REPORT ON CERTAIN BRANCHES OF ANALYSIS. 237 



In this case «' = m a, U = m a, and d ■= m c, and the second 

 equation is deducible from the first, and does not furnish, there- 

 fore, a new condition : under such circumstances, therefore, 

 the values of x and y are really indeterminate, and the occur- 

 rence of "Y in the values of the expressions for x and y is the 



sign, or rather the indication of that indetermin'ation. 



c c' . h V 



If ^- be not equal to -.-r. but if — be equal to —f, then a; = oo 

 b ^ Of a a 



and y = (X> . In this case w^e have a' = m a, b' = m b, but c' is 

 not equal to m e ; and the conditions furnished are inconsistent, 

 or more properly speaking impossible. In this case, the occur- 

 rence of the sign oo in the expressions for x and y is the sign 

 or indication of this inconsistency or impossibility, and it should 

 be observed that no infinite values of x and y, if the infinities 

 thus introduced were considered as real existences and identi- 

 cal in both equations, would satisfy the two equations any more 

 than any two finite values of x and y which would satisfy one 

 of them. We may properly interpret oo in this case by the 

 term impossible. 



c c' V b . 



If 1- = -77, but if — r be not equal to — , then x is zero and y 

 b b' a' ^ a ^ 



is finite, and therefore possible. It is in this sense that we 

 should include zero amongst the possible values of x or y, a 

 use or rather an abuse of language to which we are somewhat 

 familiarized, from speaking of the zero of quantity as an exist- 

 ing state of it in the transition from one aflPection of quantity to 

 another. 



If we should take the equations of two ellipses, whose semi- 

 axes are a and b, a' and b' respectively, which are 



f! + ^ - 1 

 «2 + b^ ~ ' 



f! 4. 1! - 1 



and consider them as simultaneous when expressing the co- 

 ordinates of their points of intersection, then we should find 



^ = //^ ^\ ^^^ y = //^ ^\ • 



V 1«2 ~ «'^/ V \w ~ b'^J 



If we suppose — = —7, or the ellipses to be similar, and at the 

 same time b not equal to 6', then .r = op and y = co , which 



