93, 94] TOP NEARLY VERTICAL. 289 



125) ^ = ~ A(&' 2 + sin 2 - $' 2 ) + H, cos ^ - Mgl cos #, 



we will convert it into terms of #, ?/, #', ?/', neglecting all terms of 

 order higher than the second. 



In the first term, since , to the order of approximation, 



r' = cos# ' = #', 



we have r t2 -f- ^ 2 ^' 2 , the square of the velocity in polar coordinates, 

 which is in rectangular coordinates x' 2 + y'*. Also we have 



11 9 , , xy' yx' 



tan ib = ~> sec 2 ^-^ f = j 2 > 



iC iC 2 



126) ^' = ^#' 

 and since 



127) cos a = { 1 - (* 2 -f 2/ 2 )} = 1 - ^jp^ 

 we have finally 



128) $ = i^' H 



We have then in the term in H, an example of the gyroscopic 

 terms of 50, in which x = q , y = g 2 , 





Forming the equations of motion, since 



dy dx 

 we have finally 



Ax" + H 2 y' + Mglx = 0, 



Ay"-H,x' + Mgly = 0, 

 or in terms of our constants, 



131) 



These equations are a particular case of a problem that is interesting 

 enough to he considered in full. If & were zero, they would be the 



WEBSTER, Dynamics. 



