DR. E. W. HOBSON ON A TYPE OP SPHERICAL HARMONICS 
479 
rt 1 Zj 
In the case m = 0, (31) agrees with Heine’s definition of the function Q„(p) for 
real values of p between i 1- Objections have been raised by Schlafli to this 
definition of Q„ (p), on the ground that the function does not satisfy Legendre’s 
equation. There does not, however, appear to be in reality auy question of principle 
involved ; it is merely a matter of convenience to give a definition of Q„ (p), which 
shall give real values of the function in the real axis, when n is real. It must, 
moreover, be remembered that although we have drawn the cross-cut along the real 
axis, it might have been drawn along any line we please joining the points i 1, and 
thus the function Q„ (p) may be regarded as satisfying the differential equation of 
Legendre for points in or near the real axis, the surface over which the function is 
uniform being a different one from that which we have hitherto postulated, and the 
function being a linear combination of the two independent integrals of Legendre’s 
equation which we have defined and used. 
19. For values of p near that part of the real axis which is between —1 and — cc , 
we see from the expression (10), that 
Q n m {p + 0 . 1 ) 
_ n(n + m)n(-H \j B1 («+!)„ v f n + m + 2 w + m + 1 
2 n+l ‘ n (n + i) ^ ^ 2 ’ 2 
n+f 
5 
Q." (m - 0.0 
e mm m ) n ( — H 
9,71+ 1' 
n (n + i) 
III £>(>1+1) 17 r 
(-/*) 
n+m+1 
n+m+ 2 % + ra + l 
,»+* - 
2 5 
where (p 2 —l)*” 1 here denotes e * nIog 'the logarithm having its real positive value; 
we thus have 
Q»“ (p + 0 . l) = e- im Q,f' 1 (p - 0 . l) .(32) 
and we may define (p), for real values of p between — 1 and — 00 , to be equal to 
either of the expressions in (30) with its sign changed, thus 
2’ 1+1 n {n + £) W 
(-/*) 
,'rt+m+I 
„ in + m + 2 n + m +1 . * 1 
F (-g— , —g— ’ 11 + 2 7 , 
where (p 2 — l)*' 11 has the meaning given above. 
To express the relation between P„'“ (p), P„ m (— p), Q„ m (p), Q,i"(— p) p is rcaZ 
and iies between ± 1. 
20. We have from (20), if 6 lies between 0 and ^ tt , 
