364 DE. PLtJGKEE ON FUNDAMENTAL VIEWS EEGAEDING ]IIEC1L\NICS. 



From my geometrical paper* we deduce that, by a proper transformation of coordi- 

 nates, the function Q may be reduced, in putting 



AD+BE+ CF 

 ^~ v'D* + E«+F«' ^ ^ 



F=x/D'+F+F% (12) 



to the simple expression 



Accordingly the general equation of the complex E assumes the form 



¥(.Ty'-a^>/) + C(z-z')=l; (13) 



and in putting 



AD + B E + CF_e 



D2^E,2^r« —pi—f^% \^V 



-^^^^=^^=j,=k^, (lo) 



may be written thus, 



(ay-.r'y)+^(2-s'-^-')=0 (IG) 



There is in a complex of right lines an axis round which it may revolve, and along 

 which it may be displaced parallel to itself, without being changed. After this double 

 movement each line of the complex occupies the place formerly taken by another of its 

 lines. After the transformation of coordinates, the axis of the complex Q, which may 

 be likewise called the axis of the complex of forces E, coincides with OZ; the origin 

 being arbitrarily chosen on OZ, and the axes OX and OY being any two right lines 

 drawn through this point perpendicular to OZ and to each other. 



The form of the last equation shows that a linear complex of forces S, like the corre- 

 sponding complex of lines Q, remains unaltered when rotating round its axis or moving 

 parallel to it, i. e. each force of the complex in its new direction and the new position 

 of the point upon which it acts, continues to belong to the complex in retaining its 

 intensity. 



5. Let 



E = Q-1=0, S' = Q'-1=0 (17) 



represent any two linear complexes of forces. Congruent forces of both complexes, the 

 coordinates of which satisfy simultaneously both equations (17), constitute a congruency 

 of forces. Their coordinates satisfy likewise the equation 



E-E = r2-Q'=:0, (18) 



derived from the primitive ones by eliminating their constant term. Hence 

 In a congruency, the forces act along right lines constituting a linear complex. 

 The forces of a congruency belonging simultaneously to two complexes, those of them 



* Geometrical Paper, p. 746. 



