184 MR. GRAVES ON A RECTIFICATION OF THE 



resulting series calculated to any number of terms, and the respective functions which they ought to 

 represent. 



Such developments have been said to be analytically accurate, notwithstanding the numerical dis- 

 crepancy in each particular case. " They serve," it is argued, " to represent their functions, and by 

 performing algebraical operations upon them, correct conclusions are attained." 



Now, it appeared to me that there was some confusion of expression in asserting universally that 

 equations were analytically true, which, numerically considered, were, in particular instances, palpably 

 false. In ascertaining the correctness of the conclusions deduced from them, and relied upon as 

 evidence of the truth of their premises, I observed that the formerly rejected test of numerical identity 

 was often appealed to. Nay further, I was induced to ascribe, in the absence of other visible causes, 

 to the intervention of such equations the limited results which were occasionally elicited where pre- 

 vious calculations would lead to the expectation of general ones, and even the conclusions absolutely 

 and unlimitedly erroneous to which the mathematician was sometimes conducted by apparently un- 

 deviating paths. 



To account for these difficulties, upon reverting to first principles, it will be found that the 

 theorems of development (such as Taylor's, Maclaurin's, &c.) are based upon hypothetic rea- 

 soning to this eflFect, viz. " if the function be developable according to certain powers, it will be deve- 

 loped in a certain form," which is assigned. Now imagine a function of x, for instance, which for 

 those values only of x that lie between certain limits, is capable of being developed according to the 

 ascending integral powers of x, such a function, it would seem, evolved by Maclaurin's theorem, 

 would afford an expansion which, when x transgresses those limits, would be illusory. 



In the treatment of developments thus partially true, when more than one of them come in ques- 

 tion, the extent of their compatibility should, in my opinion, be most carefully attended to ; for, if two 

 such developments of a function were equated, whereof the one was applicable for values of the 

 variable which would render the other illusory, the consequences derived from such equation might, 

 in proportion to the extent of those values, be partly or entirely false. An instance of the limitation 

 introduced by the caution here recommended is to be found in Appendix § 4. 



To learn how far a development was applicable, it might be useful to ascertain the error com- 

 mitted upon calculating n terms of the series, and, then supposing n an infinitely great integer, to 

 observe if there were any values of the variable wliich would prevent the expression for the error 

 from vanishing. 



Should these reflections appear dubious or unfounded, I wish it to be fully understood that they 

 may, in that case, be considered as operating on my results only, at most, by way of superfluous 

 caution. Thus, if c be deemed unnecessary to the universal accuracy of the series [18], it has, at all 

 events, the merit of ensuring its convergence. 



Since writing the above, I have been informed by Professor Hamilton that M, Poisson has lately 

 given examples of the danger of using diverging series, even when the final development to which 

 they conduct is converging. 



Note F. — This seems to prove that the logarithms of negative numbers are not in general the 

 same as those of their positives, as Jean Beenouilli and D'Alembert thought. (See Lacroix, 

 "Traite," &c. Introd. 82.) Hence also conversely by easy inference it seems to follow, that negative 

 numbers have occasionally even real logarithms, contrary to the opinion that they have none whatever, 



