OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
459 
axis. When mod. (p + 1) < 2, this expression can he evaluated exactly as in Art. 3, 
the result being obtained by writing — p for p ; we thus find at once 
- If 
(f± +i — 1 +» F—> — 1 —) 1 —n-m-1 
(J2_l )«(«-,,) dt 
— e? nm . 4 sin mr sin mrr 
II ( n ) II (m — 1) //x — 1 
II (n + m) \p + 1 
F ( — n, n -\- 1, 1 — inn 
1 + P 
’ 9! 
. (13) 
when /x is above the real axis, the exponential factor being omitted when p is below 
the real axis. 
9. 
Let L, M, N denote the values of the integral 
along loops from 0 round the three points — 1, 
directions, the phases at C being as follows : 
r 
(t* — 1 Y (t — dt taken 
1, p respectively, in the positive 
of t — 1, tt in the first figure, and — tt in the second, 
of t -f- 1, zero, 
of t — p, — (-tt — <p), where </> is the (positive or negative) angle the line joining 
C to p makes with the positive direction of the real axis. We have at once 
1 + , 1 -) 
r(M+, —i +, i —i—) 
1 )* (t - p)-'"'-™- 1 dt = N + Me- 8 ' (w+w+1) ‘ —Ne 2 ™ - M, 
— l)“ (t — p)"' 1 -” 1 - 1 dt — N + Le _2 ’ r(m+W+1> ‘ — Ne 2 ™ — L, 
the phases in the integrands being measured as just stated. 
To express 1_) (t 2 — 1 ) n (t — p)"'"' -ra_1 dt, in which, as in Art. 6, the phase of 
t — 1 at C is 'b tt, and that of t + 1 is — 277, we have for the value of the integral 
Le~ 2nm — Mu 2 " 1 , or L — M 
according as p is above or below the real axis. 
Tt follows that 
3 n 2 
