452 
DR. E. W. HOBSON ON A TYPE OP SPHERICAL HARMONICS 
-r. / \ 1 n (re + m) (u? — 1)*“ (■(;“■+. 1+) 
Pl W = SnoT * I 
(6). 
When m — 0 we have 
i /*(/*+> 1 +) i 
.(O = o=; vP~• • • 
l J -J 
(7) 
which agrees with the definition given by Schlafli. 
The only case of failure of the formulae (4) and (6) is that in which n + wi is a 
negative integer; in that case II {n + m) is infinite and the integral is zero, and 
the product can be evaluated by the rule for undetermined forms 0 X °° ; we have 
II (n -f in) = 
cosec (m + re) 77 
II (— m — re — 1) 
5 
and the limiting;’ value of 
O 
1 
sin (m + re) 
— f 
re) 77 J 
(/t +, i +, ^ i -) 
( t 2 - 1)” (t — dt 
is 
77 cos (m + n) 77 
(/x +, 1 +, fi —, 1 — ) 
(d - 1 Y(t - ix)-’ l - m ~ 1 log, (t - n) dt, 
thus 
W {ii) = 
477 sin nir 
1 
2’V cos (m + re) it II (re) II (— m — re — 1) 
(f*+. i+, i-) 
(t 2 — 1)" log, (t—fi) dt. 
If in (5) we change n into — n — 1, the hypergeometric series is unaltered, thus 
within the circle of convergence P,/“ (p) is equal to P_*_T*(p); it follows that 
the same relation holds over the whole plane; we accordingly obtain another 
expression for P,“ (p) by changing n into — n — 1 in the formula (4), we thus have 
p»*M = P-«-i w (/*) 
_ _ e ~ M,rc o" +i n (m ~ } _ iw f 
4-77 sin 7177 n(-re-l) ’ J 
= _ . e ~ nm -. 2 >t+l n ^ (» 8 —1)** f 
477 sin (re — rei) 77 ‘ II (re — m) ^ ' J 
(m +1 1 +1 M -1 1 - ) 
( t 2 — — ix) :l ~ m dt, 
(/x+i 1 +, 1-) 
(£ 2 — fx) n ~ m dt (8). 
The formula (8) will serve equally with (4), as a definition of P/' (p) ; it does not 
appear to be easy to prove directly their equivalence. 
