236 
DR. W. F. SHEPPARD ON 
The new expressions are more convenient for calculation and for tabulation, since the 
coefficients are rather smaller and are symmetrically placed about a diagonal. For 
j = 2, m =13, for instance, the formulae given by “ Fitting,” § 5 (ii.), are 
lOOlt'o = + 693£ 1 -198S 2 +22 a%, 
1001 Av 0 = -2SlS 1 + 8SS 2 -llS 3 , 
1001 A 2 n 0 = + 35*%-15£ 2 + 2.%, 
where S 1 = ^"u 0 , S 2 = 2 //2 r 0 , S 3 = 2 //3 r 0 - If we write 2, = 2 "r 0 > = 2" 2 w 1 , 
Sa = 2" 3 m 2 , these become (by (136), or by writing .% = 2 1? S 2 = 2j + 2 2 , 
S 3 = 2 1 + 22 2 + 2 s ) 
lOOliy = +5172 1 -1542 2 + 222 3 , 
1001A i’o = — 1542j + 662 2 —112,3, 
1001 A 2 n 0 = +222 1 -112 3 + 22 3 . 
r 
The symmetry of the coefficients is due to the fact that 
co. 2 " 9+1 u g in A f v 0 = X fitJ = \ gtf = co. 2 " /+1 u f in A g v 0 . 
For any particular value of m there will be only (j +1) (j> + 2) coefficients to be 
tabulated, instead of (j +1 ) 2 . 
Appendix IV.—Formulae in terms of u s. 
(i.) Formulae for A f v 0 , &c., in terms of the u s have already been given in (1 5a), 
(21), (22), (29), and (28) of “ Fitting ” ; and the results in (141)—(145) of the present 
paper can he checked by comparing the different expressions for the coefficients of the 
u’s. We should require to use the following identities :— 
(r + h, h) = (r, 0) (h, 0) + {r, l) (A, l) + (r, 2) (h, 2)+ .... , 
(r, 27?] = (0, 2A] + [r, 2) (0, 2A-2] + [r,4) (0, 2A-4J + ... , 
[r, 27? — 1) - (r, 1] [±i 2h-2) + {r, 3] [±i 2A-4) + (r, 5] [±|, 2A-6) + ... , 
{r-h 2/i-l] = [r-i, 1) (0, 2A-2] + [r-£, 3) (0, 2/?-4] + [r-|, 5) (0, 2/? — 6] + ... , 
27? 2) = [±|-, 2h — 2) + (r—7£, 2] [ + £, 2h— 4) + (r— j, 4] [±|-, 27? — G) + .... 
(ii.) Taking, for instance, the formula for S 2/ v 0 when m = 2??. +1, (21) of “ Fitting ” 
gives (replacing t by f) 
trp\ — (_ )fL nl ! 2k + 1, 2J+1) h '' / u(. /+ 1) (7< + 1 ) {2k + 1, 2h + 1} (?‘, 
KPr) * { )2 Ob2/+l) ] ~f+h+t ' (Jm,2A+l] 
2A] 
