206 The Rev. M. O'Brien on the Propagation 



+ *'('')p')(•*■,-.^■ + «;-«)]> 

 = 2f {ffi (/(;■; S a +/' (r) p S.r) + m, {<^ [r) («, - a) 



observing that by the condition of previous equilibrium we 

 have 



S {vif{r)lx + m,<^ (?•') {x, - x)} = 0. 

 Now we evidently have approximately, 



p = ~ [Ix lot. + ^ j/ S /3 + ozly), 



a"tl P'= 7- ((•■^•/-^) («/-«) + (j//-i^) {^-l3) + {~-z){y,~y)). 

 Hence our equation becomes 

 ^= Z-m|/(r)8« + -i/'(r)(8^^8« + 8^8^.8^ 



+ 8.r83.8y)| 



+ ^m, ^^ (r') (a^ _ «) + ^ c^' (,•') {{w, - xf {u-u) 



+ {^'--v) {y-y) (/3 -/3) + (.r -^0 [z-z) (r,-y))| 5 

 and aimilarly we shall obtain equations for 

 d^ d^ d^^ d^^ d^, 

 dt^ dt^ dt^ dt^ dt^' 

 We shall now substitute in these equations for 8 a. its value 

 da. ^ da. ^ da., d'^ a ^ .v^ d^ a 



dx dy '^ dx " dx^ 1 dxdy ^ 



d"^ a ^ . d"^ u htfi . „ 



and similar values for 8 /3 8 y. 



When we have made these substitutions it is evident that 

 the several differential coefficients of « /3 y a^ /3; y, may be 

 brought outside the sign of summation. The result of all this 

 will be six linear differential equations composed of the partial 

 differential coefficients of a /3 y, «; /3^ y, multiplied by quantities 

 such as 2 mfir) t x,'% mf{r) 8 x-, &c. Now it is evident that 

 these quantities must be different for different values of xj/ z, 

 Xft/i Z/ except in the cases represented by figures 1, 2, and 3 ; 



