172 PROCEEDINGS OF THE AMERICAN ACADEMY. 



vA = P^^ + '^) k. + (^^' + ^) k. + Cih, + ^Ak.. 



\ O.r^ dJs J \ 6Xx oxz J \ OXx 0X2 J 



(26) 



VxA =!(—-- + -- V- — ki23. (27) 



\ 0.^1 0x2 0X3 J 



When the quantity operated upon by V contains two or more varia- 

 bles it may be expanded in terms of its components and these scalar 

 quantities may then be differentiated in the ordinary way. We thus 

 obtain such equations as the following : 



V (^a) = </. ( V a) + a ( V <^), (28) 



V X (^a) = (^ ( V xa) + ( V <^) xa, (29) 



vx (axb) =1 bx (vxa) — ax (vxb), (30) 



By the above rules new operators may be formed from v such as a v , 

 A V , and V V or V ^ which may be applied to any scalar or vector. 

 The last is the well-known Laplacian operator and may obviously be 

 expanded by equation (11), 



a' a^ a" 



Other operations involving V twice are v(va) and v(vxa) or 

 V vxa. 



These quantities are connected by an important e(|uation which we 

 obtain by expanding according to (13), (23) and (10), namely 



V V xa =: V ( V a) — v -a. (32) 



Finally we have from (18a) and (15) the important identities, 



V X ( V xa) = 0. (33) 



vx(V(/>) = 0. (34) 



Equations (32), (33) and (34) are evidently equivalent to the fa- 

 miliar equations : ^'* 



curP a = grad (div a) — V'a, 

 div curl a == 0. 

 curl grad <^ = 0. 



" Abraliain-Foppl, 1, ofiuutions 95, »}, Ola. 



