The elements of

to 2

COR. I. Join D, E ; then is the trape-zium D A B E a minimum when the arch A Bfalls between Y and G, and also a minimumwhen G lies between the circle AB and V.This will appear evident by considering thatDYE is a constant quantity.

COR 1 II. Draw R K any where betweenV and O, cutting AV in R, and V B in K:Then, will the trapezium A R K B be amaximum when the point O falls on that partof the circumference D OE, which is convex tothe point Y; and a minimum when the pointO falls . on that part of the circumferenceDOE, which is concave to the point V;

For R Y K is a constant quantity.

COR. III. If the sides D Y, YE, (Pb 4.fig. 15.) be tangents to the circle D E inD, E; then will AV + VB + AB be a con-stant quantity when the point O falls on thatpart of the circumference D E, which is con-vex to V: (T. V. B- III.) And, hence, itappears, that of all spherical triangles havingequal perimeters, (limited as in the Theorem)and their vertical angles equal, the greatest willbe the isosceles triangle A V B.

COR. IV. Again, if the sides D V, YE,(PI. 4. fig. 13,) be tangents to the circle D Ein- D, E; then will AV + VB-AB be a.