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Book ^fLSL
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
(18371853)
Cambridge
At the University Press 1904
QA3
CTambTttise :
PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
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MATHEMATICAL PAPERS
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PREFATORY NOTE.
riiHE 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.
TABLE OF CONTENTS
PAGES
1. Analytical development of FresneVs optical
theory of crystals . . . . . 1 — 27
(Philosophical Magazine 1837, 1838)
2. On the motion and rest of fluids . . 28 — 32
(Philosophical Magazine 1838)
3. 071 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 36—38
(Philosophical Magazine 1839)
6. 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 simidtaneous 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
1 1 . 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 midtiplicity of the
roots of an algebraic equation . . 66 — 68
(Philosophical Magazine 1841)
14. A new and more general theory of multiple
roots 69—74
(Philosophical Magazine 1841)
15. On a linear method of eliminating between
doid>le, treble, and other systems of
algebraic equations 75 — 85
(Philosophical Magazine 1841)
16. Memoir on the dialytic method of elimina
tion, Fart I. 86—90
(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 — 106
(Philosophical Magazine 1844)
19. An account of a discovery in the theory of
numbers relative to the equation
Ax' + By' + Cz' = Dxyz . . 107—109
(Philosophical Magazine 1847)
20. On the equation in numbers
Ax'\ By'' + Cz" — Dxyz, and its associate system of equations . 110 — 113
(Philosophical Magazine 1847)
21. On the general solution, in certain cases, of
the equation x^ + y^ + z^ — Mxyz . . 114 — 118
(Philosophical Magazine 1847)
viii CONTENTS.
PAGES
22. On the intersections, contacts, and other cor
relations of two conies expressed hy indeterminate coordinates . . . 119 — 137
(Cambridge and Dublin Mathematical Journal 1850)
23. An instantaneous demonstration of PascaTs
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 midtijiles of a fourth nonhomogeneous function of the same 152 — 154
(Philosophical Magazine 1850)
27 . On aporismatic property of two conies having
with one another a contact of the third
order ....... 155, li56
(Philosophical Magazine 1850)
28. On the rotation of a rigid body about a fixed
point ....... 157 — 161
(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)
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 coiiicidence of a theorem
given by Mr Sylvester in the December number of this Journal with one stated by Professor DonMn hi 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 lineai'ly 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 256—264
(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 princip)les of the calcidus of forms . 284 — 327
(Cambridge and Dublin Mathematical Journal 1852)
43. On the principles of the calculus of forms . 328 — 363
(Cambridge and Dublin Mathematical Journal 1852)
44. Su7^ line propriete nouvelle de V equation qui
sert a determiner les inegcdites seeulaires
des planetes ...... 364 — 366
(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 tJie 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 sim])le 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 404 — 410
(Cambridge and Dublin Mathematical Journal 1853)
\1
54. On the calem
of invaria
(Cambrii
55. Theoreme sur
des equatio
(Nouvfa
56. Noiivelle metliode pow, ui .
superieure et une limite inferieure des racines reelles d^une equation algebrique quelconque 424 — 428
(Nouvelles Annales de Math^matiques 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 hraical common measure . . . 429 — 586
(Philosophical Transactions of the Eoyal 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 Aronhold's invariants and of a cognate theoremfi for biquadratic binary forms . 599 — 608
(Philosophical Magazine 1853)
61. On a remarTcable modification of Sturm! s
theorem 609—619
(Philosophical Magazine 1853)
rm s iding roots
620—626
lOr and , ^... 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 determinantive equation gene ralised 634—636
(Philosophical Magazine 1853)
66. On the explicit values of Sturm's quotients 637 — 640
(Philosophical Magazine 1853)
67. On a fundamental rule hi 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 volttme 647 — C50
ANALYTICAL DEVELOPMENT OF FRESNEL'S OPTICAL THEORY OF CRYSTALS.
[Philosophical Magazine, XL (1837), pp. 461—469, 537 — 541 ; xii. (1838), pp. 73—83, 341—345.]
Thk 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, ti, i,^ the angles made by the front with the optic planes, ei, 62 the angles between its due line of vibration and the optic axes, I have shown that
//b'^~p^ sin tA //b^ — p^ sint2^
cos ei = . / — — i . . —  , cos 6, = . / , , . . — ' , V Va^  c^ sm ij "V W  & sm ij '
so that all doubt is completely removed. The equation preparatorj' 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 1,0116,. and, Edinb. Phil. Mag. Vol. x. p. 336.
