110 



from F, (p, by changing « to «, . The cosines «, y, may therefore be 

 thus developed, 



d" /dip, \"+ 1 

 dct" ' \dit, ) 



» = », + 2(„," j-„ ^ 1 -|„ ,. 1 ^2^ -j- J)»+ 1 



^ da," ' da, \det, ) 



(HO 



if we put 



y, = ^ 1 — «/ • 



And since F may be thus expressed, 

 because 



dV=y. {udx, 4- yf^r,) = fx-a (z, + J) rf«, + //■ (««, + y) dz, , 



and because F becomes fi.z, — fji,WW when «^ = 0, we find, finally, 



d« /d^,Y+^ 



— = z, — fr(o) + 



_^j_±j^W_ ^-UJ . (K') 



This form (K') for the integral of the partial differential equation 

 (C), may be verified by observing that it satisfies the condition 



1 dV 



^ = -. + y, (LO 



f/, dz, 



V being differentiated for z, and a, as two independent variables ; 

 because 



d^, , =. («+*L , , »/' i!?^ _ ^ + „ 





