457 
OF UNRESTRICTED DEGREE. ORDER, AND ARGUMENT. 
Proof of a relation between Q n m (/x) and Q a m (/x). 
7. If we apply to the formula (10) the known theorem 
F (a, ft, y, x) = (1 — xf a 3 F (y — a, y — ft, y, X ) 
we have when mod /x > 1, 
o » — n ( n + n ( - I) / 9 _ 1 . F f-rn + 2 n - m + 1 ,3 1 
vr/— 9,1+1 tt^, x n vr L / x l o > o >'‘4 21 
n {n + |) 
The expression for Q re iil (/x) is obtained by writing — m for m, in the formula (10); 
we have thus the relation 
e~ mt " Q,r f) _ e™ Q ne> ^ 
IT (n + m ) IT (n — m).* ' 
which must hold over the whole plane ; it is obvious that Q n ~“ l (g) satisfies the diffe¬ 
rential equation (1), as that equation is unaltered by changing the sign of m. The 
result in (12) may also be obtained by transforming the integral in (9) by means of 
the transformation ( t — f) (t' — /x) — f — 1, which is equivalent to an inversion with 
respect to the point /x. On making the substitution, we find 
(f - If 
(-1+,1-) 
f — If (t — dt 
= - (/X 3 - 1)"*“ j (1+ ’ 1 * f 2 - l)’ 1 (; t' - 
dt'. 
Corresponding to the phase — -n of t~ — 1, the phase of t'~ — 1 is tt ; also to the 
phase — tt of t — p, in the case in which p is real and greater than unity, the phase 
of t! — p is n, hence, in order that in the integral on the right-hand side the phases 
may be measured in the same way as on the left-hand side, the factor e 2,Mrt-2(,l-m+1),rt , 
or e 2mm , must be introduced ; we thus obtain 
(/x 2 — l)* w P 1 ’ ] If — 1 f(t — f)-' 1 —'- 1 dt 
— (f — l)-^ e 2»m j ( 1 + 1 1 } (ys _ ^ 
and thus the result (12) is proved. 
Expression for Qp (/x) when mod (/x + 1) and mod (p — 1) are less than 2. 
8. It will be necessary to obtain an expression for the integral 
MDCCCXCVI.—A. 3 N 
