Remarks on the Tlieory of Vanishing Fractions. 93 



able values which the expression is competent to give, seeing 



(^ (j^ 



that it then assumes the indeterminate form . He has 



a— a 



altogether overlooked the fact that the questions, What is the 



value of ^ when x = a ? and, What is the value of 



x — a 



"2 2 



? (ouestions which have certainly been confounded be- 



a — a ^^ '' 



.r" — w 

 fore,) are perfectly distinct. The value o^ ^_ when x = a 



is not , as Mr. Woolhouse supposes; this is merely the 



x'^ — a^ 

 symbohcaiyorm which —_ assumes in that case. The value 



of ' is x-\-a; and this vaUie when x = a is 2a. The ex- 



X —a 



^•^ (i^ 



pression ' implies an operatio?i to be performed, viz, 



division, and the value is the rcsidt of that operation. When 



the form is isolated, or detached from its interpreta- 



a — a ' 



tion, it is of course indeterminate ; for the operation indicated 

 requires that we determine such a quantity P, that when it is 

 submitted to the reverse operation, {a— a) P, the result may 

 be a- — a-; and it is easily seen that an infinite variety of suit- 

 able values for P exist; lor it may be generally expressed by 

 F = (a + a) ± ]). 



X'—Q' 



But the resull of the operation '— is definite, and distinctly 



points out which of the above infinite variety is comprehended 



among the values of in the ultimate state of the hv- 



° x—a •> 



pothesis, to the exclusion of all others. 



Mr. Woolhouse fancies that when a vanishing factor, .r— a, 

 is introduced into an algebraic process an iudeterminateness 

 is ir)tro(luced at the same time. This is contrary to fact, and 

 to the doctrine of all writers on the subject. The introduction 

 of a foreign factor, whether by elimination or otherwise, can 

 7ievcr affect the analytical limitations which existed before its 

 entrance; and it is the well-known and universal practice of 

 analysis to reject these foreign I'aclors at the close of the pro- 

 cess, although they are not always discoverable without an 



