598 Analysis of Books, fyc. 



Analytical Demonstration of the Law discovered by M. Savart, 

 relative to the vibrations of Solids and Fluids. By the same 

 Author. 



IT is here shewn that the equations of motion of an elastic body, the 

 particles of which are very little moved from their natural position, 

 may, by a slight modification, be applied to the demonstration of 

 the above proposition. 



The author concludes by observing, that it may be proved in the 

 same manner that if the dimensions of a body increase or diminish 

 in a given ratio, and the initial temperature increase or diminish 

 in the same ratio, the duration of the propagation of heat will vary 

 as the square of that ratio. 



Memoir on Torsion and the vibrations of Torsion in a rectangular 

 Rod. By the same Author. 



IN this paper the author obtains some analytical formulae, from 

 which he deduces the following results : 



i. The angle of torsion of a rectangular rod, fixed at one end and 

 free at the other, when measured in a plane perpendicular to the 

 axis of the rod, is as the distance of that plane from the fixed end, 

 and the moment of the force applied at the free end jointly. 



ii. If the transverse section of the rod is variable, but continues 

 similar to itself, the angle of torsion will vary inversely as the square 

 of the area of the section. These results, similar to those obtained 

 by M. Poisson for the torsion of a homogeneous cylindrical rod with 

 a circular base, will hold equally for a circular or prismatic rod on 

 any base. 



iii. If one transverse dimension of the rod becomes very small, 

 compared with the other, the angle of torsion will vary inversely as 

 the greater dimension, and the cube of the lesser. 



iv. The tones produced by the vibrations of torsion of a rectan- 

 gular rod are invariable, so long as the breadth and thickness of the 

 rod are in the same ratio. This is confirmed by the experiments of 

 M. Savart. 



v. If one transverse dimension of the rod becomes very small, 

 compared with the other, the lowest tone produced by the vibrations 

 of torsion is directly as the least dimension, and inversely as the 

 area of a transverse section. This is another of the laws to which 

 M. Savart has been led by experiment. 



vi. If the elasticity of the rod is the same in every direction, the 

 tones corresponding with the vibrations of torsion will be directly 

 as the product of the two transverse dimensions, and inversely as 

 the sum of their squares. 



vii. When the two transverse dimensions are equal, the lowest 

 tone produced by longitudinal vibrations will be to the lowest tone 

 produced by the vibrations of torsion :: 1.9364. . . : 1. 



[To be continued.] 



