398 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
On substituting - this value for S in the value of K we find 
n\ 2\! 2 Li' 2v' 
K=(-iy— ^ 
v 'p\ q\r\\\ n'.v'. 
Finally, performing the operations with regard to the z’s, we find 
[fp„P, ; P.<2S 
47rR 2 i\ i\ 2A,! 2/jl\ 2v\ 
2i +1! 2i ! \\ W jx'. v\ v\ 
(26) 
Thi s agrees with the result obtained by Professor Adams. 
§ 12. Before leaving this case we may remark that, so far as the foregoing proof is 
concerned, the poles of all the three harmonics are not necessarily coincident. The 
proof will hold when two are coincident and the third is at an angular distance, say a, 
from them. In those circumstances the value of jJ'P^PyQx/S will be the result found 
above multiplied by P, (cos a). The same is, of course, true of the integral |J’Q /; Q^P r hS, 
provided the two Q’s have the same poles. Now let Q^, Q r/ be each expanded accord¬ 
ing to Laplace’s Formula. Then we will have the curious result that any zonal 
harmonic of degree r can be expanded in products of harmonics, zonal and tesseral, of 
degrees p and q, the quantities p, q, r being, of course, subject to the foregoing 
restrictions. If q is less thanp the number of terms in such expansion will be 2q-\-l 
| j(p, «) (q, a + fi)dS 
§ 13. Keverting to the general expression for the operator rr (l, m, n) given in § 8, 
let us leave out for an instant the numerical multipliers and the operators with regard 
to the z s and let us multiply out those with regard to ‘q and q. Let us suppose l, to, n 
in descending order of magnitude, and for the sake of brevity, instead of writing the 
differential operators, let us write only the characteristic letter, e.g., instead of ^ 
mute £j. We find 
+&v^r +n Vi + "£t m +vi m+7, & +,i v3~ m ) 
Now since any product such a3 — can be replaced by 
1 d 2 
4 dZy 
, we see that each of 
the lines just written corresponds to the sum of tesseral harmonics, in such manner 
