1888.] Mr G. H. Bryan, On the Expression of Harmonies, de. 293 
(5) On the Expression of Spherical Harmonics of the Second 
Kind in a Finite Form. By G. H. Bryan, B.A,, Peterhouse. 
1. In this paper I propose to give a handy and convenient 
method whereby both Laplace’s coefficients of the second kind, 
and the corresponding associated functions or tesseral harmonics 
may be exhibited in a finite form. In the case of the former it 
is hardly necessary to add anything to what has already been 
done in this direction, but the present method, when applied to 
the latter functions, will be found much less laborious than the 
process of differentiation by which they have been hitherto obtained. 
Taking as usual 
oo) du 
eget 
Q,, (1) n (#) i (pe a, 1) iP, (w)}? 
we know that if w>1 
where & is a rational integral function of » of degree n— 1. 
Also 
f, n! 1 (n+1)(n+2) I 
(4) 73.5... Qn +1) {aa + 2.(2n+3) pr 
(n+1) (n+ 2) (n+3)(n+4) 1 
I.4.(n+3)Qn+5) pret” 
+ 
In the first form let P,(u) be expressed as an algebraic 
function of mw, and expand coth™y, that is, tanh (1/“) in powers 
of 1/u. We thus obtain by equating to the second form of 
Q,, (#) 
 teatpet-)-R 
enh aS 
n! 1 
Hence the left-hand side cannot contain positive powers of yp, 
and therefore & must be equal to the terms of positive degree 
in # in the expansion of 
P, (x) ( 
This gives a convenient mode of calculating R, and, hence, 
of expressing @ (yw) in a finite form. 
