184 
MM phe" x i DPD: SF (&s Ys 2): (23) 
One only of the exponents, 7, m, n, can ‘usefully be odd, by 
properties of the mean function, which have been already 
stated. If all be even, and if we make 
= 2A, m= 2u, r= 2v, (24) 
the corresponding part of the general term of Wf namely, 
the part independent of zk, is by (15), (18), (19), 
(— *)* 
(2x) 
whereof the sum, relatively to A, u, v, when their sum xk is 
given, is, 
{r; My v} DOE D? f(z. Y 2) 3 (25) 
‘Got (Di + D3 + Dif (@, y, 2) = & = hi 
if my signification of < be adopted, so that 
4=1D,+jD.,+kDs; (27) 
and another summation, performed on (26), with respect to x, 
gives, for the part of M/fwhich is independent of ¢j/, the ex- 
pression, 
1 (6% +64) f(a, y, 2). (28) 
‘* Again, by supposing, in (23), 
12 ONS 1, WS Qu, BSD; (29) 
and by attending to (20), we obtain the term, 
wiD, (—w*)*, 
Gee Tyh Doro vi DEDEDE f(a, 9.2) (80) 
Adding the two other general terms correspondent, in which 
iD, is replaced by jP, and by kD,, we change iD, to q; and 
obtain, by a first summation, the term 
(wa) 
(2«+1)! 
and, by a second summation, we obtain 
I (45 Ys Z) 5 (31) 1 
