6 Proceedings of the Royal Irish Academy. 



If there are two smooth constraints at the point xyz whose specifi- 

 cations are {xyzlimi7Zi), {xyzl^niin^) the reactions are given by the 

 equations 



Fi + i?i (1 + A.11) + li2 (cos ai2 + A12) = 0, 



P2 + Ri (cos 012 + Mi) + -R2 (1 + A.22) = 0, 

 if we write in them 



x = xi = X2, y = y\ = yt, z = zi = z^. 



In this case the point (xyz) is constrained to move in a direction 

 at right angles to the lines whose direction cosines are liniiKi, Izniifh. 



If the point xyz is fixed in space, we may suppose it to he fixed by 

 means of three smooth planes drawn through the point normal to the 

 axes. The components of the initial reaction parallel to the axes are 

 obtained from the first three of the equations (8), with the two last 

 terms omitted, if we write 



Xl = X2 = X3 = X, y^=%j2 = yz= y, Z\ = Z2 = S3 = z, 



h =1 >wi = n\ = 0, l2 = m2=l W2 = 0, ^3=0 OT3=0 «3=1. 

 Making these substitutions, the equations become 



„ / M N\ ( Iz^ v'^W '>y>'XV mxz 



r+.(4-4)-i2,!^^i?2{i + ,.(^ + J)|-iJ3 



muz 



Z+m [y- 



When we have obtained the initial reactions, the initial directions 

 of the motion of any point in the body can be obtained directly by 

 finding the values of Sw, 8w, 8^^, Sw^, Sw^, Sw., and from them by means 

 of equations (1) 8i, 8^, 8s. 



The following indirect method, depending upon the fact that, when 

 the reaction due to a constraint xyzlmn is zero, the motion of the 

 point xijz is independent of the constraint, and is in a direction at right 

 angles to ^, m, w, is more symmetrical. Consider the case when there 

 are four smooth constraints 



(x\y\zilim\.n\) .... (a;4y4S4?4'«4«4). 



Introduce a constraint {xyz Imn), then we have five equations connect- 

 ing the reaction R at xyz with R^, R21 R3, Ri- If now we make ^ = 

 we can eliminate ^i, R2, Rz, Ri, from these equations. 



