504 



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



III. ^^Fundamental Views regarding Mechanics." By Professor 

 Julius Plucker, of Bonn, For. Mem. B.S. Eeceived May 29, 

 1866. 



(Abstract.) 



Being encouraged by the friendly interest expressed by English geome- 

 tricians, I have resumed my former researches, which had been entirely 

 abandoned by me since 1846. While the details had escaped from my 

 memory, two leading questions have remained dormant in my mind. The 

 first question was to introduce right lines as elements of space, instead of 

 points and planes, hitherto employed ; the second question, to connect, in 

 mechanics, both translatory and rotatory movements by a principle in 

 geometry analogous to that of reciprocity. I proposed a solution of the 

 first question in the geometrical paper presented to the Royal Society. I 

 met a solution of the second question, which in vain I sought for in 

 Poinsot's ingenious theory of coupled forces, by pursuing the geometrical 

 way. The indications regarding complexes of forces, given at the end of 

 the Additional Notes,*' involves it. I now take the liberty of presenting 

 a new paper, intended to give to these indications the developments they 

 demand, reserving for another communication a succinct abstract of the 

 curious properties of complexes of right lines represented by equations of 

 the second degree, and the simple analytical way of deriving them. 



1. We usually represent a force geometrically by a limited line, i. e. by 

 means of two points {x\ y\ z) and {x, y, z), one of which (a?', y', z') is the 

 point acted upon by the force, while the right line passing through both 

 points indicates its direction, and the distance between the points its inten- 

 sity. We may regard the six quantities 



x—x\ y—y'y z—z'y yz—y'z, zx'—z'x, xy'—xy (1) 



as the six coordinates of the force. The six coordinates of a force repre- 

 sent its three projections on the three axes of coordinates OX, OY, OZ, 

 and its three moments with regard to the same axes. Accordingly we 

 may, as far as we do not regard the point acted upon by the force, replace 

 its coordinates by 



X, Y, Z, L, M, N (2) 



in admitting the equation of condition 



LX+MY+NZ = 0, (3) 



which indicates that the axis of the resulting moment is perpendicular to 

 the direction of the force. In- making use of the primitive coordinates 

 this condition is involved in the form given to them. 



When the last condition is not satisfied, the coordinates X, Y, Z, L, M, 

 N represent no longer a mere force ; they are the coordinates of what I 

 proposed to call a dyname. 



In the general case such a dyname results when any numbers of given 

 forces act upon any given points. Here the six coordinates of the dyname 



