36 



tion of an instance of fluid motion, which seemed to offer an accurate 

 means of comparing theory and observation in a class of motions, in 

 which, so far as the author is aware, they had not been hitherto com- 

 pared. The instance referred to is that in which a vessel or box of 

 the form of a rectangular parallelepiped is filled with fluid, closed, 

 and made to perform small oscillations. It appears from theory that 

 the effect of the inertia of the fluid is the same as that of a solid 

 having the same mass, centre of gravity and principal axes, as the 

 solidified fluid, but different principal moments of inertia. In this 

 supplement the author gave a series for the calculation of the prin- 

 cipal moments, which is more rapidly convergent than one w r hich he 

 had previously given. It is remarkable that these series, though 

 numerically equal, appear under very different forms, the rath term of 



nit 



the latter containing exponentials of the forms s nVx and g™, while 

 the nth. 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 difference between theory and experiment varied in 

 different cases from the -^th to the -^st part of the whole quantity. 



December 7, 1846. 



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 especialty 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 1 — x + x* 2 — x 5 + &c. ; these 



