ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 655 
Group IV. 
ite = eyxF(-n, 4-n—-a, 1-2n-2a; Z ) 
]-—w 
Oe) o x F(1-n-2a, £-0-@, 1—2n~ 2a; : ) 
l-w 
as w 
n ») 
xF (=n, b-n-a, 1-2n-2a ;—— ) 
l+w 
0 — n+a—} 
- 
LENS FON 
— 
> 
s 
Py ‘7 2 
x (A-n-a, 1=n—2a, 1—2n -2a; 3) 
Groupe V. 
lige 2 l-w 
aera aes ) 
ap\4-e a 
) x F(1-n-2a, ie eae a 
h-a w =e ; w-l1 
ae y xF(ftnea, l+n, 3-a; mel 
— x F(4 —n—-a, 1-n-2a, 3-a; =) 
So 
i 
bo 
PN La ES aN 
— 
we 
= 
a. a a 
4 
aa 
— 
bo| + 
Ss 
Group VI. 
Ey 1 eer eer mr) 
5} 5) w—1 
| 7h ipa emeemebanaes 
That these integrals cannot all exist at the same time is evident, the convergence 
of the F-series depending on the values of n and c. It is easy, however, to find in each 
single case those series which represent convergent solutions of the differential equation. 
With regard to this point I may refer to Jacosr’s treatise, from which the conditions of 
convergence may at once be obtained. We notice that the above solutions may be 
arranged in pairs, which differ only by —w being substituted for the positive value. 
The necessity of the existence of such pairs is obvious, since the differential equation 
(3) remains unaltered if —w is substituted for +w. Since we are also permitted to 
write —n—2c for n without changing the equation (3), we have on the whole the 
following eight particular solutions, from which the others are obtained by the substi- 
tutions just mentioned. 
