Hagen.] 
B,-* By) | i. (5) 
a(r—4) 2(4)) 0 
and so on. 
Note.—The lower limit of 7 may be put == 0, because all those terms in 
which z contains a negative argument, are zero. 
§2. The equations (1) to (5) may be transformed in the following way. 
We put for pede 8 sake 
‘a n ' 
= E G) A, Bet = As. .s, (6) 
r=c 
where 
r mee (r—2)... r—d + 1) 
Ce i tie mn) 
according to the notation of Huler.* For numerical computations we 
then obtain from (6) 
yo Ay + 2A, B, +3 A; B,? + 4A, Be +... 
SA +B ALB) +6 ABS. 
,t4 A, By +... (7) 
Thus the conditions (1) to (5) present themselves in the more perspicu- 
ous forms 
x,=0 
B, : om 
rig af IFS Org oe. O (8) 
9 Rp yy > > a 
B, B, 2, ak 3," 23 = 9 
B, 3 “+ 2 (B, By +4 2 B,?) Xo ate 3 B, B, Xs + B,* Par =0, | 
The law of these series being evidenced from inspection, we deduce the 
next following conditions ; 
Bj ¥, 4 94b) 8; +B, B24 CPB, + BB) ad BPD Ty 
Bes = 0 
By 2, + 72) (BB, + B, By + - os 3, + (3) (B,B,B, +3 be Bet *) 
me B}? me a 
7 ae) 5} Fayincay “4” Pr Bee + Bi 2, =O, 
and in the general form 
Bu 3, + 2(2)(B, But +...) ¥, + 7(8)(B, . 2g te sp Bie 20, 
each term with the sign ¥, bien the factor We ) on oAac oh Be having the 
denominator z(¢). The factors of 3, are always y in number, and the sum 
of their indices y. 
There is no difficulty in solving the equations (8), except the first, 
* Acta Petropolitana, V. 1, p.89. Though his notation is not much used in 
American text-books, it is found very handy in operating on series, 
[April 6, 
