Mr. Cockle on the Method of Vanishing Groups. 117 



point of view, be valueless to indicate its traces. Regarded in 

 this light, it is the solution of a quadratic by the ordinary 

 process of completing the square that affords a first glimpse 

 of the general process of the reduction of a general quadratic 

 function of any number of variables to the form of a sum of 

 squares. For, disregarding, as for this purpose we should do 

 throughout, the signs of the squares, and considering as an 

 algebraic square every numerical quantity, the usual solution 

 of a quadratic is effected by reducing a certain quadratic func- 

 tion to the form of a sum of two algebraic squares. 



Starting then from the solution of the quadratic as its cradle, 

 we next see the same process occurring in analytical geometry 

 of two dimensions, and availing us in inquiring into the con- 

 ditions requisite in order that a given function of the second 

 degree, and of two variable quantities, may be susceptible of 

 reduction to the form of a sum of two algebraic squares (in 

 which case the function represents a straight line or lines, or 

 a point). In the present instance, by continuing the process 

 with which we commence, we might reduce the given function 

 to the form of a sum of three algebraic squares, the " algebraic 

 square " last arrived at being obtained by subtracting the 

 square of half the coefficient of j/ in the quadratic function of 

 that quantity alluded to in the third paragraph of this paper, 

 from that part of the function which is free from ij. As I 

 have already stated, it was by combining a suggestion derived 

 from the process of ascertaining the reducibility of a quadratic 

 function of two variables to the sum of two squares, with an 

 idea suggested by a contemplation of the indeterminate me- 

 thods of Mr. Jerrard, that I devised the method of vanishing 

 groups ; and I take this opportunity of mentioning, with be- 

 fitting sentiments, that it was my friend Dr. Nathaniel Lister 

 of St. Thomas's Hospital, who, some ten years since, first 

 directed my attention to Mr. Jerrard's Mathematical Re- 

 searches, and induced me to undertake their perusal, — a pur- 

 suit to which I applied myself ardently during my second 

 term's residency at Cambridge. The necessity, however, or 

 at least the propriety, of devoting myself to other sciences and 

 branches of science, prevented me from giving my exclusive 

 attention to the theory of equations, ever a favourite study 

 with me; but in the spring of 1844 I arrived at the method 

 which forms the subject of this article, and it was published 

 in the Mathematician for July of that year. It is different 

 from a process wiiich I gave in a preceding volume (xxvi.) of 

 this Journal, and to which I have elsewhere(Mechanics' Maga- 

 *-zine, vol. xlvi.) given the name of the "method of symmetric 

 products;" but I iiave employed it in this Magazine (see vols, 

 xxvii. et scq.). 



