116 Mr. Cockle on the Method of Vanishing Groups. 



sum or difference of two squares, one of which will involve 

 two, and the other one, of the undetermined quantities. And 

 this sum or difference may be decomposed into linear factors, 

 by equating one of which to zero we might ehminate x or y 

 from any other given equation involving tliose quantities. No 

 elevation of degree would ensue from such elimination. I 

 reflected on this in considering the quadratic, which presented 

 itself in the question of the before-mentioned transformation 

 of the equation of the fifth degree. 



But — how to impress tiiis form on that quadratic ? to render 

 the quadratic function of y a square? And here Mr. Jerrard's 

 indeterminate method afforded me a useful suggestion — the 

 answer was obvious. If we can render the system of given 

 equations indeterminate, and so introduce an undetermined 

 quantity into the coefficient of y and into the term free from 

 y in the quadratic function of that quantity which remains 

 after we have obtained a square by adding to and subtracting 

 from the given quadratic (multiplied by four times the coeffi- 

 cient of .r) the square of the original coefficient oi x — then by 

 the aid of this undetermined quantity, can we not reduce the 

 given quadratic to the required form, and so avoid elevation of 

 degree in eliminating x or y between it and any other given 

 equation? On trying this method I found it succeed. The 

 result, which bears traces of the manner in which I obtained 

 it, will be found at page 114 (art. 3) of the first volume of the 

 Mathematician, although the notation used in this paper is 

 different from that employed there. I have however here 

 given the notation in which the idea is most likely to have 

 suggested itself to my mind, and in which, if my memory 

 serves me aright, it actually suggested itself 



Had I to discuss the same problem again, I should pro- 

 bably employ a homogeneous function of tour undetermined 

 quantities, and, by successive operations of the same kind as 

 those which I have indicated at page 267 of the current num- 

 ber of the Cambridge and Dublin Mathematical Journal, re- 

 duce it to a sum of four squares, and then make the sum of 

 the first and second equal zero, as also that of the third and 

 fourth. The fundamental principle of the whole method, ac- 

 cording to the view which I now take of it, is, for homoge- 

 neous functions of the second degree, this reduction by a uni- 

 form process to squares as many in number as the undeter- 

 mined quantities. 



Investigations of very different classes may yet present fea- 

 tures of resemblance in their results, their principles, or their 

 processes. And wherever such resemblance exists, — be it in 

 result, principle or process, — it can never, in a philosophic 



