EQUATION AND ITS TRANSFORMATIONS. 819 
It is easy to connect these definite integrals with the series U and Y of art. 3, for 
[ (1— z 2 )~P~ 1 ef xz dz= j - (1— 2 2 ) _ ^~ 1 ^ld-axzd- fl -~+-^ -{-kc^jdz 
= 2 ( 1 -z*)-p-M 1 
a ■ Jr +kc -) dz 
r(-; ; >r(j) <f.," r(-ji)r(|) «.v t(-p)V(A) . . 
T(-p+i)“ t '2! r(-^+^)■*" 4! ry-p+f) -1 " c ' 
r(-j»)r(i)r, , a?x 
-/2,.2 1 
4,a 
cc*x 
i t 
2-2 
n-p + ij l 1+ 2 ! -p + \ ' 4 ! (-p + $(-p + f) 
-&c. 
whence 
Similarly 
v < i 
1 0 0 
arx* 
i 
4....4 
F(-i? + i)L Y-i 23 ' (p-r)(p- 3 ) 2 * 2 ! 
■&c. L 
,r^' 
-f (W)W, V; 
and we thus obtain expressions for U and A" as definite integrals, taken between the 
limits 1 and —1, for all values of p for which the integrals are finite. 
§ vn. 
Connexion ivith Bessel’s Functions. Arts. 43-48. 
43. If the differential equation (1) is transformed by putting u—x l w, it assumes 
the form 
w 
dho 1 dw „ , n3 - A 
TT+- N- cnv— “ = ° 
dx 2 x dx u x l 
(56). 
The equation of Bessel’s Functions is 
