58, 59] EADIAL AND TRANSVERSAL RESOLUTION. 61 



We have therefore, by resolving parallel to the axes 

 a cos 6 /3 sin 6 = x = -^- (r cos 6 rd sin 6) 



= (r - rtf 2 ) cos (9 - (r0 + 2r0) sin d 

 otsm0 + @cos0 = y = -r(fsm0 + r0 cos 0) 



= (r - rfc) sin + (rB + 2r0) cos ft 

 Solving these equations we find a = r - r0 2 , /3 = rd + 2r0. 



Here it is important to observe that the acceleration along the 

 radius vector is the resolved part parallel to that line of the 

 acceleration relative to the frame Ox, Qy\ it is not the acceleration 

 with which the radius vector increases. 



Further it is to be observed that the acceleration, fi, at right 

 angles to the radius vector can be expressed in the form - -r(r^6), 



T Ctt 



where the expression r 2 6 is equal to the moment of the velocity 

 about the origin. 



By expressing that, in central orbits, the resolved acceleration at right 

 angles to the radius vector vanishes, we should obtain a new proof of the 

 formula pv = h, and we should find h = r z '&. Comparing this Article with 

 Article 50 we verify the well-known formula of Differential Calculus 



x dy y dx r 2 dB. 



59. Examples. 



1. A point P describes a curve relative to axes through 0. Prove that, 

 relative to parallel axes through P, describes a curve equal in all respects 

 to that described by P. Prove also that any point dividing OP in a constant 

 ratio describes a similar curve relative to the axes through or P. 



2. The motion of a point is referred to polar coordinates r t 0, < with 

 origin at the origin of a set of rectangular axes of x t y, z, with the axis z for 

 polar axis, and the plane (#, z} for initial meridian ; prove that the resolved 

 parts of the velocity along the radius vector, the tangent to the meridian, 

 and the tangent to the parallel through the position at time t are f, rd, and 

 r sin 6<j), and that the resolved parts of the accelerations along the same lines 

 are 



r - rfc - r sin 2 0< 2 , - j (r*0) - r sin 6 cos eft, and \ ^ (r 2 sin 2 6$). 



