478 DR, E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
thus we have 
Q,r (fi) = e m " 2 m n (m - ±) n (-|) ^ ^ fj 
77 Jo 
when the real parts of n + m + 1, \ — m are positive. 
In particular 
(1 — 2 fih + 7i~y n+ i 
dh (43) 
Q» 
(/x) f: (i-s 
h n 
(1 - 2 fih + A 2 ) 4 
dh 
provided the real part of n + 1 is positive. 
Similarly we find from (42) 
(44) 
Q m / \ _ . £ -m ^ ( "' + ^ ^_ IL /.,2 _ l ^ _ .77, /AsX 
" n (n - m) n (m- -±) ^ J 0 (1 - 2/ih + 
provided that the real parts of n — m -j- 1, m + 1> are positive. 
In the formulae (43), (44), (45), change h into -p, we then find 
/h 
one qy/HT r°° 
Q»* M = e“" ■ 2 - n (m - i) n ( - i) (k- 1 ) 4 “ j (1 _ 2M + y) „,, dA (46) 
when the real parts of n + nr + 1, — nr are positive. 
Q« (p) = 
lv 
- re -1 
s (1 — 2/uA + A 2 ) 4 
dli 
where the real part of n + 1 is positive, 
Qy» (fx) = e mm 2" m n (n + 11 ^ ^ 
II (n — m) II (m — J) 
when the real parts of n — m ~)- 1, m -f- are positive. 
24. Next, consider the expression 
(47) 
^ ~ -w 4 )‘- dA • (48)> 
ra+,,- 2 -) 
(/X 2 - | W "+*(l — W )-^-” ! ( 1 
—4—m 
du. 
Suppose that the phases of u, 1 — u are zero, when the point A in which the path 
'll/ 
cuts the real axis between 0 and 1 is reached, and that 1 — — has its phase zero at 
Z* 
u 
the point B in which — is real and less than unity. 
