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



equation of any degree that we please, and the system will 

 always admit of a determination of the degree of the equation 

 so substituted. 



21. Denoting, in conformity with a preceding nomencla- 

 ture, by the term pseudo-homogeneous equation of the wth 

 degree, an equation in which the unknowns enter to the same 

 (n) dimensions in every term excepting the absolute (deter- 

 mined or known) term, there are two cases, besides those 

 before enumerated, in which pseudo-homogeneous equations 

 admit of solution. Those cases, like the others, are, where 

 the pseudo-homogeneous equations are two in number, 

 both of the same dimensions, and those dimensions less 

 than 5 ; that is to say, where they are both cubics, or both 

 biquadratics. All that we can say of pseudo-homogeneous 

 equations of the fifth, or other higher degree, is, that they 

 admit of a determination of the fifth or other respective de- 

 gree. I have noticed (Mech. Mag. vol. xlviii. pp. 538, 539) 

 other properties of pseudo-homogeneous equations, not ne- 

 cessary to be now mentioned, and I have indicated (Ibid. 

 pp. 606, 607) a process by which two general cubics, if they 

 be of the ninth order, may be reduced to a system of binary 

 pseudo-homogeneous cubics, And I have also intimated, 

 what might be shown without much difficulty, that two com- 

 plex general biquadratics, if of an order sufficiently high, might 

 be reduced to a pseudo-homogeneous binary system, and so 

 resolved by means of simple biquadratics. If, in the discus- 

 sion of questions such as these, we wish to employ the method 

 of homogeneous elimination in as pure a form as possible, we 

 must, as I have already intimated, arrange our equations in 

 systems of pairs of biquadratics or cubics, and pairs ox triplets 

 of quadratics, availing ourselves of the methods of disposable 

 multipliers and of indeterminates to effect such arrangement, 

 if indeed it be possible. The solution of two 9-ary cubics, 

 alluded to above, is sketched out by means, not simply of the 

 method of homogeneous elimination, but also of the methods 

 of vanishing groups and vanishing coefficients, one or other of 

 them. 



22. Upon the subject of si?nple biquadratics I shall address 

 but few words to you, first observing that I have made some 

 remarks upon the subject at pp. 105, 106 of vol. iii. of the 

 Cambridge Mathematical Journal, where you will find the 

 same extension given to the solution of Descartes that Simpson 

 gave to that of Ferrari. I once thought that I had obtained 

 formulae for the reduction of biquadratics to a binomial form 

 (Phil. Mag. S. 3. vol. xxviii, pp. 132, 133), but I subsequently 

 (Ibid. 395) corrected my erroneous inference, the origin of 



