[ u )5 1 
. y : 
XIII. ExtraB of a Letter from the Right Honourable Philip 
Earl Stanhope, F. R. S. to Mr. James , Clow, Prof ef 'or of 
Philofophy in the Univerjity of Glafgow. Dated Chevening, 
February 16, 1777. 
Read June 10, 1780. 
I PIAVE lately made fome curious obfervations concerning 
the roots of adfedled equations, part of which have oc- 
curred to Meffieurs daniel Bernoulli, euler, de la 
grange, lambert, and others ; but fome of them, I be- 
lieve, are quite new. I will give you one inftance of a qua- 
dratic equation, as the fimpleft. 
Let the quadratic equation 1 ixx — 1 5# + 5 = o, be propofed. 
I fay then, that if two recurring feries be formed from the 
fractions -LtiL 2 which have a common denomi- 
■zz i^-z — zz 
nator, and each feries of co-efficients, continued both ways 
(that is, as well before, as after the firft term) the fractions 
formed by dividing each term of the firft feries by the cor- 
refponding term of the fecond feries, viz. 
&c. 
21, ±2, Z±, ±2, 
14 +9 —5 +4 
• 3 2 1 
-? -? -5 
4 3 2 
1 1 
L 1, 
5 7 12 19 31 
, 2 &c . 
5° 
will converge in the f mpleft manner poffible; thofe before the bar, 
•m 
a retrograde order to the greater root ; and thofe after 
; where it is to 
22 
the bar. in a direfl order to the final led: root — ^ 5 
22 
C c % 
be 
