutic square and the
allotrious factors applicable to that case will be found contained therein.
The mode of treatment however is sufficiently dissimilar tq justify this
section being preserved unaltered under its original form.
SECTION II.
On the general solution in terms of the roots, of any two given algebraical
functions of x, of the syzygetic equation, which connects them with a third
function, whose degree in # is given, but whose form is to be determined.
Art. 14. Let / and <j> be two given functions in x of the degrees m and
n respectively in x, and for the sake of greater simplicity let the coefficients
of the highest power of x in f and </> be each taken unity, and let it be
proposed to solve the syzygetic equation
r t /-^ + ^ = 0, (17)
57] of two Algebraical Functions. 459
where S\ is given only in the number of its dimensions in x, which I suppose
to be t; but the forms of r t , t L , ^ are all to be determined in terms of
/*!, h,... h m the roots of /and 77, , 7? 2 ... T;,, the roots of $.
I shall begin with finding ^ ; and before giving a more general represen
tation of S- t , I propose now to demonstrate that we may make
* t = 2{P 9ll fc...fcX(a!-/O(*-M---(a-M (18)
where P qi , q2 ... qi is used to denote
R(h <h , h qi ...h q ) denoting any rational symmetrical function whatever of
the quantities preceded by the symbol R, and q lt q z ... </ q i+ i q m being any
permutation of the m indices 1, 2 ... m.
Suppose /= and < = 0, then x is equal to one of the series of roots
hi, h. 2 ... h m ,
and also to one of the series of roots
Suppose then that
, p ^ if , i
tX/ "CL /W 3
and consider any term of ^ t .
If in any such term a is found in the series q lt q. 2 ... q it then
But if not, then x must be found in the complementary series
and consequently -Pg,,^...^ will contain a factor ha vj^ and ?,*...* 0; m
every case therefore
Therefore ^ as expressed in equation (18) is a syzygetic function of /
and <j> ; and we have found a function of the tth degree in x, and of course
expressible by calculating the symmetric functions as a function only of x and
of the coefficients of/ and </>, which will satisfy the equation
460 On a Theory of the Syzygetic Relations [57
It will be remembered that by virtue of Art. 2 we know a priori that all
the values of S- t satisfying this equation are identical, save as to an allotrious
factor, which is a function only of the coefficients in / and <f>.
It is clear that we may interchange the h and 77, m and n, and thus
another representation of a value of S- t satisfying the equation (17) will be
j q - h m )
(v)q-hi) (rj tjn -h 2 )... ( Vqn
Art. 15. If we employ in general the condensed notation
, 7/1, ...
*
to denote the product of the differences resulting from the subtraction of
each of the quantities \, p... v in the lower line from all of those in the
upper line I, m, n...p, the two values above given for S- t may be written
under the respective forms
and $
(x -
- ^ . . . ^ - ^ ,
in each of which equations disjunctively and in some order of relation each
with each
<?i, q a , q a - q m = l, 2, 3 ...m,
and fi,| s ,f,...|=l,2,3....
These two forms are only the two extremities of a scale of forms all equally
well adapted to express S- t ; for let v and v be any two integers so taken as
to satisfy the equation
v + v i,
and let R( ...... ; ~ ..... ), where the dots denote any quantities whatever, be
used to denote a rational function which remains unaltered in value when any
two of the quantities under either of the two bars are mutually interchanged,
then we may write
. (19)
x (x - h q ) (x - h gt ) ...(x- h qv ) x (as - rj (i ) (x - 77^) ...(x-
57] of two Algebraical Functions. 461
For if, as above, we suppose x = h a = ri u >, any term of ^ t in which q lf q., ... q v
comprise among them a, or in which , ,... comprise among them &&gt;,
will vanish by virtue of the factors
(x - h <h ) (x - h gt ) ... (x - h qv ) x(x- vtl ) (x - 77 f2 ) ...(- 77^) ;
but if neither a nor &&gt; is so comprised, then a must be one of the terms
in the complementary series q v+1 , q v+s , ... q m , and w one of the terms in the
complementary series , +1 , , +s ...|f w , and therefore one of the quantities
h iv+i> V+* V wil1 ec l ual one of tne quantities 77^ , 77 f ^ ... 7; , and con
sequently the term of S- t in question will vanish by virtue of the factor
[vanishing. In either case therefore every term included
^ +1 -^ +2 --^ J
within the sign of summation vanishes when x=h a = ij ia , that is, whenever
fx = Q and </>#=0. Hence ^ t , as given by equation (19), will satisfy
the syzygetic equation r t /- t.<f> + $- t = for all values of v and v which
make v + v = t, and for all symmetrical forms of the function denoted by the
symbol R (- ; ..... ").
Art. 16. I shall now proceed to show how to assign the arbitrary
function whose form is denoted by this symbol in such a manner as to make
S- t become identical with a simplified residue to / and <. To this end I take
for R(h qi , h, h ... h qv ; %1 , %2 ... 77^) the value
R =
,A,
"Iv+z
we shall then have
(20)
-vi v rVuVfi v
XN
X {(x - h lh ) (x - h <h ) ...(x- h qv }} {(x - r, tl ) (x - -nQ ...(x- 77^)). (21)
I shall first show this sum of fractions is in substance an integral function
of the quantities h lt h 2 ...h rn ; 77], rj.,...r) n . For greater conciseness write
in general xh = E,x r) = H\ we have then, since
462 On a Theory of the Syzygetic Relations [57
On reducing the fractions contained within the sign of summation to a
N
common denominator, S\ will take the form -r , where D will be the
product of the ^m(m 1) differences of E 1} E<>... E m subtracted each from
each, and A the corresponding product of the differences inter se of
H 1} H 2 ...H n . Hence, unless the sum in question is an integral function
of the E s and H s it will become infinite when any two of the E series, or
any two of the H series of quantities are made equal. Suppose now E l = E. 2 ;
the terms in (22) which contain E l E 2 in the denominator will evidently
group themselves into pairs of the respective forms,
E,, E qt ...E, h -\ ^E tt
Hs lt Htt...Ht
...E qm
. . . H
in
and
g . . . E qm
q .
...Hi
the sum of this pair of terms will be of the form
P
Q
E,
\~EO
L E <lv+i> E <lv+2 ~ E <1mJ
P
where Q, it may be observed, does not contain H H 2 , so that ^ remains
^e
finite when H 1 = H.
The above pair of terms together make up a sum of the form
P _ 1 __ <f> (E,, E,} TJrE 2 -(j>(E 2> E,}
x
which, as the numerator of the third factor vanishes when E 1 = E 2 , remains
finite on that supposition. Hence the whole sum of terms in (22) which is
57] of two Algebraical Functions. 463
made up of such pairs of terms, and of other terms iu which E 1 E. 2 does
not enter, remains finite when E l E^ = 0, and therefore generally when
D = 0, and similarly when H l H = 0, and therefore also when A = ;
hence the expression for ^ t in (22) is an integral function of the E and H
series of quantities, as was to be proved.
Art. 17. Let us now proceed to determine the dimensions of the coeffi
cient of x l , the highest power of x in this value of S\, when supposed to be
expressed under the form of an integral function (as it has been proved to be
capable of being expressed) of h 1 , h... h m ; v)i, rj 2 ... rj n ; x.
This coefficient is the sum of fractions the numerators of each of which
consist of two factors, which are respectively of v x v and of (ra v) x (n v)
dimensions in respect of the two sets of roots taken conjointly, and the
denominators of two factors respectively of v (m - v) and v (n v) dimen
sions in respect of the same.
Consequently, the exponent of the total dimensions of the coefficient in
question
= w + (m v) (n v) v (m v) v (n v)
= (??i v v) x (n v v)
= (m i) (n i),
and thus is seen to depend only on the degree i in x of ^- t , and not upon the
mode of partitioning t into two parts v and v, for the purpose of representing
^ t , by means of formula (19).
Art. 18. I shall now demonstrate that every form in this scale (to a
numerical factor pres) is identical with a simplified residue to /, <, of the
same degree i in x. Any such simplified residue is, like ^ t , a syzygetic
function, or to use a briefer form of speech a conjunctive of/", <; and if we
agree to understand by the " weight " of any function of the coefficients of
f and < its joint dimensions in respect of the roots of f and <f> combined,
I shall prove, first, that any simplified residue of / and <j> of a given degree
in x is that conjunctive, whose weight in respect of the roots of /and <
is less than the weight of any other such conjunctive; and second, that ^- t ,
as determined above (in equation 22), is of the same weight as the simplified
residue, and can therefore only differ from it by some numerical factor.
For the purpose of comparison of weights, it will of course be sufficient
to confine our attention to the coefficients of the highest power in x (or
any other, the same for each) of the forms whose weights are to be compared.
Suppose f to be of in dimensions, and < to be of n dimensions in x ;
and let m n + e.
464 On a Theory of the Syzygetic Relations [57
Suppose Af+L(j> = Ax l + Bx - l +...+K, (23)
A = X #9 + X^?- 1 + . . . + \ q ,
L = I x<i +e + l lX i+^ + ...+ l,j +et
the number of homogeneous equations to be satisfied by the q + 1 quantities
X 0) A lt ..X ? , and the q + e + l quantities 1 , l!...l q+e will be m + qi, and
therefore q + 1 and q + e+l taken together must be not less than ra + q i + 1,
that is 2q + e + 2 must be not less than q 4- ra i + 1, that is q not less than
m i e 1 ; and if this inequality be satisfied 2q + e+2 (q + m i + 1) + 1,
that is q + t + e m+ 2 will be the number of arbitrary constants entering
into the solution of equation (23).
If q be greater than (n 1), let q = (n 1) + t ; and let
(A) = Xotf 1 - 1 + XX l ~ 2 + . . . + X n _, ,
(L) = l X n+e ~* + l^- 2 + ...+ l e+n ^
and let (A), (L) be so taken as to satisfy the equation
(A)/+ (L) $ = Ax + Bx^ + . . . + K ;
and make 5 =(A) + (f+gx + ... + has 1 - 1 ) <j>,
f, g ... h being arbitrary constants ; then
E.f + X(j> = ( A)/+ (L) <f> = Aaf- + Bx^ + . . . + K.
Now the total number of arbitrary constants in the system (A) and (L)
will be n 1 + L + e m + 2, that is i + 1 ; hence the total number of arbitrary
constants in H and X will be i + 1 + 1, that isq n + t + 2, which is equal to
q _|_ t _|_ e m -}- 2, the number of arbitrary constants in the most general values
of A and L. Hence {A = E, L = X] is the general solution of the equation
A/+ L$ = Ax 1 + Bx l ~ l + ... + K ; and consequently the most general form
of Ax 1 + Ex 1 1 + ... + K, which is evidently independent of the (t) arbitrary
quantities/, g...h, will contain the same number of arbitrary constants
as enter into the system (A) and (L), that is L + 1.
Art. 19. Let us now begin with the case of greater simplicity when
m = n, that is e = ; and let us revert to the system of equations marked (10)
in Section I., in which U and V are to be replaced by /and 0.
First, let i = n 1, then t+1, the number of arbitrary quantities in the
conjunctive, is n.
From the system of equations (10) we have, for all values of p lt p, p 3 ... p tl ,
(piQo + P -Qi + + pnQn-i)f
p n n-i
57] of two Algebraical Functions. 465
and consequently the most general value of ^ n -i in the equation
Tn-if tn-i<f> + ^fn-i = 0,
where S- n _ a = Ax n ~ l + Bx n ~ 2 + ... 4- L,
will be obtained by making
T n -i = piQo + PzQi + . . . + pnQn-i,
tn-i = PiPo P^Pi p n Pn-i ,
which solution contains n, that is the proper number of arbitrary constants.
Again, if i=n 2 > t + l=n 1, which will therefore be the number
of arbitrary constants in the most general value of *& n -z in the equation
T 2.7 *n 2<P ~^~ J n 2 == "
This most general value of ^ n _ 2 is therefore found by making
T n _ 2 = p\Q + p zQ! + . . . + p nQn-i,
/ T) P P
ti i p i-i o P i r \ P n-LnD
where p\, p *...p n are no longer entirely independent, but subject to the
equation
P\ KI + p 21 K,+ ...+ p n n-i^ = 0,
so as to leave (n 1 ) constants arbitrary.
We thus obtain X-2 = (p A + p 2l K z + ... + p n n ^K 2 ) x n ~ 2 + &c. In like
manner, and for the same reasons, the most general values of ^ n _ 3 in the
equation
T n 3J ~ tn3 Y "I" J n 3 = ">
will be found by making
T n _ 3 = p \ Qo + p" 2 Q!+ ... + p" n Q tl -i,
j. a p " p // p
i-Tis P i -L o ~~ P *! P n -LIII }
where p \, p" 2 ... p" n are subject to satisfying the two equations
p / l jr 1 -fp / il jr I +...+A-A-o J
P \ K, + P " 2 1 K,+ ...+ p" n n _,K 2 = 0,
so as to leave (n 2) constants arbitrary ; and we thus obtain
V- 3 = (p \ K, + p" 2 ,K S + ... + p" n n ^K 3 ) a" 3 + &C.,
and so on, the number of independent arbitrary constants in ^ decreasing
(as it ought) each time by one unit as the degree of ^ descends, until finally,
if TO/ t (f) +^o = 0, S being a constant, the general value for S- is found by
making
T O = (pi) Qo + (Pi) Qi+ ... + (pn) Qn-i,
t = - ( Pl ) P - (p. 2 ) P, - ... - (p n ) P n _ 1}
s. 30
466 On a Theory of the Syzygetic Relations [57
where (p^, (p 2 ) ... (p n ) are subject to satisfy the (n 1) equations
which gives ^ = K n (p\ + J n (p) 2 + . . . + n -\Kn (/>)
Now evidently the lowest weight in respect to the roots of U and F that
can be given to (p.K, + p 31 K 1 +...+ p n n-A) a?"" 1 + &c., when the multipliers
Pi, pi - pn are absolutely independent, is found by taking
/>! = !, /o a = 0, p s = 0...p n = 0,
which makes the weight of the leading coefficient in X-i, the same as that
of K-i, that is 1.
