240 UNIVERSITY OF VIRGINIA PUBLICATIONS 



Let Q — dx (Py — dy d-x. When divided by dt^ this is V^d6 where dd 

 is the element of angle turned through by V. Hence the corresponding 

 equations for moment of momentum in the hodograph. Since here V^dd is 

 ap, p the perpendicular on the tangent, this becomes the moment of the 

 forces in the hodograph. 



If the only forces acting on the system of masses are equal and opposite 

 stresses in the lines of action joining them two and two, then since the 

 sufn moments of the resultant stresses about an arbitrary axis is equal to 

 the sum moments of these equal and opposite components and this is zero, 

 then must a be zero, and its tangential and normal components zero. On 

 account of the former V is constant, and on account of the latter the radius 

 of curvature of the mass center infinite. When the mass center is at rest, 

 the equations permit of studying the functions of the relative values 

 instead of the direct values. 



7. We conclude these illustrations with an example of a similitudinous 

 homogeneous transformation of the points of a plane. Let the movement 

 be defined by the motion of three masses mi, nii, ma at the corners of a 

 triangle ABC of constant shape, the masses being projected with certain 

 initial velocities in the plane are subjected to their mutual attractions 

 /23, /si, /i2 in the sides a, b, c, these attractions being homogeneous functions 

 of the sides of , the triangle of the same degree. We consider the motion 

 with respect to the mass-center M at rest. Let X be the ratio of similarity 

 or that of each side of the triangle to its initial value. We use the sub- 

 script i to indicate the corresponding initial value, thus a = Xa,-, etc. The 

 radial coordinates of the mass-center M with respect to ABC being n, ^2, n, 

 then 



n^ = -^ {c^m2 + h-ms) - jj^ miwhc^, (23) 



= X^ (ri)i^ 



with similar values for r^, n. The resultant forces Fi, F2, Fs acting on mi, 

 m2, ms are in equilibrium and meet in a point F, the force-center, whose areal 

 coordinates x, y, z, by taking moments about F, are seen to be 



^=.^_ = _^='l. (24) 



a 7/23 6//31 C//12 c 



Since /23, etc., are homogeneous functions of the same degree in a, 6, c the 

 coordinates of F are constants. Hence the figures ^JSCMF in any position 

 is a homogeneous similitudinous transformation of any other position. 



