} 



(Q(6)) 



160 



\da.db) \da^ ^\da.db) ) 



\da.db) yda" ^ \da.db) ) 

 and reduces the expression (PW) to the following form, 



da^Kdb' \da.db) (Rf«0 



d^\ 

 I [-x-j'-rA-i— '•f" "- . I - ' 1 \- ^WIUI ^- -'101 ^- -'- • V- ' " XJ / d'<I>\*— 2 



in which the first sum contains only four terms, and the second only two, however great may be 

 the value of (t). And if we multiply this expression (R(6)) by M^''+'.r«'-2''y'r2''+3, and sum 

 with reference to (t), from t = to t = oo , we find 



_oo . 2r+l . 1— 2r ,27+3 



^0 •'"2r+l,2r+l« ^ *" '/'' ^ = 



'«■'•> 2 da^-Kdb' \da.db' 



d<i \^ (S(6)) 



»— 2 



T-j + I J— T/ ) )• We might easily extend the principles of these 



summations, but it is better to make use of the results to which we have already arrived, for 

 the solution of our second problem. 



(II.) We proposed, first, to find general expressions for the polar functions mW, iu('>, which 

 enter as coefficients into the developments (T'"), and to examine what negative powers they 

 contain of the sine of the polar angle v ; and secondly, to eliminate these negative powers, and so 

 to transform the series (T'") into others which shall contain none but positive powers of any 

 variable quantity. The I=^ of these problems has been completely resolved by the discussions 

 in which we have just been engaged. We have seen that the functions «('', lo'*', are of the 

 form 



2 ' denoting a summation with reference to i! from f = to t'=zt; U(^r, tvt/, constants, 

 which we have given general formulae to determine ; and u, w, functions, which in the notation 



