470 MR T. MUIR ON NEW GENERAL FORMULA FOR 
Putting B,=B,=B,= .... = 0, it is easily seen that 1, y2, Yes Yes Yor eee 
become 




Nagi AG 
A A, A 
A, Aa, 2 i Pe ts 2 A, A; Ay ’ 
2 3 3 4 ie A, Ae 
and writing a,, a), a3, a4, a;,.... instead of these we have from (A) 
pie date coe B 
1+ Aye + Ay? + Ag+ ...=14+ 2 _ ae a (B) 
Weer Seas Ay0.0 
cae a = 
Again, putting A,=A,=A,;=...=0 we find that y,, y2, ys, Ys, Ys;--- 
become 




1 B, B, B, B, B,; 
= Sigler . se = By BB; | , = By B, Bole 
 * i: By-By B; iB bib, 
and writing —B8,, —B,, Bs, 6:1, —s;,.-. for these, there results from (A) 

ee ee — Be — Box 
ge ae +B +. a ae eee Se ee 
—B, = 14 BBs 
By 
and .*. we have 
1 + Bix + Bix’ + B+ see =- Bye re Ne oh (C) 
ei yes 
Lastly, putting « for 2, then multiplying both sides by 2, and simplifying, 
we find 
1+ A,c~?+ A,v~*+ Agv*+... ral ae Se (D) 
2-1 + Bua *+ Bozr-°+ Bar’ +... a Ys 
Tee OE 
ae gO 
Y3 . Yuh = 
Vi, Yo, --. bearing the same signification as before. 
Gauss in the treatment of his general series,* F (a, 8; y: 2), viz. 
1+% 2 ee) AIB+1) 5 , atl) (a+2) B41) C42) 
1.2 yqy+1)” eS vy+L) (y+2) © 

+. 
+ Disquisitio cirea seriem infinitam in the Abhandl. der Gotting. Gesellsch. d. Wissensch. Bd. 
ii. 1812. 

