Okk — Extensions of Fourier's and the Bessel- Fourier Theorems. 29 



Corresponding to every value of ^i, whether characteristic or not, (35) gives 

 two values of i' ; and it will usually be convenient to distinguish between them. 

 I shall use the plus sign with the radical to denote that root whose argument 

 is half that of {gfx- - /)■ + 4cV^ and I shall denote the corresponding v by v, ; 

 the other value of v I shall denote by 1/2. We shall consider large values of fi, 

 other than characteristic ones, and for all such we have the asymptotic equations 



V, =^,/-/+cV,9', (37) 



v^ = - c^lg, (38) 



in which the errors are of the order fx~'. 



The characteristic values of fj. may be divided into two sets associated 

 respectively with characteristic values of v of the types vi, vo, and for each 

 set we evidently have asymptotic equations of the form 



,1 = + 7n.ni/(b - «,), (39) 



where m is any large positive integer. The errors in these equations are of 

 order not exceeding ?»-'. 



The successive characteristic values of v of type v^ thus tend to have 

 negative real parts which increase indefinitely, and those of type I'a tend to a 

 real negative limit. 



There are four values of fx which are branch-points for r, viz.: those for 

 which {gfx' -/y + ic-fi' vanishes. 



Taking the determinant Aa, let us form from it the elements of another 

 column by an extension of the method by which the elements (13), etc., 

 were obtained. Expressing Aa(Z') in the form 



Am MJ)) MB) 



MB) MB) MB) 



(40) 



in which 



MB) - MuB' + knB + A„ - a {c' + gB)dldx, (41) 



let us write down 



(c^ + Qv) /.. {B) - {& + gB) /„ {v) { e^ gv) MP) - jc' + gB) ^jv) 



^— _ y, + __ y, 



, (c' + gv)MD) - (c' + gP) Mv) ,. _, ao^ 



(c' + gv) MB) - (c' + gB ) Mv) (c^ + g<') MB) - (c- + gB) M") .. 



B - V y' + w- V y^ 



(c' + gv)MB) - (c' + gB) f,,{v) 



+ ft 2/3 + . ■ ., (■i'5) 



JJ — V 



etc., and afterwards replace each Dy by the corresponding v. 



