OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
49 7 
point h = z, but cannot be taken directly from — to z ; it thus appears that the 
£ 
function P„ (p) is no longer represented by the definite integral, but that the value 
of the definite integrals involves Q„ (p) as well as P„ (p); in fact, we have shown in 
(70) that in this case 
~ \ + \/fi — 1* cos cty = P„ (p) — ~ e ±nm sin mr. Q„ (p), 
where the upper or lower sign is to be taken in the exponential according as the 
imaginary part of p is positive or negative. 
The only case in which Heine’s definition is valid for all values of p is when n is a 
real integer. 
O 
Heine deduces from his definition that for unrestricted values of n, the function 
P„ (p) is represented when mod p > 1, by the series 
1 II (2n) 
2* n (n) n (n) * 
n 
9 
1 
2 
— n, 
this result, following from the incorrect definition, is erroneous, the correct expression 
being given by (23) and involving two hypergeometric series. 
It was to be expected d priori that as P„ (p) was defined by means of an integral 
taken along a path containing the singular points p and + T but excluding — 1, the 
function so defined would not in general possess any kind of symmetry about the 
imaginary axis. 
Definite Integral Expressions for P/' (p) when on is a Heal Integer. 
38. When m is a real integer, the formula (4) for P,U (p) becomes 
p.*w=d ; n( “k m> iv-jrr'V- x>-o-ft)--*—* dt. 
27 TL IT ( n ) 
MDCCCXCVI.—A. 
3 s 
