in reply to Professor Young. 19 



Young seemed to imagine that I had contented myself with 

 testing the accuracy of those principles by particular examples, 

 and appeared to be unacquanted with the circumstance that 

 my especial object in writing that essay was to show the im- 

 propriety of multiplying and dividing by zero in analytical 

 reasonings, and to establish the applications of the differential 

 and integral calculus in a more consistent and logical manner. 

 But having already given in my former communication what 

 I consider to be a complete demonstration of the general 

 principles alluded to, it is indeed strange that Prof. Young, in 

 his present letter, should have followed the same course of 

 objection and evinced the same inattention to my repeated 

 denial of the logical accuracy of processes in which multipli- 

 cations or divisions by zero are concerned. At page 396, 

 I have distinctly proved the fallacy of such processes by show- 

 ing that in the one case we may pass from a condition that 

 determines a particular value to another of ah indeterminate 

 character, and that in the other we may pass from a condition 

 that is satisfied by any value to another that would limit the 

 unknown to a particular value. But even after this Professor 

 Young proceeds entirely on the incorrect supposition that not 

 only those operations, but that analytical processes under all 

 circumstances must necessarily lead to justifiable results; and 

 this he does without condescending to adduce a single argu- 

 ment in favour of so objectionable a supposition. 



Prof. Young writes rather largely in reply to my observa- 

 tions on what I designated in my former letter " the theory 

 of analytical results," and endeavours to make out that I have 

 fallen into a mistake. I introduce the subject by first ob- 

 serving, that " I never before heard of the incompetency of 

 an analytical result to afford any positive information that an 

 investigation could admit of." The mathematical readers of 

 the Journal who have read the paragraph containing my en- 

 tire explanation of this point, will readily perceive that Prof. 

 Young through thewholeof his observations has misinterpreted 

 the meaning of this isolated sentence, which I think he would 

 not have done had he taken " a more enlarged view" of the 

 analysis. As a matter of course, I allude to analytical results 

 arrived at by a process of general reasoning, not a solution 

 obtained by imperfect means. In every analytical investiga- 

 tion I consider that the generality of the process requires that 

 each step shall hold good both directly and conversely, and 

 consequently that each successive equation shall be fulfilled 

 by neither more nor less than the several roots that are con- 

 tained in the equations that have preceded. As an instance, 

 suppose an equation with all its terms collected to one side 



D 2 



