88 Proceedings of the Royal Irish Academy. 



increases, so that K may vanish twice ; but if ^q lies between l, and S', K 

 first increases and then diminishes. In all other eases in (3) and (4) 

 K diminishes or increases during the whole of the initial sliding stage. 



■ 13. For the arrangement (2) of. the root, as Z, and Z,' lie between S and o, 

 if 00 lies between /3 and S, Q moves towards 7, and K diminishes continually 

 until = -y. Then 5=0, and the representative point moves along a line 

 incKned at tan"'yu to the axis of R, whose projection on ^ = makes an 

 angle a with the axis of F, and in such a way that the projection of the 

 representative point moves in the direction a, not in the direction - a. 

 If when d becomes equal to -y, K has not passed through zero, we now must 

 show that K wUl become zero while the representative point moves along 

 this line. "We can prove more than this, for we can show that if P'Q'R' is 

 any point on the same side of the plane if = as the origin, or one for which 



K, + gF ^fQ' -cR 

 is positive, then by proceeding from it along a line inclined at tan" '^ to the 

 axis of R, whose projection makes an angle a with the axis of P, we shall 

 meet the plane K = Q at a finite point. We have then to show that we can 

 get a finite positive quantity R such that the point whose co-ordinates are 



P' + fiR cos a, Q' + fiR sin a, R' + R 

 satisfies cR-gP -fQ = K,. 



If so, K, + gP' +fQ'-cR' 



c - fig cos a - ^/ sin a 

 The numerator of the fraction is given to be positive, and the denominator 

 is positive, as the point ft cos o, n sin a lies inside the ellipse, and therefore 

 on the same side of Z = as the origin ; or indeed we proved that it was 

 positive in Section 11. Xote, it cannot be zero for the arrangement (2). 



14. In the arrangements (3) and (4) the line of no sliding is reached 

 when becomes equal to a or y'. The line of no sliding is given by 



a a 00 



so it passes through the point 



&'o cos 6l„ /S„ sin B^ 

 a b 



in the plane ^ = 0, and its projection on ^ = is parallel to the common 

 chord of H and E. It meets the plane of no compression in a point for which 



AT, + — <S„ cos 00 *• ■^— "^0 sin i)^ 

 R= " I 



c- ^ - ^ 

 a 