Again, when one equation,
p\ K + p z i-STi + + p n n-1%1 = 0,
exists between the (/a) s, the lowest weight will be found by making
p i = i#i, (/-- JTi, P s = 0, p 4 =0... P w = 0,
which makes the weight of the leading coefficient in X_ 2 depend on
l K i jttT 2 ~ M !**
which is of the weight 1 + 3, that is 4, in respect of the roots of/ and 0.
Similarly, ^ n _ 3 will have its lowest weight when its leading coefficient
is the determinant
I M, 2 -n-o, 2&
the weight of which is 1 + 3 + 5 = 9 ; and finally, the lowest weighted value
of ^ is the determinant represented by the complete Bezoutian square ; the
weight in general of X-; being 1 +3 + ... +(2i -1), that is i\ or which
is the same thing otherwise expressed, the weight of the leading coefficient
of the lowest-weighted conjunctive of / and </> of the degree i in x is
(n -C)(m- i)*. It will of course have been seen in the foregoing demon
stration, that the weight of r K s [which means 2 (a r b s - a s b r ), a r , a s being the
coefficients of x n ~ r , x n ~* in /, and b r , b s of the same in <] has been correctly
taken to be r + s in respect of the roots of /and < conjoined.
* n and m are supposed equal and i = n-i.
57]
of tivo. Algebraical Functions.
467
Art. 20. If now we proceed in like manner with the general case of
ra = n + e, it may be shown, in precisely the same way as in the preceding
article, that the most general value of any conjunctive of f and < will be a
linear function of e functions,
X H + !#
* + a 2 # w + ... +a n ,
?l+l 1 n
w \^ Ci/i t*.
iW 1 /f /ii?l 1 _l L t /^ //i
|^ L/ .1 1^/ \^ ^^ U/ /j tX/ j
4P-. + M
; w +l _[_ d^x n + + (X i2? 2
and of the n functions,
&c.
K n ,
&c.
&c.
and that consequently, if the degree of such conjunctive in x be (n i),
it will be of the lowest weight when it is a linear function of the entire
e upper set of functions, and i of the lower set; and consequently, the
coefficient of the highest power of x in such conjunctive will be the
determinant
fl,
A
1 n n n n
1
the weight of which is evidently that of
#! X i^Tj, X 2J fiT 3 ... X i-xTf; X (<Xi) e ,
that is 1 + 3 + 5 4. . . . + (2i 1) + ei,
that is i 2 + ei, or z (e + i), which is (w - t) (m t) if i = n i.
302
468 On a Theory of the Syzygetie Relations [57
Hence the weight of the leading coefficient in the lowest-weighted
conjunctive of/ and < of the degree i in x is (m i)(n i), m being the
degree off and n of <.
From this we infer that any conjunctive of f and <f> of the degree i,
of which the leading coefficient is of the weight (m t) (n i), all the
coefficients being of course understood to be integral functions of the roots
of f and (f>, must, to a numerical factor pres, be equivalent to any other
of the same weight ; and furthermore, any supposed function of x of the tth
degree which possesses the property characteristic of a conjunctive of vanish
ing when f and < vanish simultaneously, but of which the weight of the
leading coefficient would be less than (m i) (n i), must be a mere nugatory
form and have all its terms identically zero *.
Art. 21. We have previously shown, Art. 16, that S-, as defined by
equation (21), is an integral function of the roots f and <, and vanishes
when f and < vanish. Moreover, its weight in the roots has been proved
to be (m i) (n i), and consequently, if by way of distinguishing the several
forms of ^ t we name that one where i in the equation above cited is supposed
to be divided into two parts, v and v, &,, we have for all values of v and v,
such that v+ v is not greater than n, ^f ViV to a constant numerical factor pres
identical with the (v + v)th simplified residue to (f, (f)), so that the form of
*& ViV depends only upon the value of v + v.
Art. 22. It must be well borne in mind that this permanency of the
value of ^ Vji - v for different values of v has only been established for the case
where i can be the degree of a residue to / and <, that is to say, when t
is less than the lesser of the two indices m and n. When t does not satisfy
this condition of inequality, the theorem ceases to be true. It, is clear that
when m = n and v + v = m = n,^r V}V , which always remains a conjunctive of f
and (f>, can only be a numerical linear function of / and </> ; and I have
ascertained when m = n on giving to v and v the respective values succes
sively (0, w), (1, TO - 1), (2, (w - 2)) ... (n, 0) that
Thus, by way of a simple example, let
f = x* + ax + 1) = (x Aj) (x h 2 ),
</> = a? + ax + ft = (x - &j) (x k 2 ),
* And more generally it admits of being demonstrated by precisely the same course of
reasoning, that the number of arbitrary parameters in a conjunctive of the degree t, and of the
weight (m-i)(n-i) + e in the roots, cannot (abstraction being supposed to be made of an
arbitrary numerical multiplier) exceed the number e.
57]
of two Algebraical Functions.
469
(x hi) {x n 2 ) f,
h.i - 2 (X A!> (X k^ r . -. .-, -.
1 X M
_A 2 J [kt]
_ v X hi ^X h ,, _] f \(l t J,\]
~h 1 -h,kl-k, [(nl
1 rA w ^11 -* / \ *^ *v 1 / V M " **" 1 / \ ^ 2 " *^*> /
that is = ^I Tit. 7T *.\/i l.\/Jl
Ai-j "~" it 2 ("^l ~~ *^2 \ ^~ \ ^ 2/ \ 1 ^~" 2/ \ 2 ^~
S*^ ^1 f/f 7\ .r/7 ,7\7 /?; .7;
-j j- {(Aj ii 2 )x + L(^i + *) A 2 (AjA, + ^y
ft] A< 2
^~* j w ~~ \ *i "T" !<>>) X ~i~ /ti iZ-o t ~*~ t*^- "~~ \ *^i ~i 2/ ~^ A/iA
= (a- 2 + a# + 6) + (x 2 + a# 4- /3)
-/+*;
so we find also ^r 2>0 = 0.
Art. 23. The expression ^ v<v , which is universally a conjunctive of/
and <f), continues algebraically interpre table so long as v + v has any value
intermediate between and m + n ; when v + v = we must of course have
v = and v = 0, and ^ 0>0 becomes the resultant off and < ; when v + v = m + n
we must also have the unique solution v = m and v = n, and ^ m> n becomes
necessarily / x <f>, which we thus see stands in a sort of antithetical relation
to the resultant of / and <, say (f, <). Nor is it without interest to remark
that f x (f> = implies that a factor of f or else of </> is zero ; and (f, <) =
implies that if a factor of the one of the functions is zero, so also is a factor
of the other, that is that a factor of each or of neither is zero. As L increases
from to n or decreases from m + n to m 1, the number of solutions of the
equation v + v = i in the one case, and the number of admissible solutions
of the equation v + v = i in the other case, which is subject to the condition
that v must not exceed n, continues to increase by a unit at each step ;
there being thus n + 1 different forms *$t VjV when v + v = n, and the same
number when v + v = m 1. For all values of t, intermediate between n and
(m 1) (both taken exclusively) it is very remarkable that v>v will vanish,
as I proceed to demonstrate.
470 On a Theory of the Syzygetic Relations [57
Art. 24. The weight of the coefficient of the highest power of &_,,
(v + v being equal to t) is (m i) (n t), and consequently, when t is greater
than n, and less than m, *b v>v would contain fractional functions of the roots
of / and <f>, if there were in it a power # , but ^r v< has been proved to be
always an integer function of the roots. Hence the coefficient of # l will be
zero, and so more generally the first power of x in *& ViV) of which the coefficient
is not zero, will be a?~ w , subject to the condition (since evidently the weight
of the several coefficients goes on increasing by units as the degree of the
terms in a; decreases by the same) that <o be not less than (m t) (i n) ;
let then o> = (m t)(i n), $>, becomes of the form Ax 1 --" + Ex 1 " 1 + &c.,
where A is of zero dimensions ; but this is impossible if i u> < n, for then
Ax l ~ M + &c. is a conjunctive of weight lower than the lowest-weighted
simplified residue of the degree t a>. Hence <u is not greater than i n,
that is (m t) (t n) is not greater than i - n, that is m i cannot be greater
than 1, that is i when intermediate between m and n cannot be less than
m 1, otherwise ^r VfV will vanish identically. Moreover, when i = m 1,
&&gt; = i n, and i u> = n, and accordingly ^ ,_!_ is not merely, as we might
know, d priori an algebraical, but more simply a numerical multiple of < for
all values of v. The same is of course true also, m being greater than n, for
every form *& Vin _ v , since this is always a conjunctive of /and <, of which the
former is of a degree higher than the ^ in question, so that the multiplier
of/ 1 in this conjunctive must be zero*.
Art. 25. To enter into a further or more detailed examination of the
values assumed by ^ v>v for the most general values of m, n, i, would be to
transcend the limits I have proposed to myself in drawing up the present
memoir. What we have established is, that to every form of % v>t - v apper
taining to a value of t between and n, there is a sort of conjugate form for
which i lies between m + n and m ; that for t = m 1 or i = n, ^r v>t - v becomes
a numerical multiplier of < ; and that when i lies in the intermediate region
between n and m 1, <b v>t - v vanishes for all values of v. I pause only for
a moment to put together for the purpose of comparison the forms corre
sponding to i and to m + n i. By Art. 16, making i = v + v,
* t = 2 (X - h q ) (X - h q ^ ...(X- h qo ] X (X - 1J J (X - ^) ...(*- y^)
, ** VI
-^
...h qv
* It thus appears that if the indices m and n do not differ by at least 3 units, S- will have an
actual quantitative existence for all values of t between and m + n; or in other words, the
failure in the quantitative existence of the forms ^ t only begins to show itself when this difference
is 3 ; thus if m = n + 3, ^ n exists, and ^ n+2 exists, but ^ n+1 = 0.
57] of two Algebraical Functions. 471
The conjugate form for which t = m + nc and m v, n v, vv take the
places of v, v and (m v) (n v), will be got by taking
^ = 2 (x - h qv+l ) (x - 7 W ) ,..(*- h qm ) x (x - 17, ) (x - rj; ) ...(*- ^ n )
. . _ . *u
hq
which it will be perceived are identical, term for term, in the fractional
constant factor, and differ only in the linear functions of x, which in S- t and
in $v ai> e complementary to one another. Our proper business is only with
those forms for which L < n.
Art. 26. It will presently be seen to be necessary to ascertain the
numerical relations between ^ 0>l and ^ Ij0 when i <n, and this naturally brings
under our notice the inquiry into the numerical relations which exist between
the entire series of forms *b v>l - v for a given value of i, corresponding to all
values of v between and i inclusive.
In order to avoid a somewhat oppressive complication of symbols, I shall
take a particular numerical example, that is m = 7, n = 6, t = 4, and compare
the values of S- 0i4 ; Si, 3 5 ^2,2; ^3,1 5 ^4,0.- all of which we know to be identical
[to a numerical factor pres] with one another and with the second simplified
residue to / and <j>, that being of the fourth degree in x ; our object in the
subjoined investigation is to determine the numerical ratios of these several
forms of S- to one another.
First, let v = 0, v = 4. The leading coefficient ^ M is
which we know a priori (it should be observed) to be essentially an integral
function of the h and the 77 system. In this, the term containing ij 6 s will be
evidently
r* l
\JiihJt a hJi 6 h 6 h 7 ]
the 77 system to which the latter summation relates being now reduced
to consist of 7/1,772,773,774,775. In this expression, again, the coefficient of 77 5 3
is evidently 1. Hence, therefore, the leading coefficient in S- 0)4 contains the
term 7 5 3 7 6 3 .
472 On a Theory of the Syzygetic Relations [57
Secondly, let v = 1, v = 3. The leading coefficient in S- 1>3 becomes
" x p?4%/6
J [AaMA^A \
.
i x
In this, the factor affecting 77 6 3 will be
x
Ji^h 3 hJi, 5 h 6 h 7
774775
7; 6 being now understood to be eliminated out of the 77 system included within
the above summation. Again, in this latter sum the factor affecting ^
will be
/4
J-M.M.M, x ^j
775 and 77,; being now both eliminated out of the 77 system. This last sum can
of course only represent a numerical quantity.
So in like manner, again, if = 2, v= 2, the coefficient of 77 6 3 77 5 3 in $- 2i2
will be similarly reducible to the form
S^r 1
So, again, when v = 3, v = 1, the coefficient of 77 6 3 77 5 3 in $- 3il will be
(D)
Lfi J
and finally, the coefficient of 77 6 3 77 5 3 in % 4 will be
(E)
out of all which sums it is to be remembered that 775 and 77,, are supposed
excluded from appearing. All these several coefficients being numbers
in disguise, we may determine them by giving any values at pleasure
to the terms in the h and 77 system.
57] of two Algebraical Functions. 473
Let now ^ = h lt ^ 2 = h 2 , ?) 3 = h 3 , rj 4 = h A , then in (B) it will readily be seen
that all the terms included within the sign of summation vanish identically,
except the following, namely,
A 4 I
*
^] x |~173
J ^hjijishshi]
MAMAl
J
x
In each of these expressions the first factor of the numerator is identical
in value (by reason of the equations /*I = T; I , k 2 = rj 2 , h s = r) 3 , /i 4 = ^ 4 ) with
{ ) 3 x the second factor of the denominator, and the second factor of the
numerator with (-) 6 x the first factor of the denominator; hence the
coefficient of y^t]/ in S- 1)3 is 4.
