E l 95 3 
XIII. Ext raff of a Letter from the Right Honourable Philip 
Earl Stanhope, F . R. S. to Mr. James Clow, Profeffor of 
Philofvphy in the Univerfty of Glafgow. Dated Chevening, 
February 16, 1777* . 
1 j 
Read June 10, 1780, 
i ; ; v 
I 
HAVE lately made forne curious obfervations concerning 
the roots of adfedted 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 15X+ 5 — 0, be propofed. 
I fay then, that if two recurring feries be formed from the 
fractions -ii— , 2 — ^ which have a common denomi- 
zz 
z — zz. 
nator, and each feries of co-efficients* continued both ways 
(that is, as well before* as after the firft term) the fra&ions 
formed by dividing each term of the firft feries by the cor- 
refponding term of the fecond feries, viz. 
t> —11 +7 —4 +3 —1 —3 2 1 I 3 4 7 11 18 29 Q . 
&C. 5 — 5 — > — -> — ^ 5 -5 5 M — ) — > — > -- &C4 
-14 +9 -5 +4 -i -4 3 2 I 5 7 12 *9 3 1 50 
will converge in the fimpleft manner poffible; thofe before the bar, 
in a retrograde order to the greater root i and thofe after 
the bar, in a direft order to the finallefr root -VrAll ; where it is to 
C c % be 
