OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT 
469 
16. In (25), change n into — (n + I), we then find, after some transformation of 
the numerical factors, 
Q-n-r (p) 
_ _ lg(*+!i) 2 o m _ 
2 . n—m 
sm-7r 
n + m fm+n—1 \ _ . .. 
cos-^-tt n/—-—)n(—i) 
n 
71 — 7ft 
(/x 2 -1 F 
7ft+ 77-fl 7ft — 77 
’ b b 
— ie (m+M) 2 . 2’ 
. 77 + 7/7 
sm-7r 
2 
n — m n—m 
COS - 7T n 
n/5±?W ( -i) 
^ ' vrW- 1) 1 ’/*!’ (“I-' -=f +1 
Q O 
On substituting these values of Q,”‘ (/x), (/x) in the formula 
P« w (/a) =- {Q.,r (y) sin (71 + m) tt — Q_ n _r (/a) sin (77 — m) tt}, 
TT COS 777T 
we find, after some reduction, for the case in which the imaginary part of y is 
positive, 
77 + 7/7 
P n m (jit) = e“ ,nT ‘. 2 m cos ——- tt 
n 
77 + 7/7—1 
(/x s —l) s "F 
m + 77 +1 
7/7 — 77 
-5 
0 
+ e- 
2 m sin 
77 + 777 
n 
TT. 
77 + 7)7 
9 
n 
77 — 7/7—1 
)n(i) 
{/x 2 -l)^>F 
777 + 77 + 2 777 — 77 +1 
2 > /A' 
(27). 
When the imaginary part of /x is negative, we obtain in a similar manner a formula 
which differs from (27) only in having the exponential factor e mm instead of e~ mm . 
From (27) it is seen that when m + n is an integer, only one of the two 
hyper-geometric series is required to express P n m (/x), the first or the second according 
as n + m is even or odd. 
Definition of the functions P„ nl , Q, n m for real values of /x which are less than unity. 
17. The functions P„ m (/x), Q/' (/x) have been defined as uniform functions of /x for 
all points in the plane of /x in which a cross-cut is made along the real axis from 
1 to — co ; at points indefinitely close to one another on opposite sides of the cross-cut 
the values of P/ 2 (/x) or of Q,” (/x) will in general be different. We shall consider first 
