ON DOUBLE REFRACTION. 275 
Denoting for shortness the partial differential coefficients of —q with 
respect to , ap ee by 7 7a) tq) &e., we have 
du\ . du du 5 ae 
rice Ta)? da tf (783 dy* 
du dou Jas 
aes (i) we +#(%) ay 
whence 
=| dee dy dz =|\\r(z FS du a de dy aer(\rG f' i) \ae a ti ee. 
=({+( i) du dy dz + Wr (=) a ) au dz dx \r(z A ) bu dx dy 
Wr aE bu dy dz+ &e. 
(du du a:)+ fav 
-\\\{ sa af (G+ 3, dys \a 7) +8" af (a)+ av af r(x) 
+60} de dy dz. 
We must now equate to zero separately the terms in our equation involving 
triple and those involving double integrals. The result obtained from the 
former further requires that the coefficient of each of the independent quan- 
tities éu, dv, dw under the sign lie yanish separately, whence 
Pu du du du\ ) 
dt =i z x) +5 tay f a) +z: Tez i ) 
du d du du, d (dv “ 
dé ~ dx t(a)+% t(G;,)+ Taz r(z) “ue rahe 
Pw id ,f(dw\ d ,(dw\ d (dw ; 
we ata 1 (da) + ay (ay) +a Fae) 
equations which may be written in an abbreviated form as follows :— 
Cu ” dg| uv oe do} Cw_ do 
EA a At ele Eat nF Nd 
where the expressions within crotchets denote differential coefficients taken in 
a conventional sense, namely by treating in the differentiation the symbols 
aq@dd 
da’ dy? dz as if they were mere literal coefficients, and prefixing to the whole 
term, and now regarding as a real symbol of differentiation, whichever of 
these three symbols was attached to the w, v, or w that disappeared by differ- 
entiation. 
The equating of the double integrals gives 
{feas—(['r( =) du dy dz ala a) du dz da+c&e, 
=(j| [i “+m mf" i; +f =z ae) ut (oe. dv+[&e.] au | as 
* These agree with Professor Haughton’s equations (5), : 
T 
