84: Proceedings of the Royal Irish Academy. 



The path followed by the representative point can now be described 

 exactly. In the initial stage it ascends a curve inclined at tan"' n to the axis 

 of R, and such that the projection of the tangent line on i? = tends con- 

 tinually to become parallel to that root of F{d) = Q which is adjacent to 0^ 

 and makes F' (6) negative. Thus in (1), no matter what the original value 

 of 0g may he, it tends to become equal to a, but never assumes that value, 

 and the impact terminates before sliding ceases. In (2) if 0^ lies between 

 a and (3 or o and a, 6 tends continually to become equal t-o a, but never 

 attains that value, and so sliding does not cease : but if is between /3 and y 

 or y and S, 6 tends to become equal to y, and if the impact is sufficiently 

 prolonged will attain that value, and then S=0, and the representative point 

 will move along a line inclined at tan-' n to the axis of M, such that its 

 -projection on 5 = is inclined at o to the axis of P. In (3) tends con- 

 .tinually to become equal to a, attains that value if the impact is sufficiently 

 prolonged, then S = and the representative point proceeds along the line 

 of no sliding. In (4) if 6o is between a' and /3' or S' and «', tends con- 

 tinually to become equal to n' ; if 0„ is between ^ and y' or y' and S', 6 tends 

 continually to become equal to y' ; and if the impact is sufficiently prolonged, 

 9 attains these values, and at that instant S = 0, and afterwards the repre- 

 sentative point proceeds along the line of no sliding. 



By drawing planes through the axis of B, inclined to the axis of P at 

 angles equal to the roots of F (6) = 0, we divide space on the positive side 

 of JS = into two or four departments, and we see that during the initial 

 stage of sliding the motion of the representative point is confined to one of 

 these departments. In section 7 we saw that if 0, = a root of F{d) = 0, then 

 6 remains constant and the representative point moves along a line inclined 

 at tan"' /u to the axis of It in one of the planes just drawn. It now appears 

 that when 0,, is equal to a or 7 or a or y' such motion is stable, but when do 

 is equal to ;8 or S or j3' or S' it is unstable, because a small variation in 0„ 

 would cause d to tend towards the root at the other end "of the department 

 into which it enters. 



During the initial stage of sliding the value.s of P, Q, If are given by 



P = fJc{ cos 0xiS)dd, Q = fik\ sin exi6)dd. 



11. We shall now show that the representative point always gets on to 

 the plane of no compression, or, in other words, that .£" always becomes zero. 



The values of P, Q, R for which X= lie on the plane gP+fQ- cR = Ko. 

 This plane meets the axis of iZ at a distance Kjc from the origin, which is 



