4 Proceedings of the Royal Irish Academy. 



But suppose the geometrical character of the constraints only give 

 five relations connecting the velocity constituents, 8m, &c., from them 

 we can only get five equations connecting the six constituents of 

 reaction. Therefore the additional relation necessary to determine 

 them can only be obtained from the dynamical nature of the con- 

 straints. In general, if the geometrical character of the constraints 

 give us m relations connecting the velocity constituents, where 

 m is not greater than six, we must have Q - m relations between 

 the constituents of reaction due to the dynamical nature of the con- 

 straints. For instance, if a point fixed in the body is constrained to 

 move on a smooth surface, we get one geometrical relation, from the 

 fact that the point cannot move in the direction of the normal to the 

 surface at the point of contact, and the five dynamical relations neces- 

 sary by expressing that there is no force of reaction tangential to the 

 surface, and no moment of reaction round any line through the point. 



Smooth Consteaints, 



"We will now consider the case when the constraints are of such a 

 nature that at each point of constraint the motion of the point is pre- 

 vented in a certain direction and the reaction at the point is a force in 

 the same direction. This is so when the points are compelled to move 

 on smooth surfaces or are connected to points fixed in space by per- 

 fectly fiexible strings. "We may call constraints of this description 

 smooth constraints. Such a constraint is specified by the co-ordinates 

 X, 2/, z of the constrained point, and the direction cosines, I, m, w, of 

 the line of no motion. 



Suppose there are five smooth constraints, 



ociy\Z\l\min\, X2t/2Z2kmin2, x^y^^hm-yni, 



and that the initial reactions are ^i, R2, . . . It^, then if we write 



Pi ^ ymi - zimi, q\ = z\li - x\n\, n = x\m\- yih, 



&c. &c. &c., 



we have 



Zi = -ZRih, Fi = S.Riini, Z\ = 2i?i«i, 



ii = :ZEiPi, Ml = 2iJi?i, iVi = 2-ffiiri. 



The geometrical equations are 



hBu + miSv + niSw + piSux + ^iScoj, + riSuz = \ 



hdu + WsSv + n^Sw + psSwz + f sSwj, + r^Sooz = 



(7) 



