Mr. J. Cockle's Analysis of the Theory of Equations. 495 



Part of a projected series of papers ' On the General Prin- 

 ciples of Analysis ' by our friend Professor J. R. Young, late 

 of Belfast. This first Part is the only one that has as yet ap- 

 peared. The obstacles to the further publication of the work 

 are stated on the cover of Part I., and will, I sincerely trust, 

 be speedily obviated by its more extended circulation among 

 mathematicians. It will richly repay perusal, and gives new 

 and efficient means of analysing an equation. In explanation 

 of the principle involved in it, I shall venture to say, that, 

 some two or three years ago, in the course of my Horce Alge- 

 braicce, I employed Mr. Gompertz's process for porismatizing 

 expressions, and I porismatized the surd factors of a quadratic 

 for the purpose of ascertaining whether in certain cases, the 

 roots of the given quadratic being k?iown 9 the surd factors so 

 porismatized admitted of being made to vanish when the 

 natural and obvious signs were given to the radicals. [See 

 Mechanics' Magazine, vols, xlvii. pp. 409, 410, and xlviii. 

 pp. 181-183.] By reversing this process, and changing its 

 object, Professor Young has been enabled, from the exami- 

 nation of the porismatized surd factors of equations of all de- 

 grees, to arrive at a knowledge of the nature and limits of the 

 unknown roots. This statement neither implies, nor is it in- 

 tended to imply the slightest claim to Professor Young's re- 

 searches. I neither have, nor do I wish to be considered as 

 having, any. This appears to be the proper place for adding, 

 that some of the topics included in the algebraic theory — such 

 as elimination and transformation — enter likewise into the 

 other theories in certain cases. The numerical process of 

 solution is in fact, in certain cases, a series of transformations. 

 And in this, as in most other similar instances, we find one 

 department insensibly encroaching upon another. For ex- 

 ample, it is difficult to refrain from considering the alge- 

 braic solution of a cubic as it bears, both upon the nature, and 

 upon the numerical evolution of the roots. Such considera- 

 tions are the common ground upon which two or more de- 

 partments coalesce, but it will serve no purpose to allude 

 further to them. 



7. A basis for the classification of algebraic equations is 

 presented to us, either in the dime?isions to which the unknown 

 quantities enter into a given equation, or in the number of 

 those unknowns, or in a combination of considerations derived 

 from both sources. An equation of n dimensions is said to 

 be of the n\\\ degree. An equation between n unknowns is 

 said to be of the wth order. I have now for some time adopted 

 this word order in exclusive reference to the number of un- 

 knowns contained in an equation. And I shall continue to 