2 Analytical Development of [1
Analytical Reduction of Feesnel's Optical Theoey of Crystals.
Index of Contents.
In Proposition 1, a plane front within a crystal beiag given, the two lines of vibration are investigated.
In Proposition 2 it is shovm that the product of the cosines of the iacHna tions of one of the axes of elasticity to the two lines of vibration, is to the same for either other axis of elasticity ia a constant ratio for the same crystal; and the two liaes 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 Hne 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^, v, be the two normal velocities with which any front can move perpendicular to itself, and ij, i„ the angles which it makes with the optic planes, then
v^ = a ( si
sm'L^j+c' cos^'^
% = a ( sm — — ^ 1 + & I cos — „
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 11th 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.
'if — h° = (a — c") cos 6i cos 62. In the 12th Proposition Cj, 62 are separately expressed in terms of ti, ta in the Appendix I have given the polar or rather radioangular equation
to the wave surface, from which the celebrated proposition of the ray flows as
an immediate consequence.
IJ FresneVs Optical Theory of Crystals.
Proposition 1.
If lx + my + 7iz = 0 (a)
be the equation to a given front, to determine the lines of vibration therein.
It is clear that ii 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
I m. n '
and the forces acting at x, y, z are arx, b^y, cz parallel to x, y, z, so that the equation to the line of force is
X.^Yy^Z^_ ■
ax by &z
From (2) we obtain
b'yX — ax Y =(b — a^) xy (3)
czY  b'yZ = (c=  b^) yz (4)
a^xZ — c^zX = (a^ — c^) zx. (5)
Hence
(&^ — a^) xyn + (c^ — 6^) yzl + (a^ — c'^) zxm
= % {nX  IZ) + (fz (lY mX) + a'x (mZ 7iY) ; but by equations (1)
lZnX = 0, mXlY=0, nYmZ=0 therefore
(&' a') + (c' b^) + (a=  cO  = 0. (6)
' z ^ ' X ' y
Also we have
nz \lx + my = 0 (a)
therefore
(6=  a^) n^ + (c=  &2) P + nl Uc  6^  + (&'  «') 7) = (»'  c') »i' or
(c^  ^) (3' + ^ [{C^  h^ Z= + (6=  a=) n^  (a=  c) «i=} ^ + (6^  «0 = 0. And in like manner interchanging b, y, m with c, z, n
(b". _ c') gy + _!. f(6= _ c=) Z^ + (c=  a^) «,=  (a^  60 «^} f + (c=  a^) = 0.
1—2
A7ialytical Development of [1
Hence if (^, — ) (— , — ) be the two systems of values of , , then
\K x^' X Xi' \X x„ ' X xj are the two lines of vibration required.
Proposition 2. By last proposition it appears that
x,x. 
h' 
& 

z^z^ 
6= 
~a^ 

x,x. 
c^ 
Jf 

2/12/2 + ^■1^2 
c^ 
b" 
(c)
and ~ = ^zi:^ (d)
therefore _
aJiflj, b — c
therefore
^1*2 + 2/12/2 + •^1^2 = 0
And therefore the two lines of vibration are perpendicular to each other. N.B. Equations (c) and (d) must not be overlooked.
Proposition 3.
A line of vibration is given ( that is ~ , — are given ] and the position of the front is to be determined.
Let Ix + my + nz=0 be the front required, then Ix^ + my^ + nz^ = 0, and
(6=cO+(c=a=) + (a— 6=) = 0. «i 2/i «i
Eliminating 7i we get
I ((a^  ¥) I  (6^  cO I) + m ((a^  &0  " (0^  o?) ) = 0
therefore
I _Xi (a^ — b") y^ — (c^ — a^) z^ m 2/1 (6^ — &) z^ — (a^ — 6^) ajj^
_ X, a (.«i^ + 3/1^ + giO  (ctV + 6 V + cV)
~ yi &' («i' + 2/1' + ^i)  ("'W + b'yi + cW) '
1] FresneVs Optical Theory of Crystals. 5
If now we make x^ + y^ + ^f = 1
a?x^ + }pyx~ + c^z^ = v^
and therefore
I _Xi a^ — ■yi" m y^' b^ — v{^
and in like manner
Z _ a?! a — Vi _ m 2^1 " c" — Vi^ '
therefore
(a^ — «!) «!« + {h" — vf) y^y + (c' — ■Wj^) ^i^ = 0
is the equation required.
