Mr. J. Cockle's Analysis of the Theory of Equations. 507 



nation will be converse terms, denoting inverse operations. 

 Now, let us consider two general tertiary quadratics. Elimi- 

 nate one unknown between them, and we have a binary bi- 

 quadratic; and, if I mistake not, we may so determine one of 

 the unknowns, as to reduce this biquadratic to a simple bi- 

 quadratic, of such a form as to be capable of solution by means 

 of quadratics only. Regard as known the process of adlimi- 

 nation by which we pass, from the binary biquadratic, to the 

 system of two tertiary quadratics. So that, in point of fact, 

 if we had been asked, — What must be the order of two qua- 

 dratics in order that they may admit of quadratic solution? 

 — we might answer thus. Find the least number of indeter- 

 minates by which the elevated equation (a biquadratic), which 

 is the result of ordinary elimination, may be depressed to a 

 quadratic ; and to this add unity, and as many quantities as 

 may be necessary to form the original binary system by adli- 

 mination. If it were required to find the order of two cubics 

 capable of cubic solution, I should proceed thus. Find how 

 many indeterminates are required in order that an equation of 

 the ninth degree may be depressed to a cubic. To this num- 

 ber add unity and the quantities required for adlimination. 

 Although you may find this paragraph at present somewhat 

 vague, I do not think that it will eventually turn out to be 

 meaningless. More on the subject I shall not say now. 



29. The apathy of which I have complained above is not 

 limited to the present time. When, generalizing Descartes' 

 method of representing curves, Pacent, John Bernouilli, and 

 Clairaut had represented surfaces by an equation between 

 three variables, and when, moreover, the general reduction 

 of the equation of curves of the second degree had been 

 effected, all the materials were at hand for effecting a trans- 

 formation which is among the most striking that Mr. Jerrard 

 has presented to algebra — that of the equation of the fifth 

 degree to a trinomial form. And, even now, this general 

 reduction, as effected by vanishing groups, is the simplest way 

 of performing the transformation in question. But the step 

 was never taken until the transformation had been effected by 

 other means. Had the general reduction of a tertiary qua- 

 dratic presented itself in its full force to earlier investigators, 

 it would have given them & prima facie, but illusory, transfor- 

 mation of a biquadratic to a binomial form, and would have 

 involved them in a fallacy that the algebra of that time might 

 not readily have exposed. But, attention once drawn to the 

 point, we might now have been surveying a richly cultivated 

 garden, instead of what is almost a barren waste. And yet 

 we must not complain that those mighty minds were idle, 