In like manner the only effective terms of ^ 2 ,2 will be
["^i^
[A 3 A 4
n^hthshihj I
IM J
.\n
h 2 h 4
P/i^ri x r^s
\Jhh 3 ] \_hihjis
Any other term will necessarily contain in the numerator a factor, whose
symbolical representation will contain one of the quantities 77], 772, 773, 774, in the
upper line, and one of the quantities h 1} h%, h 3 , h 4 , having the same subscript
474 On a Theory of the Syzygetic Relations [57
index, in the lower line, and which will therefore vanish ; the number of
effective terms being evidently the number of ways in which four things
can be combined 2 and 2 together, and the value of each term is evidently
(~) 22 ( 1) 2S 1> so that the entire value of the coefficient of rj^Tj/ in Sr 2i2
is +6.
Precisely in the same manner, we shall find that the leading coefficient
in Sy 3;1 will contain the term 4<rj/ rj^, the ( 1) resulting from the operation
( I) 1 3 ( ) 34 , and in S- 4(0 the term + < r) 5 3 r) 6 3 , the + 1 resulting from the operation
( I) 43 . Hence it appears that ^- 0) 4; ^"1,3; ^"2,2$ ^"3,1 5 ^4,0 are to one another
in the ratios of 1; 4; 6; -4; 1; and so in general for any values of
?n, n, i (t being less than m and less than n) it will be found that
C\. C\. CV CL.
^O, I) ^1,11) ^2, i 2 Ji,
will be in the ratios of the numbers
1; (-1)^; (-!) < *> t -~-; (- !) < *> i -"- , ... ; (-!) < -".
Art. 27. The method employed in the preceding investigation will
enable us to affix the proper sign and numerical factor to ^ ,<. or ^i,o> or i n
general to Sv, t _,,, in order that it may represent the Bezoutian secondary
of the degree i in x. This latter has been already identified with the
fhtJC
simplified residue obtained by expanding ^- under the form of an improper
jx
continued fraction. For this purpose, it will be sufficient to compare a
single term of any such Sr with the corresponding one in the Symmorphic
Bezoutian secondary. Let us first suppose that m = n, f and < being of
the same degree. A glance at the form of the Bezoutian square will show
that if we form the Bezoutian secondary of the degree (n i) in x, the
G -D-
coefficient of its leading term will contain the term ( ) 2 (0, i} 1 ; (0, i)
as usual denoting the product of the coefficient of x n in f by the coefficient
of x n ~ i in (f>, less the product of the coefficient of x n in </> by that of x n ~ l
in /; and as we suppose the first coefficients in / and < to be each 1, if
we term the other coefficients last spoken of a t and a respectively, this
said coefficient of the leading term of the iih Bezoutian secondary will
contain the term ( ) ^ (a f o. 4 -) : , and consequently (1) 2 &? and
.i+i
(-) ITaA
Now by the like reasoning to that employed in the preceding article,
the coefficient of the leading term in ^ m _i )0 , that is
v
57] of ttvo Algebraical Functions. 475
will contain the quantity 2 (/h h z h s ... A,-)*, and therefore will contain a
term {^(/tjOs ... /* t -)}\ that is ( ) *a/, which is equal to ( ) l a/, since
(i 1) i is always even. Hence ^ m _,- j0 = ( )* 2 x the corresponding Bezoutian
secondary.
Art. 28. The above applies to the case where we have supposed m = n.
When this equality does not exist we may proceed as follows. Prefix to
<f>x, the first coefficient of which is still supposed to be 1, a term ex m , where
e is positive and indefinitely small, and let (f>x so augmented be called <!> (x).
Then if 77!, r} 2 ... rj n are the roots of <j>x, 7/1, rj., ... 77 W , together with the (in ri)
values of f-V"*. will be the roots of 4>(a?).
w
But it has already been proved that when (as here supposed) the first
coefficient of fa is 1, the Bezoutian secondaries to/ and </> will be identical
with those to / and <l> respectively ; at least it has been proved that these
latter, when e = 0, but the form of <E> is preserved, become identical with the
former, and consequently the same is true when e is taken indefinitely small.
Now if we call the (ni n) roots of <I> which do not belong to </>, 77,1+1,
77 n+2 f) m , and make
q m ri. /.
J, , tq 2
L" q i+l > "7t+2
we have
VTA . _yp//, h I \\ h q * "]
^_M- ^V-v)^,^...^"
where
P (A 7i , A, 2 . . . h qi ) = (a- - h qM ) (x - h qi j ...(x- h q j
But since ?7, l+ i, r) n+2 ... rj m are infinite in value,
[h fj h q ...h q ~\ . \H/ 1 ^ ^
= (- ?fi) (- *)n+J (- if*)}* -
L^n+i, 7lH--"?J \/
/l\ f
Hence ^ m _,-, = f - SP (h qi , h qt ...h q .)
jn t,o>
and X-t,o = e i ^, rt -f,o.
But by what has been shown antecedently, taking account of the fact of the
476 On a Theory of the Syzygetic, Relations [57
leading coefficient of <& being e in place of 1, which introduces the factor e 1 ,
we have
where E{ is the Bezoutian secondary of the (m i l)th degree in x to/ and
<f> ; but Bi has been proved = B { , the Bezoutian secondary of the same
i 1
degree to /and $ ; hence ^ m _; )0 = ( ) % 2 B t .
Art. 29. If now we return to the syzygetic equation, rf t(f> + ^r = 0,
^ may be treated as known, having in fact been completely determined
as a function of the roots, as well in its most general form, as also so as to
represent the simplified residues to / and <f> in the preceding articles ; it
remains to determine the values of T and t as functions of the roots corre
sponding to any allowable form of ^, but I shall confine the investigation
to the case where ^ is the lowest-weighted conjunctive or, which is the same
thing, a simplified residue to /and <f> of any given degree in x\ each value of
T rh
- will then represent one of the convergents to ~ when expanded under the
/
form of a continued fraction. If ^ be of the tth degree in x, T is of the
degree (n t 1) and t of the degree (m i 1). This being supposed, and
calling n i 1 = v, m t 1 = p, I say that t will be represented by G and
T by F, where
h qi , /,,... ^j
G = ( V ^(x h Mx h \ (x h 7 ?! > ^2 In J
= \ ) ** V* ft qj M* l qi> - V* ri q
and r is an analogous form F; A 1? h 2 ...h m , as heretofore, being the roots
of/ and ?7i, % ...7/ n of <. To fix the ideas and make the demonstration
more immediately seizable, give m and n specific values ; thus let m = 5,
n = 4>, 6 = 2, so that ^,= 5 2 1 = 2. Put S- under the form ^ l)0 , so that
^ in the case before us
94
Now make x = h lt then/= 0, and "^ becomes
7 , -i
9., "f, f,
9, ^9,
57]
that is
of two Algebraical Functions.
[!M[" M * 1
fAj h 2 h 3 j
IM
477
A A,
/?! being kept constant in the above sum, but A 2 , /i 3 , /?,, ^ 5 being partitionable
in all the six possible ways into two groups, as into A 4 , h s ; h 2 , h 3 in the term
above expressed. This sum is evidently identical with
A
-7 -1 I
_, , that is
";
b 4 A,
Again, < becomes
Hence t = - becomes
9
G
[TO
LAi*J
But, when x = h lt ^ becomes
[hji 3
that is
[h 2 h 3 TA 2 h a ~\
h, [Ma]
Thus when x = h li t= G. In like manner, when x = h 2 , or A 3 , or 7 4 , or /t s ,
i always = G ; but and G are both functions of x of the same, namely
of only two, dimensions in x. Hence t is identical with G. So in general
it may be proved, that whenever x = h l or h 2 or h 3 ... or k n , t and G, which
are each of only (m 1 t) dimensions in ac, are equal. Hence universally
t = G, as was to be shown. To find r we must avail ourselves of the sym-
morphic, or as we may better say (it being at the opposite extremity of the
scale of forms), the antimorphic, value of ^ represented by S 0> t . taking care
to preserve S- strictly identical under both forms of representation, in point
of sign as well as quantity. That is to say, we must make
478 On a Theory of the Syzygetic Relations [57
ffl
9f*i>%t+f" fai
LVftl tyfe **J
- ,
**H %%,
fc. 17ft ffcj
where w = i (m i) + m (n t),
SO that ( ) w = (_)i-i+m?i-u _ /_\
\mi-t
and consequently the same reasoning as was applied to t to prove t = G, will
serve to show that T = F, where
or
T;
where w = inn 1 mv = inn I m(n i I)
= mi + m 1.
Art. 30. I have not succeeded in throwing t and T under any other than
the single forms for each above given, and it is remarkable that whilst
apparently t and r admit only of this single representation, S- admits of the
variety of forms included under the general symbol ^ )t _ B for a given value
of i ; and it ought to be remarked that these forms, although the most
perfectly symmetrical and exactly balanced representations, and for that
reason possibly the most commodious for the ascertainment of the allotrious
factor belonging to them respectively, by no means exhaust the almost
infinite variety of modes by which the simplified residues, that is, the
hekistobarytic, or if we like so to call them, the prime conjunctives, admit of
being represented as functions of the roots of the given functions ; for if in
Art. 16, instead of writing
R =
:J
57] of tivo Algebraical Functions. 479
we had made
P(W--V> -.1
H =
where P represents any function symmetrical in respect of h qi , h qt ... h q>
and also in respect of 77. , 17. . .. 17. , (the interchanges, that is to say, between
one h and another h, or between one 77 and another 77, leaving P unaltered),
it might be shown that the value of ^r VjV resulting from the introduction
of this more general value of R would (as for the particular value assumed)
always be expressible as an integral function of the roots ; and consequently,
if P be taken of the same dimensions in the roots as the numerator of R
previously assumed, that is vv, *b v>v would continue to be (unless indeed it
vanish) identical (to some numerical factor pres) with the . corresponding
simplified residue. If, on the other hand, P be taken of less than w
dimensions in h and 77, we know d priori that <& VjV must vanish, as otherwise
we should have a conjunctive of a weight less than the minimum weight.
When P is of the proper amount of weight vv, it is I think probable that
another condition as to the distribution of the weight will be found to be
necessary in order that *b v>v may not vanish, namely, that the highest power
of any single h in P shall not exceed v, nor the highest power of any single
77 exceed v. But as I have not had leisure to enter upon the inquiry, the
verification or disproval of this supposed law, and more generally the evolu
tion of the allotrious numerical factor introduced into ^ v>v by assigning anv
particular form to P satisfying the necessary conditions of amount and
distribution of weight, must be reserved, amongst other points connected
with the theory of the remarkable forms (19) Art. 15, as a subject for future
investigation.
Art. 31. A property of continued fractions, which, if known, I have not
met with in any treatise on the subject (but which has been already cursorily
alluded to in these pages), gives rise to a remarkable property of reciprocity
connecting T and t severally with ^ in the syzygetic equation rf
Let the successive convergents to the ordinary continued fraction
JL JL J_ i 1
qi+q* + q* + "qi-i+qi
be called
_n_ j^ "ii H
m l ra 2 " mi_j m {
respectively ; it is well known that
JltM^ffif <_!(-}(- 1;
480 On a Theory of the Syzygetic Relations [57
but I believe that it has not been observed that this is only the extreme
case of a much more general equation, namely
where /t 1( yu 2 .../^ denote respectively the denominators to the convergents
to the continued fractions formed with the quotients taken in a reverse order,
that is, the continued fraction
_1 __ 1 __ 1_ j_ _1
"
This is easily proved when p = 1 ; /A O is of course (as usual) to be considered 1.
So more simply for the improper continued fraction,
of which the convergents are supposed to be
_^i_ >*_ hi
m 1 m 2 " mi-!
and the reverse fraction
of which the convergents are supposed to be
we have the more simple equation
liVHi^p li-ptHi + (JLp-i = 0.
And it is well known, or at all events easily demonstrable, that
k_i = J __ 1 __ 1 1
k qi - qi-i - ?i-a " q s
m-i 1 1 1 1 1
Art. 32. If now we use subscript indices to denote the degree in x of the
quantities to which they are affixed, we have the general syzygetic equation
KTn-^f m ~ Kt^-i $n + K* L = 0,
where K, a constant (which I have given the means of determining in the
first section), being rightly assumed, Kr^^, Kt m ^^ become the numerator
and denominator respectively of one of the convergents to j , expressed as
* See London and Edinburgh Philosophical Magazine, " On a Fundamental Theorem in the
Theory of Continued Fractions," Vol. vi., October, 1853. [See below.]
57] of two Algebraical Functions. 481
an improper continued fraction, and K^^ becomes the denominator to one of
the convergents to 7 l , or, which is the same thing, to -^ *. Conversely,
/ r
it is obvious that if we adopt as our primitive functions cf m and t m -i,
c being the value of K when i = 0, we shall obtain as the general form of
our syzygetic equation, bearing in mind that (in 1) now replaces n,
GJ\. T 7l _ t _iy 7/ j, J\. rJjH-.,! &,._! -\- Ji. t t = () J
and similarly, if we adopt as our primitive functions T n _j and c</> n , we obtain
for our general syzygetic equation, observing that (n 1 ) now replaces m,
^T X-.-iT,,-! - cK tm^fn + K r, = ;
so that (making abstraction of the constant factors and looking merely
to the forms of the several functions which enter into the equations) we see
that on the first hypothesis, namely of t m - l being substituted for <f> n , the con
junctives of each degree in a; change places with the second conjunctive
factors, that is the original multipliers of <f> of the same degree in x, and
vice versa ; and in the second hypothesis, where r n -i takes the place of f m ,
the conjunctives of each degree in x change places with the first conjunctive
factors, that is the original multipliers of f of the same degree in x, and
vice versa ; ^ OT _ a and T n _! being respectively multipliers of </> and /, such that
the difference of the respective products is independent of x. These results
ought to be capable of being verified by aid of our general formula; for t, T, &,
and as this verification will serve to exhibit in a clearer light the nature
of the reciprocity between the conjunctives and the conjunctive factors,
it may be not uninteresting to set it out.
