470 Mr. H. W. Chapman on 



Therefore neglecting 4 and substituting in above equation 



we get 



b 2 t 

 y= — + constant, 



/. 6 = \/ _^ 2 _ sin ( A V ~ ° 4 yfr + constant). 

 V — C 4 \ b 2 I 



This is a well-known equation, and when 6 2 = A V — C 4 we 

 see that the curve traced out by the axis is a circle on one 

 side of the vertical. 



10c There are some cases where the equation (15) can be 

 solved in elliptic functions. 



The first is the degenerate case when 



^ + A + C^ + 2C ^ + ( C - A K is a P^ect square. 

 The condition for this is 



^ = (^+ c S)< c -^> 



This necessitates C>A, giving a flattened top, not an egg. 



The case is real and, it is interesting to note, depends only 

 on the structure of the top, not on the initial conditions. 



11. Another case is when a is small so that a 3 can be 

 neglected. In this case we must remember that \ 2 , n 2 , K 

 have a 2 in their denominators, while the coefficient of /jl 2 has 

 no term with a denominator of order higher than a?. 



The only terms with a 4 in the denominator occur in the 

 constant term, and these must disappear since they only occur 

 in terms independent of /a, and must disappear when /a = /jl I 

 for the equation (17) is then satisfied identically, so that the 

 right-hand side must reduce to //, , and there being no term 



in — ; in the denominator there cannot be in the numerator. 



a 1 



This reasoning will not apply to the terms containing -^ 



for some of them occur in the coefficient of /*. 



These are on the left-hand side of (15) 



,*fo«*: r -,* + Xl * c ^ 2 +l) 



