37 



limits being arrived at by the continuous variation of x from some 

 inferior value up to x=l, and from some superior value down to 

 a?= 1 . It is shown however that the series 1 — 1 + &c. has no equi- 

 valent among the individual cases of 1 — x + x' 2 — &c, with which 

 latter, indeed, it has no connexion whatever. 



By properly distinguishing between the real limits, and what is 

 generally confounded with them, the author arrives at several con- 

 clusions respecting the limiting values of infinite series directly op- 

 posed to those of Cauchy, Poisson, and others. And to prevent a 

 recurrence of errors arising from a neglect of the distinction here 

 noticed, he proposes to call sueh an isolated series as 1 — I + 1 — &c. 

 independent or neutral; and the extreme cases of 1 — x+x q — &c, 

 dependent series : the difference between a dependent and a neutral 

 series becomes sufficiently marked, as respects notation, by introdu- 

 cing into the former what the author calls the symbol of continuity, 

 which indeed is no other than the factor, whose ascending powers 

 Poisson introduces — and, as here shown, unwarrantably — into the 

 successive terms of strictly neutral series ; thus bringing such series 

 under the control of a law to which in reality they owe no obedi- 

 ence. 



An error somewhat analogous to this is shown to be committed 

 in the treatment of certain definite integrals, which are here submit- 

 ted to examination and correction, and some disputed and hitherto 

 unsettled points in their theory fully considered. The author is thus 

 led to what he considers an interesting fact in analysis ; viz. that the 

 differentials of certain forms require indeterminate corrections, in a 

 manner similar to that by which determinate corrections are intro- 

 duced into integrals ; and he attributes to the neglect of these the 

 many erroneous summations assigned to certain trigonometrical 

 series. This is illustrated by a reference to the processes of Poisson. 



The paper concludes with some observations on what has been 

 called discontinuity ; a term which the author thinks is sometimes 

 injudiciously employed in analysis, and prefers to treat discontinuous 

 functions as implying distinct continuities ; and by considering these 

 in accordance with the principles established in the former part of 

 the paper, he arrives at results for definite integrals of the form 



x~P dx totally different from those obtained by Poisson. Two 



■m 



notes are appended to the paper ; one explaining what the author 

 denominates insensible convergency and insensible divergency, and the 

 other discussing some conclusions of Abel in reference to certain 

 trigonometrical developments. 



March 1, 1847. 



On the Theory of Oscillatory Waves. By G. G. Stokes, M.A., 

 Fellow of Pembroke College. 



The waves which form the subject of this paper are characterized 