Proposition 4.
— ,  having each only one value, shows that only one front corresponds
to the given line of vibration. Let x^, y^, z^, v^ correspond to (c^, y^, z^, v^ for the conjugate line of vibration, then the equation to the front may be expressed likewise by
(a^ — v/) x^ + Qf — Vo') y^y + (c^ — v^') z^^z = 0,
so that
(g^  v^^) x^ ^ Qf  Vi"^) yi ^ (c^  vj") z^ {o?  vi) «2 Qf  vi) 2/2 (&  vi) z^ '
Proposition 5.
To find CO, ^, ■xjr, the angles made by the front with the 'planes of elasticity in terms of vijV^.
By the last proposition
, „ {aviYxi
(cos ft)/ = '^—, ^;r^ — ., , ,7, :;^ „ , , , ^„ — 7.
^ ' {a""  v^y X:c + {b  V:) yc + {c  viy z^
__ (g^ — vf) (a — vi) x^x^
~ (a^  vi) (a^  vi) x^x^ + (6  vi) (b"  vi) yiy._ + (c'  vi) (c  vi) z^z^ '
Now, by Proposition 2,
a^A ^ yxyi ^ z^z^ c^ — b a^ — & ¥ — a^
6 Analytical Development of [1
therefore (cos &>)
{a'v^''){a''vi){c¥)
~ {d'v;') {a'vJ'){c'b')+{b'  v^'){b'  vi){a?  &) + (c  v,^) {c^v^'){b'  a')
(a  Vj') (g^  Vj) (c  b)
~ a' {c  6^ + 6" (a  c=) + c' (a  b")
Similarly,
Proposition 6.
To find Vi, V.2 in terms of co, <f), •\r.
By the last proposition (cos my
therefore
a"  ^1= (a=  60 («'  c") ' ■ (a'  b'} (a^  c^
(cos (^) 6^ 2 1
b^v,' ~ (6  a'O {b"  c=) "'' ■ {¥  a') {¥  c")
(cos v^) _ C" _ ., 1
c  Vi' ~ (c^  b') (c  a'O ''''■(c^a)(c'6')
(cos o))' (cos (/>)^ (cos ■y^y _ a'' — 1)1^ ¥ — v^ & — v^
Just in the same way
(cos aif (cos <^y (cos •^) _ . ci^ _ vo^ '^ b^ vi '^ C v^^ ~ '
so that Vi^, V2 are the two roots of the equation
(cos of (cos <^) (cos v^)'
— ; r "I — TT, r H ; 5~ — "•
a — V' 0 — ■« c — w
COR. Hence the equation to the wave surface may be obtained by making
(cos co)x + (cos 4>)y + (cos ■y^)z = v,
1] FresneTs Optical Theory of Crystals.
or if we please to apply Prop. 5, we may make
or, if we please *,
/(aVl)(a'^0 /(6V1)(6^^)
/{chi?  1) {d"  ■wO
■/i
.2 = 1.
id)
{c^  a') (c^  h")
Proposition 7. To find when Vi = v^.
By Prop. 4,
Hence whea Vi = v.2 we have, generally speaking,
Xl 2/2 Z2
Now a;ia;2 + 2/12/2 + ^1^2 = 0 ;
therefore x^ + j/,^ + zf would = 0, which is absurd.
The only case therefore when v^ can = ii^ is when one of those terms of
0 a;, ^ 0
equation {&) becomes ^ : thus suppose v^ = h, then we have ~ = ~ = k > ^'^^
we can no longer infer — = — . x„ 2/2
Let now (coi, ^1, \rj)((U2, <^^, ■<if^ be the two systems of values which oj, (^, •\r assume when Wi = ■«„ = &, then applying the equation of Prop. 5 we have
la?—h' cos Ui^=^ . —: r 
/a? 
h" 
cos 0,2 ^ ^, 
& 

cos </>i = 0 
cos ^2 = 0 

, /6^  c'' 
cos^ /^' 
& 
COSt2^^2 
& 

so that h must correspond to the mean 
axis. 

