208 On Fundamental Views regarding Mechanics. [June 14, 



In putting x\ y', z as constants, those forces of the complex are obtained 

 which pass through the given point {x , y\ z). In this supposition the 

 equation of the complex becomes the equation of a plane. This plane is the 

 locus of the second points, by means of which the forces of the complex are 

 determined. Hence in a linear complex there are acting on each point of 

 space forces in all directions, the intensity of each force being the segment on 

 its direction between the point acted upon and its corresponding plane (16). 

 Hence we derive the following theorem : — 



In a complex of rotatory forces any given plane of space is acted upon 

 by an infinite number of forces, each line within the plane being an axis of 

 rotation. The rotations round all axes are determined by second planes 

 passing through a fixed point, the position of which depends upon the 

 given point. 



I showed in the geometrical paper the way of discussing linear complexes 

 of right lines. The properties of linear complexes of forces, either ordinary 

 or rotatory, may be developed in a similar way. 



6. Right lines belonging simultaneously to two complexes constitute a 

 single congruency, and accordingly intersect two fixed lines ; if belonging 

 to three complexes, they constitute a double congruency, i. e. one gene- 

 ration of a hyperbolic or parabolic hyperboloid. Forces, either ordinary or 

 rotatory, belonging simultaneously to two, three, four complexes, consti- 

 tute a single, double, or threefold congruency of forces. In admitting 

 these denominations, the following results are immediately obtained. 



In a linear congruency of ordinary or rotatory forces, the directions of all 

 ordinary, or the axes of all rotatory forces, constitute a linear complex ef 

 lines. All ordinary forces acting on any given point of space are confined 

 within the same plane ; the intensity of each force is equal to the distance 

 between the point acted upon and the point where its direction meets a 

 fixed line within the mentioned plane. The axes of all rotatory forces, 

 acting upon any given plane of space, meet in a fixed point within that 

 plane. There is a fixed line passing through the fixed point, round which 

 the second planes of all forces turn when their axes turn round the fixed 

 point in the given plane acted upon. 



In a double congruency of ordinary forces there is one force of given 

 direction and intensity acting upon any point of space, as there is in a 

 double congruency of rotatory forces one force acting upon any given plane. 



In a threefold congruency of ordinary forces the directions^ in a three- 

 fold congruency of rotatory forces the axes of all forces constitute one of 

 the two generations of a hyperboloid. 



These few indications may be sufiicient here ; but before concluding I 

 must, in referring to the original paper, make a last remark. Forces acting 

 along a given right line may either be regarded as the same, whatever may 

 be the point of the line acted upon, or they may be regarded as varying in 

 intensity according to the position of the point. There is an analogous 

 distinction to be made with regard to rotatory forces. Accordingly two 

 different kinds of complexes must be distinguished. 



