DR. PLUCKEE OX TUXDAMEXTAL VIEWS EEGARDIXG MECHANICS. 365 



passing through a given point meet the right line along which the two conjugate ^ilanes 

 of the point in the two complexes intersect each other. 



In a congruency, there act on every point of space an infinite number of forces along 

 right lines constituting a plane, their intensity being given by the distance of the point 

 acted upon from the points of a given right line confined within that plane. 



6. Let 



EeeQ-1 = 0, E'^a'-\.=0, H"eee:O"-1=0 (19) 



represent any three linear complexes of forces. Forces, the coordinates of which satisfy 

 simultaneously the three equations, constitute a double congruence of forces. Hence we 

 derive immediately the following theorem : — 



In a double congruence of forces there is passing through each point of space one single 

 force of given direction and given intensity. 



The intensity of the force is equal to the distance between the point acted upon and 

 the point where the three planes conjugate in the three complexes meet. 



7. We may derive from the equations (19) the two following: 



Q-O'=0, Q-Q"=0. (20) 



The coordinates of forces of the double congruency satisfy likewise both equations (20), 

 the system of which represents a congruency of right lines. 



The forces of a double congruency act upon right lines which constitute a congruency. 



I proved in the geometrical paper that all lines of a congruency intersect two given 

 lines. Hence 



All forces constituting a double congruency meet two fixed lines. 



8. In following our way we meet congruent forces of four complexes constituting a 

 threefold congruency. Their coordinates satisfy simultaneously the equations of the four 

 complexes, 



E=0, E'=0, S"=0, E'"=0, \ . (21) 



as well as the equations 



E-S'=0, E-E"=0, S-E"'=:0 (22) 



derived from them, the system of which represents a rectilinear hyperboloid. Hence 



The forces belonging to a threefold congruency act along the generatrices of a hyper- 

 boloid*, the points of which are the points acted upon. There are conjugated to such 

 a point in the four complexes of forces (21) four planes meeting in another point of the 

 same generatrix. The distance between the two points represents the intensity of the 

 corresponding force, varying if the point acted upon move on the generatrix. 



9. There are only two forces belonging simultaneously to five complexes, i. e. there 

 are two right lines, on each point of which one single force of given intensity acts along 

 its direction. Indeed by means of the five equations of the complexes, we may deter- 

 mine, by elimination, five of the six coordinates, which, for simplicity, may be denoted 



* Geometrical Paper, p. 757. 



