374 DR. PLtJCKEE ON FUIS'DAMENTAL VIEWS HEGAIIDING MECHANICS. 



rotatory dyname, are obtained hy adding tlie coordinates of the given rotatory forces. In 

 the case of eguiUlrium the six sums obtained are equal to zero. 



Accordingly the given rotatory forces (or rotations) being represented by the general 

 symbols (t'u'v', tuv), their coordinates are 



t—f, u—v!, v—v', wJ—u'v, vt—'dt, tu'—tlu, 

 and 



X{t-i!), -^u-v!), -^v-v'), X(ui/-u!v), %(vt'-^t), t{tu'-t'u), 



the coordinates of the resulting force or dyname. 



9. The theorem of the last number embraces the statics of rotatory forces as the 

 analogous theorem of Part I., No. 2 involves the statics of ordinary forces. We gave this 

 theorem as the expression of known statical laws. Inversely we might, having previ- 

 ously stated the theorem in a direct way, deduce from it the theorems of statics. 

 Indeed the theorem follows from the mere consideration that the corresponding coordi- 

 nates of forces, — the three first of which, X, Y, Z, are represented by segments of 

 right lines, the three last, L, M, N, by areas, — indicating homogeneous quantities, may 

 be added, and after addition the sums obtained interpreted in the same way. 



The following numbers will show the application of the new theorem, and of its 

 inverse, regarding decomposition of rotatory forces or dynamos. 



10. Any number n of rotatoiy forces acting simultaneously on the same plane (if, u', i/) 

 may be represented by symbols, tf, u', v' being the same in all. By adding their coordi- 

 nates, the six sums obtained (2) may be written thus, 



%t—nt', %u—nu', Xv—nv', 



'2,u.v'—'2,v.u', 1v.t'—%t.'d, Xt.u'—Xu.t'; 

 or in putting 



Xt^'ii^, '2,u=n§, 5lt)=«ff, 



thus 



«(9— ^), n(q — u'), n{(r—v'), 



n(§v'—m'), n{(Tf—^v'), n{^tt!—§t'). 



These expressions are the coordinates of the resulting rotatory force ; &, f , <r are the 

 coordinates of a plane, replacing in the theory of rotation the centre of gravity, which 

 may be called the central plane of the given planes {t, u, v), by which, the plane acted 

 upon (f , u', v') being given, the rotatory forces are determined. The resulting axis of 

 rotation is the intersection of the given plane (tf, u', t/) and the central plane (&, §, o-). 

 The intensity of the resulting force is 



n^i^-ty+is-^'r+i^-vj. 



In the case of equilibrium, 



^=f, §=u\ ff=.v\ 



i. e. the central plane is congruent with the plane acted upon hy the given rotatory forces. 



11. A rotatory force, 



(^mV, tuv). 



