CASE III.

Given the angles B and C and side a, that is, two angles, and their adjacent side: to determine the angle A and the sides b and c.

On the horizontal plane take the side a, of which Ol, Om are the edges.

Take the plane xy at right angles to Ol

The vertical trace of the side c on this plane forms the angle B with xy, which angle being known this trace can be drawn.

On a plane x1y, at right angles to Om, the vertical trace of side b forming the known angle Cwith x,y, can be drawn.

By the aid of the elevation on xy determine the vertical trace of side c on plane x,y,; the two vertical traces on the latter plane intersect in a point i, whence i is found in x,y, and ii' is a point in the third edge OI, of which draw the indefinite plan Oi.

The faces b and c can now be readily 'constructed' on the plane of a by rotation on Om and Ol respectively. Angle A is found by Problem 15, Chapter I. as before, by a plane through point P or I and perpendicular to OI.

CASE IV.

Given two sides a and b, and the angle A (that is two sides, and an angle opposite to one of them): to determine the angles B and C, and side c.

Assume the given side b on the horizontal plane, edges Ol and Om and side a 'constructed' into that plane about Om.

On plane xy at right angles to Ol make the given angle A with xy, the second line of this angle is the vertical trace cf side c on this plane.

On a second plane x,y, assumed at right angles to Om determine the vertical trace ep of the side c by the aid of the former elevation: cut this trace in p by the circular arc described on plane x,y, by point P, on the third edge: join f, it is the vertical trace of side a; obtain pin x,y,, then pp is a second point determined in the third edge OP of the spherical triangle. Draw the indefinite plan Op and obtain the third angle B as before by Problem 15, Chapter I.