[* See below, 
p. 27. Ed.] 
8 Analytical Development of [1
Proposition 8.
ti, t2 heing the angles made by the front with the optic planes, to find ti, t2 in terms of Vi, v^.
By analytical geometry
cos li = cos ft) . cos (Uj + cos (p . cos ^1 + cos yjr . cos iri
_ /(vi"  a) {vj  g") /a^6^
/(t;i'  C) (z;^"  cO l&\P^ V (c^  a=) (c^  6^) ■ V c'  a
" a'c^
and similarly
cos (2 = cos ft) . cos (1)2 + cos cf> . cos <f)o + cos v/r . cos ^fr.^
V{(di^  g^) (v^  a°)}  V{(^i'  cQ (^2  cQ} a — c
Proposition 9.
To find Vi, v^ in terms of h, u. By the last proposition
cos ., . COS . = fa«)fa;)fac)Wc°)
^{a'c'){a'&){v^ + v;)
_ (g + c')  (^Ji" + vj) {a^  c^) therefore
v^ + vi = g^ + c^ — (g^ — c^) cos tj cos i^.
Again, (sin i^f . (sin ^2)^ = 1 — (cos ti)^ — (cos tj)^ + (cos i^^ (cos ta)"
+
(g=  &f (g^ + c^f  2 (g^ + c°) (w^^ + v^) + («i= + ^aO'
(g=  c=)= (g=  c^Y
1]
FresneTs Optical Theory of Crystals.
therefore
but therefore
t>i^ — vi = {or — c") sin ti . sin tj Vi + •Va^ = (a^ + C") — (a — c') cos ti cos i^
a' + c^ a— c , ,
Wi = ?;^ ?; COS (ti + lo)
= a' Sin ,
2 ti 4 toA"
+ C^ COS
*1 + '2\'
COS (ti — ta)
a" + c" a
= a=(sin^)Vc^(cosVT.
Thus for uniaxal crystals where tj + ta = 180° v^ = a^ v^ = a? (cos t)^ + c^ (sin i)l
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 Fig. i
pose IGH to be one of the refracted fronts intersecting OA, OC in 0 and H, then
(a + c') (a'' c') cos (G + H)_ (sin PIFf
2 (vel. in air)= (sin PIGHy "
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 Aiialytical Development of [1
Proposition 10.
To determine the portion of a line of vibration in terms of the two velocities of its corresponding front.
We have here to determine the quantities — , — (of Prop. 1) in terms of Vi, v^,OT on putting a;i^ + 3/1^ + ^1^=1, x^ , y^ , z^ are to be found in terms of Vi, v^.
By Prop. 3
I m n
Xi.yi.Zi
' a=  vi' ' b^Vi"' (f Vj'
: m^ : 71^ : : (b — c^) (a — vc) (a^ — V2') : (c  a") {¥  Vi") (b^  vi)
a — v^ b^ — Vi' c^ — Vi^
Let a, /3, 7 be the angles made by the given line of vibration with the elastic axes, then
™ 2 (cos a)^ =
and by Prop. 5
therefore
= (b"  c") {a?  v.?) (6  v^) (c=  z/iO divided by
(6  C) (a"  Vo^) (b  v^) {c  Vi^) + (c  a) (6=  v/) (c  1;/) (a^  vf)
+ (a — 6^) (c — v.^ {a — ■Wi^) (b — Vi) and therefore
_ (i!)^  c^) (g^  vj) (6'  zii") (c'  •j^i")
~ (Vi^  ■ya^) (a  6^) {¥  c^) {(!'  a')
(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" — b^), (b — c'), (c — a'), a simpler instance of which was seen in Proposition 5).
In fact the coefficient of v* . v"
= (&  C) + (c=  a") + (a^  b^)
= 0
1] FresneTs Optical Theory of Crystals. 11
that of v^'' . v«} = (c + 6) . (c^  Jf)
+ (a + cO . (a^  C) + {b^ + a') . (6=  a")
= 0.
