and Infinite Series. 405 
otherwife their product zy will not be = -~p, as it ought 
to be. 
42. Or if we put a = \p y and b - the fame three 
roots will be 
sft> + ^ b~ + a 1 + sj b - y/b z + a? — the ift root or r, 
- i ^/b+ ^F +7 . I - 3-i <¥b- VV+Ji . 1 + -/ — 3 the 2d root. 
“ i v^+ ‘Ib z + a 1 . 1 + V — 3 — § s/ t>— V'b z - S * ra‘ . i + v" — 3 the 3d root. 
43. Or again, the ift root r being 
•y b + \/tf- + a 1 + v' b — ^ b z + a*, the other two will be 
— |r+ ~~~ t> + v 'b' L + a 3 4 — — — v 7 b — ^b z + <? Et the 2d root, and 
2 2 
— Jr — V b + ^ ^ + a* ^ b ~ ^ rz the 3d root* * 
44. Or, if we put J = \/£ + ■+ <z 5 , apd « =• 
3 '-Yj £ + <z 3 , the roots will be 
4 + </ = r the i ft root, 
$ + d s — d 
+ s . — f%/— 3 = the 2d root, 
2 
- - iz_f v/— 3 == the 3d root. 
Jfli 
45 .The firft of thefe roots x or r ~ s + d-^/b + v b z + fiy 
+ sj b — y/b z + a\ is that which is called cardan’s rule, 
by whom it was firft publifhed, but invented by fer- 
REUS. And this is always a real root, though it is not 
H h h 2 always 
