389 
Hence, equating and arranging terms, we get a linear partial differential 
equation in r and ¢ as dependent variables, and z, y, %, p, g as indepen- 
dent variables, namely, 
dg Lares ec te a) ear ss ar : Ad dr 
ar as dy I~ PG ) dz \°” ar dp dr } dq 
ds(dt dt | dt dt) _de de ds de («) 
~ at Farley dp * dq de dz? dp” dq 
ds ds ds de 
in which, of course, the co-efficients a my Be ap’ &c., are all known 
functions of x, y, £, p, g, 7, and ft. 
The second condition gives the corresponding equation 
dt ds dt do\ de (,_ jds\dt ,ds\ at 
dz ae dy" \P~ 1 ae jaz * at }dp \" "at a | 
calle dr ages ds ds) ds as, 
\ dy Iz dp dq dy dz q dp dq J 
; : ; ds d. 
in which, as before, the co-eflicients 7 “, &c., are known functions 
of x, ¥, &, P, q, 7,.and t. 
This system of simultaneous partial differential equations may be much 
simplified by writing 
d d a a 
real ear daa a 
d d 
d = 
i ee 
dp 
ds ds 
TS dea aw 
By making these substitutions, the given system of simultancous partial 
differential equations becomes, simply, 
Vio tle Nut PN ih Fi 
Me bo Tee hae var= Ff, | 
Hence, in gencral, we have a system of two simultaneous linear par- 
tial differential equations to determine r and ¢ in terms of p, q, x, y, 8. 
Supposing these found, it remains to substitute their values in 
