22 Mr.Woolhouse on the Theory of Vanishing Fractions, 



trary doctrine, that a result can be received as general only so 

 far as the reasoning employed in deducing it is fairly applica- 

 ble to the particular instances. In the present case consider 

 the general equations given in my former letter, viz. 



— {.r—a)" ^ x — y{x—a)^ (^x ... (p) 



0= {x — aY-^^x—yi^x {q) 



If we proceed from the first of these equations to the second, 

 we divide by the factor [x—a)^ ; and if we proceed from the 

 second equation to the first, we multiply by the same factor. 

 The logical accuracy of either of these processes must neces- 

 sarily fail for the particular case in which x is equal to a, as 

 I have shown in my former letter ; for in this case the first 

 equation is evidently satisfied by any value of y while the lat- 

 ter limits it to a particular value. The complete solution of 

 the first equation [p] must comprehend both systems of values 

 obtained from the two separate equations indicated by the two 

 factors, viz. 



= x — a (r) 



= {x—aY-^^x—y(^x ... (s) 

 and as the former does not contain the quantity y, that vari- 

 able may obviously possess any value whatever in the first 

 system when x fulfills the equation o = x—a. The system of 

 values given by the condition (s) will determine the curve 

 M N described at page 28 of my essay, while the former sy- 

 stem will determine the indefinite straight line R S; and both 

 the curve and straight line will complete the geometrical re- 

 presentation of the equation [p), while the curve alone is the 

 representation of {q). It would be just as improper to reject 

 the system of values furnished by (r) in the present case, as it 

 would be to reject the system of values furnished by the equa- 

 tion ()3) in page 20, in resolving the equation (a). If in the 

 equation (a) we were to adopt the hypothesis that the quan- 

 tities must not fulfill the condition (|S), the whole of the solu- 

 tions would be comprised in the result obtained by the ge- 

 neral resolution of (y) ; but if this hypothesis were removed 

 the result so obtained would evidently fail to represent all the 

 solutions to the proposed equation. In the same manner, if 

 in the equation ( j) ) we were to adopt the supposition that the 

 condition ( r) must not be satisfied, or that x must not = «, 

 the whole of the solutions would be comprised in the resolu- 

 tion of (s) alone; but the result would necessai'ily fail to be 

 complete when the supposition becomes removed*. 



• The celebrated Bishop Berkeley, in his able work entitled the " Analyst," 

 has, with singularly acute and convincing aiguments, amply elucidated 

 this fallacy of shifting the hypothesis. 



