251] INITIAL MOTIONS OF RIGID BODIES. 285 



3. This shows that &amp;lt;/&amp;gt; must be zero. Again, if &amp;lt; iv is finite the equation 

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



. . 9a .. 



givmg * = 



Again, taking equation ( 1 ) and observing that cos 6 = 1 - + ... 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 &amp;lt;j&amp;gt; contains no term in t 3 

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

 we have 





 gmng =2VS-- * -4 



Now, in the figure, taking as origin the initial position of B, 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, 



x= a (I cos 6} + r sin &amp;lt;, 

 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 2ab/(3r 26) unless r=%b. 



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

 we must expand to a higher order. &quot;We find 



AA vi $ _ m l &amp;lt;?L ^O 3 /6 



&quot;^^ 6!&quot;m 1 + 2w 2 b 480^ 



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

 equation 



(y - &)3 = 



