ON THE VORTEX THEOET OF THE LUMINIFEROUS iETHEE. 487 



4. Let xav, xzav, xyzav denote space-averages, linear, surface, and 

 solid, through infinitely great spaces, defined and illustrated by examples, 

 each worked out from (ti), (7), (8), as follows, L denoting an infinitely 

 great length, or a very great multiple of whichever of m~\ }i~\ q~^ may 

 be concerned : — 



xav«= -!.[ dx 2i='S,^^I.a^^ff] cos (ny+f) cos (qz + <j) . (10), 



xyzavM= f_j ds dij dx u — a ^f^o^'';^] . . . (12), 



xavu'-=^2^i:Z^:^la^^;^-^;l]]cos^,>y+f) cos'(q::+g) . . (13); 



this with the exceptions that 



in the case of m=0, e=0, we take in place of h, 

 and in the case of m = 0, e=^7r ,, 1 5; » • 



xzav u'-=i i:2:s2S2 K;-;;;;J cos^ (>'y+f) ■ - ■ (1^), 



xzav uv=i ih^^ [a''-"^'"] 13';' ^' f. 



— «(«,«,7) Am,«,,)] COS (ny+f) sm (ny+f) . (lo) ; 



witb the exceptions for (14) that 



in the case of m^=0 and e=0 "I , i a • j. i c i 



J . , , c A 1 1 ^ '^ve take instead of + : 



and m the case or g=v and g=^Tr J 



in the case cf m=0 and e=|ir"l j^ 2. 



and in the case of q=0 and g'=0 j " a " >' 4 ' 



in the case of 7n=0, e=i7r, ?i=0, /=v~ ,, 1 ,, >, 4 ; 



and analogous exceptions for (15). 



xyzavH^=|MvsS3:[a;;-;;;g'. . . (IC), 



with exceptions for zeros of m and q, analogous to those of (14). 



5. As a last example of averagings for the present, take xyzav of (5). 

 Thus we find 



-xyzavv^i.=i^2v^Es|.m;:;-:-;+<;^;;;;+<?y;:;^^^ 



=0 by (9). 



The interpretation is obvious. 



6. Remark, as a general property of tlais kind of averaging, 



xav'^=0 (18), 



ax 



Q be any quantity which is finite for infinitely great values of x. 



(17). 



