105 



( [«]« (» + 2) {z, + .4)" ^-2 + [n]» (n + 2) (2, + B)" l-« 3 



"♦■ (», »■) 0,' (-„ ^ J-]„+l |-„' _^ l-|„'+l (z, + J)M 1 [z, + /?)»■+' 



_2 °°. f p + ^ + ''^ + . ) du.J'd&r'\dx,d^\d»,) \dli,) ) (■s^ 



(n, n'l 0, V n J. 1 ~ m' _U 1 7 ■ — . ' ' '— \_JXiI 1 • l-'; 



and, by (L), 



f d^ /^\n+l • a ^ /(/», X" + 1 1 



(«)0 ^ |-„ ^ j-]„^.l (^^ ^ ^)„^.l i--|-^ ^ ]-]„.l 1 (2^ ^ £)» + ! j 



"^ *' "(».«') 0.' [-^p [•„, ^ l|„. 1- 1 (z^ ^ ^)» -t-i (2^ 4. £)»' + 1 



d^'J-"- / d'(p, .d(l>,.n+ l/cf^N"'X 

 , 3 _ 0. 0. d»,'^dli«\du.,d^\dcc,) W0,) / . (T) 



while the development of 



»", + M 



may be deduced from the general formula (K) by changing 



F («, /3) to 7 = -/ 1 — «' — /3S 



and 



, F,toy,= -/l-«/_,3/. 



To compare these several developments, and to examine whether 

 they satisfy the condition QK), we are to observe, that from the 

 nature of the function <p, we have by the foregoing number. 



yj_. ^'^1 -n ,^yj_. ^f> - ^°y/ . \ 



'», ' rf/3,rfz, '^''^d^/ d»,dli,dz,- d»,dfi, ' J 



da, dz, di 



VOL. XVI. B 



(U) 



