ON FARAD AYS LINES OF FORCE. 



201 



then 



a =a 



dV 

 dx' 



satisfy the condition 



V 7, dV 

 ' = b -j i 



dy 



,_ dV 

 C ~~ dz' 



da' db' dc' A 



-T- + -J-+ j- = 0; 

 ax dy dz 



and therefore we can find three functions A, B, C, and from these a, ft, y, so as 

 to satisfy the given equations. 



THEOREM VII. 

 The integral throughout infinity 



Q = /// (ii + &.A + c .yi) dxdydz, 

 where 0,6,0,, oj8,y, are any functions whatsoever, is capable of transformation into 



Q = + JJJ{4"7>/>, - (o,a a + &b, + y c a )} dxdydz, 

 in which the quantities are found from the equations 



da, db, dc, 



! + + y 1 

 dx dy dz 



af e determined from a,6,c, by the last theorem, so that 



dfr_dK + dV. 



dz dy dx ' 



a,6 s Cj are found from oj8,y, by the equations 



and p is found from the equation 



' ' ' 



VOL. I. 



26 



