DYNAMICS OF A POINT. 283 



1358. The accelerations of a point describing a curve an- 

 resolv^l into two, along the radius vector and parallel to the 

 prime radius respectively; prove that these accelerations are 



cot0 d f a d6\ /dO\ 9 1 d d 



dt ( f dt) + dt*- r (dt)> and ~7 



1359. The motion of a point is referred to two axes x, y, of 

 which the axis of x is fixed and the axis of y revolves about 

 the origin ; prove that the accelerations in these directions 

 are 



d*x Id/ S d0\ i cot0 d ( a d0\ _ (dO\* 



dt 3 " ,, MII f> .It \ J dt) ' ~7~ 'dt V dt) y \dt) ; 



6 being the angle between the axes at a time t. 



1 360. If PQ be a tangent to a curve at <?, a fixed point on 

 the curve, QP = r, arc OQ = 8, and < the angle through which the 

 tangent has revolved from to Q; the accelerations of /' in the 



.11 V/'. and at right angles to this direction, are rev 



A/<\ f 1 .1 </6\ d<> ds 



-- 



1361. A point describes a curve of double curvature, and 



. -ordinatesat time t are (r, 6, </>); prove thai it> acceler- 

 ations, (1) along tin- radius \< otOP, ('2) pi-rpriidii-ular to the ralius 

 vector in th-- plan.- of 0, and (3) p-rj.eiidirular to the plant? of 0, 

 are respectively 







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