REPORT ON CERTAIN BRANCHES OF ANALYSIS. 303 



be allowed, however, to consider it apart from such considera- 

 tions, it would appear to be complete and satisfactory, and 

 very carefully guarded against any approach to an assumption 

 of the proposition to be proved, a defect to which most of the 

 demonstrations of this class are moi*e or less liable*. It extends 

 to equations whose dimensions involve different or successive 

 orders of parities, nearly in the same manner as in the demon- 

 stration which we have given above. 



The demonstration given by Mr. Ivory is different from any 

 other, and the principles involved in it are such as naturally 

 present themselves in such an investigation ; and it will be re- 

 commended to many persons by its not involving directly the 

 use, or supposing the necessary existence of, imaginary quan- 

 tities. It is not, however, altogether free from some very serious 

 defects in the form under which it at present appears, though 

 most of them admit of being remedied without any injury to 

 the general scheme of the demonstration, which is framed with 

 great skill, and which exhibits throughout a perfect command 

 over the most refined and difficult artifices of analysis. 



Lagrange has devoted two notes to his great work on the 

 Resolution of Numerical Equations to the discussion of the 

 forms of the roots of equations. In the first of these notes, 

 after examining the very remarkable observations of D'Alem- 

 bert on the forms of imaginary quantities, he proceeds to con- 

 sider the case of an equation such as f (x) + V = 0, where 

 y (x) is a rational function o£ x ; if for different values a and b 

 of the last term of this equation, where a ^b,we may suppose 

 a root which is not real for values of V between those limits, to 

 become real at those limits, he then shows that for values 

 of V between those limits, and indefinitely near to them, the 

 corresponding root of the equation must involve -/— 1, or 

 •v/ — 1, or \/ — l, and so on; or, in other words, that the roots 

 of the equation in the transition of their values from real 

 to imaginary (whatever may be the affection of magnitude 

 which re nde rs them imaginary), will change in form from a to 

 7W -f- « -/ — 1 . He subsequently shoAvs that the same result will 

 follow for any values of V between a and b, and consequently, 



* I do not venture to speak more decidedly; for though I have read it en- 

 tirely through several times with great care, I do not retain that distinct and 

 clear conviction of the essential connexion of all its parts which is necessary 

 to compel assent to the truth of a demonstration. It is unfortunately fre- 

 quently the character of many of the higher and more difficult investigations 

 connected with the general theory of the composition and solution of equa- 

 tions to leave a vague and imperfect impression of their truth and correctness 

 even upon the minds of the most laborious and best instructed readers. 



