462 Mr. H. W. Chapman on 



.-. integrating,— x.<*>s i (- + cos0j = W A^^ + sin ^ ) + constant, 



.'• »3 2 {4?r+Asin^.+ c(- +cos6>y | =n 2 

 " I « 2 M \a J > 



= »3 2 { ^ + A dn*0, + c(~- + cos ^) 2 } . (13) 

 We get from this 



O)o = + 





V^+Asin^ + cg+cos*/ 

 n I sin { A cos 0-cft + cos ) j 



.'. -|o) 3 | decreases as 6 increases if 



Acos0>C/" + cos d\ 



i.e.(A-C)cGs6>C-. 



v y a 



We have then four cases : 

 (i.) A>Cand/i>9, 



If (A — - C) cos 6 > C — we shall have | a> 3 1 decreasing as 



a h 



increases until (A — C) cos 6 = C — , when it will be at a 



v a 



minimum, and then as the top gets further down it will 

 increase again. 



If (A — C) cos O >C— and the top is rising, | o> 3 1 will 

 increase as long as the top rises. 



If (A — C) cos 6 < C— and the top is descending, | g> 3 | will 

 increase as long as the top descends. 



If (A— C)cos O <C— and the top is rising, | <o s i will 

 decrease until (A — C) cos = C— , and then increase again 



7 J CI 



if A — C>C— ; if A — C<C- it will decrease all the way up. 



