( 



LEWIS. — ON FOUR-DIMENSIONAL VECTOR ANALYSIS. 179 



= //l2kl2 + //l3kl3 + //23ko3 + En^u + E^^Ki + ii'34k34. (75) 



By our fundamental equation (69) we have 



CM = q. 



Expanding this equation by (50) with the aid of (75) gives four 

 equations 



\ 3^-2 dxs dXi ) c 



\ dxi dxz dd\ J c 



dflsi 57/32 dEsi \ ^ _ P ^, jj 



dxi 3^2 dXi J ^ c ^ ^' 



fdE.r , 3^42 , dE,A , ., 



\ d.Vi 6x2 dx3 J 



Collecting the first three equations into one, with the aid of (25), (63) 

 and (64), and writing id for a\ gives 



and the last equation by (24) and (64) changing E^ to — En, etc., 



gives V e = p. (78) 



By a similar expansion of equation (70) 



OxM = 0, 

 we find with the aid of (49) 



1 5h ^ 



V xe + - — = 0, (79) 



c at ^ ^ 



and vli = 0. (80) 



Equations 77-80, in the more familiar notation, are the well-known 

 equations, 



curl h — ~ — = ~v, (a) 



(r) 



(S) 



