530 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
/ IT 
Q-*+jh (cos 0) — - P_i +iM (cos 0) = Q_ i+J)l (cos 0 + 0 . i) 
cos pu , cos vu , 
= W W" , o /) du — le p / ,-- 0 . du, 
o v 2 cosh u — 2 cos 0 J 0 v 2 cosh w -f- 2 cos 0 
lienee, changing p into — y>, and adding the two equations, we find 
Q-j+jh ( cos #) + Q-i-i* ( cos #) — t7r P-h-i* (cos 0) 
n C cos pu ■, n , 
= 2 ^ ■ . du — 2i cosh vtt 
J o \/ 2 cosh u — 2 cos 0 1 
hence 
cosh pir ( ^ ^ 2 cosh pir [ 
-(cos 9) + (cos 0)} = 
U 
cos pit 
cosh u + 2 cos 0 
e£w, 
7T 
7T 
cos pu 
J o \/2 cosh xi — 2 cos 0 
= (— cos 0). 
Thus we can use K,, (cos 0), K /; (— cos 0), as the two independent functions. 
It thus appears that the expressions given by Mehler and Heine for K„ (cos 0), 
K,(- cos 0), are particular cases of the general formulae we have obtained above. 
Potential Functions for the Bowl. 
60. It has been shown by Mehler that for potential problems in which the 
boundaries are spherical bowls with a common circular rim, the functions K p (p) can 
be used, p being in this case real and greater than unity, say p = cosh xp. 
We find from (lit), that 
2 f 
P_i +pi (cosh i Jj) = K p (cosh xp) — - cosh pn 
7T Jo v 2 
cos pu 
cosh u + 2 cosh xp 
du, 
and from (113), we find 
k *(cosW) = ;£;/5 
also from (115), 
cos pv 
7T J o \/2 cosh xp — 2 cosh V 
2 f°° sinpw 
civ, 
K p (cosh *) = ~ ooth pv | 4 y 2eosh T-hcoshf dm ’ 
these formulae are all proved by Heine* by other methods. 
From (112) we have 
YT . . ,. 2 (— l) Hi II (n + m) . . _ , f 
Kf (cosh xp) = ———-— -— -: smh m xp - 
p v r 7 2"‘ IT (— A) II (m — vV) II (n — m) r J 0 0 
cos pu 
(2 cosh xp — 2 cosh w) i “ 
* See ‘ Kugelfunctionen,’ vol. 2, p. 220. 
