32 
Proceedings of Eoyal Society of Edinhurgli. [dec. 5, 
21. Going back to (29), taking Q great enough to allow W/Q to 
be neglected, and simplifying by (46), (43), and (44) we find 
Q = " 
ii‘ 
a (d y 
(53); 
and the problem (§17) of determining Z, m, w, subject to (5) and 
(6), to make P'ja + + n^^jy, a maximum or minimum for given 
values of A, /x, v, yields the equation 
Z m 71 
■3S-X — •st7 h 0 ; - 7!r'm H = 9 > (^^)> 
33-, denoting indeterminate multipliers; whence 
i2 ^2 
73 - — 1" "W “1 
a (d y 
r-2 = l^\ 
73X — I 
■srix = m 
P rrP n^\] 
^ 7/ 
P 1 - 
/ Z2 m- 
OTj/ = ^^ I )- 
\ a 
^2 
’ 1 - 
) 
(55) , 
(56) , 
(57) . 
P y 
These formulas are not directly convenient for finding Z, m, %, 
from A, /X, 7/, given (the ordinary formulas for doing so need not be 
written here) ; but they give A, /x, v explicitly in terms of Z, m, n, 
supposed known ; that is to say, they solve the problem of finding 
the wave-front of the simple laminar wave whose direction of vibra- 
tion is (Z, m, n). The velocity is given by 
P 
„ P 7}P 
v^ = H - + -^ + — 
a (d y 
(58). 
It is interesting to notice that this depends solely on the direction 
of the line of vibration ; and that (except in special cases, of partial 
or complete isotropy) there is just one wave-front for any given line 
of vibration. These are precisely in every detail the conditions of 
Fresnel’s Kinematics of Double Eefraction. 
22. Going back to (35) and (36), let us see if we can fit them to 
double refraction with line of vibration in the plane of polarisation. 
This would require (36) to be the ordinary ray, and therefore re- 