The term in which neither Vi nor v« enters
= a%c' {{b"  C) + (c'  aO + (a  &^)) = 0. The coefficient of
 vi' = aP.{¥ c^) + b^ . (c'  a') + c" . (a*  b*) and that of
v/ = b^c^ . (c  b) + c'a^ . (a=  c') + a%^ . (b^  a") each of which
= {a''b).{b"c^).(c'a'). Hence
^ ^ Vi' vi ' (a^  b^) (a  c") '
in like manner (cos /3)^ = &c.
, / V «i'6' (c'vi)(a''vi) and (cos y) = . ^, ff4 r .
Proposition 11.
6i, 62 being the angles between any Line of vibration and the optic axes, required the velocity due to that line in terms of ej, €3.
By analytical geometry,
cos 61 = cos a . cos ^1 + cos 7 . cos i^r^ cos 62 = cos a. . cos ^1 — cos <y . cos iIti therefore cos ei . cos e^ = (cos a)^ (cos c/),)^ — (cos 7)^ (cos 'y^if
Vj^  6^ Ua^  vi) . (c^  vi)  (c^  vi) . (a"  Vi"))
vi vi' { (a^  c^y
^ vi  b^ (a^c^){vivi) vivi' (a^c^f
_b^ — vi a^ — c^ '
Hence vi = 6^ — (a^ — c^) cos 61 cos e^,
and in like manner, for the conjugate line of vibration
vi = b' — (a — C) cos e/ cos 63'.
12 Analytical Development of [1
Proposition 12. To find €i, 62 in terms of ti, ta
(cos 6])^ + (cos 62)^ = 2 (cos a)= , (cos (/)i)= + 2 (cos 7) . (cos 1/^1)
== 2 ^^'~^' [(g^  i;/) . (c°  2;i^) + (c^  •i)^'') ■ (cr  v^)] v^^  vi \ (cC  dj J
but by Prop. 9
v^ = a I sm —  — ) + c ( cos — ^ — 1
V = a" I sin — g— 1 + c (cos — 2" therefore
(cos 61) + (cos 62) = —^
{a? — C) sin c^ . sin i^ multiplied by
2 (a^  c^^ [(cos ^j^ (sin 'i^j^ + (cos ^)^ (sin '^^^
(a=  c'f b"  Vi'
{a^ — c^) sin tj . sin i^
and we have seen that
b  V'
{(sin iif + (sin lof
cos fii cos 60 =
therefore
therefore
cos fii + cos eo =
a"— c
/Z) — tijN sin ti + sin i
cos e, cos 62=^ (^,^,^2 j
a^ — cV ' V(siiiti sinta) 6^ — ^i^N sin ti — sin I2
V(sinti.sint2)
sm t, sin tj and in like manner
{b^ — v^ sin ti ■ c^ " sin t.
, / (6^ — v„^ sin ti
cos e., = . / ^ „ ' . . — ,
V {O' — c sm tij
where v^, v^ for the sake of neatness are left unexpressed in terms of ti,
1]
FresneVs Optical Theory of Crystals.
13
This is the simplest form by which the position of the lines of vibration can be denoted.
COK. From the last proposition it appears that
cos 61 _ sin ti cos 62 sin to "
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
cos PK _ sin KE cos PI " sin IE '
Draw FEG 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 
sin KG sin KE sin FI " sin IE 
\ 
therefore 
^. 

sin KG 
sin IF 

sin KE ~ 
' sin IE 
Fig. 2.
therefore
sin KEG = sin lEF
therefore KEG=IEF or 180° lEF. But PEF=PEG, therefore EP bisects either the angle lEK 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 aif (cos 4>y^ (cos ^y _
a} _ j;2 ^2 _ ^2 g2 _ ^2
we chancre a, b, c, v into  , y ,  ,  , and consider v to be the length " a b c V ^
of a line drawn perpendicular to the plane
cos 0) .X + cos 4>.y + cos Tr . ^ = 0,
the equation to the extremity thereof must be
a^r^ (cos a>y b^r^ (cos <^)^ c^r^ (cos i/r)^ a^ — r 6^ — r'^ c^ — r^
where w, ^, yjr denote the angles between the radius vector r, and the axes of X, y, z, so that the equation may be written
a^a? y^y^ c^z^
+ iir^ 1 = 0,
a? — r^ 6' which is that of the wave surface. But we have seen that
v = c \ cos
mvH"^)]'
therefore the equation to the wave surface may be written
/ ti + i^ [ . ti + u
where (j, i^ denote the angles between the radius vector v and the^two lines
which would be the optic axes if a, b, c were chan£;ed into  , ^ , so that ^ ° a b c
if e be the inclination of either to the meaa axis of elasticity '1 _ r
_c //d—b\
^b\/w^')
a //b"c^\ b\/ WcV'
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 Hnes normal to the fronts of single velocity.
