VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 535 



tion of equations, we have n distinct equations from the several co-efficients in 

 succession; viz. 



a _|, /) _[_ c _|_ d &c -\- n in number n = p, 



n i 



ab + ac + ad &c n X — 37* = 7, 



abc + abd kc w X — J— X -~- = r, 



abode 8cc. 77, or the product of them all, being the co-efficient of the last term. 

 Now, as we haven equations, and n indeterminate quantities, it is evident, that by 

 employing each equation successively to determine one quantity, the whole will be 

 determined. But the equations are not all of the same degree: the first, is a 

 simple equation: the 2d, being composed on one side wholly of products by two 

 is in degree a quadratic: the 3d, for the 6ame reason, a cubic: and so on. If the 

 first of these equations be used to determine a, we shall have a = p — b — c — d 

 & c# — n; inserting that value for a in the 2d equation, it becomes the quadratic 

 pb — /} 2 4. p c — c 2 -f- pd — d 2 — be — bd &c. = q. If that quadratic be solved to 

 determine b, and the values of a and b be inserted in the 3d equation, it becomes 

 the cubic c 3 &c. . . = r. Now, the quadratic having 2 roots, its solution will have 

 introduced a quadratic surd. Before therefore we can proceed to employ the 3d 

 equation to determine c, it must be squared to clear it of that surd, and of course 

 will then rise to the 6th degree. The solution of such a dimension, if admitted 

 for the present to be equally possible, must introduce higher radicals; and, by the 

 intrusion of these superfluous roots at every stage, our labour increases, instead of 

 diminishing. This is the difficulty alluded to before; and, as we have appropriated 

 already all our subordinate equations, we have nothing to oppose it. It therefore 

 seems hopeless, to expect to make any general impression on indeterminate equa- 

 tions, without more help, beyond the mere knowledge of the constitution of the 

 co-efficients. 



1 1. This difficulty however is wholly removed by the least circumstance that dis- 

 closes any particular relation among the co-efficients of an equation, independent of 

 the general law of their construction. This of course, whenever it occurs, fur- 

 nishes new conditions and means of comparing the terms. Every particularity in 

 the co-efficients that gives specific varieties to the forms of equations, must, from 

 the nature of their construction, have its source in some particular relation between 

 2 or more of the roots, and therefore, as far as that relation extends, detects them 

 infallibly. The observation of the forms and relations of the co-efficients under 

 different species of equations, and the correspondent inferences to be drawn, as to 

 the connection of their roots, would form a curious and very useful part of a com- 

 plete treatise on the whole doctrine of equations, which is a work much wanted. 

 The most striking of these relations will be obvious, or familiar, to the reader who 

 has at all considered the nature of the subject; such as, that equations deficient in 

 every alternate term arise from pairs of equal roots with opposite signs, ± a, ± b 

 &c. ; that those whose terms on both sides the middle term are alike, which are 



