84 
whence - 5) - 2 (ti- 3) + (w- 1) = 0. 
(^-5)-3(?^-3) + 3(?^-l)-(^^ + l) = 0. 
Also 
(71 - 5)(^ - 4) - 3(?^ - 3)(?i - 2) + 3 (ti - 1> - (?^ + l)(?^ + 2) = 0, 
whence 
{n - 7)(n - 6) - 4(?2 - 5)(n — 4) + 6(n - 3)(?i - 2) - 4(7^— l)n 
+ (n+l)(n + 2) = 0. 
and the general null value of the expressions can be inferred. 
TIL 
In general, we may find the terms in U and V correspon- 
ding to any particular term in u_i as in a similar way, 
it will only be necessary to consider a few. 
As before, we obtain consecutively 
^ ^ + 1 
I 0 02 „ 2 „ _ V X , o - 1 ) V X «(» -!)(»+ 1 )(« + 2 )m„ 
\AApU^S = ^-g 
4 = - + 3 
-3(7^ - l){n - 2)n(n + 1) vXi 7^(7^-l)(7^ - 2)(7^ + l)(7^ + 2){n + 3)Un 
1 4 2 - 4 (»»-2)(>t-3)V%„ g «(m-l)(m-2 )( »-3)v%„ 
I — ^ ® j-w— 3 — 1 j^w + 1 
» 4 - l)(7^ - 2)(tz - 3){n + 1)(72 + 2) V ^ 
+ »(» - 1)(» -i){n- 3)(m + l)- (w + 4K 
+ 5 
whence the consecutive coefficients can be found. 
Let us write %i=fi{x)fi{yy z\ so that 
V X = V ^fi{x)fiy, z) +f,{x) V J,{y, z) 
where /^.{y, z) is supposed homogeneous of the 'nth order. 
Substituting for y and 0 and writing for - x^- 
z) = p” I Asin nd + Bcos?i0 + &c. + constant j- 
where the constant will be zero if n be odd ; and may be 
considered according to the last paragraph as independent 
