251] INITIAL MOTIONS OF RIGID BODIES. 285 



3. This shows that < must be zero. Again, if < iv is finite the equation 

 can be reduced, by picking out the terms of order 2, to 





 giving ^-B'-'-s 



Again, taking equation (1) and observing that cos = 1 H : . . . we see 



that the lowest power of t in this series is the fourth, and then it appears from 

 equation (1) that the lowest power of t in 6 is the fourth, so that the series 

 for 6 begins 



Going back now to equation (2) it is clear that $ contains no term in t 3 

 but there is a term in t*. In fact, picking out the terms in & in equation (2) 

 we have 



, 

 ** 



Now, in the figure, taking as origin the initial position of j5, and taking 

 the axes of x and y horizontal and vertical, we can write for the coordinates 

 of a point of BC distant r from B, 



expanding these we have approximately 



gvng 



correct as far as t*. Hence the initial path of the point is approximately a 

 parabola 



and the radius of curvature of the path is 2a6/(3r 26) unless r=$b. 



If however r=&, in order to get an approximate equation to the path 

 we must expand to a higher order. We find 







correct as far as Z 6 , and thus the initial path is given by the approximate 

 equation 



