1866.] 



regarding Mechanics, 



207 



locus of points, or an envelope by planes. In my geometrical paper I 

 introduced the right line, depending on four constants, as the element of 

 space. An equation between these constants, if regarded as variables, re- 

 presents a complex of lines. Each line of the complex miaybe regarded either 

 as a ray described by a point, or as an axis enveloped by planes. In order 

 to render the developments symmetrical, I adopted as coordinates of the 

 right line the same expressions (1) and (4) (connected by an equation of 

 condition) made use of in the determination of ordinary and rotatory forces. 

 At the same time, general equations between the six coordinates were re- 

 placed by homogeneous ones. In doing so, A, B, C, D, E, F denoting any 

 six constants, 



+E(y^'-2/'.r)+F(^?/'+^y) = O=0, J ^ ^ 



A.{uv —u'v)-^-^{vt' —v't)-\-C(J;u' — t'u) 1 ^ 



-f D(^-0 + E(w-w') + F(^-O = '^ = ^ j ^^"^^ 

 represent the same linear complex of lines, which may be regarded as 

 a complex of rays as well as a complex of axes. 



When general equations between the same six coordinates are admitted, 

 the complex of rays becomes a complex of ordinary forces, the complex of 

 axes a complex of rotations or rotatory forces, both ordinary and rotatory 

 forces depending upon five constants. Here we meet a reciprocity be- 

 tween both kind of forces corresponding to the reciprocity between point 

 and plane. 



Finally, in omitting the equation of condition hitherto made use of, we 

 pass from forces to dynames, depending upon the six independent constants 



X, Y, Z, L, M, N, (14) 



or 1, % a ^ (15) 



A complex of dynames is represented by an equation between six variables. 

 Here again, as in the case of right lines, the same complex may be repre- 

 sented in a double way by means of the two sets of coordinates. 



In order to complete these general considerations, we add the following. 

 A dyname may be resolved into two variable forces, either ordinary or 

 rotatory. These variable forces constitute a linear complex, either of ordi- 

 nary or extraordinary forces. A homogeneous equation between the six 

 independent coordinates (14) or (15) represents a complex of two coupled 

 variable right lines. 



5. I will conclude by giving some details regarding complexes of the 

 first degree. 



A linear complex of ordinary forces is represented by the linear equation 



12=1, 



which may be expanded thus : 



(A + Fy— E^')a7- 

 + (B-F.r'-D/)2/ I , . 



+ (C + Ea?'-D2/> ^ ^ ^ 

 = A^' + By'-1-C/. 



