OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
503 
therefore 
— e 
WM = 
4:7r II (n — m) 
H ( " } . 2" +1 1 — 2t X dt + 2i. | X dt 
Substituting for Q_ ;t _T l (p) its value in terms of Q(p), P/ 1 (p) given by (18), 
we have 
Q,“ (p) sin (n + m) 77 — 77 cos utt . e mn (p) 
1 n (n) 
. _ . . 2" +1 1 2 cos W77 X dt — 2 cos wtt X dt 
4t II (ti— ra) 1 fo -V 
On substituting the value of P„ TO (p) in this equation, we have 
Q’M = 2" • e ~“" i r ^vh )-(^ - c-fy 8 -1)-"” 1 (< - O” - ”*. 
which holds for all values of n and m, such that the real parts of n + m + 1, 
n — m + 1 are both positive. 
In this formula, when t is just greater than p, the phase of t — 1 is the same 
as that of p — 1, but the phase of £ + 1 is less by 277 than that of p + 1, hence if 
we wish the phase of £ 2 — 1 to be that of p 2 — 1, the result must be multiplied 
by e 2 ”‘ ; again the phase of £ — p is that at A, and this is less by 77 than the phase 
of \/p 2 — 1, hence, in order that the phase of £ — p may be that of \/p 2 — 1, we 
must multiply by • the formula now becomes 
Q.*M = 2 "• e"" n ^ (P - i) ! »fy 5 - 1)—■'(« - rf-dt-. 
on substituting £ = p + \/p 3 — 1 . e w , 
where u is a real quantity, we have 
which gives us 7 
t 3 — 1 
2 (£ - p) 
= p + \/p 2 — 1 cosh 
«, 
QA (p) = . 
IT ( n) 
r. 
n (n — m) ‘ 2 J -oolp + \/p 2 — 1 cosh u) n 
+1 
du, 
or 
QA (p) = . 
IT (n) 
cosh mu 
IT (n — m) J 0 (/x + v/p 2 — 1 cosh p) n 
+1 
chi. . . . (91), 
where the real parts of n + m -j- 1 , n — m + 1 are positive. 
In (91), the phase of p + v' / p 2 ™ 1 cosh u is equal to that ol p -f \/p 2 — 1 
when u = 0. The formula (91) is a generalization of the formula given by Heine ;* 
* ‘ Kagelfunctionen,’ vol. 1, p. 223. 
