344 Sir W. Thomson on the Propagation of Laminar 



xav ^= 2L 1 dx w=SSSS« ( ^;/ 2 f cos (ny +/) cos {qz+g) . (10), 



xzav m = {-^ J J cZsr/fo w =22*^^ cos (n?/ +/) . (11), 

 xyzav ?«= (jl) | J I dzd ydxu = u^^ . . (12), 



(«,/,ff) 



xav W 2 = iSSSSSS[ a ^;%] cos a (ny+/) coa*(qz+g) . (13); 



this with the exceptions that 



in the case of m=0, = 0, we take in place of |, 

 and in the case of m=0, e—jTr „ 1 „ „ . 



xzav« 2 =i^l22S[«;: / ^ ) ] 2 cos 2 (^+/) • • (H), 



mn q *■»»»-' 



xzav wt? = i SSSS2 [« (m> n> £ (TO> M> ff) 



m n q 



-$£»% 'ffcit cos (ny+f) sin (»y+/) . (15) ; 

 with the exceptions for (14) that 



in the case of m = and e=0 ~) ' . 



. j* we take instead of ± ; 

 = £71- J 



and in the case of q = and 



in the case of m = and 6 = ^7r 1 



and in the case of q =0 and g = J " ¥ " " ¥ ; 



in the case of m=0, e— \ir ) 7i = 7 /=^7r „ 1 „ „ \ ; 

 and analogous exceptions for (15). 



xyzavw^isll^Sra^'^] 2 . . . (16), 



mn q L ( m > n > 2) J 



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

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

 xyzav of (5). Thus we find 



nZ */£ v ^f ie,f,g) (e,f,g) (e,f,g)\ 



— xyzavV^:> = £2,2,2,2,2)2; S ma + n/3 + qy > , 



^ „ 5 l (m,n, 2 ) (m,»,«) *'{m,n,q)J S (17). 



= 0by (9). 

 The interpretation is obvious. 



