( 
1883, } 97 (Hagen. 
which is of the mth degree and will be considered presently. The other 
equations give the following solutions : 
1 
B, pad + i 
ra | 
1 
— 5’ 
B,=-- Vee { 
{ » (9) 
oe . sliteibe a. 32 b Maen 
Bs af V6 (? - “1 =3) 
nad 
1 
ia * R ys PR ST LOT ST yay ste 
By ya © xy — 5 21 An As aj 44), ete. 
eg! 
The formulas show, first, that in general we shall have p= ®, and 
secondly, that the series of the coefficients B decreases the faster the 
larger 2, is. For the quotient Diy By is of the same order as 1 + 4,. 
Fence a few terms will suffice to compute y as often as the coefficient A, 
is large in comparison to the following coefficients. 
Now as to the condition 
r=m 
-_— yj’ _ 
3, = 2 A, BS BS = 0, 
Hy 10) 
it is evident that its cwact solution is impossible as often as m >4, except 
in one case, viz.: when A, = 0, in which we have also B, = 0. The 
approximate solution of the above equation by development will be ex- 
plained in Part II. 
§8. In the special case A, = 0 we have B, = 0, because % and y are 
zero at the same time. Consequently we have 3 = A, and the formulas 
Dp Ag ee 0 
By AG cs 1 
B, A, + BY A, =0 
B, A, + 2 B, B, A, + B,? A, = 0 gr 
B, A, + 2 (B, B, + $B,") A, +8B?B,A,+ Bsa, =0/:° 
B.A, 9 (B,.B, + By B,) A, + 8 (BY By+ BY B,) Ay + 
4B, B, B, -+ BY At = 0 
ete., their solutions being 
a 
By 
1=+ a | 
a 
Be = A 
Chae 
t ) (on 
B= + it QQ Ag — AVA.) 
B= — a (6 A,? —B A, A, A, + Aj? A,), ete. 
UAC Dis 
§4. When A, is not zero, the given series may be written in the form 
r=m 
a SA, ee 2 AL yt (10) 
rea] 
PROC. AMER. PHILOS. 800. XXI. 114. M, PRINTED JUNE 28, 1888. 
