294 Mr G. H. Bryan, On the Expression of Spherical [Oct. 29, 
Examples.— 
5 — 3 
OR). 
also 
5 — Bu (1 1 I \-o=- rae 
5 é -- 3a! + 5a “+ re ion aw negative powers of pu; 
eee Op es are 
see eat Z , and Q, (u) sip ait coth® wu — }oiitents 
6 2 6 
63° — 70m + 15 
(2) P,(@)= —. 
Gey = (0G sp lyn ld ) 
also aie Tis ailelaaed ( Pe awe oe 
pon A PA 
we ES (apes + negative powers of pu, 
120 
hence 
63u° — TOw? +15 > 945" — 735u? + 64 
01) = I 
results which agree with those obtained by the usual methods. 
9. The associated function or tesseral harmonic of the second 
kind of order n, and rank s is proportional to 
(w'-1)" D’Q, (H), 
where the symbol D stands for the operator d/dy. 
Now taking Q, (#) in the form (1) and differentiating s times 
using Leibnitz’s theorem, it is evident that the result can be put 
in the form 
DQ, (u) = coth ™wDP, () 
where R’ is a rational algebraic function involving only positive 
powers of pu. 
Hence (u?—1)* D’Q, (u) =coth™ w. (u?— 1)" D'P, (xz) 
TEE (VPP TD) 2 ae babe sbeebs a das-6s> 3e- (5), 
and (u2—1)* DQ, ()=coth* ». (u?—1)° D* P, (u) — B’ ....(6). 
From this formula, R’ may be obtained in a similar manner 
to that which we have exemplified in the case of the zona- 
harmonics. For taking the form (2) for Q, (#) and differentiating, 
