OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
50 7 
77 II (— n — 1 ) 
II (m — n — 1) 1II (m + n — 1) II ( — m — n) 2 n+1 cos nir II (n + J) II ( — 4) 
(^-1 )*> 
—n—m—l 
„ fn + m + 2 n + m + 1 , « 1 
F(- 5 -,-—.» + !, Ji 
n( m»-i) _ F -zJ!. i _ I 
II (n — m) II ( — J) r ' \ 2 2 /x“ 
_ Z2 p (n+m)m o« 7T ' _L_ ^ ^ ^ _ ll ( ,,2 _ 1 \l m u] l ~ m JT 
- e z n (n}U ( - 71 - J) j ^ 
m — n 4- 1 m — n . 1 
2 5 2 ’ 2 ” ffi ^2 
The ratio of the coefficients of the last two terms can easily be shown to be — 1 , 
thus the result reduces to 
77 
II (m—n—1) Tl(m + n — l) 
or to 
(-w-l)...(-w-m + l) 1 Oi a iyn» 
n(-j)n( K +j)^ 11 
9»+l 
COS W77 
(n + 7ii + 2 % + m + l . o 1 
F (-,-y—, w +1, j, 
A 6 " 
(- 1) 
»+i 
n (m — — i) n (% + to ) n (?i) 
QA 04 
This result must hold whether mod /x is greater or less than unity ; hence when 
m and n are real integers 
1 e -m t (t-+)± m u _ ( - 1)» +1 1 0 „/ W96 n 
277 Jo {/x + \//d — 1 cos (<p — + «<)} K+1 C ^ H (??r — n — 1) II (n + m) II ( n) ^ ' 
when m > n, and is equal to zero when m < n. 
The case in which m and m + n are negative would require special examination, 
but the result in that case may be deduced from (96) ; change xfj, u into -- xp, — u, and 
<p into 277 — <p, we thus find 
2 r2rr g mi (c f> - 1 ji) ± mu ( — \ y/i + 1 ^ 
277.1 o [fx + vV - 1 COS ((f) - T[r ± iu )} H+1 C ' ( 1 J ~ n (rn - n- 1) n (n + m) II (n) Q* (p) 
when m > n, and is equal to zero when mY n . . . . . . (97). 
The results in (94), (96), (97) agree with those of Heine, # the more general formulae 
(93), (95) are not given by him. 
45. Results such as those in Arts. 38, 43, 44, could be foreseen by a consideration 
of the fact that (z + ax + PyY satisfies Laplace’s equation V 2 Y = 0, provided a, /3 
are any constants such that a 3 + = — 1 ; this holds for complex values of n, and 
when x, y, z are not restricted to be real. Let a——i cos (ifT®), /3= —tsin (xp^fut), 
then, since 2 = ry, x = D'v//x 3 — 1 cos <p, y = ir\/ /x 3 — 1 sin <p, we have 
* ‘ Kugelfunctionen,’ vol. 1, p. 211. 
3 t 2 
