536 PHILOSOPHICAL TRANSACTIONS. [ANNO 17QQ. 



generally called recurring equations, arise from pairs of roots, of which each pair 



11 

 contains a quantity and its reciprocal, a, -, Z>, ^, &c. ; together with Maclaurin's 



demonstration of the particularities of the co-efficients when an equation has equal 

 roots.* And the extent to which these notices might easily be carried, from ob- 

 servations of the effects of the different sorts of proportion, and all other relations, 

 is prodigious. But my present concern is merely with the result, supposing from 

 any means a relation to be previously discovered affecting any number of the roots. 

 For example, — suppose, in the above given equation, of 1 — px*~ ' -f- qx"~* — rx?~* 

 + sof* &c. = 0, whose roots we called a, b, c, d, &c. . . . ?i, we happened to know 

 that 2 of the number, a and b, were equal; then, since they might both be ex- 

 pressed by the same character, the n roots of the equation might now be repre- 

 sented by only n — 1 distinct characters; and therefore, of the subordinate equa- 

 tions derived from the construction of the co-efficients, 2 might be employed to 

 determine one root, a and b being equal, the equation furnished by the value of 

 the co-efficient jb, and also that furnished by the co-efficient q, may be both toge- 

 ther used to determine the same quantity. But, if any quantity a be a root of an 

 equation, the simple equation x — a = O must be a divisor of that equation;-}- 

 therefore here a; — a must be a common divisor of the 2 equations furnished by/> 

 and q, and consequently may be found, without resolving either of them, by con- 

 tinual division or subtraction, according to the ordinary rule for finding the com- 

 mon measure.;}; 



12. Any other relation from the knowledge of which one character may be made 

 to represent two or more roots, evidently answers the same end. Indeed all rela- 

 tions of that kind may be converted into equality itself, by taking, instead of the 

 given equation, some other properly derived from it. Thus if, instead of a and b 

 being the same, b had been supposed the negative of a, or — a, and then, instead 

 of the former equation, that of the squares of the roots were taken, the relation 

 would be made equality; for a and — a have the same square. If arithmetical 

 proportion was known to be the relation of any number of the roots, by taking the 

 equation of their differences, it would also be converted into equality. 



13. If 3 or more roots, or any number of parcels of roots, are known to be re- 

 lated, and their common relation be used to represent them, of course the number 



* Vide Maclaurin's Algebra, chap. 4, p. Ifj2, et infra. f Vide Sanderson's Algebra, vol. 2, 



p. 679, 6*80, art. 432, and all algebras on the method of divisors. — Orig. 



\ Vide Sanderson's Algebra, quarto ed. vol. 1, p. 86", 87, S8, where the rule is well given; and 

 Maclaurin's Algebra, p. 2, cap. 4, p. J 62; or Mr. Hellins's Essay on the Reduction of Equations having 

 equal roots. But of the last it should be observed, that some qualification must be made to the assertion, 

 that the reduction may be carried on till a simple equation is obtained. In cases where there is only one 

 pair of roots equal, that proposition is undoubtedly true ; but, if 2, 3, or more pairs of roots are equal, 

 the reduction can only be carried down to a quadratic, cubic, &c. for, every pair of equal roots being 

 equally to be found by the method, of course the final or resulting equation must be of a dimension as 

 great as their number. — Orig. 



