Mr. A. J. Robertson on the Positive Wane of Translation, 515 

 (DB-A.r)(BC + A*,) = DBxBC, 



a -or* ^- x nr( Ax 



••• A *= BC DB^U= BC DB' 



since Ax is exceedingly small ; but 



/3 



BC 



(a + k) + x' 

 and 



DB = a + k + x; 



therefore -r^, and ultimately 



^/ = £ 



dx {a + k + x}* 



which is the velocity in the horizontal direction when that in 

 the vertical is unity. 



Inertia in vertical direction = — ; 



3' 



8 f S 2 1 

 inertia in horizontal direction = —< £- -, > ; 



3 la+£ + <2?) 4 J 



/. total inertia ?il+ ; £ — ^ I. 



3 L (a + £ + t r) 4 J 



Moving force = weight of column 



DG=—f— . x; 

 a + fc + x 



.'. accelerating force 



(3x x 



a+k+x r a+k+x 



3l H (a + k + xrJ 1 + 



(a + & + #) 4 J (a + ^ + ^) 4 



The column being exceedingly thin, (3 is exceedingly small, 



and a fortiori - j . 4 . This quantity may therefore be 



neglected 5 and when a? is small in comparison with a-\-k, 



x d?x 



as the effect of fju is to diminish x. 



Integrating, remembering that when x = k the velocity is 

 nothing, and that x diminishes as t increases, 



/a + Jc ,x . . 



. = X /_cos-- (3.) 



2 L 2 



