366 



Dr. J. W. Nicholson on the Scattering 





where 



~ + cotha^ +(o) 2 cosh 2 a-X)A = . 

 a ex." da y 



(3) 





^ + cot /3~+(\-co 9 cos 2 f3)B = 0; . . 

 dp" dp 



(4) 



the summation being for all possible values of X, which have 

 been determined by Mven from the consideration that ty 

 must recur on moving round any confocal spheroid. They 

 will here be chosen to make i|r finite everywhere, the two 

 methods being really equivalent. 



When (0 = 0, X=n . n + 1, n being a positive integer ; and 

 therefore when o) is small, 



\=n.n + l + e/ + S )1 &) 4 + (5) 



Writing cos/3 = ^, and H n (/i) for a typical function B(/3), 



U 1 -^}**-****' - ■ ■ (6) 



H„ is a non-hypergeometric type of zonal harmonic, and 

 cannot be simply expressed unless co is small, a condition 

 valid in this discussion. 



Calculation of the Harmonic Functions. 



Writing H 7l = P ?l (l + o) 2 E ;i ), and neglecting &> 4 , it is readily 

 shown that 



E -=j"l = *Tvj* ( *'"'- >P "'*' ' ' "> 



In order that E may be finite when //,= +1, 1—fi 2 must 



be a factor of ~ — e /f, and therefore 

 o 



e =l E (m) = -^. 



The lower limit has been ignored, for any value may be 

 given to it, so far as the order o> 2 is concerned. 



This indefiniteness is to be expected, and does not vitiate 

 the harmonic expansion subsequently used, for it only repre- 

 sents multiplication by a constant series in powers of co. 



In all subsequent work, the harmonics are all so chosen, 

 for clear definition, that the lower limit may be ignored. 

 The corresponding associated functions are then selected in 

 accordance with these harmonics. 



Thi 



