206 



Prof. J. Pliicker on Fundamental Views [June 14^ 



theorems constituting the base of statics. In a similar way as d'Alembert's 

 principle is derived from the "principe des vitesses virtuelles," both 

 theorems may be transformed into fundamental theorems of mechanics. 



In starting from the coordinates of ordinary forces, the equations of 

 equilibrium are 



=0, =0, =0, | , 



which, by replacing these forces by rotating ones, become 



2(2f-0 =0, ^{u-u') =0, -Liy-v') =0, 1 

 ^{uv'—uv)^0, ^{vt'—vt)=Oy =0. J ^ 



We likewise may express the conditions of equilibrium by the following six 



equations — 



2(.r-.r') = 0, 2(y-y') = 0, 2(^-y) = 0, 1 



^{t-t') =0, 2(m— m') = 0, 2(v-t?') = 0, J ^ ^ 



three of which are taken amongst (9), three amongst (10) — and expand 

 these equations in the analytical way, starting either from the consideration 

 of ordinary or rotatory forces. The interpretation of these equations 

 immediately gives the following two theorems. In the case of equilibrium — 



I. The centre of gravity of the points acted upon by the forces coincides 

 with the centre of gravity of the second points, hy means of which the 

 forces are determined (No. 1). 



II. The central jplane of the planes acted upon by the rotatory forces 

 coincides with the central plane of the second planes, by means of which 

 the rotatory forces are determined (No. 2)*. 



If we introduce the notion of masses, both theorems hold good, only the 

 definition of both kinds of forces and therefore their unity is changed. The 

 points acted upon become centres of gravity corresponding to masses ; the 

 planes acted upon, central planes corresponding to moments of inertia. 



If equilibrium does not exist, we get, in the general case, one resulting 

 force, determined by the two centres of gravity, and one resulting rotatory 

 force determined by the two central planes. We easily obtain the six 

 differential equations of the movement produced. 



I shall think it suitable further to develope the principles merely indi- 

 cated in the paper presented. A Treatise on Mechanics, reconstructed on 

 them, will assume quite a new aspect. 



4. In making use, within a plane, of point- or line-coordinates, we repre- 

 sent, by means of an equation between the two coordinates, a plane curve 

 described by a point, or enveloped by a right line. In making use, in 

 space, of point- or plane-coordinates, by means of an equation between the 

 three coordinates, a surface is represented which may be regarded as a 



* The coordinates of tbe central plane are obtained in the same way by means of the 

 coordinates of the given planes as the coordinates of the centre of gravity are obtained 

 by means of the coordinates of the given points. 



