106 



and therefore 



J_ ( <p<p, /dip, Y + 1 (d^Y + M - '^y^ f^Y + ' f^Y + ' 



dz, \d»,dfi\d», J \dlij ) ~ dx,d^\d»J {'//ij 



+ r+rf«,;u^j dfr \d^) ^v'^wJw "^/w 



by which means the difference of the developments (S) and (T) 

 becomes 



d v_ 



5^---(««/+^^. 



i d^//d^Y+^dy,\ d^f /d^Y+^dy^\ ) 



^/)=r/ + 2^,j o" j dccr\[d»J d^j </A"VW ^ y ( 



C [ji4- 1]»J 1 (j, ^ ^/'+i + [n + l]''+i (2, + B)''+i" > 



+2 „ ". » rf<rf^,"' \du,dii\d»j w>j_jj»\d/3j ^/ {d»J_;^dfiXd» ) T»Sd k, ) \ 



[n + l]Mi [n' + Ijx'+i (z, -f J)»+i (2^4- .O-'+'i^ ' 



and the series in this second member being exactly that which would 

 result as the development of 



y=. ^ i — cc^—fi^. 



from the formula (ZT), we see that the condition (Q) or (R) is satis- 

 lied, and the sought verification is obtained. 



Another verification of the foregoing developments may be 

 obtained by applying the general expresssion in series (A^), for any 

 function F of the cosines «, /S, to the case where this function is = 

 ■£-• We find, first 



dp dip, , A j(l//f[?^y' + >f^A d^_i ,d<p.n^\ d'tp, ^-^ 



^'^".S '^''^' (d<l>y'^'fdP,Y'^\d'<p,/d<pY+\d fd<p,Y'^\ d^<p, ,d<p.-^'id ,d<p.'"H-) 

 + 2 "."d u-dfirld'c^'dfiXdJ \d^J '^dcc^\d/ 3j d/3/ [dj -^d^\,dZ') di ' {d^} \ 



