Cambridge Philosophical Society. 137 



the latter containing exponentials of the forms e"** and e " , while 

 the «th term of the former contains exponentials of the second form 

 only. In conclusion, the author referred to some experiments which 

 he had performed with a box, such as that described, filled with 

 water, employing the method of bifilar oscillations. The moment of 

 inertia of the fluid about an axis passing through its centre of gra- 

 vity (?'. e. the moment of inertia of the imaginary solid which may 

 be substituted for the fluid), was a little greater as determined by 

 experiment than as determined by theory, as might have been ex- 

 pected, since the friction of the fluid was not considered in the cal- 

 culation. The diff"erence between theory and experiment varied in 

 difi^erent cases from the Jgth to the -J^st part of the whole quantity. 



Dec. 7. — On the Principle of Continuity in reference to certain 

 results of Analysis. By Professor Young of Belfast College. 



The object of this paper is to inquire into the influence of the law 

 of continuity, as it affects the extreme or ultimate values of variable 

 functions, more especially those involving infinite series and definite 

 integrals. 



The author considers that this influence has hitherto been impro- 

 perly overlooked ; and that to this circumstance is to be attributed 

 the errors and perplexities with which the different theories of those 

 functions are found to be embarrassed. He shows that every parti- 

 cular case of a general analytical form — even the ultimate or limiting 

 case — must come under the control of the law implied in that form ; 

 this law being equally efficient throughout the entire range of indi- 

 vidual values. Except in the limiting cases, the law in question is 

 palpably impressed on the several particular forms ; but at the limits 

 it has been suffered to escape recognition, because indications of its 

 presence have not been actually preserved in the notation. 



It is in this way that the series 1 — 1 + 1 — 1-|- &c. has been con- 

 founded with the limits of the series \—x-\-x°—x^+ &c. ; these 

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

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

 x=si\. It is shown however that the series 1 — 1 + &c. has no equi- 

 valent among the individual cases of 1— a? + a?^— &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 such an isolated series as 1 — I + 1 — &c. 

 independent or neutral; and the extreme cases of l—x+x''^ — &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 



