OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
The particular case m = 0, gives as the expression for the zonal function, 
529 
P „_1 (cosh cr) 
Ii(n — Y) 
no-i)n(-i) 
2n(M) 
(n-h)a 1 | l- n 2 -2<r j I °)(n h) ('ll _2f ,i- 
l)a- 
.1 
+ 
ri ( n) n (- i) 
log(4<? <T ) e (m+i)<r F (71 + g, i, n + P € 2 ' T ) 
, i _(■„+!) a / i \ n (% + s — i) n (s - b) 
+ — « ( J J («*+, + “• -j^-ry— - e 
This particular case has been obtained by other methods by Basset, and by W. L). 
Niven. 
The case in which in is fractional has really been included in the above investiga¬ 
tion ; the simplification of the coefficients in the expression (2) does not apply to the 
genera] case. 
Mehler’s Functions for the Cone. 
59. The normal potential functions for problems in which the boundaries are 
coaxal circular cones* are spherical harmonics of complex degree — g + T n > ^ ' s 
therefore desirable to consider the forms which the functions P„ (cos 6), Q u (cos 6) 
take when n is of this form ; P_ J+2)t (cos 6) will be denoted by K ;J (cos 0). 
We find from (103), (110), (111), 
K p (cos 6) = 
cosh pit 
2 c e 
7 r J o \/ 2 cos u — 2 cos 0 
du, 
2 f 
= — cosh P7T — 7 = 
7T x J 0 V i 
cos pv 
2 cosh v + 2 cos 6 
dr; 
these formulae have been proved by other methods by Mehler and by Heine.! 
From (103), we obtain the new formula 
K p m (cos 6) — (— 1)" 
2 II (n + m) 
2™ II ( — i) IT (in — i) IT (n — m) 
sin 
e 
cosh pit 
(2 cos v — 2 cos 0) 
h ill 
where m is any positive quantity. 
From the above formulae, we see that P_i +pt (cos 0) = P_x_ pi (cos 6). 
From (117), we have, 
* See Mehler’s paper in Crelle’s ‘Journal,’ yoI. 68. 
t See ‘ Kugelfunctionen,’ vol. 2, p. 221. 
3 Y 
MDCCCXCVI.—A. 
