Fey — Impact in Three Dimensions. 91 



no compression will be crossed once. If it is crossed during the sliding 

 motion, it will not be crossed again. If it is not crossed, it will be crossed 

 during the motion along the line of no sliding. (See section 14.) 

 For (X very large and <Sg not equal to zero 



dS a cos' 6 + b sin' ad sin 6 b d cos 



S (a ~ b) sin cos a - h sin 6 a -b cos 



.: S^sJ'-^rl^^.]-'-', where A ' 



dR = 



<'\cos6'„; Vsin(^J ' a-V 



Sd0 ^s„ (cos 9)^-' (sin 8)-^-^ d0 



{a-b)^, sin 9 cos {a-b)n (cos Q,f (sin «„) -"-' 



_ S, (tan e,)" sin g„ (cot 0)^'' sin'^g^^g 

 {a-b)fx 



As (ii? is positive, so is dd, or in every case 6 increases. 

 During the initial stage of sliding 



It-': 



, 4. /JNA w • u\-s m 1 7 'S'o(tan (*„)^ sin 9„ 

 (cot f/)'^"! fsm W) " ff^, where k = — _ " 



a - b 



re 

 cos 6 dB = k 



Q = 1^ 



sin e dR 



TT 



(cot Qf cosee' f?0 



= ^{(cot0or'-(cot0r'j 



(cot 0j^- ' cosec' c?0 = - I (cot d^f - (cot 6») . 



When = |, S = 0, i? is finite, P = ^i^^, Q = S^m^„^ ^^ ^^^^^ 



ought to be from the equations 



5^ cos = Sa cos 0„ - aP - gR, S sin = S^ sin 6o- bQ - fR, 



because S = 0, and i2 is negligible in comparison with F and Q during the^ 



sliding stage. During sliding the projection on i? = of the path described 



by. the representative point is therefore 



I i_ 



^S„ cos 0„ - alY /<S„ sin 0„ - bR\!> 



/S„ cos 00 - aP\« 

 \ cos 0„ J 



cos 00 y \ sin 00 

 which would also follow by integrating 



dP cos So cos 00 - aP_ 

 dQ sin <So siu 0o - bQ 



K.I. A. PEOC, VOL. XXXni., SECT. A. _ F^'^l 



