184 
yim 
Mime x = DD" Dt f(a, y, 2). (23) 
m\n! 
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 
1-2. — an, — 2y, (24) 
the corresponding part of the general term of Mf namely, 
the part independent of 7k, is by (15), (18), (19), 
(2x) ! 
whereof the sum, relatively to A, u, v, when theer sum x is 
given, is, 
(Ay ps vj} DX DY D3 f(@, Ys 2) 3 (25) 
Gat (Di + Dz + D3)" f (& Ys Z) = cares (2; > z), (26) 
if my signification of < be adopted, so that 
4 =tD,+jD,+kD;; (27) 
and another summation, performed on (26), with respect to x, 
gives, for the part of Mf which is independent of 7h, the ex- 
pression, 
(e044 64) (0, y, 2). (28) 
‘«¢ Again, by supposing, in (23), 
l=2A +1, m=2u, n=2p, (29) 
and by attending to (20), we obtain the term, 
wiD, (—w*): 
“Gesiyt ety) DPDEDY Say). (0) 
Adding the two other general terms correspondent, in which 
iD, is replaced by jD, and by £D;, we change iD, to q; and 
obtain, by a first summation, the term 
(wa jee 
(2e+1)! 
and, by a second summation, we obtain 
AERO ELA © (31) ; 
