7.8 Proceedings of the Royal Irish Academy. 



that the commonly accepted solution of the problem of collision between very 

 rough bodies is unsound. I give the correct solution at the end of this paper. 

 It is not very much less complicated than the general solution. 



6. Continuation of the general theory. In the equations (2) we take S to 

 be positive, its direction being given by Q. It is most important also to keep 

 in mind that E increases continually during the impact, so that it forms a 

 useful independent variable in terms of which to express S and Q. 



Differentiating equations (2) 



d {H cos 6) = - adP - hdQ - gdR 

 d (S sin 0) = - hdP - hdQ - cdR. 



Now, while sliding is taking place, the increment of the impulsive force of 

 friction on A' is in the direction of sliding and equal to fidB ; 



.•. dP = n cos MR. dQ = fi sin QdR ; 

 .•. cos -y^ - S sin -p^ = - {au. cos 6 + hfi sin H + g) = - Ui 



sin —- + S cos d -jY^- = -(h/j. cos 6 + b/x sin +f) = - U.; 

 dR dR 



.: S 4^ = U, sin - ?7, cos = F(e) 

 dR 



^^ - U, cos - Cr, sin y = - ^(0). 



dR 



7. If initially F {6„) = 0, and S is not zero, 6 will remain constant, and 

 therefore the representative point will move along a straight line inclined 

 at tan"'/x to the axis of ii. 



For, as F {6^) = 0, and S is not zero, -j— = initially. 



dR 



Differentiating 



d^ dS_ dd_ _ dFjd) dd 



dR '^ dR dR ~ dd ' dR' 

 d'ti 



initially. Similarly all the derived functions of d with respect 

 dRr 



to R vanish initially, so that remains constant. 



There are two or four values of which make F{6) •= 0; for by putting 



X = fi cos d, 2/ = ju sin 6 



in F{6), we see that such values of 6 are given by the intersection of 

 the rectangular hyperbola 



{a-b}xy + h (y^ - n^) + gy -fx = 0, 

 which we shall call If = with the circle x^ + if = n, whose centre is on the 



