120 



Reviews and Notices of Books. 



We have entered thus into detail in order to introduce to our 

 readers the following very beautiful proof of the *' equable descrip- 

 tion of areas — 



" When the acceleration is directed to a fixed point, the path 

 is in a plane passing through that point ; and in this plane the 

 areas traced out by the radius-vector are proportional to the 

 time employed. . . . Evidently there is no acceleration per- 

 pendicular to the plane containing the fixed and moving points, 

 and the direction of motion of the second at any instant ; and 

 there being no velocity perpendicular to this plane at starting, 

 there is therefore none throughout the motion ; thus the point 

 moves in the plane. Again, if one of the components (of the re- 

 sultant velocity of a particle) always passes through the point 

 (about which the moments are reckoned), its moment vanishes. 

 This is the case of a motion in which the acceleration is directed 

 to a fixed point, and we thus prove, that in the case supposed, 

 the areas described by the radius-vector are proportional to the 

 times ; for the moment of velocity, which in this case is constant, 

 is evidently double the rate at which the area is traced out by the 

 radiuS'Vector." 



We have only time, before leaving this division of the subject, 

 to request our readers' attention to the novelty and beauty of the 

 authors' proof that the evolute of a cycloid is a similar and 

 equal cycloid ; and with this remark we pass on to the second 

 part of the pamphlet, in which dynamical laws and principles 

 are discussed. And first, we note that the term centre of inertia 

 is advantageously substituted for centre of gravity. 



" The centre of inertia of a system of equal material points 

 (whether connected with one another or not), is this point whose 

 distance is equal to their average distance from any plane what- 

 ever. The centre of inertia of any system of material points 

 whatever (whether rigidly connected with one another, or con- 

 nected in any way, or quite detached), is a point whose distance 

 from any plane is equal to the sum of the products of each mass 

 into its distance from the same plane divided by the sum of the 

 masses. 



"The sum of the momenta of the parts of the system in any 

 direction is equal to the momentum, in that direction, of the whole 

 mass collected at the centre of inertia." 



The following is the authors' statement of the second law of 

 motion, and its corollary the parallelogram of forces :- — 



** When any forces whatever act on a body, then, whether the 

 body be originally at rest or movivig with any velocity and in any 

 direction, each force produces in the body the exact change of mo- 

 tion luhich it would have produced if it had acted singly on the 

 body originally at rest. 



