48 On Functions connected with Tesseral Harmonics. [May 24 r 



a right angle, and can be expressed by a single series. When n r is 

 even, the series 



n r,n + r 



w-_r - j } ml ml 



_ 2 l ? - - - 



. ( n + 2mr2t\n + r2t\ 



is the value of u mr ( T/2) when m is even, and of v M r(w/2) when m is 

 odd ; the series being continued until one of the factorials in the 

 denominator becomes negative ; and n being supposed greater than 

 2m. When n is less than 2m, the lower limit of t is m ^( + r). 



A similar series gives the values of , rtr (7r/2) when m is odd, and 

 f 2w(""/2) when m is even for the case when n r is odd. The values 

 of u mr (IT 1 2) when m is even and of Hir (7r/2) when m is odd, are in 

 this case equal to zero. 



When nr is even, the values of w wr (7r/2) when m is odd, and of 

 t- WJr (5r/2) when m is even, are also equal to zero. 



The value of u mr is in all cases equal to u rm , and the value of v mr is 

 equal to v rm . This result gives several algebraic identities, using 

 general values of u mr . Since 7t ,r = P(w,r,yw,), we have by this result 

 u mi0 = P(,m,/t), whence we get the result that 



f P (,m,;/) cos w/dj = 2 7rP,j (v) . P (n,m,fji) cos 7717. 



Thus the line integral of a Laplace's function referred to the first 

 pole along a small circle described about the second pole at angular 

 distance from it is the value of the function at the second pole 

 multiplied by 27rP n (i>) . sin 8, where v is cos 8. 



Equations can also be obtained connecting _,.,. and u n+ i m ,r, where 

 the ns are different. The most useful result is 



n (nm + 1) (nr+l) u n+ i >m>r (2 



. .... (45), 



and a similar equation obtained by interchanging u and v. From this 

 equation a table of the functions for different values of n can be cal- 

 culated, and is given from n = to n = 4. Since v m , = v 0im = 0, the- 

 number of the functions for any given value of n is (w + l) 2 + w 3 . 



Two physical applications of the results are given. The first is an 

 application of the result of a line integral of a Laplace's function re- 

 ferred to one pole along a small circle described about another. The 

 result is employed to establish that the law assumed by Boltzmann and 

 Maxwell for the number of particles which have a given velocity in 



