Tarleton — On Laplaceh Coefficients, 17 
where Yi is a spherical harmonic of the degree and comparing 
co-efficients of cos ntfi', it is easy to show, as in Laplace, Mecanique 
€eleste, book iii., section 17, that 
"y-lf^D'P,fdi. = ^^, (1) 
except n^Oj in which case 
2e+ 1 
(2) 
If we put ^\''{jyP,)Hii = A, 
4 
we have, then, <?„A^ = -— — -, 
except w = 0, in which case 
2 
2i + 1 
"We shall now prove that 
A„+i + - + ?^ + 1)A^ = 0. (3) 
This readily follows, from the differential equation satisfied by P,-, 
viz. : — 
If we operate on this equation with D**, we obtain 
B^"^^ {uBFi) - i{i + 1)7)'* Pi = 0. 
Expanding the first term by Leibnitz' theorem, and remembering that 
B'^u = 2, D^w = 0, 
we get 
uB^nPi + + \)BuB^^^Fi + [n{n + 1) - i{i + 1)) i)"P, = 0, 
or 
p. + + l)i)wi)«+ip. = {i ^n){i + n + l)i)'*P,-. (4) 
Prom the definition of A„ we have 
A„+i + (^ - w) (e + ?J + 1)A„ 
■ ^n+l p.)2 + > + ^ + 1 ) W« P.)2 ) f?^. 
-1 
E. I. A. PROC, SEE. in., VOL. I. C 
