OH the Theory of Vanishing Fractions. 95 



exist it must be perfectly unique ; anil I trust that Mr. Wool- 

 house will make it known to tlie public, in order that con- 

 noisseurs may possess themselves of so singular a specimen of 

 scientific absurdity. 



From what I have now said it will be seen that I deny, 

 in toto, the justness of Mr. Woolhouse's charge of bad logic 

 in the common processes of the doctrine of vanishing frac- 

 tions; and I have moreover very briefly shown how those 

 processes ought to be interpreted. It is only upon this as- 

 sumed bad logic that Mr. Woolhouse rests the stability of his 

 remarkable "principles " ; if then the logic is shown to be 

 sound, but that Mr. Woolhouse has, unconsciously, misinter- 

 preted the steps, what becomes of these said " principles " ? 



I shall now take my leave of this subject; I have carefully 

 examined my former letters, and do not find a single remark 

 which I wish to recall, nor a single mathematical statement 

 at variance with received and well-established principles. The 

 only alteration I would wish to make is, that the word " may" 

 be substituted for "will" immediately after equation (2.) at 

 page 517 of last volume; the "will" occurring for " may" 

 justifies Mr. Woolhouse's foot-note at page 24.. 



Ample as are the materials which Mr. Woolhouse's last 

 letter supplies for coumient and objection, I shall in conclu- 

 sion merely notice two points, more imnjediately concerning 

 myself. At page 21, Mr. Woolhouse says that I "deny the 

 competency of the results of the ellipse question to furnish the 

 requisite values, and at the same time agiee to receive them 

 from the original analytical conditions," Mr. Woolhouse is 

 again at fault; let him read what I really do deny, instead of 

 attributing to me his own imaginings. I have said at page 

 298 (last volume) that" the /ac/ of the problem admitting 

 multiple solutions is information which the analytical result is 

 incompetent to supply ;" this is very different from asserting 

 that these multiple solutions, if they exist, could not be fur- 

 nished by the result; the question is— do they exist or not? 

 and on this question the result supplies no information. 

 Again, Mr. Woolhouse appears to have views different from 

 other people on the subject of singular solutions, or else he 

 uses technical terms in reference to this topic, in an unautho- 

 rized sense. It is sufficient for me here, without inquiring into 

 his peculiar notions, to show that my remark, in reference to 

 this subject, is in strict accordance with the received language 

 of analysis. I have said that " singular solutions, though not 

 comprised in the resulting integrals which furnish the general 

 solutions to (01 tain dillcnntial cfjualions, have neverllicless tlie 

 proi)crty of satisfying the proposed conditions." Lngrange 



