296 Mr G. H. Bryan, On the Expression of Spherical [Oct. 29, 
4. From this result the general form of Ff’ can be obtained. 
For since #’ is the portion involving positive powers of mw in the 
product 
(n+ s)(n+s-1) 
N+ _ n+s—2 
At 2.(2n —1) a 
, @ts)(nts=1) (ues aes) ee 
DAL Cp =Ly pS) xi 
1 1 a 1 i 1 + 
Bu Bye Tp 
therefore 
/ = = —¢ 
R = wer PaO) Tee FS cease Dye 2rtl4 ; 
where 
Ry ss) @aeSa I) 
a= Ala (Xj) 2r —3 
4 (mts) (n+ s -1)(n+s—2)(n+s—8) eee | 
2.4. (2n —1) (2n— 3) 27 
5. The principal applications of the functions which we are 
considering are those relating to potentials of spheroids, and the 
forms discussed in the present paper are more convenient for 
calculation in most cases which present themselves, than the 
well-known expansions in descending powers of ». In potential 
problems which relate to oblate spheroids w is imaginary, but 
the present methods are equally applicable with the requisite 
modifications, while the expansions in series become divergent 
when applied to an oblate spheroid whose eccentricity is greater 
than 4,/2. 
For convenience of reference, I have given in the following 
table the expressions in a finite form of 7\*(u), and U,*(u) for 
values of m up to 5 inclusive, 7*(u) and U,*(u) being defined 
as follows: 
7” (#) = (w* — 1) De P, (uv), 
n—s! 
- d 
Ue (2) a is (1) i (2 a 1) (Te () e 7 (= 1)’ n+s! (u*— 1)*DsQ),, (#4). 
