487 



circles is bisected in the point of contact ; therefore, either 

 of the spherical triangles whose base is the tangent arc so in- 

 tercepted, and whose other two sides are the directive circles, 

 has a constant area ; because, if we suppose the tangent arc 

 to change its position through an indefinitely small angle, 

 and to be always terminated by the directive circles, the two 

 little triangles bounded by its two positions and by the two 

 indefinitely small directive arcs which lie between these posi- 

 tions, will have their nascent ratio one of equality, so that 

 the area of either of the spherical triangles mentioned above, 

 will not be changed by the change in the position of its base. 

 But in each of these triangles the angle opposite the base 

 is constant ; therefore the sura of the angles at the base is 

 constant. 



From this reasoning itappears that if a spherical triangle 

 have a given area, and two of its sides be fixed, the third 

 side will always touch a spherical conic having the fixed sides 

 for its directive arcs, and will be always bisected in the point 

 of contact. 



§ 7. The intersection of any given central surface of the 

 second order with a concentric sphere is a spherical conic, 

 since the cone which passes through the curve of intersection 

 and has its vertex at the common centre, is of the second 

 order. The cylinder also, which passes through the same 

 curve and has its side parallel to any of the arcs of the given 

 surface, is of the second order ; and the cone, the cylinder, 

 and the given surface are condirective, that is, the directive 

 planes of one of them are also the directive planes of each 

 of the other two. This may be seen from the equations of 

 the different surfaces ; for, in general, two surfaces, whose 

 principal planes are parallel, will be condirective, if, when 

 their equations are expressed by coordinates perpendicular 

 to these planes, the differences of the coefficients of the 

 squares of the variables in the equation of the one be pro- 



VOL. II, 2 T 



