10 



ART S.— S. X AK AMUR A. 



Lot us suppose that we take a right circular cylinder of radius 

 r with its axis horizontal. Draw a curve with u as the abscissa and 

 x as the ordinate. The scale of the abscissa may be any whatsoever, 

 but that of the ordinate must be so taken that when the curve is 

 wrapped round the cylinder, it covers just half its surface. Let OA'A 



Fig. 4. 



and the initial meridian be d, then evidently 



s 1 = lu. r. cos d.dd 



o 



be the initial meridian and 

 OB'B perpendicular to it. 

 Project the curve APBC 

 orthogonally on the plane 

 OB'B, then we shall get a 

 curve A"P"B"C." Denote 

 the area bounded by A"P" 

 B"C"A" by. V If the angle 

 between the plane OP'P 



But since 



x: H=i 



7TX 



6 = 



if' 



we have, therefore. 



u 



T7Z (' 7ZX 



irJ UC0S ir 



âx 





H 2 



Similarly if the scale of the abscissa he so taken that the curve 

 covers the surface of the cylinder from O^o to d=p- and the area 

 obtained by projecting it on the meridian at ^——r- be denoted by 



s„. then 



