8 



of these different values. This principle is the one which was adopted 

 by Leibnitz in his well-known explanation of the meaning of 1 — 1 

 + 1 — 1 + ... Without assuming that anything like proof can be 

 given, the author notes, as instances in which the thing asserted is 

 true, algebraical series, trigonometrical series, Fourier's integral, 

 Poisson's expression of a function between any limits by an infinite 

 series of trigonometrical integrals, and also the sine and cosine of 



infinity. Assuming / (pxdx: (b — «) to represent the mean value 



J a 



of <p x between a and b, the author tries what ought to be the value 

 of tan oo , if this principle be true, and finds + V — 1, which on trial 

 is found to satisfy the fundamental equations of trigonometry. 



Section 4. — Series of alternately positive and negative signs stand 

 upon a much safer basis than those in which all the terms have the 

 same signs, and that whether their divergence be finite or infinite. 



It has long been observed, that when the terms of an alternating 

 series begin by diminishing, even though they afterwards increase, 

 the converging portion may be made effective in approximating to 

 the arithmetical equivalent of the series. The error committed by 

 stopping at any term is not so great as the first of the rejected terms. 

 In many alternating series this has been proved to be true, and it 

 seems never to have been supposed that the theorem was anything 

 but universal. In this section instances are produced in which the 

 theorem is not true ; and at the same time various proofs of it are 

 given, each of which applies to very extensive cases, and the tendency 

 of which is to show that it is only under definite and unusual condi- 

 tions that the theorem can fail. Still, however, no positive criterion 

 is established for ascertaining whether the theorem be true or not in 

 any particular case. 



Section 5. — On double infinite series, in which the terms are in- 

 finitely continued in both directions. 



It seems, in many different ways, that the series 



... + <p(x-2) + ([>(x-l) + <l>.r + (p(x + l) + <l>(x + 2) + ... 

 can be resolved, by analytical transformation, into + 04-0 + + . , . 

 When there is no discontinuity whatever in the relation between 

 <p x + <p (#+ 1) + . . . the value of the preceding is 0. But when dis- 

 continuity dees exist, the value of the series may be some other so- 

 lution of ^(#+1) = \px. This assertion, derived from observation 

 of instances, is here discussed in the case of 

 _ 1 



the value of the double series is obtained, and some corresponding 

 products of an infinite number of factors are deduced. 



