46 U. S. COAST AND GEODETIC SURVEY. 



for all values of 7 and 8, and the sections will therefore be 

 plane sections of a sphere. Therefore, there are two series 

 of circular sections made by two systems of parallel planes, 

 and both systems are parallel to the plane ^ = 0. 



The trace of the cone upon the plane y = has for its 

 equation : 



(fz-lxy-aKz-'hy = 0. 



This is, therefore, the equation of the two generating lines 

 which lie in that plane. The equation of the two planes 

 in opposite systems giving the circular sections is 



(s - 7) WJix + (a^ -f + Ji^) 2- 27ia^ -d] = 0. 



By adding these two equations we get an equation of the 

 form 



x^ + z-' + A'x + B'y+C' = 0. 



This shows that the four points in which the two generating 

 lines in the plane y = meet the planes forming the circular 

 sections lie upon a circle. Hence the first system of 

 planes makes the same angle with the one of the generating 

 lines that the second system makes with the other. We 

 will now show that the mapping plane fulfills the conditions 

 for the second system of circifiar sections. The mapping 

 plane is evidently perpendicular to the plane of the great 

 circle ALOK, and it thus fulfills the first condition. The 

 further condition is that it must, make the same angle with 

 one of the elements of the cone lying in the plane of the 

 great circle that the plane of the circle on the sphere makes 

 with the other element in this plane. In figure 11 



Z CBO = i arc OLAC== ^(arc OLA +arc ^(7) = |+ i arc ^(7 



Z J^FO=^ (arc OZ+arc LAC) =| + i arc AC, 

 Therefore 



and 



lCBO= AKFO 

 IBC0= LFGO, 



It is thus seen that the p)oints 5, C^ F, and O lie upon a 

 circle and all the conditions are fulfilled for a circular 

 section. 



Construct the tangents BD and CD, draw EM parallel 

 to CDy and draw EH parallel to BD, 



