86 PHILOSOPHICAL TRANSACTIONS. [ANNO 1781. 



they ever perceived, when the shocks in question have happened, any falls of the 

 loose and shattery strata, in which the last especially work ; yet the earthquakes 

 have had violence sufficient to terrify the inhabitants of the surface. Neither 

 were these local ; for, excepting the first, all may be traced to very remote parts. 

 The weather was remarkably still at the time of every earthquake Mr. P. 

 had felt. 



XIII. On the Roots of Equations, in an Extract of a Letter from the Rt. Hon. 



Philip Earl Stanhope, F. R. S., to Mr. Jas. Clow, Prof, of Philos., Glasgow. 



Dated Chevening, Feb. ]6, 1777- p- 195. 



I have lately made some curious observations concerning the roots of adfected 

 equations, part of which have occurred to Messieurs Daniel Bernoulli, Euler, De 

 La Grange, Lambert, and others ; but some of them, I believe, are quite new. 

 I will give you one instance of a quadratic equation, as the simplest. 



Let the quadratic equation lixx — 15a - -f- 5=0, be proposed. I say then, 



l_l_ Or O _1_ *%■? 



that if two recurring series be formed from the fractions — , *" — — 



1 — Z — ZZ 1 — Z — ZZ 



which have a common denominator, and each series of co-efficients, continued 

 both ways (that is, as well before, as after the first term), the fractions formed 

 by dividing each term of the 1st series by the corresponding term of the 2d series, 



j, -U +7 -4 +3 -1 -3 2 _l I 3 4 _7 11 18 29 » .„ 



viz. sc. _ i4 , + g , _ 5 , +4 , _ 1? _ 4 , -, -, i 5 , 7 , i2 , -, — , — , &c. Will 



converge in the simplest manner possible ; those before the bar, in a retrograde 



" + - "" 



22 



order to the greater root — — — ; and those after the bar, in a direct order to 



the smallest root — — — ; where it is to be observed, that the greater root is 

 affirmative, notwithstanding the sign — being prefixed to some of the terms, 

 because in each fraction the numerator and the denominator are affected by the 

 same sign, whether + or — . 



The chief improvement I have made, consists in approximating to two roots 

 at once, by one and the same series, continued backwards as well as forwards. 

 I have not time to enlarge on this subject at present ; but the little I have said 

 will be a specimen of the method to be used in higher equations. 



