596 



where V represents the same function of (v) as U oi {x)\nïovm. [Q). 

 The coefficients A are then determined by the expression: 



00 V. 



s-i = 



-00 X 



n ! m ! 



2'"+" 



(16) 



Substituting again for x and ?/, R sin and RcosO, then, by 

 integration svith respect (0 <9, all uneven polynomia vanish and, 

 because 



27t 



/ 



(2,0' 



da ^=z , 



cos^^ a 2'^" . n ! n ! 



we find 



2ti 



ƒ 



2jr (2n)!(2m)! „,,„ 



"(i/i2)2('«+«; 



(//fl)2C'«+»-l) 



{>u-\-n)\ 



(HRfOn+n-i) 



(17) 



etc. 



(m + n — 1)! ' '■■* (m + n — 2)! 



i.e. the same expression as 13'', found in a different way. 



As to the determination of the A coeflicients, it is expedient to 

 consider first the case that a and b are equal to zero. 

 It is then easily found that 



(2n)! (2n)\ ^ 



2" . n ! 

 and similarly for the V function 

 (2m)! 



2n.Ml 



'So.2n = 



2"* . m ! 



(2m)! ^ 



2A* 



il/'^ = 



2'" . m ! 



1 

 2A^* 



The arbitrary constant H now can be given such a value that 

 P or Q = 0; putting P=0, then H z:zz h, and in the development 

 only the V functions remain. 



If a and b are different from zero, then it appears that (for P:=0) 



v) S^ — 3Q'^6 Irb'Q + h'b* 



I 5, z=15 Q« + 45 h'b'Q' + 15 h'b'Q + A'è' 

 or, generally : 



