58 MAJOR P. A. MACMAHON: MEMOIR ON THE 



This may be written 



{o- 4 + 2 (X + /*) 03+(X 2 + /t s ) <r,+ 2\po f a +2\p. (X + /t) 0-, + XV 2 }", 

 0-4 = pa-^paPi + 4 (p + j) + 2 (pj 2 + 7T, 2 ) - 8p, + 3, 



0-3 = ^2^1 - ( P2 + 7I- 2 ) - ( p^ + TTj 2 ) + 5pi - 2, 



o- 2 =j>! 2 - 4^ + 2, 



where 



For the uneven order 2>i + l it is easy to show that the coefficient is symbolically 



2X^ (X + ya) o-^XV}" x ( ^- 

 It is easy to calculate the values of 



KpiVV 



for small values of <t, />, c, d. 



Some results are, omitting the obvious result 2^1 b ', 



