268 PROFESSOR TAIT ON THE 



that there will be, in the expression for the relative speed in the direction of 

 the line of centres at impact, an additional term 



(« x — a 2 )cos \Js , 



where xfj is the inclination of the line of centres to the axis of x. Thus to the 

 impulse, whose expression is of the form 



2PQ , x 

 (u-v), 



P+Q 



as in § 19 of the First Part of the paper, there must be added the term we seek, 

 viz., 



~F^fP (ai_a2)cos ^' 



This must be resolved again parallel to x, for which we must multiply by 

 cos \jj. Also, as the line of centres may have with equal probability all 

 directions, we must multiply further by sin \jjd\jj/2, and integrate from to it. 

 The result will be the average transmission, per collision, per P 1? of translatory 

 momentum of the layer parallel to x. Taking account of the number of impacts 

 of a Pj on a P 2 , as in § 23, we obtain finally 



p 4 /•n-iK + h) P i P 2 / \ n\ 



x-jnpf V-^- 1 p^p^-^ • • < 7 -> 



where s is the semi-sum of the diameters of a P x and a P 2 . 



49. To put this in a more convenient form, note that (2), in the notation of 



(5), gives us the relation 



Id&i 1 dG, 



h-y dx h 2 dx " ' 

 whence 



G 1 /h, + G 2 /h, = 2px (8.) 



We have not added an arbitrary constant, for no origin has been specified 

 for x. Nor have we added an arbitrary function of t, because (as will be seen 

 at once from (3)) this could only be necessary in cases where the left-hand 

 members of (6) are quantities comparable with the other terms in these equa- 

 tions. They are, however, of the order of 



dt* dxdt Ul ' &C -' 



and cannot rise into importance except in the case of motions much more 

 violent than those we are considering. 

 From (8) we obtain 



f/*'+f/'«°=°- • <w 



