1890.] and the Curvatures of their Trajectories. 41 



which gives, to the first order, 



7 7 cos 7 ., K . 



h = -h —± (17), 



cos/y v ' 



to the second order, 



/„ k cos co\ f n , 7Q TO cos 2 7 



h = 9 ( 1 -= ) ( 2ak cos 7 — k* — Tc 



2a cos (3 V a cos /3/ \ cos 2 /8 



— ~h ~ + ~— - , ~ (cos 2 /3 + cos 2 7 + 2 cos /3cos 7 cos co). . .(18). 



cos p 2a cos p 



Now by (8) we have to the same order 



c 2 V c/ 



-'*+^* (19)- 



Comparing these, we have 



b cos /3 = — ccos 7 (20), 



c 3 

 6c — be = 3-5(cos 2 /3 + cos' 2 7+ 2 cos /3 cos 7 cos co)...(21). 



Substituting in (10), 



p _ (|0 2 cos 2 /5 + p' 2 cos 2 7 — 2pp' cos /3 cos 7 cos co)- 99 . 

 pp sm co (cos p + cos 7+2 cos p cos 7 cos co) 



where /3 and 7 are the angles which BO makes with. the curve 

 of inflexions at B and C, and co is the angle between the tangents 

 to the curve at those points ; so that the curvature is expressed 

 in terms of purely geometrical quantities. 



It is easy to verify that this theory leads to the correct results 

 for uniplanar motion. For in (4) we have now w = 0, co = 0, co\— ; 

 the curve of inflexions is therefore a circle passing through the 

 central axis /. If now the special chord PI meet this circle 

 again in Q, we may apply (14) if we write in it 6=0, b = to 2 IC; 

 thus we obtain (and still more directly from (22)) the correct 

 result 



R.PQ = IP 2 (23). 



The theory for uniplanar motion may also be reduced to 

 simple kinematic considerations as follows. There is one point 

 connected with the solid which has no acceleration ; by taking 

 this point for origin and reducing it to rest in the usual manner, 

 we see that the acceleration of any other point with respect to it, 



