1894.] connected ivith Jesseral Harmonics, with Applications. 47 



All the relations connecting u,,,,-, v lltr , &c., are duplicate ones, similar 

 relations being obtained by interchanging u and v. 



The differential equation satisfied by either function is of the fourth 

 order, the two functions being different solutions of this equation. 

 'The two remaining solutions of the equation have also been obtained, 

 and called " functions of the second kind." The equation of finite 

 differences satisfied by the functions is also of the fourth order. 



The general value of u Mr is 



2U, nr i (^n. 



( 



\ 



Y .rm (m + r2k) P (n,m + r2k, 



,m k + sl ! m + rk 1 ! n + ks ! r k+s l! k 1 ! 

 m k ! m + r k s ! n k + s ! r k \ s 1 ! k s ! s ! 





^?\ 

 2k) 



n + m ! n 



-,-,., o7 ^ 



P(n,m + r 2k,ft) 



*~^~ , ^ 



k lln+rkslr 



mk\ m+rkHYnr+k+s ! r k >-s ! s ! k 1 s-^1 ! 



, , > -r-,, -. 



-f(-l) r . - P (,m-r,/t), 



nm I n + m r ! 



where I(r/2) is the greatest integer in r/2. 

 The value of v mr is given by 



v mr .sinp = ( : 



fc=0 





- 2k 1) P(n,m + r 2k 1,) 



I ! m+ r k 1 ! n+k s ! r k + s 1 ! k ! 



mk 1 ! m + r k s 1 ! n k + s ! r k 1 ! s \ ks ! s ! 



w m! n+m 



l ! P( +r _ 2fe _ 1 } 



^Q m A; i ! m+r /c s -1 ! r + k + s + 1 ! r A; T i UT^l 



Simpler values for u mr v mr are given for general values of m from 

 r = to r = 6 inclusive. 



The values of the functions u mr , v mr are of a simpler form when ft is 



