88 Mr smith, on the INVESTIGATION OF THE EQUATION 



Multiply (5) by A, (6) by B, and subtract the sum froiu (-i), making 

 the coefficients of each differential equal to zero. We thus obtain 



x = Al +-^ (7). 



I- - a 



. Bm ,o\ 



y = Am + , _ ., ■••• (o;» 



t'- 



% = A)l +— :. 19)> 



V — c 



-«"{(^rH.^)'+fe)l "">• 



(7)/+(8)»« + (9)m gives 



Ix + my + Mz = A {I' + m' + n) + B \;^—. + p-rjs + -:^_c 



which by equations (1), (2), (3), is reduced to 



v = A ; 

 (7)' + (8)^ + (97 gives 



af + y' + z' = A'(l'' + m'^n') + 2AB {- :. + -r-r. + -;r~3 



•^ ' Vi)"^ - a- r- — »■ V- — c I 



which by (2), (3), (10) is reduced to 



af + y- + x^ = A'' + —, 



putting r' for .r' + y'+as^ and for A its value r, this becomes 



B = v{r-v'). 



Substituting these values of A and B, equation (7) becomes 



r^ — v' 





or x(d'-v'') = fv{a'-r'). 



