458 
DR. E. W. HOBSON" ON" A TYPE OP SPHERICAL HARMONICS 
(P -1)*» [ 
(n + . -l + i M — > -1-) 1 
')n 
(< 8 -l Y(t-!>.)- 
-m—1 
dt, 
analogous to the corresponding integral round the singular points p, L, obtained in 
Art. 3. To define the phases of the integrand we shall distinguish the cases in which 
the imaginary part of p is positive, and is negative. 
We suppose p to move from a point in the real axis for which its value is greater 
than unity, up to its actual position, the path of integration being drawn as in the 
figures ; it will he observed that as p moves from a position on the positive side of the 
real axis to one on the negative side, the path cannot be displaced from its first 
position to the second one without crossing the singular point + 1, it is therefore 
necessary to distinguish the two cases. 
In the first figure the phase of t — 1 at A is + tt, and in the second figure it is — n, 
in both cases the phase of t + 1 at A is zero, and that of t — p is measured as before. 
Put t -f 1 = (p + 1) u, the expression then becomes 
jji — Lyi«z r(l +, o +, 1 0 —,) 
p + 1/ 
u” 
\n —it-m—1 
1) (u — 1) du, 
now we put 
or, 
P + 1 1 ‘V, p + 1 
—— u— 1 = e (1-y— u 
P f 1 . p + 1 
n U — l— e 1 - -r— u 
according as the imaginary part of p is positive or negative, in both cases the phase of 
1-— u is zero at A, and then i 1 — —-— u) will have that value which is given 
by the expansion by the Binomial Theorem. 
We have for the integral 
the upper or lower sign being taken in according as p is above or below the real 
