Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 267
constants entering only by their mutual ratios, Lévy finds the following 4. co5 solutions of
the proposed problem:
a2 u 2ut a2u a2u a2v v a2w
2 = a ~+c2-b+ b2 2 + 2 (-C2) +-Qy + -b2)
( watg ôc 2 3ay2 + a2 a axay x / x- a
a2 a2v a2v a2v v v w X (C) a2u
(A ) =c2 + + a2 ( - -2)a2) - C2)
at 2 y2 az2 ayaz ayax
a2w a2w a2w a2w X a2, a2v
= b2 +a2 -+ - ( b+- 2) -a- + - (,2 a2)
3t2 ax2 ay2 a92 ^ v zox v azaôy
a2u a2u 2a2u a2u. a2V V a2w
d2 =ac-2 +c2y2 +b2) + (cac)aa +x( axa)8z'
C 2 - + b2, 2+ ( 2) + (a - b2)
ata z ay2 ax h axay X axaz
(a2 + — 2 + + (+ - C2) + (/ - C2)
(B) = 2 ~j+a + a2)+ +( + - ( c) aya
aw a2u a2W aw x' a2 a_2v
at= =b22 +ay a2 + azax v azay'
1 ~a 2 u a 2 u + 2 ( a U a u + / ( 2 ) a -V, ~ ( a 2. a 2 w
at2 ax2 ay2 azx axay X
ac v a2a v 72 a2V\ a a aVv a2./
(C2 = j-2 V + b2 ay2 + (7 - b2) - + - ( - b2)
2c) aX Wzay2-^ ~a^a^)+ a^~ ayax'
ura h of line etry. This m r vw a2
tions of ontat transformations to optic as.
at2 az2 X2 ay2 axay v azy
C(D) oueves to od a sp o a2V e eie X a2p
a2w a2w (^ a9-v a2w\? a2î u,1z, a2v
However, these half-dozen possibilities or tentatives towards the solution of the
problem of finding contact transformations which leave the family of wave surfaces
invariant have so far yielded nrio further result than the trivial one formed by the
repetitions of the reciprocal apsidal transformation.
13. Assuming the rectilineal propagation of light the theory of optics becomes a
branch of line-geometry. This familiar view opens up other possibilities in the applications of contact transformations to optics.
Confining ourselves to ordinary space for convenience of expression these applications
may be made either by means of the contact transformations which change straight
lines into such, or by means of other correspondences set up by contact transformations
between two spaces such that straight lines are changed into the elements of some
other four-dimensional manifoldness.
34-2