in reply to Professor Young. 23 



In speaking of dividing by zero, it must be understood that 

 in deducing the equation 



_ {x — aY^x 



^ ~ {x-af<^x ^^^ 



from {p\ we do not in this case actually perform the opera- 

 tion of division so long as the fraction is not reduced to 

 lower dimensions. The equation (^) in its present shape is 

 synonymous with the condition {p), being merely the same 

 condition differently written down. I have had occasion to 

 say before that the mere placing of a quantity in the denomi- 

 nator of a fraction cannot be strictly recognised as an actual 

 performance of division. 



From the foregoing observations the effect of the fault com- 

 mitted in dividing by zero is very evident. We proceed from 

 the condition (j9), which contains an hifinite number of values 

 of y, to another that contains only one value. The fault is 

 similar to that which would be committed in dividing the 

 equation (a) in page 20, by F {x,y), and then taking only the 

 resulting equation (y) for the solution of («), which would 

 necessarily exclude from the results the solutions contained 

 in (/3). But the error in the present case would be greater as 

 we should exclude from the result the infinite number of 

 values contained in (r). If we were permitted to multiply and 

 divide by zero we should be at liberty to play many curious 

 tricks with analytical equations. As an instance, if any equa- 

 tion were proposed for solution, of the form ^ {x,y) = 0, we 

 might first multiply by any arbitrary function Q {x,y) which 

 would give <P{x,y) $ {x,y) = 0; we might next divide by 

 4> {x,y\ which would give fl {x,y) = 0, that is, we should be 

 able to make any function whatever of the same quantities 

 equal to zero, which is a manifest absurdity. 



On the whole we may remark, that when the value of a 

 symbol i/ which ought to be determinate comes out in a va- 

 nishing fraction (t), the fault alluded to has occurred in 

 the process. The symbol j/ and its corresponding vanishing 

 fraction as they appear in the equation ( t ) cannot in this case 

 represent the solution of the original condition, but a solution 

 of the equation which is antecedent to the final result and 

 which possesses an infinite number of values; and consequently 

 the symbol given in the final result is not strictly synonymous 

 with the same symbol as it appears in the original equation. 

 It is therefore vain to refer to the original condition for the 

 interpretation of this vanishing fraction, for in that case we 

 interpret the root of the first condition, whereas the fraction 



