Mr. J. Cockle's Analysis of the Theory of Equations. 501 



may give an arbitrary value to one of the unknowns and 

 then apply, to the resulting tertiaries, the processes just dis- 

 cussed. So, we may make the fourth unknown subservient to 

 a simplification of those processes. If we employ the method 

 of homogeneous elimination, we may (see Mec. Mag.vol.xlviii. 

 p. 512) so use the fourth unknown as to avoid the superpo- 

 sitions and transformation prescribed in («.). If we employ 

 the method of vanishing groups, we may, by expelling the 

 first power of one of the unknowns, avoid one of the super- 

 positions ; or, by expelling the first power of all the unknowns 

 simultaneously, we may abridge all the superpositions. In 

 applying the method of vanishing coefficients we should also, 

 most probably, materially shorten our operations, xlnd, pos- 

 sibly, in all three methods an advantageous relation might be 

 established between the two given complex quadratics. If we 

 have given for solution two quadratics of the fifth order, Mr. 

 Jerrard's process becomes explicitly and immediately appli- 

 cable. I would here add, that, by a combination of the method 

 of disposable multipliers with that of vanishing coefficients (to 

 which latter method the former may, in innumerable instances, 

 be made subservient), we may, as I have shown (Mech. Mag. 

 vol. xlix. p. 10), solve two tertiary quadratics without any de- 

 composition whatever. 



16. And here, with another remark or two, I shall leave 

 the subject of the quadratic solution of complex quadratics. 

 You will understand, by the term " quadratic solution," solu- 

 tion without the occurrence of equations of a degree higher 

 than the second. You will not, however, fail to remark that 

 the subject is inexhaustible, and that there are infinite num- 

 bers of systems which admit of a quadratic solution. The 

 method of vanishing groups, for instance, enables us to obtain 

 quadratic solutions of m simultaneous quadratics of the 

 (2 m — l)th order. It must be borne in mind, that, in attempt- 

 ing the application of the method of homogeneous elimination 

 to the quadratic solution of more than two complex quadratics, 

 we must, by the aid of superposition, disposable multipliers, 

 and a sufficient number of undetermined quantities, endeavour 

 to form a number of distinct systems, each system consisting 

 of two equations in each of which tw r o unknowns are homo- 

 geneously involved ; the unknowns being different in each 

 system. Or we may combine, as circumstances may suggest 

 or require, the three different general methods considered in 

 the 14th article of this letter. Let me here add an important 

 remark connected with the method of vanishing groups. 

 When the squares of all the unknowns are absent from a 

 given complex quadratic, or when any square or squares 



