•wilh Mr. T. S. Davies's Islotes on some of the Topics. 3.57 



the equation of its second term, antl so have recourse to 

 a transformation, and that many important particulars re- 

 specting equations depend upon a knowledge of the number 

 of their roots. How a knowledge of equations of the second 

 order is necessary in Cardan's solution of a cubic, I have 

 already remarked; but it remains to be observed, that in the 

 transition from equations involving one to those involving two 

 unknowns, we have the subject of ehmination, which is inci- 

 dental to that transition, forced upon us. The present, then, 

 would seem to be the proper place to discuss the subject. But 

 such discussion is not now my object, except to the extent of 

 remarking that, when the unknown quantities are sufficiently 

 numerous, the obstacles to the solution of a number of simul- 

 taneous equations involving those quantities do not arise from 

 elevation of degree incurred by elimination. That such ele- 

 vation of degree may be avoided, has been shown by Mr. 

 Jerrard in his Mathematical Researches : as may be expected, 

 the number of unknowns must, in order to the application of 

 his processes, considerably exceed in number the given simul- 

 taneous equations. The methods of Mr. Jerrard are in fact 

 indeterminate, and he has thrown open the gates that lead to 

 the higher parts of this species of analysis. 



Results similar to those of Mr. Jerrard may be obtained by 

 means of the method of vanishing groups. The latter method 

 will enable us to satisfy three homogeneous and simultaneous 

 quadratics between six undetermined quantitieswithout making 

 those quantities equal to zero or decomposing tticm, and without 

 having to solve other equations than a biquadratic, two qua- 

 dratics, and two linear etjuations. And for this system of 

 homogeneous quadratics we may of course substitute three 

 ordinary quadratics between ^V^ unknowns, and arrive at the 

 same results by the same means. 



The present appears to be a proper place for noticing the 

 subject of equations that have no root. An instance of such 

 an equation is given by Garnier at p. 335 of his Analyse 

 (Par. 1814). The subject of "congeneric surd equations" 

 was however first, I believe, discussed in anything like detail 

 by Horner in a letter to yourself, published in the year 1836 

 (Phil. Mag. vol. viii. pp. 43-50). In the last edition of Wood's 

 Algebra, doubts have been expressed as to the fact of the non- 

 existence of roots of a surd equation; and it is suggested that 

 possibly the method of solution is at ihult. But it may 

 easily be shown that no quantity other than a root of the 

 rational product of the surd congeners can satisfy any of those 

 congeners. Hence, when a congener is not satisfied by a root 

 of such product, it has no solution whatever; and I have 



