498 Mr. J. Cockle's Analysis of the Theory of E quations. 



contraction, the contraction by depression, and, lastly, a simple 

 transposition furnishes us with the required roots. 



11. In the case of a quadratic the superposition is of the 

 simplest kind. A determined quantity is added and sub- 

 tracted. But, if we suppose the quantity superposed to be 

 undetermined in its value, we give our analysis a greater range. 

 Under this new form I shall term it the subsidiary method, as, 

 in my ' Notes ' above alluded to, I have previously done, in 

 allusion to the subsidiary quantity so introduced. The sub- 

 sidiary method is not, however, limited to the use of one new 

 undetermined quantity. Simpson's solution of a biquadratic, 

 as given by Murphy at p. 54? of his Theory of Equations, is 

 an instance of a subsidiary process involving three undeter- 

 mined quantities B, C, D, besides the unknown x. As there 

 employed, the quantity m appears to be used, rather to deter- 

 mine the subsidiaries, than as a subsidiary in itself. But by an 

 alteration of the process, which I shall not dwell on, m might 

 either be dispensed with altogether, or be made a strictly sub- 

 sidiary quantity, and, thus, the means of observing uniformity 

 of method. So, by a slight change of form, my solution of an 

 imperfect cubic (Phil. Mag. S. 3. vol. xxii. pp. 502, 503) may 

 be presented as effected by a quadratic subsidiary function of 

 x and p. Again, Ferrari's solution of a biquadratic is a sub- 

 sidiary one. And there is a certain form of equations of the 

 fifth degree (that which I proposed for solution in the Diary 

 for 184-9) which admits of a subsidiary solution. 



12. The case, that I have just alluded to, is that in which 

 we superpose a subsidiary quantity or quantities foreign to the 

 expression in its primitive form. But suppose that we have 

 before us a quadratic function involving, in its original state, 

 m undetermined or unknown quantities besides determined or 

 known ones. Then, in general, by means of successive su- 

 perpositions we may reduce the given quadratic function to 

 the form of a sum of m squares of linear functions of the un- 

 knowns, together with the square of a known quantity. This 

 remark I shall bring to bear presently. 



13. But, before doing so, let us proceed to examine the 

 species of complex equations that belong to the genus qua- 

 dratics. The first question that now presents itself is, — Can 

 we solve two simultaneous binary quadratics by means of equa- 

 tions of no higher degrees than quadratics ? I call this the 



Jirst question, because it is sufficiently obvious that, if we are 

 required to solve n linear and one quadratic, all of the (n-\- l)th 

 order, or of some lower one, then no elevation of degree will be 

 introduced by elimination. Nor will such elevation be intro- 

 duced by any number of linear equations in combination with 



