77 



theorem, that two sides of a triangle are equivalent to the third, when 



direction, as well as magnitude, is taken into account) proposes an 

 elementary step in symbolization which consists in representing the 

 Translation of a Directed Magnitude by a Product. Any magnitude 

 which is drawn or points in a particular direction, such as a force, a 

 velocity, a displacement, or any of those geometrical or physical 

 quantities which we exhibit on paper by arrows, he calls a directed 

 magnitude. By the translation of such a magnitude he means the 

 removal of it from one position in space to another ivitJiout change 

 of direction. 



U representing any directed magnitude and u any distance, the 

 translation of U to any parallel position in space, in such wise that 

 every point or element of U is caused to describe the distance is 

 termed the translation of U along u. 



This translation consists generally of two distinct changes, one the 

 lateral shifting of the line of direction of U, and the other the motion 

 of U along its line of direction. The former is called the trans- 

 verse effect, the latter the longitudinal effect of the translation of U 

 along u. 



Both these effects are shown to be products of U and u ; the 

 transverse effect is represented by wU, and the longitudinal by z^.U, 

 inserting a dot between the factors in the latter for the sake of 

 distinction. 



The author then goes on to apply the principles established to the 

 proof of the Parallelogram of Forces, and the determination of the 

 effect of any set of forces on a rigid body. In doing this a remark- 

 able symbolization of the point of application, as v/ell as the direc- 

 tion and magnitude of a force, is obtained, namely, that the expres- 

 sion (l4-w)U represents a force U acting at a distance u from the 

 origin. 



The principles of statics are deduced with remarkable facility 

 from the symbolical representation of the translation of a force along 

 a given distance. 



2. On an Air-Engincc" By James Prescott Joule, F.R.S. (S:c. 

 Received May 13, 1851. 



The air-engine described in this paper consists of a pump by 

 which air is compressed into a heated receiver; and a cylinder, 

 through which the air passes again into the atmosphere. The dif- 

 ference between the work evolved by the cylinder and that absorbed 

 by the pump, constitutes the work evolved by the engine on the 

 whole. Two tables are given ; the first of which contains the pres- 

 sure, temperature and work absorbed, for various stages of the 

 compression of a given volume of air. The second table gives the 

 theoretical duty of the air-engine described, worked at various pres- 

 sures and temperatures. The temperature recommended to be 

 adopted in practice is as little below the red heat as possible, which 

 would involve the consumption of only about one-third the amount 

 of fuel consumed by the best steam-engines at present constructed. 



