1866.] 



regarding Mechanics. 



205 



are the sums of the six coordinates of the given forces (x', y', z, y, c). 

 If between the six sums thus obtained, 



^(x-x'), ^{y-y'), \ /4) 



^{yz-yz\ ^{z3(^—s'x), ^{xy'—x'y),\ ' ' " ^ ^ 

 an equation analogous to (3) takes place, there is a resulting force. In 

 the case of equilibrium the six sums become equal to zero. 



2. In quite an analogous way as we have determined an ordinary force 

 by means of two points in space, one of which is the point acted upon, we 

 may represent a rotation, or the rotatory force producing it, by means of 

 two planes, 



t'x-{-u'y-\-v'z=\, tx-{-uy-\-vz=lf 

 the coordinates of which are u', v and t, w, one of the two planes 

 (t', u'y v') being the plane acted upon.. The right line along which both 

 planes meet is the axis of rotation. The plane acted upon may in a 

 double way turn round the axis of rotation in order to coincide with the 

 second plane (t, u, v) ; but there is no ambiguity in admitting that during 

 the rotation the rotating plane does not pass through the origin, and con- 

 sequently its coordinates do not become infinite. 

 Let us regard the six quantities 



t—t', u—u'y V — v'y uv'—uvy vt' — v'ty tub — t' u (5) 



as the six coordinates of the rotatory force. As far as we do not regard 

 the plane acted upon, we may replace them by 



^e, % 3. ^ (6) 



in admitting the equation of condition, 



SX+9}l?)+^3 = (7) 



(For the geometrical signification of these new coordinates I refer to the 

 original paper.) 



When the last equation of condition is not satisfied, the coordinates 36, 

 5), 3> ^> ^ longer represent a mere rotatory force ; they are the six 

 independent coordinates of what I called a rotatory dyname. 



In the general case, such a rotatory dyname results when any number of 

 given rotatory forces acts on any given planes. Here the six coordinates 

 of the resulting rotatory d3mame are the six sums of the six coordinates of 

 the given rotatory forces. 



^{u-u'), ^{v-v% I 

 ^(uv'—u'v), ^{yf—vt), ^(tti—t'u)J ^ 



If between these six coordinates the equation of condition subsists, there is 

 a resulting rotatory force. In the case of equilibrium the six sums become 

 equal to zero. 



Any movement whatever may be referred as well to ordinary as to 

 rotatory forces ; consequently an ordinary dyname and a rotatory dyname 

 mean the same. 



3. From the notions developed we immediately obtain two general 



