94< Prot. Young's concluding Remarks 



" appeal to the original analytical conditions." In the case, 

 however, of vviiat has been called a vanis]iing factor, the ex- 

 pulsion is always easily effected by the usual method of vanish- 

 ing fractions, and Mr. Woolhouse will find upon examining 

 anif modern author that nothing more is effected. 



Mr. Woolhouse appears to fall into great inaccuracy of 

 reasoning at page 22, when he infers that the introduction of 

 the factor x—a into the equation 



= {x—a)''~^Lv—yi^x, ((/.) 



must introduce new values for i/; or when he supposes that 

 the result of this introduction, viz. 



= {x-a) {{x-af-^ Lv-y^x}, {p.) 



is equivalent to the two independent equations 



= x—a (;•.) 



== {x-ciY~^ ^x — y^x (s.) 



If Mr. Woolhouse regard this as sound logic, the logic of 

 Lacroix must seem to him " palpably inconsistent" indeed. 

 I beg to suggest to Mr. Woolhouse that his equation (r.) 

 does not exist independently of, but only simultaneously with, 

 the equation (.?.); and he will not be able to point out any 

 writer who argues otherwise. He is also wrong in affirming 

 that the equation (/>.), due regard being paid to the circum- 

 stance of a foreign factor entering it,- becomes indeterminate 

 for X = a. That equation, as to number of admissible values, 

 is identical with (5.). If this be satisfied for x = a, then 

 x = a will be an admissible solution of (p.), but not else, 

 .seeing that (]>.) resolves itself, when .r = a, into the two si- 

 mvUaneons equations (r.) and (5.) ; and it is distinctly in re- 

 ference to this circumstance that the solutions of (p.) are to 

 be viewed. 



Such is, in substance, the doctrine of ail modern analysts, 

 but it is very different from that which Mr. Woolhouse has 

 gratuitously condemned. I regret that he did not examine, 

 with more attention, the process of Lacroix, in solving the 

 problem of Clairaut, to which I referred him in my formei' 

 letter. I think if he had done this, instead of passing over in 

 silence so decided an argument against his own " principles " 

 as that process furnishes, he surely would have spared the 

 truisms with which his " Reply" abounds. 



It is no doubt possible that Mr. Woolhouse may have met 

 with some obscure work in which his zero piocesses may oc- 

 cur. I certainlv have never seen any such work ; and if one 



