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for calculation. If, for abridgment, we write them thus, 



«{»,«'), toC«.«'), denoting the terms of «<,,/, w,,t,, which correspond to any given values of the 

 integers n, n' ; we have, by (O^), 



„g.nO = tO]-. [0]-[0]-«.[0]-.'.2»' ^,,^/+n-l^;«-l ' ^ ^^,,^ 



^(^„0 = [0]-*. [0]-". [0]-. [0]-"'.2"' ^,>^^.'+n-l^^,n'-l = 



and if in (N(S)) we change^ <p, to their values (IW), 

 we find the following developments, 



nn.n') — 2.«".»') ; ,(".») =:2.«("'»'> ; (S('>) 



in which 





(TC5)) ■ 



„(».«' = |.[0] . [0]-" . « ^ ^ 1- . «".« » da-^M^ 



a &, being supposed to vanish after the differentiations, gnd »i, «', being any integer numbers, 

 but the only values of these integers which do not make the partial differentials 



vanish when « = 0, 2 = 0, are those for which, in ?^_'^;), 



ts= 2« + m' + l — 2(n + n'), <' + n — 1 = w, 



and, in /«■"?, 



< = 2m + wi'+2 — 2(n 4- k'), ^'+m — 1 = »«; 



we have, therefore, when < = 0, «sr_0. 



