135 



2d 0= — r. sin. D + \, ,^ . (/-"".a+V 



rfa* ^ ^ ' 



In order that these two equations should be identically satisfied, we must have, in the first place, 

 for the exponents of tlie lowest powers of r in the developments (T'") 



?«= 1, w = ^; 

 and for the corresponding coefficients, (m, w), 



/i\ « , dx' , , d^x' 



{1)...0 = -. COS. «+-^.„ + x ___.,„., 



(2)...0 = _sin.«+i.-^. r«% 



-that is, in the notation of the preceding paragraph, 



25i5 J. cos. V + 28{a. M = -Bto% (U'") 



2g2'.sin. c = C.w*. 



In a similar manner we find for the next greater exponents, m' = |, n' = 1 ; and for the cor- 

 responding coefficients, u', to', 



r da da.db db* 6 db^ 



I uu uu.au ao 



And so proceeding, we can find as many of the exponents and coefficients of the developments 

 (T'"), as may be necessary ; the exponents forming the two following series, 



m=\, m' = |, m"=2, >«(')=: *+^ 



2 

 «= I, n' = 1, n"=| mW = 



VOL. XV. 



