TRANSACTIONS OF THE SECTIONS. 



sin 1= 



For segments of (rue distance. 



sin Mm . sin M sin Ss . sin S 



sin bn sin Is 



. sinSs.sinS . sin So . sin ZM 



Bm ls = sinMmsinM sul lm = sin Mm. sin ZS s,n lm 



Ss . cos A 



- M^o7a sin(ws - Is) - 



On (he Figure of an imperfectly Elastic Fluid. 

 By Professor Hennessy, F.R.S. 

 It appears that the shape of a mass of such a fluid is dependent on its volume in 

 such a way that any abstraction from or addition to that volume will in general be 

 attended with a change of figure. This proposition, when applied to the case of a 

 mass in rotation, shows that if the earth has gradually passed into its present state 

 from one of complete fluidity, the figure of the inner surface of the consolidated crust 

 must be more elliptical than the stratum of fluid out of which it was formed. The 

 actual amount by which the ellipticity would be so increased would depend upon 

 the law of density of the fluid, but the general result of an increase in the value of 

 the ellipticity is completely independent of the law of density, and of any hypothesis 

 as to the constitution of the interior fluid mass. 



Note on Hie Calculus of Variations. 

 By Professor Lindelof, of the University of Helsingfors. 



In the problems with which the calculus of variations is concerned, it becomes 

 necessary to regard the form of certain unknown functions as variable and capable 

 of passing from any given form to any other in a continuous manner. The only way 

 of rigorously establishing this indispensable continuity appears to be that followed 

 by the illustrious Euler, that is to say, by the introduction of an arbitrary and vari- 

 able parameter. The function which ought to vary, or rather change its form, is 

 then regarded as a particular value of a more general function, which, besides the 

 principal variables, also involves an arbitrary parameter, and the variation of the 

 function is nothing more than its differential taken with respect to this parameter. 



The calculus of variations has thus become a simple application of the differential 

 calculus to the case where the function to be differentiated is a definite integral. 

 The first problem which presents itself is to find the variation of the given integral, 

 in other words, to differentiate the integral with respect to a parameter which it may 

 contain in any manner whatever. This first problem has gradually received its 

 complete solution through the researches of Euler, Poisson, and Ostrogradsky. But 

 the moment we attempt to pass to the applications of the calculus we encounter a 

 second, and more difficult problem, which for a long time has resisted the efforts of 

 the greatest mathematicians ; it consists in preparing the variation so that all its 

 parts may be examined, in other words, so that the equations relative to the limits 

 of the integral may be found. This preparation necessitates a series of partial in- 

 tegrations. Now the limits of the variables which, after the integrations, must be 

 substituted for the variables themselves are so mixed up with each other, and with 

 the variables themselves, that it appears impossible to take them into account by 

 means of the system of notation in general use. 



This difficulty has at length been happily overcome by an expedient, at once in- 

 genious and simple, due to M. Sarrus of Strasburg. In a memoir to which the 

 Parisian Academy of Sciences awarded a prize, M. Sarrus has introduced a new 

 symbol to indicate the substitutions to be made in any expression, and by means of 

 this symbol he has been able, not only to find the variation of a multiple integral, 

 but to examine the same completely. We may add that Cauchy, in a memoir on 

 the calculus of variations which he was never able to finish, adopts the innovation 

 introduced by M. Sarrus, after slightly modifying the symbol of substitution in order 



