20 Mr. Woolhouse on the Theory of Vanishing Fractions, 



should resolve itself into two factors, and become of the form 



F (^ji/)/(-^>i/) = («) 

 Its complete solution will evidently comprehend the whole 

 of the solutions that can be obtained from the two separate 

 equations 



F(^,3/)=0 0) 



fi'r,y) = (y) 



each of which will determine a separate system of values. It 

 is hence clear that if we had to resolve only one of the two 

 equations (/3), (y), that we should arrive at only a. part of the 

 solutions to (a), and therefore that neither (/3) nor (y), sepa- 

 rately considered, can be regarded as a complete deduction 

 from the preceding equation. This principle, which applies 

 to any number of factors, is so well understood by mathe- 

 maticians that it would be a waste of time to discuss it at 

 length. When an investigation is conducted in the general 

 manner, just described, it is obvious that the final result 

 must furnish every value capable of satisfying the original 

 analytical conditions. Prof. Young must have overlooked al- 

 together the general natui'e of complete analytical processes 

 when he mentions " singular solutions" that occur in the in- 

 tegration of differential equations, as a case in opposition to 

 my remarks. He must be aware that a direct process of 

 integration of a differential equation, that admits of a singular 

 solution, will always determine that solution at the same time 

 with the general solution. The operation will lead to an equa- 

 tion consisting of two factors, similar to the foregoing equation 

 («), and when decomposed into the two separate equations 

 (/3)j [y\ the one will give the general while the other will give 

 the singular solution, and both solutions taken together will 

 form the complete solution. I should have thought that Pro- 

 fessor Young would have better examined the matter before 

 he favoured me with the gratuitous compliment that I had 

 done myself a " wanton injustice." We might similarly refer 

 to the case of a general solution of a functional equation, 

 usually obtained by the assumption of a particular solution : 

 with very few exceptions the general solution so elicited does 

 not comprehend all the forms, but only represents one of an 

 infinite variety of classes of solutions that will satisfy the pro- 

 posed equation ; and it cannot, therefore, be recognised as a 

 perfect result. It would, however, be wrong to adduce in- 

 stances of any kind in which the results are arrived at by in- 

 direct and defective means ; for in that case they would no 

 more be the results of investigation than the mere anticipa- 

 tion of a value by trial in the original equations : we cannot 



