90 Proceedings of the Ro)jal Irish Academy. 



of the roots of F [H) = 0,, in which a = '-, |3' = tt, 7' = -^ , S' = 2t, 



and the roots of c - gu cos ^ -fu sin 9 are given by ^ cos ^ +/sin 6 = 0, and 

 so by an angle ^' in the second quadrant such that (/- + g-)^ sin Z' = 5', 

 (/' + 9^)- cos r = - /, - gy- cos 9 - fu sin = fi f/^ + (7-)^ sin (^ - Z'), and by an 

 angle 2=1" + ?' in the fourth. If 6^ is the first or second quadrant, 



6 moves towards -^; if it is in the third or fourth, 6 moves towards — ; and 



o 



when 6 becomes equal to ^ or -— , <S = 0, and the representative point 



moves along the line of no sliding. If 6^ lies between f and -, K first 

 diminishes and then increases, so that bj' properly arranging ^0, K,,, and Sa the 

 plane of no compression may be crossed three times during the impact. For 

 every other position of do it is crossed once. If 6,, is in the first quadrant or 



between — and ^, Z" increases during the initial stage of the impact, and so to 



cross the plane of no compression the representative point must proceed 

 along the line of no sliding, and the solution is that usually given. If 9^ is 



Stt 

 in the third quadrant or between — and Z', K diminishes during the 



initial stage. If d is between ^'and 2w, K first increases and then diminishes. 

 16. These results also appear when we integrate the equations, which we 

 now proceed to do ; but in order to avoid troublesome complexity of notation, 

 we shall measure from the axis of /' in that quadrant in which 9^ is. This 

 means that having made an arrangement of axes for which h = Q,a is greater 

 than 6, and /and g are each negative, we reverse if necessary the direction of 

 P and if necessary the direction of Q, so that the initial direction 9^ shall lie 

 in the first quadrant for the new axes. This quadrant will be the first, 

 second, third, or fourth quadrant for the original axes, according as the signs 

 of/ and g are -, - or -, + or +, + or + -. Accordingly, using /'and g with 

 their original signification, we have 



/S-j^ = {a -h) fismQ cosQ + gsin9 -fcos,6. 

 j^ = - (ttfi cos- 6 + hfx sin" 9 + g cos 9 + /sin 0), 



when /I is very large, Vp = initially, when 6„ = or ^ or — or27r,andthe 



subsequent direction of sliding is along the corresponding axis, and <S 

 diminishes until S = 0. These special cases present no difficulty, either for n 

 very large, or in the general case when the direction of sliding remains con- 

 stant, because the representative point moves along lines, and so the plane of 



