94 The Law of Distribution [CH. v 



Let us suppose that for any single sphere the directions of the axes , 77, 

 are connected with those of the fixed axes x, y, z by the following scheme of 

 direction-cosines : 



I 17 



00 k* ^2, "3 . t (223). 



y 



, w 2 , ra 3 



The condition of the sphere at any instant may be regarded as determined 

 by u, v, w, the velocities of the centre of gravity parallel to x, y, z\ t ar 1) r 2 , 

 the angular velocities of rotation about the axes f , 77 ; and the nine direction- 

 cosines of scheme (223) of which three only are independent. 



There are therefore eight independent variables necessary to determine 

 the condition of a sphere. In discussing a collision between two spheres, 

 it is necessary to know not only the conditions of the two spheres, but also 

 X, /JL, if, the direction-cosines of the line of centres at impact referred to the 

 fixed axes x, y, z, Ihis introduces two more variables, so that a collision 

 requires eighteen independent variables for its complete specification. 



108. Let us examine first the changes in the eight variables of a sphere, 

 which are produced by an impulse / acting along the radius of which the 

 direction-cosines are X, /A, v. 



Referred to , TJ, % axes, the point of application of this impulse may be 

 taken to be 0, 0, r. The components of the impulse will be 



(^X + m l p, + n^v) I, etc. 

 and the components of the resulting couple 



+ n^v}I, 0. 



Hence, if variables after the impulse are distinguished by a horizontal 

 bar, the new velocities of translation will be given by 



u = u-\ -- , etc ........... ........... (224), 



ra 



and those of rotation by 



r (/ 2 X 



p 



(225). 



Let us next regard this impulse as arising from a second sphere, of which 

 the condition is determined by accented variables u, v',w . . . , the line of 

 centres at impact having direction-cosines X, /*, v. 