Art. 33. As usual, let /^ , h 2 . . . h m be the roots of fx, and rj l , r) 2 ...r) n
the roots of <#; the last conjunctive factor to <, which is of the degree
(m 1) in x, will be represented, neglecting powers of ( ), by m _i, where
*m 1 2i \X tlqj \X h>q 2 ) . . . (SC >
An
If now we for greater simplicity make ,_! = t (x), and call the roots of t,
i , ^ 2 rfm-i any such quantity as
i > v 2 n m i
* Since i is always supposed less than n (n being the degree of the lower degreed of the two
functions /and <p), the fact of the last quotient to ^=- 1 being wanting to J will not affect the
i
accuracy of the statement in the text above, since this latter will contain as many quotients as
can in any case be required for expressing & t .
s- 31
482 On a Theory of the Syzygetic Relations [57
R denoting a constant independent of the root h qm selected, in fact the
resultant of the two functions fx and <J>a?, that is to say,
...4(*>
But by our general formula the simplified residue to fx and t(x) of the
tth degree in x will be represented by
therefore
or
the relation which was to be obtained. So conversely, in precisely the same
manner, calling t\ the conjunctive factor of the degree * in x to *(*)in
the syzygetic equation which connects fx and t(x) with a corresponding
simplified residue, we have
the conjugate equation to the one previously obtained*.
And evidently the same reasoning serves to establish the reciprocity,
or rather reciprocal convertibility, between the * series and the r series,
when in lieu of the original primitives fx and ** we take as our
primitives T (a) and &, r (x) being the function which satisfi
equation
* M Hermite, by a peculiar method, first discovered one of these ^ I *^*T"
reciprfdty, applicable to the case of Sturm s theorem, where **=/ *, and I am mdebted to him
for bringing the subject under my notice.
57] of two Algebraical Functions. 483
Art. 34. It may be remarked that if n = ra 1, the last syzygetic
equation being thus tm-^m-i. Tm-zfm ^o = 0, when t m ^ and f m are taken
as the primitives, the corresponding equation will be of the form
* m i t-mi T" mzjm ~r -J o == " >
these two equations must therefore be identical, and consequently t m-\ = </>m-i
(to a numerical factor pres), so that t m ^ and <f>m-i are reciprocal forms ; this
is also obvious from the consideration that ^V-i must, by the general law
of reciprocity (established above), be a residue to (f m , < m _i), which the
latter function itself may be considered to be. Or the same thing is obvious
directly, by writing
t_, -(()- 2 (,- ;o ( - h g ) ...(- / Wl > ^
\ n q m ~ n
and then making
f -2<X-h}(x-h} (x-h ) _ *(*.
c - 4 J ^* rl ---^ "-
-
i ( i i \ /; /, \
v^m ~~ n qj V l q m ~ n q m -i)
_ _
- , qi ... qn- ( , _7 v /, _, s,
\ n q m n qi> \ n q,n n 9wt-i)
or finally,
/ _ Pm-2 fL
^ ml " <p,
as was to be shown.
SECTION III.
On the application of the Theorems in the preceding Section to the expression
in terms of the roots of any primitive function of Sturm s auxiliary
functions, and the other functions which connect these with the primitive
function and its first differential derivative.
Art. 35. The formulae in the preceding Section had reference to the case
of two absolutely independent functions and their respective systems of roots :
when the functions become so related that the roots of the one system
become explicitly or implicitly functions of the roots of the other system,
the formulae will become expressible in terms of these latter alone, and in
some cases the terms (of which the sum is always essentially integral) will
become separately and individually representable under an integral form.
Such, as I shall proceed to show, is the case for two functions, of which one
312
484 On a Theory of the Syzygetic Relations [57
is the differential derivative of the other. When / and < are thus related,
so that </> = ^f- , calling as before h lt h 2 ... h m the roots of/ and ih, i? a ... i?-i
the roots of <, we shall have in general
+i i
_ i , * fc ... VI
Consequently
x
X *
-
1
ij
-J
i-J
Hence
-iJ
**
9 - , ,
l M *ft-lJ
the denoting the operation of taking the product of the squares of the
differences of the quantities which this symbol governs. Hence the Bezoutian
secondary to /and/ of the (m-i- l)th degree in ao, namely
becomes
(-) *- 2?(V A ft . . . ^ 9i ) ( - Vx) ( - *) ("* V)
fc ... A ft ) (x - h qi+l ) (x - h qi j ...(x- h g j,
57] of two Algebraical Functions. 485
since ( )* (i ~ l) 1 ; this gives the well-known formulae (enunciated* by me
in the London and Edinburgh Philosophical Magazine for 1839) for expressing
M. Sturm s auxiliary functions in terms of the roots of the primitive, and
which I therein stated were immediately deducible from the general formulae
(also enunciated in the same paper) applicable to any two functions. These
more general formulae appear to have completely escaped the notice of
M. Sturm and others, who have used the special formulae applicable to
the case of one function becoming the first differential derivative of the
other.
Art. 36. In precisely the same manner, if we form as usual the ordinary
syzygetic equation
tf x - rfx + ^ = 0,
we m^y find the different values of t given by the complementary formulae ;
and using ti to denote the multiplier of the degree i in x, that is appertaining
to the residue of the degree (m i 1) in x, we have
, ^--Ai "1
^7i > f)i ^lin
q\ &&gt; *** q\
n n It
- 9H-1 9i+2 <lm J
= S (h qi , hg^ . . . hg.) (x hg) (x hg^) ... (x- hq t ).
Art. 37. Thus, if we make i=m \,
hq-< ^9m-i) ( X ~ hq) ( X ~ ^ 2 ) (*-
It is evident from the form of fas that it possesses relative to fx, the same
property as fas, I mean the property that when x is indefinitely near to a real
root offu, and is passing from the inferior to the superior side of such root,
/ x . f x
"T- like*^- will pass from being negative to being positive, or in other
jx jx
words, fx and f x have always the same sign in the immediate vicinity
to a real root of fx. Hence it follows that fix might be used instead
of f x, to produce, by the Sturmian process of common measure, a series
of auxiliary functions, which with fx and fix would form a rhizoristic series,
that is a series for determining (as in the manner of M. Sturm s ordinary
auxiliaries) the number of real roots of fx comprised within given limits.
The rhizoristic series generated by this process will, it is easily seen, be (to a
constant factor pres) the denominators (reckoning + 1 as the denominator
f x
in the zero place) of the successive convergents to^r- thrown under the form
/*
[* p. 45 above.]
486 On a Theory of the Syzygetic Relations [57
of a continued fraction --- . . . --- ; M. Sturm s own rhizoristic
0i - & - ?n-i - q n
series, on the contrary, will be (to a constant factor pres) the denominators
f x
of the convergents to the inverse fraction ^4- , which will be of the form
jx
accordingly these two rhizoristic series will be
q n - q n -, - q,- q
equivalent as regards the number of changes and of combinations of sign
(afforded by each) corresponding to any given value of x, of which of course
the q s are linear functions. This result agrees with what has been demon
strated by me* by a more general method (in the London and Edinburgh
Philosophical Magazine, June and July 1858), where it has been proved,
by means of a very simple theorem of determinants, that the two series
1 1 !J__JL! _L _L 1 1
qi ?i - & #i - ?2 - q " qi-q*-q " q n
and
i JLJI^ _j __ i __ i_ _i __ i __ L 1
q n q n - <?-! q n - q n -i - q n -t " q n - q n -i - q n -z qi
always contain (for real values of q lt q a ,q... q n ) the same number of positive
and negative signs.
Art. 38. Having now determined the general values of ^ and t in
the equation tf x - r/a? + S- = as explicit integral functions of the roots
of fx, the more difficult task remains to assign to T its value similarly
expressed. This cannot readily be effected by means of substitutions in the
general formulae, the method we adopted for finding t and ^ ; but all the
other quantities except T in the syzygetic equation being integral functions
of the roots, it is evident that T also must be an integral function of the
tf x + ^
same, and to obtain it we may use the expression T = -- ^ .
jx
To obtain the general form of T by direct calculation from this formula
would however be found to be impracticable ; the mode I adopt therefore
to discover the general expression for T corresponding to different values
of ^, is to ascertain its value on the hypothesis of particular relations
existing between the roots of fx, and then from the particular values of T
thus obtained to infer demonstratively its general form, as will be seen
below. The demonstration of r is unavoidably somewhat long, T being in
fact represented by a double sum of partial symmetrical functions.
Using the subscript indices of each function as the syzygetic equation
to denote its degree in x, we have in general
tiniij & TmizjX -p i3{ U,
[* See below pp. 616 and 621.]
57] of two Algebraical Functions. 487
where if we make
h l -x = k l , h 2 -x = kt ... Ji m -x = k m ,
so that
hi h u -ki k^,
and therefore
S(he lt **.V""^V ** *V
we have in effect found
^i = 2& 9l fcg a . . . k q . (kq M , kg M . . . k q j
and
m_t_i = S&fe&fe &gm-i-i f(*i *fc %n-i-l)
we have also / (#)= (, ) m " 1 1k qi k qii . . . k qm _ r
Let us commence with the case where i = ; we have then
*o=?(fci, k,...k m \
t m _, = %k q jc qt . . . k qm _ l (k qi , k gi . . . kg^) ;
we have thus
(-) m T m _ 2 k, ^ 2 ... k m =(k l} k z ... k m )
^.k q jc qt . . . k gm _ t x Sk^kfr . . . k qm _ l %(k qi , k q2 ... k qm _j.
It may easily be verified that the negative sign interposed between the two
parts of the right-hand member of the equation has been correctly taken, for
f (&!, k a ...k m ) contains a term k^ m ^ /c 2 2 < w ~ 2) ... fc^-aAV-i,
^k q k q ^ ... k qm _ l contains a term k^ 2 ... fcm-a^m-u
and
^ fll * ft ... ^^ ?(^ *h- *W-,) contains a term kf" ****...*<++**+,
and thus the term k^ m ~^ k^ m ~^ ... k\ n ^k^ m ^, which does not contain
&!&,...&, will (as it ought to do) disappear from the right-hand side of
the equation.
Now suppose
/ lr
n/i ft/2,
then
b (A*i > ^2 "^rn/ =: ^>
and also
except when one or the other of the two disjunctive equations
<?i> <? 2 , q 3 9m-i = 1 3, 4...m,
?i, ?, ? 3 <Z-i = 2, 3, 4 ... w,
is satisfied (by a disjunctive equation, meaning an equation which affirms
the equality of one set of quantities with another set the same in number,
each with each, but in some unassigned order).
488 On a Theory of the Syzygetic Relations [57
Hence
= Ki/e 3 ... K m ^\rCi, K 3 ... K m ).
Hence when ki = k, ( ) m r, n _ 2 becomes
2
r- 2%% . . . A^J (&! , A; 3 . . . & m ),
*i
that is
*(>C"^i MS km) l*i i^r,*^ "Vm_i ~t~ *^|A?4 ... ,},
the S referring to r s , r 4 ... ?* m supposed to be disjunctively equal to 3, 4 ... m.
Now r m _2 is of (m 2) dimensions in a?, and whenever more than one
equality exists between the k s, ^ and t m _ l both vanish (in fact every term
^ + t _ f x
in each vanishes separately), and therefore T m _ 2 , which = -t-v r >
fl?! A? 2 ... fl? m
will vanish.
Hence ( ) m r m _ 2 must be always of the form
^ denoting some integral function of (m 2) dimensions in respect of the
system of quantities k qi , k q3 ... k qm . The result above obtained enables us
to assign the value of
^F (KI, K S ... tc m , K 2 ),
when ki=k 2 , namely
KI Zi \K rs , K rt . . . K rm _ l ) + 2K 3 K4 . . . K m .
Now for a moment suppose, selecting (m 1) terms k l} k 3 , k t ... k m out
of the m terms of the k series, that
O (If I- Jf Jf k\ k m ~ 2 Jf m -
-L ^/l/i , /tg, A/4 ... A/ m , ti%J ft/2 " 2
+ ... + n^s^ m I ("^1, MS " W/ i ^^77^2 (" i i " S " m/t
where Si means that the quantities which it governs are to be simply added
together, 8 2 denotes that their binary, S 3 that their ternary, and in general
S r that their r-ary products are to be added together.
When ki = k 2 , H becomes
- ki m ~ 5 [kiS z (k,,k t ... k m ) + S,(k s ,k 4 ... k m )} ...
ki {kiS m -4 (k s , k 4 . . . k m ) + m _ 3 (k s ,k 4 ... k m )} 2S m _ 2 (k 3t k t ... k rn ),
which evidently equals
[2>Sf m _ 2 (k t , k 4 ... k m ) + k&n-s (k,,k t ... k m )},
that is {ki^ (k ra , k r4 . . . A;^.,) + 2k 3 k t . . . k m }.
57] of two Algebraical Functions. 489
Hence when k 1 = k 2 , M* = H, and
and so in like manner, when k^ is equal to any one of the (m 1) quantities
ft* 2 > & 3 ... &WJ, the form of T TO _ 2 above written will have been correctly assumed.
But T m _ 2 maybe treated as a function of (m 2) dimensions in &j, and
consequently any form of (m 2) dimensions in k lt which fits it for (m 1)
different values of ki, must be its general form, and accordingly we have
universally,
... / x a? - A-
&c.
Art. 39. With a view to better paving our way to the general form of
T for all values of i, let us pass over the case of i = 1 and go at once to the
equation
tmsf x T m if & + ^"2 = ^ 5
and to better fix our ideas let m = 7, so that the equation becomes
we have then, preserving the same relation as before, that is, using h to
denote any root of fx, and k to denote h x, the equation
+ KiK2K 3 K 4 k 5 K G K 7 T 3 = *^fi"fc{\*f l " f|"*f ( " ft ff
^>Kq l kq t kq t Kq 4 Kq l Xq t X 2 l&ft&fc&fc&fc (% "^^"WJ
now r 3 will vanish whenever more than three relations of equality exist
between the k s, for then each term in both of the two sums in the right-hand
member of the equation above written will separately vanish ; and of course
three relations of equality between the same are sufficient to make all the
terms in the first of these sums vanish. This relationship between the
different k s corresponding to a multiplicity 3 may arise in different ways ;
the multiplicity 3 may be divided into 3 units corresponding to 3 pairs of
equal roots, or into 2 and 1 corresponding one set of 3 equal roots, and a
second set of 2 equal roots, or may be taken en bloc, which corresponds to
the case of one set of 4 equal roots. I shall make the first of these supposi
tions, which will sufficiently well answer our purpose in the case before us.
