MancJiester Memoirs, Vol. xlv. (1901), No. \\. 



XI. The Representation on a Conical Mantle of 

 the Areas on a Sphere. 



By C. E. Stromeyer, M.Inst.C.E. 



Received and read February ^th, igoi. 



This problem is solved as soon as it is shewn how to 

 project zones of latitude from a sphere on to a cone, so 

 that the areas of the two zones are equal. 



Solution. — Place the conical mantle over the sphere 

 so that the two touch each other tangentially : then the 

 areas of two zones on the sphere and on the cone, produced 

 by their intersection with two spherical surfaces whose 

 common centre is at the apex of the cone, are equal. 



This can be proved when the difference of length of 

 radii of the two intersecting spherical surfaces is infini- 

 tesimally small, and by summation can be shewn to be 

 true for wider zones. 



Let C be the centre of the sphere, while A is the apex 

 of the conical mantle, which touches the sphere tangen- 

 tially along the latitude BK. Let LiF be the radius of 

 a zone whose width is LjN, being infinitesimally small. 

 With A as centre, draw the arcs L1L2 and NP, prolonging 

 the latter to M, on AL produced. Also draw LjF and L.H 

 normal to AC, then it is required to prove that 



LiN.LiF = LoP.LoH. 



To prove this, prolong ALj through M to D ; draw 

 CD normal to ALD, and join LjC. 



Comparing the triangles LiMN and CDLi, we have 

 the angle LiMN = LiDC, being right angles. LjN is normal 



September lot/i, igoi. 



