82 



KINEMATICS. 



[i 60. 



tangents of the curve, is the angle of contingence da. at P. 

 This angle being equal to the angle between the normals at P 

 and P', we have pda = ds, where p is the radius of curvature 

 at P. 



Resolving the elementary acceleration, i.e. the infinitesimal 

 vector FF', along <9Fand at right angles to OF, we find the 

 components FF' cos^ = ^, FF' sin ^r = vda vds/p. Dividing 

 by dt and observing that ds/dt=v, we finally obtain 



dv 

 dt 



da 



By composition we have 



(2) 



(3) 



160. When rectangular Cartesian co-ordinates are used, we 

 may resolve the acceleration j into two components j x =j cos <j>, 

 j 9 =jsm<l> parallel to the co-ordinate axes Ox, Oy\ <f> being the 

 angle made by the vector j with the axis of x. We obtain an 

 expression for j x by projecting the infinitesimal triangle OVV f 

 (Fig. 37) on the axis Ox and denoting, as before, the projections 

 of the velocities OV, OV by v x , v' x . This gives 



VV cos ^ v ! x v x = dv# 



^, 



whence, dividing by dt, j x = dvjdt. Similarly, we fi nd j\ 

 Hence, by formulae (i), Art 141, 



dt dt* 



Jy ~df dt* 



dvjdt. 



(4) 



These so-called equations of motion offer the advantage that 

 the curvilinear motion is replaced by two rectilinear motions, 

 thus avoiding the use of vectors. 



