SOLIDS ON SUB* W l>. >t,'l 



(Jons vanish alt gether, as all the terms arc infinitesimals oi 

 the second order, and the other four become (36) 



A I = At— v) AR-\-S,AQ : 

 Ay, = dn,— t AP+lAB; 



A% = d-i,—dr,ll{. 

 tftr. = A?ri,-\-AlAR\ 



where £ becomes a constant, and equal to a — y. 



These equations are to be taken in connection with the equa- 

 tions of motion, and. as will presently ho seen, will, along with 

 these equations, assume the form of eight linear equations in 

 (i.l>. i.r. i.r. £„ >?„ with constant coefficients, reducible 

 to four, by means of which the motion of the body will be 

 completely determined, and the elements of its position as- 

 signed in finite and explicit functions of the time, 



It would lie easy to show, as Lagrange has done in the ease 

 of a body revolving and oscillating aboul a fixed point, that the 

 centrifugal force of a body revolving on a surface nearly hor- 

 izontal will throw its vertical axis ton finite distance from the 

 fixed vortical, unless when either the rotation round the bodj 's 

 vertical is very small, in which case the distorsive moments 

 of inertia /'and C may he any whatever, or else when /'and 

 G are very small, and then the rotation round the vertical 

 may he what we phase. In both cases the form of the body 

 and the distribution of its density may be such that the third 

 distorsive moment of inertia H (which is brought into action 

 only by the velocities p and y. and enters into the values ol 

 /'. /'and /('. multiplied by these velocities only, or by their 

 rates of increase) may he indefinitely great without affecting 

 the (ruth of the solution. 



Supposing then in the first place that /• is very small, the 



values of p and q already found become p = Ab\ q = — <Ui . 

 ami the four equations last given (omitting here. titer the in- 

 ferior accents of \ ami j . as no longer wanted) arc reduced 

 to (37) 



