1889.] A. Mukliopadhyay— ii/Z/Zpfic Functions and Memi Values. 215 

 which, gives 



z' dx dy = \/r3-p2 p dp dta, 

 and, by the same substitution, (5) is transformed into 



Similarly, to integrate z" dx dy, put 



- = pcosa), f=psinw, 

 a b ' 



which gives 



z" dx d.y = aho \/ 1 ~ p dp dw, 

 and the same substitution transforms (5) into 



p"= / S ) 



V I «2cos2a)+62sin2o.-cS \ C^) 

 By these two substitutions, the formula (4) becomes 



J^J^A/r2-p8 p dp do, 



In the first double integral, the limits are 



P=0 ) 



w = 0 



p=p' 5 









the second 





P = 0 ) 





p=p" 5 









Xp' 

 v/)^2— 2 p ^„ = _ £! ^ a^Z'^ - '-^f?'^' cos2 (0 + a2 sin^ <o) ^ I 

 3 3 ^ a^i^ - c2 ( 62 cos^ « + a2 sin« co) 5 



Xp" 

 — 1 C a2cos2co + 62sin2«,-r2> * 



The formula (8) reduces to 



y=^ (r3-a&c)-ic5A+ia6cB 

 where the values of A and B are given by 



A= i "^^^-''^( ^^^ cos2(o + a2 sin^o)) ^ 'S" 



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