438 PROFESSOR YOUNG, ON THE PRINCIPLE OF CONTINUITY, 



,T~' becomes infinite for x = ; so that we have two continuous series of values, each terminating, 

 or each commencing at .r = ; which value of x however is united to one series by the sign plus, and 

 to the other by the sign minus. Hence, integrating over the former series, we have log ?j - log ; 

 and integrating over the latter, we have log (- m) - log (- 0). Consequently 



,+" 

 / ,f "'</,(■ = {log ti - log Oj - jlog (- m) - log (- 0)J 



''-in 



n m n 



— log log; — = log — . 



" " ° m 



There is of course nothing new in thus dividing a definite integral into portions : but the 

 treating of these portions, when their boundaries are infinite, as distinct continuities, allowing the 

 influence of each continuous law to operate throughout the entire range, the limits included : — this 

 mode, I say, of treating what are called discontinuous functions, is not that generally adopted ; 

 though the neglect of it has occasioned a difficulty that has appeared to interfere with the clearness of 

 the idea of a definite integral when considered as the limit of a summation. Moreover, from this 

 same neglect, Poisson and others have been led to very erroneous values for the definite integrals 



included in the form / x'' da. Thus Poisson affirms that 



f a'-' dx = - (fin + 1) ir\/- 1*, 



.*' 

 an imaginary quantity, instead of zero as above: and the value of / .t ^ dx he states to be —2, 



•'-1 

 instead of infinite, as it is found to be by the method here proposed, which gives 



/ x-"dx = (+ CO - 1) - (- CO + 1) = 2 cc - 2, 



and many other such errors might, if necessary, be adduced from his writings. 



But tile examples already given of the influence of the principle of continuity in extreme 

 or limiting cases of general forms, and of the mistakes committed by analysts from disregarding this 

 influence, will, I think, be considered as sufficient to invite more general attention to this matter: 

 and I shall rejoice if the brief and imperfect sketch I have here attempted to give of the views and 

 principles, by conforming to which such mistakes may be avoided, meet with acceptance from the 

 Cambridge Philosophical Society. I have been induced to submit it to the indulgent consideration of 

 that distinguished body, chiefly because the topics embraced in it have already furnished matter for 

 two Papers printed in tlie Cambridge Transactions : — one by Professor De Morgan, and the other 

 bv the Rev. Mr. Earnshaw. I have ventured to entertain the opinion that the views and investi- 

 gations of these excellent analysts do not preclude the necessity for a further consideration of the 

 interesting and somewhat delicate points of analysis which they have discussed : an opinion which is 

 strengthened by the fact, that the Papers referred to are in a considerable degree opposed to each 

 other, both in principle and in result. It is scarcely necessary to add, that in the present communi- 

 cation I have contemplated the subject under an aspect more or less different from that in which it 

 has been considered either by Mr. De Morgan, or by Mr. Earnshaw ; and I think it probable, 

 from the study of the three Papers, that the truth may be elicited ; and something like consistency 

 and stability be at length given to a portion of analytical science which has long been affected with 

 much uncertainty, vagueness, and perplexity. 



• Journal de I'Ecole Polytechnique., Cah. xvui. p. 318. 