Thus I shall suppose
T~ _ / l- If Jf k
li l rt 4, ft 2 ftsj fts 1-6 }
then, as above remarked,
490 On a Theory of the Syzygetic Relations [57
for all values of q 3 , q t , q 5) q 6 , q 7 , and therefore
q^) = U ;
also S&a^ft&ft&ft&ff&ft becomes
1 2 3 1123 "* ^"- 7 \*^1 *^2 ~T~ " l ^S ~T" *^2 3/ I J
and ^(kqkqjcqjcq) vanishes, except for the cases where q 1} q 2 , q 3 , <? 4 represent
respectively, q l the index 1 or 4, </ 2 the index 2 or 5, q 3 the index 3 or 6, and
q 4 the index 7.
Hence 2&gAA 3 ^
and consequently T S becomes
+ 8 (k^k-jk?) x (A^&g + 2A- 7 (^^ + k^ + k^)}.
Hence we are able to predict that the general expression for our r in the
case before us will be
X {( kq? + V + V) - (V + V + V) (^ + *<h + ^3 + * T )
+ (^ 4 + & 3j + A; g6 ) (A gi ^ s + k qi k qt + k q k qi + k q2 k q3 + k q k qi + k q jc q ^
4 \k qi kg 2 /Cq 3 + kq l kq li Kg 7 + &f t &ft"9, + "^*fc"f I )|-
For in the first place, the fact that the T vanishes when more than three
relations of equality exist between the k s, proves that we may assume T 3
of the form
* \fCq t Kq t kq t Kq 7 ) X <f> [kqfiqJCq^K^] KqJCq t Kq t } t
the semicolon (;) separating the k s into two groups, in respect of each of
which severally < is a symmetrical form. But if in the expression last
above written for T Z we make y
" I = "^4 J "^2 == "*5 ) "*3 == ^6 )
it becomes
.ksk,} x {(A?! 3 + ^ 2 3 + ^ 3 3 ) - (^ 2 + k.? + k*} (k, + k* + k 3 + k 7 )
+ h + h> + k 3 (k^ + kik s + k 2 k 3 + k t k r + k z k 7 + k 3 k 7 )
Now in general if
o- r = aS +aj + a 3 r + ... + a t r ,
and S r = (a-i^as . . . a r ),
then cr r OY^^ + o- r _ 2 -S 2 + ... + rS r = 0.
Consequently the sum of the terms constituting the second factor in the
above expression
= (3 - 4) k,k z k 3 + (2 - 4) k 7 (kjc* + k,k 3 + k 2 k 3 ).
57] of two Algebraical Functions. 491
Hence the above expression becomes
8(&i& a fcs&7) {kik 2 k 3 + 2 (k^ 2 + &!&,+ k 2 k s ) k 7 }.
Thus, then, whenever k 1} k 2 , k 3 are respectively equal to any three of the
quantities k 4 ,k s ,k 6 ,k 7 , which may take place in twenty-four different ways
(twenty-four being the number of permutations of four things), our r 3 will
have been correctly assumed ; but %(kqjcqjc q jcq^ being replaceable by
^(hqjtqjiqjiq^, the T 3 may be treated as a cubic function in k lt Ar 2 , k 3 , and
arranged according to the powers of A^,^,^ will contain only twenty terms ;
hence, since the assumed form is verified for more than twenty, that is, for
twenty-four values of h lt h 2 , h s , it follows that the assumed form is universally
identical with the form of r, which was to be determined.
Art. 40. Now, again, in order to facilitate the conception of the general
proof, let us suppose fx to be of only five dimensions in as, i still remaining 3:
it will no longer be possible when we suppose a multiplicity three to prevail
among the roots, to conceive this multiplicity to be distributed into three
parts, for that would require the existence of three pairs of roots, there
being only five. But we may, if we please, make A 1 = A 8 =A, I and ^ 4 = ^5
or else h, = h z = h s = h 4 , or in any other mode conceive the multiplicity to be
divided into two parts, 2 and 1 respectively, or to be taken collectively
en bloc. As a mode of proceeding the more remote from that last employed,
I shall choose the latter supposition. Then we obtain (T now becoming
r s _ 2 _2, that is TJ)
K l h. 2 K 3 /C 4 K 5 T l = + ^Mq l Kq t Kq t Kq t X ^Kq l K
and %(k q k q ^) will vanish, except in the case where q l represents the indices
1 or 2 or 3 or 4, and q 2 the index 5 ; also
Hence our equation becomes
and r becomes 4(/i 1 & 5 ) (^ + 4& 5 ).
If, now, we assume for the general value of T in the case before us
T = ^(k q kq 5 ) {(kq 2 + k q , + k^
when ki = k 2 = k 3 = k 4 , r becomes
that is 4 (k\ k s ) (h + 4>k 5 ).
Hence then for the two systems of values of h lt h. 2 , h 3> namely
h. 2 = h 4 or j h. 2 = h 5
\h 3 = h 5 ,
492 On a Theory of the Syzygetic Relations [57
the form of T will have been correctly assumed. But since the derived form
is a linear function of h lt h 2 , h 3 , this is not enough to identify the assumed
with the general form, since for such verification four systems of values must
be taken, four being the number of terms in a function of three variables
of the first degree. If, however, we had adopted a separation of the multi
plicity three into two parts, and had started with supposing ^=^2 = ^3,
& 4 = k s , we should have found that T would have become
Moreover, when these equalities subsist,
becomes 2 1 3 & 5 + 3/V& 5 2 , and the common factor /V& 4 disappears in the course
of the operations for finding T, and eventually we have to show (in order to
support the universality of the previously assumed form for T) that
becomes 2k gt 3k 5 when
1, ._ 7. ._ k _ /-.
faq^ q 3 tiq i /tj ,
and k qt = kq s = k 5 ,
which is evidently true. Hence then T will have been correctly assumed for
the following cases,
7, _ 7, _ 7, _ 7,
A<1 lB| "5 t-3
7, _ 7. _ 7, _ 7.
ft/j "~ A/2 - **^5 ~~~ " / 4 ;
and also for the cases
ki = k z = k 3 and k s = k^}
A/1 = ^ A/5 ^ = A7g tllH I rC-2 ^ A*4 f
k 2 = k s = k 3 and k^ = k^ j
k^ = k 2 = k 4 arid k 6 = k s \
&! = k s = k and k z = k s > ,
k 2 = k s = k and k l = k z )
that is, for eight cases in all, whereas four only would have sufficed. Hence,
ex abundantid demonstrationis, the form assumed for TJ is in the case before
us the general form.
Art. 41. We may now easily write down the general form which T
assumes for all values of i and prove its correctness. If the roots be
h lt h a ,h-,... h m ,
and t m -i-if x - r m -i-ifx + ^i = 0,
57] of two Algebraical Functions. 493
we shall have
+ T m _ _ 2 = 2 { (fcgAA. ^m--i) X ["m-i- 2 ~ m-i-*Si + ^m-i-A + &C.
+ (_)m-<-3 o-^^ + (_)m-i- 2 (o - fl + !) S m _ { _ 2 ]},
where cr r denotes in general the sum of the rth powers of the (i+l) quantities
- ^mJ> O ~ ^m-i-n) (- V,)
and $ r denotes in general the sum of the products of the complementary
(m i l) quantities
(x - h q ), (x - h qt ) ...(*- /^ m _ i _ 1 )
combined r and r together. It will of course also be understood that
o- = t + l, so that o- +l=t + 2.
Art. 42. To prove the correctness of this general determination of the
form of T m _;_ 2 , let us suppose in general that i + l relations of equality
spring up between the m quantities k 1} k. 2 ...k m ; we shall then easily
obtain (N representing a certain numerical multiplier)
,... m _
m-i-l
k lt kz^.k^i-! being what the k system becomes when repetitions are
excluded, and being respectively supposed to occur //, 1; yu, 2 ... yu, OT _i_! times
respectively, so that
the fractional part of the right-hand member of the equation immediately
above written will be readily seen to be equivalent to
^Mflm-i-i *i*<h ^Om-i-2-
To establish the correctness of the assumed form, we must be able, as in
the particular cases previously selected, to prove two things ; the one, and
the more difficult thing to be proved is, that when the series of distinct
quantities k 1} k 2 , k 3 ...k m become converted into /^ groups of ^ ; /* 2 groups of
& 2 , ... ya m _i_! groups of & m _i_i, then that
2/A0J % 2 &e 3 &e 4 ^e m _i-i
or in other terms
i
2 ***** *Wi 2 M,
m i-l
becomes identical with
(T,,^ - 0- m _ t -_ 3 8 l &C. + (- 1)-*-* (0-0 + 1) S m _i_ 2 .
The other step to be made, and with which I shall commence, consists
in showing that the number of terms in the expression last above written,
considered as a function of (m-i- 2)th degree of (i + l) variables, is never
greater than the entire number of ways in which (i+l) quantities out of m
quantities may be equated to the remaining (m-i 1) quantities, namely
each of the first set respectively to all the same, or all different, or some the
494 On a Theory of the Syzygetic Relations [57
same and some different; in short, in any manner each of the i+ 1 quantities
with some one or another (without restriction against repetitions) of the
m i1 remaining quantities. This latter number being in fact the
number of ways in which (m il) quantities may be combined (i+I)
together with repetitions admissible, by a well-known arithmetical theorem,
A* _i_ i\ /Y i 2) (tn 2)
is (m i iy +1 , and the first number is - ^ , " : - which is
1.2... (m i 2)
always less than the other. It remains then only to prove the remaining
step of the demonstration*.
Art. 43. To fix the ideas let m= 10, i= 5, and consider the expression
"T" \" 5 i A^g i A/ 7 i A/g i *^9 *^10/ \^1 **^2 i ** 1 * 3 i A7j A/4 "T" AJgA/g ~T~ A^A^ ~T~ A/gA^)
Now suppose the six quantities A- 5 , &&lt;;, Ay, A 8 , & 9 , & 10 to become respectively
equal each to some one or another of the four quantities k 1} k 2 , k s , k 4 , as for
instance, I shall suppose
i. .- i. /. _ i.
n-5 n/g n/ 7 /tj
A^g ^ fC 9 - = A?2
" W = 3
Then ^ = 4, /^ = 3, ^ =2, /x 4 = 1,
and the formula of Art. 41 becomes
/O7/ 3 _i_ O Z" 3 i 7/ 3\ _ / QA-> 2 i v/" 2 i /* 2\ / ju i Ai i / i A \
l OA/I *i ^AO ~T" A/ 3 / ^^ V "- " 1 "T" ^ * / 2 *^ *v 3 / V*^! i^ **^2 ^^ 3 i^ 4/
~T" \ OAVi "^~ /i . "T" *^3/ VA jA/g l" A/j A/g ~| A/j A>4 |~ fvi^iVQ ~\ A/2 AV^ ~f" A 3 A 4 J
^ = O I j /Ci ~"~ A/J \** / 2 i" 3 "i *^4/ i" 11 i" *^l I \ ** 2 3 "i 2 4 *~ 3 4/ * 4 \ 2 i" 3 4/ j I
~T~ j A/o ^~ *^2 \**/j "T" A/3 ~}~ A/4y "T" A/oi ~t~ A/t> j \ A/i A/a "T" A/j A/4 "T" A/3A/4) "y" ( A/2/t j j" IVQ) ~j * - / 4(
"i JA^ A3 \A/j "i A/2 "i *^4/ ~r A/3[ "j~ A/3 j\A/jA/2 ~r A/jA/4 i* A/2 4/ " 3 v" ! "i 2 I " / 4/J
J5- A- A- t I^ 1 -i- ^- 2 ^ ^- 3 4- 4 l
A/i/> 2 /i/3ft/4 -^j -p v TT IT f *
I A/1 A/ 2 A 3 A 4 )
* If this first step of the demonstration appear unsatisfactory or subject to doubt, it may be
dispensed with, and the result obtained in the succeeding article (the demonstration of which is
wholly unexceptionable) being assumed, it may be proved that the formula there obtained on a
particular hypothesis must be universally true, in precisely the same way and by aid of the same
Lemma in and by aid of which the formula obtained in the Supplement to this section for the
f x
simplified quotients to upon a like particular hypothesis is shown to be of universal application,
that is, by showing that otherwise a function of 2i - 1 variables would contain a function of 2i
variables as a factor.
57] of two Algebraical Functions. 495
In the above investigation the quantities which with their repetitions
make up the k s system, are k 4 , k lt k 2 , k 3 , appearing respectively 1, 2, 3, 4
times, that is to say repeated 0, 1, 2, 3 times; 7 is one more than the sum
of the repetitions + 1+2 + 3, and the numbers 1, 2, 3, 4 arise from sub
tracting from 7 the sums 1 + 2 + 3; O-t-2 + 3; + 1 + 3; + 1 + 2; respec
tively, so that the remainders 1, 2, 3, 4 denote respectively one more than
the number of repetitions of k, k 1} k 2 , k 3 , that is, are the number of appear
ances of & 4 , k lt k 2 , k 3 ; and thus with a slight degree of attention to the
preceding process the reader may easily satisfy himself that the preceding
demonstration (although not so expressed) is in essence universal, and the
form of r as an explicit function of x and of the roots of fx is thus com
pletely established for all values of m and of i.
Supplement to SECTION III.
On the Quotients resulting from the process of continuous division ordinarily
applied to two Algebraical Functions in order to determine their greatest
Common Measure.
Art. (a)*. We have now succeeded in exhibiting the forms of the
f x
numerators and denominators of ->- developed into a continued fraction in
/*
terms of the differences of the roots and factors of jfc. It remains to exhibit
the quotients themselves of this continued fraction under a similar form.
