A treatise on the theory of Bessel functions, by G. N. Watson.

5-73]

MISCELLANEOUS THEOREMS

159

namely
d2y 1-y-a' 1-3-3 1-7- dy
d Z2 z- a + - b + - c dz
x - 0,
+ aa' (a - b) (a - c) /' (b - c) (b - a) 7y' (c - a) (c - b)
z- a z-b z-c )
(z - a) (z - b) (z - c)
we see that the latter reduces to the former if
a= O, a = -, =- v -,
while b, c, /3, 1', 7, 7' tend to infinity in such a way that / +/ ' and 7 + 7'
remain finite (their sum being 2, + 1) while //' = y77' - b2 and b + c = 0.
We thus obtain the scheme
f 0. 2,/3, - 2zi, -
lim P v-, P, /,, z
-1-/k, -/3, 7 J
where 7, ' = + )2 + + )2 + /3}.
Another similar scheme is
( 0, i/3, 00 \
lim P v-,, /, 7, Z
- v-, -/, / 7',
with the same values of 7 and y' as before.
A scheme for J, (z) derived directly from Hansen's formula is
0, oo, -4aR, 8
lim P v - v, - 0, z
(a') 00 ^, /3 -, v+l-a-/3.
Olbricht has given other schemes but they are of no great importance and
those which have now been constructed will be sufficient examples.
NOTE. It has been observed by Haentzschel, Zeitschrift filr 2fath. utnd PAys. xxxI.
(1886), p. 31, that the equation
d2y = v2 - -- 2/,
whose solution (~ 4'3) is u~2v (liu), may be derived by confluence from Lame's equation
dd2'=[(^2 - 4) { (%)-1l} - 2] y,
when the invariants g2 and g3 of the Weierstrassian elliptic function are made to tend to
zero.