Mr. J. Cockle's Analysis of the Theory of Equations, 503 



of the unknowns from the remaining two given quadratics, 

 both of which may be solved by means of a biquadratic and a 

 quadratic. And we may effect the biquadratic determination 

 of three quadratics of the sixth order without any transforma- 

 tion or decomposition of the unknowns, but by the direct ap- 

 plication of the method of homogeneous elimination (Ibid. 

 p. 512). By a combination of the method of disposable mul- 

 tipliers with that of Mr. Jerrard (similar to the combination 

 alluded to in article 15 of this letter), three 5-ary quadratics 

 admit of a biquadratic determination (Ibid. vol. xlix. pp. 10 r l 1). 



19. A system of four quadratics of the eighth order is ca- 

 pable of biquadratic determination, by a combination of su- 

 perposition, disposable multiplication, and the method of 

 vanishing coefficients. I have pointed out two methods of 

 effecting this determination. [Ibid. vol. 1. pp. 33, 34. The 

 'second solution' of the last- mentioned page, however, re- 

 quires the corrections which I subsequently gave at p. 106 of 

 the same volume.] This system of four 8-ary quadratics is also 

 capable of cubic determination (Ibid. p. 34, ' Third Solution'). 

 And here I leave the subject of the determination of complex 

 quadratics, throughout which you will have borne in mind the 

 distinction between a determination of equations, and that of 

 the variables which they contain. The terms 6 quadratic,' &c. 

 prefixed to * determination,' indicate the sense in which the 

 term determination is used ; and it may, at some future time, 

 be desirable to give a still more precise system of nomencla- 

 ture in reference to that term. By regarding their determi- 

 nations we shall often be able to obtain an advantageous com- 

 parison of problems. You will remark that, whenever a sy- 

 stem of equations admits of a simple determination of less 

 than five dimensions, such a * determination' amounts to a 

 solution of the system. In other words, when n is less than 

 5, a simple n-\c determination of the system gives us the 

 means of solving it. 



20. Of simple cubics I shall say nothing. When complex 

 quadratics and a complex cubic are presented to us simulta- 

 neously for solution, it will always be possible to solve them, 

 provided that the system would admit of quadratic solution, if 

 in place of the cubic we had a quadratic, For, if no elevation 

 of degree be introduced into a quadratic by elimination, then 

 none would be introduced were a cubic substituted for it. 

 Linear equations in fact introduce no elevation, whatever be 

 the degree of the equation in which the elimination takes 

 place. And I may here observe — once for all — that, when- 

 ever a system of quadratics admits of a general quadratic so- 

 lution, then we may substitute for one of the quadratics an 



