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XVI. Fundamental Views regarding Mechanics. 

 By Dr. J. Plucker, of Bonn, For. Ifemb. U.S. 



Received May 29,— Read June 14, 1866. 



Beixg encouraged by the friendly interest expressed by English geometricians, I have 

 resumed my former researches, which have 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, translatory and rotatory movements with each other by a principle in 

 geometry analogous to that of reciprocity. I proposed a solution of the fii'st 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," involve 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. 



I. 



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

 points (;r', 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 dii-ection, and the 

 distance between the two points its intensity. We may regard the six quantities 



x—x\ y—r/, z—d, y^—ijz, zci^—z'x, oc^—x'y (1) 



as the six coordinates of the force. The six coordinates of a force represent its three 

 projections on the three axes of coordinates OX, OY, OZ, and its three moments with 

 regard to the same axes. By means of the three first coordinates the intensity P and 

 the direction of the force ; by means of the three last the resulting moment E and the 



R 



direction of its axis ; by the quotient p the distance of the force from the origin, and 



therefore its position in space is determined. 



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

 replace its six coordinates (1) by 



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



MDCCCLXVI. 3 D 



