BesseU Clifford Function, and its a} 'plication*. 509 

 and put p 2 r 2 = 4^, 





• • (2) 



»=6C|(*)«BC|(ipV)., 



subject to the surface condition 



du 



dv 



?>hi '6p 2 ku 



= —i — t = — ---o- 5 at ? J =0, 

 A: -i- in n-p 



■ ■ (3) 



and 6 = for a solid sphere, to make u finite at the centre 

 r=Q. Then 



du du 



c «( i/)V)= M'c f (i jW v) ) 



leading to the period equation 



n 2 p 



■ w 



tan pa 

 pa 



1 



|£^a 2 

 n 2 p 



ip 4 ka 2 



(5) 



l-i^v- 



/ry9 



9. The differential equations of Hiccati and Bessel were 

 compared in the ' Quarterly Journal o£ Mathematics/ xvi. 

 1879, p. 294 ; and starting from the canonical form (4) § 5, 



1 dhj 



y dz* ~ L ~ L ~ ' 



(1) 



this changes into Riccati's form 



— +bu 2 = cz m , on putting - -£ '= u, and lc = -be ; (2) 



or into 



dv 

 z — —v-\-bv 2 = c£P, putting v = uz, m + 2 ==p ; . . (3) 



or into 



dv 

 x-j av-t b'v 2 = c'zw, putting c = .v a , b' = ab y c' = ac : (4) 



Boole's form, 