LEMMA. An equation being supposed of an arbitrary degree n, there
exists no function of n and of less than 2i of the coefficients^, which vanishes
for all values of n whenever the n roots reduce in any manner to i distinct
groups of equal roots ; or in other words, any function of n and the first 2i 1
coefficients of an equation of the nth degree, which vanishes for all values of n
in every case where the roots retain only i distinct names, must be identically
zero.
To render the statement of the proof more simple, let i be taken equal
to 3. And let the roots be supposed to reduce to p roots a, q roots b, and
* The articles in this and subsequent sections to which Latin, Greek and Hebrew letters are
prefixed, although in strict connexion with the context, are supplementary in the sense of
having been supplied since the date when the paper was presented for reading to the Koyal
Society. All the articles marked with numbers (from 1 to 72), and the Introduction, appeared
in the memoir as originally presented to the Society, June 16, 1853.
t In the proposition thus enunciated the coefficient of the highest power of x is supposed to
be a numerical quantity.
496 On a Theory of the Syzygetic Relations
[57
r roots c. And let s r in general denote the sum of the rth powers of the
roots. Then we have evidently
p +q +r =s ,
pa +qb +rc = s ly
pa? + qb* + re 2 = S 2 ,
pa 3 + qb s + re 3 = s 3 ,
pa 4 + qb 4 + re 4 = s 4 ,
&c. &c., ad infinitum.
Eliminating p, q, r between the first, second, third and fourth equations,
we obtain
1, 1, 1, s
a, b, c, j
a 2 , b 2 , c 2 , s 2
a 3 , b 3 , c 3 , s 3
In like manner eliminating ap, bq, cr between the second, third, fourth and
fifth equations, we have
1, 1, 1, 8 l
a, b, c, s 2
= 0.
a 2 ,
2 , c 2 ,
b s , c 3 ,
= 0;
and so in general we have for all values of e,
1, 1, 1, s e
a, b, c, s e+1
n 2 7) 2 r 2 <?
U/ , U , (, , "g-)-2
= 0;
a , o , c , s e
whence it may immediately be deduced, that, upon the given supposition of
there being only three groups of distinct roots, we must have the following
infinite system of coexisting equations satisfied, namely,
s t + s-iU + s 2 v + s 3 w = say L = 0,
S}t + s 2 u + s 3 v + s 4 w = ^ = 0,
s 2 t + s 3 u + s 4 v + s 5 w = L 2 = 0,
s 3 t + s 4 u + s 5 v + s 6 w = ^ 3 =0,
s 4 t + s 5 u + s 6 v + s 7 w = 1/4 = 0,
&c. &c. &c. &c. ;
57] of two Algebraical Functions. 497
and conversely, when this infinite system of equations is satisfied the roots
must reduce themselves to three groups of equal roots.
Let now <f> be any function of s , s lf s 2 ... which vanishes when this is
the case. Then </> must necessarily contain as a factor some derivee of the
infinite system of equations above written, that is, some function of s , s lt s 2 ...
which vanishes when these equations are satisfied, that is, some conjunctive
of the quantities L , L 1} L 2 , L 3 ...; but it is obviously impossible in any such
conjunctive to exclude s e from appearing, unless by introducing some other s
with an index higher than G, and consequently <f> cannot be merely a function
of s > *n S 2> s s, s *y s &&gt; nor consequently of n and the first five coefficients ; or if
such, it is identically zero. And so in general any function of n and only 2i 1
of the coefficients which vanishes when the roots reduce to i groups of equal
roots, must be identically zero ; as was to be proved.
Art. (6). It ought to be observed that the preceding reasoning depends
essentially upon the circumstance of n being left arbitrary. If n were given
the proposition would no longer be true. In fact, on that supposition, the
n roots reducing to i distinct roots would imply the existence of n i
conditions between the n roots; and consequently ni independent equations
would subsist between the n coefficients, and functions could be formed of i
only of the coefficients, which would satisfy the prescribed condition of
vanishing when the roots resolved themselves into i groups of distinct
identities.
Art. (c). Let .A-,,r a ...r, be used in general to denote the determinant
then the simplified iih Sturmian residue jR^ may be expressed under the form
7") . n.nil T) rgOi i 2 i T) ^ni3 4. 7)
L/ l,2, 3... I " - L - 2,3...l+l ^ T -L/3, 4 ... 1+2 ^ I - L n-l,n I 1 ...n,
which is easily identifiable with the known expression for such residue.
Now obviously the necessary and sufficient condition in order that the n
roots may consist of only repetitions of i distinct roots is, that Ri shall be
identically zero, that is to say, we must have
*%...< = "> -^2,3.. .1+1 = " J^ni,nii...n ~ "
But the reasoning of the preceding article shows that although these equa
tions are necessary and sufficient, they are but a selected system of equations
of an infinite number of similar equations which subsist*, and that, in fact,
* But quaere whether any other sufficient system can be found of equations so few in number
as this system.
s. 32
498 On a Theory of the Syzygetic Relations [57
whatever be the value of n, we may take r 1} r 2 ... r; perfectly arbitrary and
as great as we please, and the equation
must exist by virtue of the existence of the ni equations last above written.
Art. (d). I now return to the question of expressing the successive
f x
quotients of J -y- as functions of the differences of the roots and factors ; that
jx
they must be capable of being so expressed is an obvious consequence of the
fact that the numerators and denominators of the convergents have been
put under that form, since, if
AT. AT. AT-
-"t 2 -"11 *<
are any three consecutive convergents of the continued fraction
_JL 1_ _!_
we must have
It would not, however, be easy to perform the multiplications indicated in
the above equation, so as to obtain Qi under its reduced form as a linear
function of x. I proceed therefore to rind Qi constructively in the following
manner.
Let Ri-z, Ri-i, Ri be three consecutive residues, f x counting as the
R-- R- p p
residue in the zero place, then Qi = ^ l , and is of the form - x + , .
Now in general if we denote the n roots of fx, where the coefficient
of x n is supposed unity, by AJ, /& 2 ... A n , and if we use Zi to denote
9, h 0l ...h e l)* } with the convention that Z 1 = n, Z = l, we have, employ-
ing (i) to denote -{(-
i_3--- (i)
5 Z
* f it will be remembered is the symbol of the operation of taking the product of the squares
of the differences of the quantities which it governs.
57] of two Algebraical Functions. 499
The part of R^ within the sign of summation is
Z ia r* - 2 (A em + h^ +...+ h e>t ) C (h di , h 9t . . . h, t ) *-*- + &c.,
say ZiX 11 1 - Zjap-*- 1 + &c.,
and the part of R t _ 2 within the sign of summation is
Zi_ lX n -^ - Z i^X^ + &C.,
and
Z^~7-~^ ~ 7 Z ^^ = Zi-JiX + (Zi_,Z{ - ZiZ ,^) + an algebraic fraction.
/iJ^OCr ~~" /* , JU
H o . i z^ 3 z\_ 5 ...z\. (Z\_ z z*i_t...z\ ]
+
Ti denoting Z-^Z.x + (Z^Z{ - ^Z ^).
Art. (e). If the process of obtaining the successive quotients and
residues be considered, it will easily be seen that each step in the process
imports two new coefficients into the quotients, the first quotient containing
no literal quotient in the part multiplying x and containing the first literal
coefficient in the other part, the second quotient containing two literal
coefficients in the one part and three in the other, and in general the iih
quotient containing 2i - 2 of the letters in the one part and 2i 1 of them
in the other. Hence Ti being made equal to L^x + M i} Li contains 2i 2
and Mi contains 2i 1 of the literal coefficients of fx.
Moreover, we have Zi of the form
** p
L i i
where P { _, = ^(h 6i , h e ,... h
and P i} which is the ith simplified residue, vanishes when the n roots in any
manner become reduced to only i distinct groups.
I proceed to show that if we make
AiX + Bi = Ui = A\, (x - hj + A\ a (x -h 2 ) + ...+ A\ n (x - h n ),
where in general
A {> e represents 2$(h ei , h^ . . . A w ) (h e - h 6i ) (h e - /^) . . . (h e - h^),
then will
T- U-
LI u t .
322
500 On a Theory of the Syzygetic Relations [57
It will be observed that A { _ e is identical with what the simplified
denominator of the (i l)th convergent becomes when we write h e in place
of x, and consequently, when arranged according to the powers of h e , will be
of the form
cA i - 1 + cA < ~ 2 + ... + d,
where c 1; c 2 ... Ci are functions of the coefficients, but containing no more of
them than enter into QJ_I, that is, containing only 2i 2 of them.
Now Ai is made up of terms, each consisting of some binary product of
Cj , C 2 ... Ci,
combined with some term of the series
and any one of this latter set of terms expressed as a function of the coeffi
cients of fx contains at most 2i 2 of them.
Hence only 2t- 2 of the coefficients enter into A i} and in like manner
only 2i - 1 of them into B{.
The number of letters, therefore, in A { and in B t is the same as in Li
and in M i} namely 2i - 2 and 2i 1 respectively.
Now let the roots consist of only i distinct groups of equal roots, so that
Ti becomes =^ 2 ^.
-i i i
I shall show that in whatever way the equal roots are supposed to be
grouped upon this supposition, there will result the equation
p.
where T { =
and Hi = A^^ + A*^ + ... + A n ^ n ,
A e meaning 2 {(r) e - ^e,) (n e - Ve.) (Ve~ ^-i) ?(^, Ve t ?)}
and ?;, meaning x h u .
Let the n factors be constituted of m-^ factors %, w 2 factors
factors iji. Then
where
57]
and
of two Algebraical Functions.
501
&c. &c.
,
where
Hence
u,
Again, in Ui the term containing ^ will be
{2 0?! - 772) Oh - 773) . . . Oh - ??;) 0/2, ^ *7t)
x (m 2 m s . . . m*) 2 x Oh - ?7 2 ) 2 Oh - ^ 3 ) 2 0?i ~
it
^
7/t j
Hence
Hence, therefore, Ui Ti vanishes whenever the roots of fx contain only i
distinct groups of equal roots, and it has been shown that Ui and Ti each
contain only 2i 1 of the coefficients of fx, so that Ui Ti is a function
only of n and these 2i 1 letters, and consequently, by virtue of the Lemma
in Art. (a), Ui Ti is universally zero, that is, Ui is identical with Ti, as was
to be proved. In the same manner, as observed in a preceding note
[p. 494], the expression given in the antecedent articles for the numerator
of the tth convergents, having been verified for the case of the roots consist
ing of only i distinct groups, could have been at once inferred to be generally
true by aid of the Lemma above quoted.
Art. (/). Since the coefficient of x in Ti is Z i _ l x Z i} we deduce the
unexpected relation
2C(Ai, ti ... *i_i) x SC^, h, ... hi) = P^ + P 2 2 + ... + Pn 2 ,
where P e = 2 {(h e - h 6i ) (h e - hj . . . (A. - A,_i) Z(h 6l , h e . 2 . . . Vi)l-
fx
So that every simplified Sturmian quotient to J -j- , when the n roots of /x
J x
are real, will be the sum of n squares. But the equation is otherwise
remarkable, in exhibiting the product of the sum of - ^ - /._ , ,
squares by another sum of 1 ^-~ * " \ n . - squares under the form of
1 . 2i . . . 1
the sum of n squares.
502
On a Theory of the Syzygetic Relations
[57
If we denote the iih simplified denominator to the Sturmian convergents
to -r- by DiX, and if we call the ith simplified quotient X{X, we have
J x
If we construct the numerators and denominators of the convergents to
according to the general rule for continued fractions, as functions of Q 1} Q 2 , Q 3>
&c., so that calling the denominators AJ, A 2 , A 3 ... Af,
, = Q, A 2 = Qt Q, - 1 . . . A, = Q t A^ - A,_ 2 ,
we have
Z 2 i- 1 Z 2 i_ 3 ... Z 2 (
A^a; being in fact the multiplier of f x in the equation which connects fa
and/"# with the (i l)th complete residue, and consequently, retaining Q(x)
to designate the complete iih quotient, we have
72 74 74 74
. t-1 i 3^ is & (t) f i
74
fJ ,
. ^8.
i-1 ^ i-3
which equation gives the connexion between the form of any quotient and
that of the immediately preceding convergent denominator of the continued
f x
fraction which expresses J -s- .
fa
Art. (g). I have found that the coefficients of the n factors of fx in the
expression above given for the quotients possess the property that the sum
of their square roots taken with the proper signs is zero for each quotient
except the first (the coefficients for the first being all units), that is
Di^ + Dihz + ... D { h n = for all values of i except i=i. Moreover I find
that the determinant formed by the n sets of the n coefficients of the factors
of fx in the complete set of n quotients is identically zero, that is, the
determinant represented by the square matrix
= o.
1,
1,
l,
1,
(A^) 2 ,
(A/O 2 ,
(A^s) 2
.-(A^n) 2
(DAT,
(A^ 2 ) 2 ,
(A^s) 2
..(AM 2
(B^-M
, (A-iW
(D n _^ 3 ) 2 .
..(D^/^) 2
57] of two Algebraical Functions. 503
Art. (h). It should be observed that Ui is the form of the simplified
quotients for all the quotients except the ?ith (that is, the last), for which
the simplified form is not U n> but U n -r- %(h lt h 2 ... h n ), which arises from the
circumstance of the last divisor, which is the final Sturmian residue, not
containing x; it being evidently the case that the division of a rational
function of x by another one degree lower, introduces into the integral part
of the quotient the square of the leading coefficient of the divisor, subject
to the exception that when the divisor is of the degree zero, the simple power
enters in lieu of the square. The general formula gives for the reduced nth
quotient the expression
which equals
Z(h lt h 2 ... h n ) 2(*2, h 3 ... h n } (x - h,).
Rejecting the first factor, we have
which is equal to the penultimate residue, which residue is (as it evidently
ought to be) identical with the simplified last quotient.