1] FresneVs Optical Theory of Crystals. 15
be the two values of r corresf
Cor. 1. If ^1, r^ be the two values of r corresponding to the same values of t] , ia we have
; . = ! I COS ^
1
a sin ti . sin tj,
2 )
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 Ly + tj becomes constant, and there fore rii + rta = 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 sphericoellipses or sphericohyperbolas become great circles :
(1) When t] + t., = the angle between the two poles, in which case the curve of intersection is the great circle which comprises the two poles.
(2) When l^ — u= 0, when the locus is a great circle perpendicular to the former and bisecting the angle between the optic axes.
(3) When t, + 12= 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 radioangular 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 0 be the centre of the wave surface, OA, OB the two prime radii, OP any radius vector. Let OP = v, POA = i^, FOB =1,2, and let the in clination of the planes POA, POB = fi;
then  = — — H —
(taking only the positive sign for the sake of brevity).
16
Analytical Development of
[1
Let OQ, OR be the two adjacent radii vectores, so assumed that
QOA = POA , QOB = POB + hu,
ROB = FOB, ROA=POA+Bi„
and let p, q, r, a, b be the projections of P, Q, R, A, B on a sphere of which 0 is the centre, then it is clear that
qpa^ 90", rpb =90\
draw qm perpendicular to pb, then pm = 8u, and therefore
_ pm _ pm _ 8t.j "' sin pqm sin a/)6 sin /i '
In like manner
Now the angle QPO
sin n
also
, _, r.POQ , .
d.\
dr du,
±L
Bi,
therefore therefore
rrf., ^i''U «.)«'" ^"^''>
cot QPO = ^ (^  Jj sin (i, + «,) sin fi. In like manner
col KPO = — I Jsind, + <.)sina,
4 Vc* av
therefore
QPO = RPO.
Also it is clear that rpq = apb = fi. Ai '^ to find the inclinati.>n of OP to RPQ. wo ha ^^ only to describe a sphere of which P is tl'® centre, and intersecting PQ. PR, PO in ^ • /2'. 0'.
Then iJ'O'Q' = /*. and
Fig. 4.
OQ' = O'R' = cot'
{i(^a')''°^''+*'^''°^'
1] FresneVs Optical Theory of Crystals. 17
Draw O'N perpendicular to R'<^, then O'N measures the inclination of the radius vector to the tangent plane*.
And qO'N = ^,
therefore cos  = tan O'iV . cot O'Q',
, . r..^^ cot O'Q'
therefore cot 0 iv = ^^— ,
and therefore
Let A OB the angle between the optic axes = 2e, then by mere trigonometry
t
= ^'"'(i'^)^'"^"''''
cot O'N = i' . (—  ^J sin I . sin ((, + 1^). between the optic axes = 2e, then b /sin (e + *' ., ^) sin (e ' ., 2 ~ V' ~in(,.sinij
itiun between tl
,.(.+!^)si„(."rH)
sm „ =
therefore tlie tangent of the inclination between the ra.iiiis vector and ihe normal
sin (, .sin (.
In like manner the tangent of the inclination between tho sj»iik ratiius ■ ^ctor and the normal at the other point of the wavesurface pierced by it
" ,,J' M, , /'K''^)'"('n')
a, =i('')'U'"ciJ''"^*'~*'*\/ sin ...sin,. '
We may, in the .same way, find the inclination of the tfingLnt plane
'■'< either of the prime radii, and to the plane which contains them both,
0' terms of t, and «,; the former by a remarkably elegant construction ;
,t the final expressions do not present themselves under the same simple
pect.
If we call 6 the angle between the ray and the front, we may still
further reduce by substituting for r" its values in terms of «,, u, and we
shall obtain
•2 ( r  r/ • )
cot d) = —
'i I h . i 'i + 'j
c' tan — g + u' cot X A /]■'''" (^ + '9 J "'" (e*'^].cosect, .cosec t,/.
* O' is the projection of thi: rav and R'O'