REPORT ON CERTAIN BRANCHES OF ANALYSIS. 237 



In this case «' = in a, U = m a, and c' = 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 -r- in the values of the expressions for x and y is the 

 sign, or rather the indication of that indetermination. 



If 4- be not equal to -,-t, but if — be equal to —f, then x — co 

 b o, a 



and y = 00 . In this case vfe have a! — m a,h' =■ mh, but c' is 

 not equal to m c; 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 go in this case by the 

 term impossible. 



c e' b' b 



If -^ = ^7, but if —7 be not equal to — , then x is zero and y 

 h V a a 



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

 should include %ero 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 affection 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! J. ^ - ] 

 „2 -f- ^,2 - i' 



^ ^ it - \ 



and consider them as simultaneous when expressing the co- 

 ordinates of their points of intersection, then we should find 



X = //6^ W\ and y = /f^ ^ . 

 V la^ ~ ci'^S V 16^ ~ b'^S 



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

 same time b not equal to b', then or = op and y = cc , Avhich 



