l6o Dr. J. BOTTOMLEY 071 



periments in the laboratory ; the first enquiry may be 

 pursued in perfect independence of the latter. In the 

 present case the area BEF in terms of c may be obtained 

 as follows : 



BEF - 2BEL = 2(0GL - EBGK - OKE) 



Let be the angle OEK, ^ the angle OLG, then 

 EBGK = arcKG.^ - - ( ^ - </>), KG = OG - OK 



OK = a — ^ + alog{tan^ + v/ i + tan*^} 



OKE = 4.W^ + /i^^i£^ 

 3 ^ 2 



also, we have the following equation connecting ^ and c 



2a = ccos(p, 



from these equations by elimination of (j> we obtain 



area BEF ^ A + ^ .^ ' -Yc+2adoe, ^~ 



ba ° 2a 



ria-cos-^") 



wherein A stands for the area, and P for the perimeter of 

 the curve GOH ; if / be the length of the normal LG, then 

 the values of c in the last equation will extend from 2^: to / ; 

 in the latter case the area BEF vanishes, and this corre- 

 sponds with complete dissolution of the cylinder. If^be 

 less than 2a for the area BEF we should have the value 

 A-Vc+rQ,. 



In what precedes the figure has been supposed to 

 represent a section of a cylinder, if we suppose the figure to 

 revolve round OL, the values of x and y deduced from 

 equations (4), and the equation to the parabola, intro- 

 duced into the expressions V = tt I j/^dx, would serve 



to find the volume undissolved at any time of a surface 

 of revolution generated by the solution of a paraboloid, 

 the action being restricted to the curved surface. 



