190 Dr, E. W. Hobson. On a Type of Spherical [Jan. 1(5, 



known as the differential equation of Legendre's associated functions ; 

 the degree n, the order m, and the argument jx are not, as in the case 

 of the ordinary system of spherical harmonics, restricted to be real 

 and such that n and m are integral and jx is a proper fraction, but are 

 supposed to have unrestricted real or complex values. The investiga- 

 tion is undertaken with the object of bringing the various types of 

 harmonics, such as toroidal functions, conal harmonics, &c, under 

 one general treatment. 



The two particular functions P ra m (,a), Q B W W, which satisfy the 

 above differential equation are first defined in such a manner that 

 they are uniform over the whole /i-plane, which, however, has a 

 cross-cut extending along the real axis from fx = l to fx == — oo. 



The definitions obtained are the following — 



__ e- n " L . n(^+m) 7 .j_i\hni[ < J i+ > 1+,/*- i~) 



Q» m O) 



dt 



e -(n+i)«r n(n + m) , ■ N , f(-i + ,i-) /0 v 



where in (/* 2 — the phases of — 1, /t-f 1, are both zero when is 

 real and greater than unity, and each varies between the values + tt 

 for various positions of the point /x. Precise definitions are given of 

 the meanings to be attached to the integrands. The path of integra- 

 tion in the case of P^'"^) consists of a loop described in the positive 

 direction round the point t = ji, followed by one in the positive 

 direction round the point t = 1, then a loop in the negative direction 

 round the point t = fx, and finally a loop in the negative direction 

 round the point t = 1, the whole forming a closed path, i.e., one for 

 which the integrand attains its initial value after a complete descrip- 

 tion. In the case of Q» a 0*) only two loops are required to form the 

 closed path, one described positively round t = — 1, followed by one 

 described negatively round t == +1. These definitions are so chosen 

 that in the case of real integral values of n and m, the functions 

 coincide with the ordinary well-known Legendre's associated functions. 



From these definitions the following representations of the functions 

 by series are deduced — ■ 



7re m7rt 1 r + 



. J gTHTTl l ! 



sin (n-j-m) tt ][( — m) \ \fi—l/ 



