Johnston — Initial Motion. 7 



The resultant is an ecjuation of the form 



la + m^ + wy = 0, 



where 



a,^ ax + by + cz + d, 



(8 = a'x + h'y + c'z + d' , 

 y = a"x + h"y + e"z + d" ; 



a, h, &c. being functions of the specifications of the four constraints 

 and the system of applied forces. 



This equation is the relation connecting the co-ordinates of a point 

 in the body with the direction cosines of a line- perpendicular to its 

 direction of initial motion. If I', m', n' are the direction cosines of 

 another line perpendicular to the direction of initial motion, we have 



also 



I'a + m'fi + n'y = 0. 



Therefore, if I", m", n" are the direction cosines of the initial direction 



of motion, we have 



— - Jl — J- 

 l" m" m"' 



which give at once the initial direction of motion of any point in the 

 body. 



We may note that the locus of points in the body which initially 

 have no motion in the direction I, m, w, is the plane 



la. + mj3 -f «7 = 0, 



that the locus of points which move initially in the direction I, m, n, is 

 the right line whose equations are 



and that the point which initially has no motion — in other words, the 

 initial position of the acceleration centre — is the intersection of the 

 planes 



a = 0, /3 = 0, 7 = 0. 



This method can also be employed to get the initial direction of 

 motion of any point in the body when there are no constraints. In 

 this case the initial direction of motion I", m", n" of the point xi/z is 

 given by the equations 



Z + m^.--y-j i^m[x--z-) Z^m[y--x-) 



