Mr. Cockle on the Method of Vanishing Groups. 115 



manner in which, to the best of my recollection, the method 

 occurred to my mind, may, as well as the subsequent portion 

 of this article, prove to be not entirely uninteresting to some 

 of the readers of the present Journal. 



I was engaged upon the subject of the transformation of the 

 general equation of the fifth degree into another of the same 

 degree, in which the second, third, and fourth terms should 

 be wanting. This transformation had already been effected 

 by the peculiar processes of Mr. Jerrard ; but another method 

 of arriving at it had occurred to me, suggested as follows. 



In analytical geometry of two dimensions, when we desire 

 to ascertain whether or not a given equation of the second 

 degree between two variable quantities x and y represents a 

 system of (in general) two straight lines, our object may be 

 attained thus: multiply the given equation by four times the 

 coefficient of .r^; add to and subtract from the equation, as it 

 will nowstand. the square of the coefficient of a? in the equation 

 as it originally stood ; the right-hand side of the equation 

 being supposed zero, the left will then consist of the square 

 of a linear function of x and y, together with a quadratic 

 function of j/ only; now if four times the coefficient of y^ in 

 this quadratic function be multiplied into the part free from 

 _y, and this product be found to equal the square of the coeffi- 

 cient of j/ in the quadratic function, that function is a square, 

 and the given equation represents two straight lines, if the 

 latter square be negative (or one only if the quadratic func- 

 tion disappears in tfie first instance, or a point if both the 

 squares be positive). 



This part of my studies at the University of Cambridge 

 flashed across my mind in comiexion with the problem re- 

 specting the equation of the fifth degree. The problem in 

 question requires the solution of a linear, a quadratic, and a 

 cubic equation ; or rather, in the point of view from which I 

 was considering it, of a quadratic and a cubic only. I now 

 saw that there might exist a quadratic equation between two 

 unknown quantities, which yet should involve us in no eleva- 

 tion of degree when one of those quantities was, by means of 

 such quadratic, eliminated from another equation involving 

 them. This favourable case of the quadratic occurs when 

 that equation consists, or may be reduced to the form, of one 

 or two s(|uares, which, or one of which at least, involve both 

 the unknown quantities. 



Now whenever the quadratic function of y vanishes, or is a 

 perfect sijuare, the given quadratic is of one of the forms just 

 alluded to. In the latter case, which alone attracted my at- 

 tention, the last-mentioned equation may be exhibited as the 



12 



