249-250] INITIAL MOTIONS OF RIGID BODIES. 283 



geometrical quantity the value 0, and for every geometrical 

 quantity the value that it has in the initial position, we shall 

 obtain the relations between the initial accelerations of the various 

 geometrical quantities involved. Thus if x, y are the coordinates 

 of any particle whose acceleration is required, and 0, &amp;lt;f&amp;gt;, . . . are a 

 series of geometrical quantities which define the position of the 

 system, there will be certain values , &amp;lt;/&amp;gt; , ... for these quantities 

 in the initial position. Now the geometrical equations provide 

 the means of expressing the x and y of the particle in any 

 position in terms of the values of 0, &amp;lt;f&amp;gt;, . . . for that position. Let 

 x =/(#, &amp;lt;/&amp;gt;,...) be the form of one of the equations we can obtain. 

 On differentiating we have 



-!+$*+.... 



Reducing, as explained, we obtain 



where x , ... denote the initial values of x, 6, ..., and [^4), 



wv/o 



] ,... denote the values of 53 , ^ , ... when 6 = , &amp;lt;/&amp;gt; = &amp;lt;/&amp;gt;o&amp;gt; 



0&amp;lt;p/Q CU 0&amp;lt;p 



Now this process can be carried further, and arranged as a 

 process of approximation for expressing the values of x, y, ... as 

 series in ascending powers of the time. We have in fact as a first 

 approximation x = ^x^f 2 , y = \y Q t 2 . 



It will be easier to understand how this process is carried out 

 after studying its application to a particular problem, and it will at 

 the same time be seen how simplifications may at times suggest 

 themselves. The problem chosen is intentionally complicated. 



The method here sketched is of course applicable to systems of 

 particles with invariable connexions as well as to rigid bodies with 

 such connexions. 