Art. (i). We have thus succeeded in giving a perfect representation
of -j.- , that is, of
A
1 1 1
I r~ ~r . . . H
7 r~ ~r . . . 7 >
x Aj x ri z x n n
under the form of a continued fraction of the form
11 1
m,! (x e-i) m 2 (xe. z )" " m n (x e n )
where m 1} m 2 ...m n ; e 1} e 2 ...e n are all determinate and known functions
of h 1} A 2 ... h n .
We may by means of this identity, differentiating any number of times
with respect to x both sides of the equation, obtain analogous expressions for
the series
1 1 1
(*-*,) (*-*t>* + (x-h n ) t
But to do this we must be in possession of a rule for the differentiation of
continued fractions whose quotients are linear functions of the variable.
I subjoin here the first step only toward such investigation.
Let the denominator of
504
On a Theory of the Syzygetic Relations
[57
where q 1} q 2 ...q n are any n arbitrary quantities, be denoted by [q 1} q. 2 , q s ...q n ],
so that the entire fraction will be equal to
[ft. ft...gn]
Any such quantity as [q { , q i+1 ...q n ] may be termed a Cumulant, of which
qi, qi +1 ,..q n may be severally termed the elements or Components, and the
complete arrangement of the elements may be termed the Type. The
cumulant corresponding to any type remains unaffected by the order of the
elements in the type being reversed, as is evident from any cumulant
being in fact representable under the form of a symmetrical determinant,
thus, for example, the cumulant [q lf q 2 , q 3 , <? 4 ] may be represented by the
determinant
ft, 1, 0,
1, ft, 1,
0, 1, ft, 1
0, 0, 1, q t
and [5-4, q 3 , q 2 , q^ will in like manner be represented by the determinant
ft, 1, 0,
1, ft, 1,
0, 1, q 2 , I
0, 0, 1, q,
which is equal to the former.
Art. (j). Let it be proposed in general to find the first differential
coefficient in respect to x of the fraction
[ft, ft, ft ...ftj
where each q is a function of one or more variables.
I find that the variation of F { may be expressed as follows :
-SF i = {S[q 1 ,q 2 ... q { _ 2 , q n ] + 8 [ft, ft . . . ^-_ 2 , gy_,] q n *
+ 8 [ft, ft, ft qi-*, fti-*] [fti, q n -i] 2 + ...
+ 8 [ft, ft, ft ... ft-_ s , ft-i] [ft, q n -i, q n -v ft]
Kft* ft- ft Q n] 2 .
57] of two Algebraical Functions. 505
Art. (&). Suppose i=2, and q l =a 1 x + b 1 , q 2 = a 2 x + b 2 ... q n = a n x + b n ,
we shall have by virtue of the above equation,
d v ^ . . d ( I 11 1
-y- .r o, that is -r- { ...
ax dx (^ q 2 q 3 q n
= j^- -,2 t a 12 + a n-i<ln + n-2 [?n, ?n-i] 2 + &C.
>i2?
If we call F 2 = -~- every such quantity as [q n , ^n-i <?&] represents to a
jx
constant factor pres the (i l)th simplified residue (fas counting as the first
fDT*
of them) to - f - , and making certain obvious but somewhat tedious reductions,
jx
and rejecting the common factor - , , we obtain the expression
\J X )
G7? 2 7? 2 7? 2 7? 2
j JVj ^1/2 -*-3 -^^71 i / / /*
X <p XJX,
where ^, R%...R n represent fa and the successive simplified residues
to/#, ^a;, while (7^ means the coefficient of the highest power of x in Ri, and
(7 the first coefficient in/a;*.
Art. (I). If we take $w of the same degree as fx, and for greater
simplicity make the first coefficients in fx and gx, each of them unity,
* This result may be obtained directly as follows :
Let fx, <f>x and the (m - 1) complete Sturmian residues be called p , p t , p 2 . . . p n ; let the n
complete quotients be called q lt q z ... q n , and let the allotrious factors to the residues p. 2 , p 3 ... p n
be called /j^, /x 3 ... n n ; then
Po=1iPi-P2 Pi = ^P-2-p3y P2 = l3P3-Pi> &c - <
hence Pl d Po - p 3 Pl = p? Sq 1 + (p., S Pl - Po dp. 2 )
= Pi 2 S 1i + Pa 2 5( /2 + (Pa s P-2 ~ P-2 s Pa)
= &c.
= Pl 2 S 1l + Pz 2 5 ?2 + />3 2 5 ?3 + + Pn 5 1n
but we have in general p i = n i E i ,
hence 5q i= -p ^ 8x,
^i /"t
and tfafc-%**^,^**!
t
but it may be easily seen that
Mt-iMt^Tw > except when t = l, for which case ^-.j/Li^l,
t-i
1 C
hence P?$1i = n ^r^i ^x, when i>l, and = ~ R^dx when i = l,
^i-i^t c i
which proves the theorem in the text.
506 On a Theory of the Syzygetic Relations [57
the successive simplified residues to ~ will be identical with the simplified
fvr* _i_ ny*
residues to -* (including amongst them the quantity gx fx itself),
gx
and, since
{fx - gx} g x - {fx - gx} gx = g xfx -f xgx,
the right-hand side of the equation above written, when the residues, instead
of referring to / and </>, are made to refer to / and g, taken of the same
degree in x, becomes equal to f xgx fxg x ; and if we now agree to
consider /and g as homogeneous functions each of the nth degree in x and 1,
the equation becomes
7? 2 7? 2 7? 2 7? 2
J M . -^2 , - tt 3 , , -""tl
ri " fift*ftfi*"**n n
L/l V^iOo L/oU q
23
= 9 (x, 1) /O> !) ~f( x > l ^
1 / d d \ / d ^ 1
I /y /* I j~f \ t * \
\ Us , U ^" ~j~ U I I ^ / I
W\ ri T fi \ I \ n W /
\ w/u/ (ju L / VUwv / it
n \dx dl dl dx]
where J indicates the Jacobian of the given functions / and g in respect to
the variables x and 1, meaning thereby the so-called Functional Determinant
of Jacob! to / and g in respect of x and 1, which equation also obviously
must continue to hold good when we restore to the coefficients of x n in/ and
g their general values.
It may happen that for particular relations between the coefficients of
/ and g certain of the residues may be wanting, which will be the case
when any of the secondary Bezoutics have their first or sucpessive terms
affected with the coefficient zero; the equation connecting the residues
with the Jacobian will then change its form (as some of the quantities
Cj, C 2 ... C n will become zero) ; but I do not propose to enter for the present
into the theory of these failing, or as they may more properly be termed,
Singular cases in the theory of elimination.
Art. (ni). The series last obtained for / (/ g) leads to a result of much
interest in the theory, and of which great use is made in the concluding
section of this memoir, namely the identification of the Jacobian (abstraction
made of the numerical factor n) with what the Bezoutiant becomes when in
place of the n variables in it, u lt w 2 w > we write x n ~ l , x n ~ z ..,x, 1. Thus
suppose / and g to be each of the third degree, and let
Ax" 1 + Hx + G,
+ Fx + C,
57] of two Algebraical Functions. 507
be the three primary Bezoutics ; if we make
X 2 = U, X = V, 1 = W,
these may be written under the form
Au + Hv+ Gw = L,
Hu + Bv +Fw = M,
Gu + Fv + Cw = N,
and if the Bezoutiant be called 3 , we have
L = d M= dS N- d
du dv dw
The simplified residues to / and g are L, (L, M}, (L, M, N}, where (L, M)
means the result of eliminating u between L and fit, and (L, M, N) the result
of eliminating u and v between L, M, N: and by a theorem (virtually implied
in the direct method* of reducing a quadratic function to the form of a sum
of squares), if we call the leading coefficients of these quantities C lf C 2 , C 3 ,
we have
If } (L,M? | (L,M,N)^ S
Hence, when ?i = 3, /(/, g} = 8 when in ff", u, v, w are turned into # 2 , x, I ;
and so in general for any values of n, the Bezoutiant correspondingly modified,
becomes - /(/, g), as was to be shown -f.
?i
f x
Art. (n). The expressions obtained for the quotients to J -j- may be
jx
cf)i^
generalized and extended to the quotients to V , where <j>x and fx are two
/*
functions of x of any degrees ra and n, whose roots are respectively, k ly lc z ...T<; m ,
and h lt h 2 ... h n . If we suppose
fx ~Q(x)-q 2 (x)-q 3 (x)- " q m+l (x)
where Q (x) is of n m dimensions, and q z (%), q 3 (x) . . . q m +i (#) each of one
dimension in x, it may be proved that on writing
111 Ni(x)
* Namely, that of M. Cauchy, adverted to in Section IV. Arts. 44 45. [p. 511 below.]
t Compare Jacobi, De Eliminatione, 2. The general expression for the allotrious factor,
I may here incidentally mention, is given under the head Theorem a, 16, which comes quite at
the end of the same paper.
508 On a Theory of the Syzygetic Relations [57
we shall have
m ( fl. \
Cq i+1 (x), (A)
=i
2 {( DM $ (x - h e )\ = C q i+1 (x), (B)
0=1 ( J n e )
where C C = 0, (E)
Cq i+1 (x) being the (i+l)th simplified quotient. When Q (x) is a linear
function of x, in finding q-jc from the formula (B), we must take D x = 1. The
proof of this theorem being generally true, may easily be shown to depend
upon its being true in the special case*, when m = //, + i, and n = p + i
(m being supposed less than n), and h 1} h 2 ... h n become 1 1} 1 2 ... Z M , h lt A 2 ... A f <,
while k lt k 2 ... k m become 1 1} 1 2 ... Z M , k lt k 2 ... A^; and the truth of the theorem
for this special case (if for instance we wish to prove the formula (B)) depends
upon the expression
h 2 . . . hi _j\ ^ fh-i , h 2 ... h^.
k z ...k m J \hi , hi +l ...h n
hi, h 2 ...}
,. .
( * ~ ^
^ I T J
n. o
being identical with the expression
At, A, Af_
as it may readily be shown to be. And the formula (A) may be verified
in precisely the same manner. There is no difficulty in finding the values
of C and C , which are products of powers, some positive and some negative,
of the leading coefficients in the simplified residues, and recognising that
they satisfy the equation (E) ; when <f>x is of one degree below fx this equation
is of the form C+C = 0.
Art. (o). When $x =f x, this expression for the (i + l)th simplified
quotient becomes (D i h) 2 (a; h ), as previously found; the correlative ex
pression will be
* By virtue of the Lemma, that when (px and fx are two algebraical functions, no function
of the coefficients vanishing identically when i roots of fx coincide with i roots of <px respectively
can be formed, in which there are fewer of the coefficients of / and <f> respectively than appear
in the leading coefficient of the (-t + l)th residue of j.
57] of two Algebraical Functions. 509
k being any root of f x 0, which is equal to the former expression. The
general expressions above given for the simplified quantities are of course
integral functions of h and k, although given under the form of the sums
<\J,
of fractions, by virtue of the well-known theorem that 2 ^TT , where S- is an
/ "
integral function of h, and the summation comprises all the roots (h) of
fh = 0, is always integral.
Art. (p). It will be found that for all values of i greater than unity
e=i
and that (D
0=1
The theorem of Art. (n) is in effect a theorem of cumulants of the form
[Qi 0), fc (as)... qt (as) . q n (as)],
where the elements are all independent of one another, and
fx = [Q, (x), q, (x), q 3 (as)... q n (as)], $x = [q z (x), q 3 (as)... q n (as)],
n being any number whatever greater than i; this makes the theorem still
more remarkable. The urgency of the press precludes my investigating
for the present the more general theorem which must be presumed to exist,
whereby q i+l can be connected with [q lt q. 2 , q 3 ... q t ], or [q. 2 , q s ... q { ], and with
[<?i> <?2, <?s qi+e] and [q 2 , q 3 ... qi +e ], when each q represents a function of an
arbitrary degree in x. The theorem so generalized would comprehend the
complete theory of the quotients arising from the process of continued
division, without exclusion of the singular cases (at present supposed to be
excluded) where one or several consecutive principal coefficients in one or
more of the residues, vanish.
Art. (q). The complete statement of two twin theorems suggested by
and intimately connected with the biform representation of the quotients
fD^T"
j-~ , given in the preceding article, is too remarkable to be omitted.
J x
f x
Suppose <f>x=f a;, and let the successive convergents to ^ be called
/*
L]ib i/ft _ 2 ** t"n, _ i w
5V T& "T^v ~T^
where the subscript index to t or T indicates the degree in x. Then if we
call the roots of fx, h 1} h 2 ... h n> the theorem already cited in a preceding
510 On a Theory of the Syzygetic Relations [57
article, concerning the denominators of the convergents, may be expressed
as follows :
<av
= 0,
where it will be observed that the first line of terms consists exclusively
of units, since f x = $x by hypothesis.
Correlatively I have ascertained that preserving the same assumption
1/7 f"lf
that $x=f x, so that consequently ^.- means - , the following theorem
/* J c
obtains, namely that if k lf k. 2 ... &_! are the (n 1) roots of <f>ac,
/^-
It may consequently be conjectured, when $ and f are independent functions
ft)//*
of # and respectively of the degree n I and n, and -^- is expanded under
jx
the form of a continued fraction, of which, as before, -, -^-. ..-Sf- 1 are the
*i *i *n
successive convergents, that we shall have analogous determinants to the
twin forms above given, each separately vanishing, these more general
determinants differing only from their model forms in respect of the upper
most line of terms in the one of them, being each multiplied by certain
functions of h lt h z . . . h n respectively (all of which become units when <f>x =f x),
and in the other of them by certain functions of k lt k 2 ... k n .
The exact form, however, of such functions, and even the possibility
of such form being found capable of making the determinants vanish, remains
open for further inquiry.
57] of two Algebraical Functions. 511
SECTION IV.
On some further Formulae connected with M. Sturm s theorem, and on the
Theory of Intercalations, whereof that theorem may be treated as a
corollary.
Art. 44. As preparatory to some remarks about to be made on the formulae
connected with M. Sturm s theorem, it is necessary to premise two theorems
of great importance concerning quadratic functions, one of which, notwith
standing its extreme simplicity, is as far as I know very little (if at all)
known, and the other was given in part many years ago by M. Cauchy, but
is also not generally known. The former of these two theorems is as follows.
If a quadratic homogeneous function of any number of variables be (as it may
be in an infinite variety of ways) transformed into a function of a new set of
variables, linearly connected by real coefficients with the original set, in such
a way that only positive and negative squares of the new variables appear in
the transformed expression, the number of such positive and negative squares
respectively will be constant for a given function whatever be the linear
transformations employed. This evidently amounts to the proposition, that
if we have 2?i positive and negative squares of homogeneous real linear
functions of n variables identically equal to zero, the number of positive
squares and of negative squares must be equal to one another, so that
for example we cannot have
Mi 2 + W a 2 + + u n + W 2 n+i ~ V|-a ~ u *n+ ~ ~ U 2 m
identically zero when n of the variables are linear functions of the remaining
n ; and this is obviously the case, for if the equation could be identically
satisfied we might make
Un =
and we should then be able to find u n+1 as a real numerical multiple of u n ,
and consequently should have the equation u n 2 {1 + & 2 } = 0, which is obviously
impossible; d fortiori we may prove that in the identical equation existing
between the sum of an even number of positive and of negative squares
of real linear functions of half the number of independent variables, there
cannot be more than a difference of two (as we have proved that there cannot
be that difference) between the number of positive and negative squares.
Hence there must be as many of one as of the other ; and as a consequence,
the number of positive squares or of negative squares in the transform of a
given quadratic function of any number of variables effected by any set of
real linear substitutions is constant, being in fact some unknown transcen
dental function of the coefficients of the given function. I quote this law
(which I have enunciated before, but of which I for the first time publish
the proof) under the name of the law of inertia for quadratic forms.
512 On a Theory of the Syzygetic Relations
[57
Art. 45. The other theorem is the following. If any quadratic function
be represented in the umbral notation* under the form of
where a^, a 2 ...a n are the umbras of the coefficients, aud x l} x z ...x n the
variables, then by writing
0,
,
i
ar + ttl
ar +
a,
ar + + ttl
fll
a 2
a 3
a 4
a n
Cti, 2
/r _1_ 1> 2
^ 2 ~
*3 +
a,, a,
, , &i, a z
a lt a 2
a lf a 3
a lt a 4
i, a n
1 > ^2 ) 3
,3 +
;; ;; ;
Oi, 2 , a 3
a?4 + +
&c. &c. &c.
a^ + a 2 ^ 2 + ... + a w ^n) 2 will assume the form
02... a n
a 2 ... a n
a ls
aj,
, a 2 , a 3
and consequently the number of positive squares in the reduced form of the
given function will always be the number of continuations or permanencies
of sign of the series
i;
j, 02
i> o 2 , a 3
!, a 2 ... a n
the several terms of this progression being in fact the determinants of what
the given function becomes when we obliterate successively all the variables
but one, then all but that and another, then all but these two and a third,
until finally, the last term is the determinant of the given function with
all the variables retained. This comes to saying that if we call the function
(suppose of four variables) f, and write down the matrix
dxf
// /v rj nr* ri qp /y ,-** fi /v rj rp rj /y* 2
* For an explanation of the umbral notation, see London and Edinburgh Philosophical
Magazine, April 1851, or thereabouts [p. 243 above].
57]
of tivo Algebraical Functions.
513
(where all the terms are of course coefficients of the given function expressed
as above for greater symmetry of notation), the inertia of f will be measured
by the number of continuations of sign in the series formed of the successive
principal minor coaxal determinants fin writing which I shall use in general
d 2 f
(r, s) to denote -
1, (1, 1),
(1, 1), (1, 2) ;
! (2, 1), (2, 2)
(1, 1), (1, 2), (1, 3)
(2, 1), (2, 2), (2, 3)
(3, 1), (3, 2), (3, 3)
(1, 1), (1, 2), (1, 3), (1, 4)
(2, 1), (2, 2), (2, 3), (2, 4)
(3, 1), (3, 2), (3, 3), (3, 4)
(4, 1), (4, 2), (4, 3), (4, 4)
and in like manner in general*.
Art. 4G. Reverting now to the simplified Sturmian residues, since by
the theory set out in the first Section these differ from the unsimplified
complete residues required by the Sturmian method only in the circumstance
of their being divested of factors which are necessarily perfect squares and
therefore essentially positive, these simplified Sturmians may of course be
substituted for the complete Sturmians for the purposes of M. Sturm s
theorem. The leading coefficients in these simplified Sturmians, reckoning
f (x) as one of them, will be
m% (h, , /z 2 ), 2 (/*!, A 2 , h 3 ) . . . (k,, h 2 . . . h m ),
which it is easily seen, as remarked long ago by Mr Cayley, are the successive
principal minor coaxal determinants of the matrix
O" 2 ,
* I have given a direct a posteriori demonstration in the London and Edinburgh Philosophical
Magazine, that the number of continuations of sign in any series formed like the above from a
symmetrical matrix, is unaffected by any permutations of the lines and columns thereof, which
leaves the symmetry subsisting, that is to say (using the umbral notation), if 0j, 2 , 3 ... t - are
disjunctively equal, each to each, in any arbitrary order to 1,2,3 ... i, the number of continua
tions of sign in the series
1,
is irrespective of the order of the natural numbers 1, 2, 3 ... t in the arrangement lt 2 > #3 fy
s. 33
514 On a Theory of the Syzygetic Relations [57
where in general o> = A/ + A/ + ... + h m r , and of course o- = m. M. Hermite
has improved upon this remark by observing, what is immediately obvious,
that if we use cr r to denote, not the quantity above written, but
H
AT r . n ; ,
! X fl 2 X h m
the successive coaxal determinants of the above matrix will become re
spectively
(x Ai) (x A 2 )j
i , A 2 , A 3 ) (hi , A 2 . . . A m )
(x - Ai) (x - A 2 ) (x - A 3 ) " (x - AJ) (x - A 2 ) . . . (x - h m )
that is to say, these successive coaxal determinants, when multiplied up by
fx, will become respectively
~Z(hi, A 2 ... h m ),
that is to say, will represent the simplified Sturmian series given by my
general formulae. M. Hermite further remarks, that the matrix formed
after this rule will evidently be that which represents the determinant of
the quadratic function (which may be treated as a generating function)
2 - r {uj, + Ma + V^j + . . . + h- l u m }*,
in which, since only the squared differences of the terms in the (A) series
finally remain in the successive coaxal determinants, we may write (x hi),
(xh 2 ) ... (x h m ) simultaneously in place of A a , A 2 ... h m without affecting the
result ; consequently the generating function above may be replaced by the
generating function
1
the corresponding matrix to which becomes
1
Vmi > "m 1
57]
of two Algebraical Functions.
515
1 f x
where { denotes ^(x hY, and 2 1~ ~T Hence every simplified
SC ~"~ /Z/1 ./ ^
residue is of the form
+/XX
0,
6,
The residue in question will be of the degree ra r 2 in x, and consequently
we have, according to the notation antecedently used for the syzygetic
equations
j, #2 ... ^r
2, #3 ... # r+1
0,
Elegant and valuable for certain purposes as are these formulae for t r +i
and r r , they are affected with the disadvantage of being expressed by means
of formula of a much higher degree in the variable a; than really appertains
to them, the paradox (if it may be termed such) being explained by the
circumstance of the coefficients of all the powers of x above the right degree
being made up of terms which mutually destroy one another; upon the
face of the formulae, t r+ i and r r which are in fact only of the degrees r + 1
and r respectively in x would appear to be of the degree
that is of the degree r 2 .
Art. 47. I may add the important remark, which does not appear to
have occurred immediately to my friend M. Hermite when he communicated
to me the above most interesting results, that in fact, by virtue of the law
of inertia for quadratic forms, we may dispense with any identification of the
successive coaxal determinants of the matrix to the generating function
with my formulae for the Sturmian functions, and prove ab initio in the
most simple manner, that the successive ascending coaxal determinants
332
516 On a Theory of the Syzygetic Relations [57
(always of course supposed to be taken about the axis of symmetry) of the
matrix to the form above written, or to the more general form (which I shall
quote as (G), namely)
2 (p - htf {</>! (AO Ul + <j> 2 (h,) u a + . . . + <f> m (A,) w m ] 2 , (G)
(where 1} < 2 ... <j> m are absolutely arbitrary integral forms of function with
real coefficients), will form a rhizoristic series in regard tofx (that is a series,
the difference between the number of the continuations of sign between
the successive terms of which corresponding to two different values of p will
determine the number of real roots of x lying between such two assumed
values), provided only that q be an odd positive or negative integer. Nothing
can be easier than the demonstration, for whenever p is greater than any
one of the real roots as hi :
Firstly, any pair of imaginary roots will give rise to two terms of the
form
(I + m V- 1) ? (v + w V- I) 2 and (l-m*J- l) q (v - w*J-l}\
or more simply
(L + M V- 1) (w a - w 2 + 2vw x/- 1)
and (L-M*J-l)(v 2 -w 2 -2vw\/-l),
where v and w are real linear functions of u lt u 2 ... u m . The sum of which
couple will be
o
2 {L (v 2 - w 2 ) - 2Mwv] = j {(Lv - Mw) 2 - (D + IF) w z ] = p-- q*;
so that each such couple combined will for every value of x give rise to one
positive and one negative square.
Secondly, any real root of the series h lt h 2 ... h m , when p is taken greater
than such root, will give rise to a positive square of a real linear function
of Ui, u 2 ... u m .
Thirdly, any real root of the same series, when p is beneath it in value
(q being odd), will give rise to the negative of the square of a real linear
function of the same. Hence the number of real roots between p taken
equal to one value (a), and p taken equal to any other value (6), will be
denoted by the loss of an equal number of positive squares in the reduced
form of the expression (G) when p is taken a and when p is taken b;
that is by virtue of Art. 45 will be denoted by the difference of the number
of permanencies of sign in the successive minor determinants of the matrix
corresponding to the quadratic form (G)* (which we have taken as our
* The inertia of the quadratic form (G) is the measure of the number of real roots of fx
comprised between oo and p, and may be estimated in any manner that may be found most
convenient. If p be made infinity, and <p t h be taken equal to ft*- 1 , and the inertia of the corre
sponding value of (G) be estimated by means of -the formulae in ordinary use by geometers for
57] of two Algebraical Functions. 517
generating function) resulting from the substitution respectively of a and
b in place of p, which gives a theorem equivalent to that of M. Sturm,
transformed by my formulae, when we choose to adopt the particular
suppositions
q = -I, <f>Jt = I, <f>.Ji = h , <M = h 2 > <M = hm ~ 1 -
This method of constructing a rhizoristic series to fx by a direct process
is deserving of particular attention, because it does not involve the use of the
notion of continuous variation, upon which all preceding proofs of Sturm s
theorem proceed. It completes the cycle of the Sturmian ideas. Happily
this cycle was commenced from the other end, for it would have been difficult
to have suspected that the root-expressions for the terms in the rhizoristic
series could be identified with the residues, had the former been the first
to be discovered, and much of the theory of algebraical common measure
laid open by means of this identification would probably have remained
unknown.
Art. 48. I proceed now to consider a theorem concerning the relative
positions of the real roots of two independent algebraical functions as
indicated by the succession of signs presented by their Bezoutian secondaries ;
this more general theory of intercalations or relative interpositions will be
seen to include within it as a corollary the justly celebrated theorem of
M. Sturm.
Let the real roots of fx taken in descending order of magnitude be
h lt h z ...h p , and the real roots of <f>x taken in the like order ^, ^...^q,
so that
fx = (x- h^ (x - h 2 ) ... - h p ) H,
(f)X=(x TJj) (X T? 2 ) ... (X 1]q) K,
H and K being functions of x incapable of changing their signs. Now, as m
M. Sturm s method, let us inquire what takes place in respect to the sign of
^?. which I shall call the Indicatrix, as x descends the scale of real
/(*)
magnitude from + oo to - oo . If between + GO and h 1} i real roots of <f>x are
contained, it is obvious that as x travels from + x> to the superior brink
of h ly the Indicatrix will change its sign from + to - and from - to + alto
gether i times, so that at the moment when x is about to pass through h lt it
determining the nature of a surface of the second degree, the criteria of the number of real roots
in fx will be, or may be made to be, symmetrical in respect to the two ends of the expression fx.
This system of criteria, however, is not so good as that given by the Bezoutiant to the two
differential coefficients of / (x, 1) taken with regard to x and 1 respectively, which will also
possess the like character of symmetrical indifference, and be one less in number than the
former.
518 On a Theory of the Syzygetic Relations [57
will be positive if i is zero or even, and negative if i is odd ; but the moment
after x has passed through the value h lt the indicatrix will be negative
on the first supposition, and positive on the other supposition. Hence
immediately after the passage of x through ^ the indicatrix will have been
once oftener negative than positive on the one supposition, and as often
negative as positive on the other. Again, in like manner as x traverses
the interval between A x and the inferior brink of A 2 > if n V or an even
number of ?? s occupy this interval, the sign which the indicatrix had at the
beginning of this interval will have been reversed once oftener than restored;
but if there be an odd number of 77 s so interposed, the number of reversals
and restorations will have been identical ; and so for each successive interval,
reckoned from a value for # immediately subsequent to one real root of fx,
down to a value immediately subsequent to the next less real root of the
same ; and it is evident that the effect upon the sign of the indicatrix at
the end of every such interval depends, not upon the number of tj s grouped
together in such interval, but upon the form of the group as regards its
being made up of an odd or even number of terms, the first interval being
of course understood to extend from + oo to a value immediately inferior
to h lt and the last from a value immediately inferior to h p to oo . Hence
as regards the relation of the sign of the indicatrix at the beginning to the
sign at the end of every such interval, nothing will be altered by taking
away any e 
