2 therefore cos p (sin m, − sin m) + 2 sin p sin m sin m1 = cos ẞ sin (mm). But by Arts. 89 and 90 we have m = 8 − a, and m1 = s; therefore by the aid of Art. 45 we obtain where n has the meaning assigned in Art. 46. In like manner if we eliminate siny between (3) and (4), for 8-C, and m3 for s-b, we obtain putting m Α cos p (sin m, + sin m.) - 2 sin p sin m, sin m ̧ cot 2 that we are sure of a possible value of cos ß from (9). It remains to shew that when p and ẞ are thus determined, all the four fundamental equations are satisfied. It will be observed that, p and ẞ being considered known, cos y can be found from (1) or (2), and 'sin y can be found from (3) or (4): we must therefore shew that (1) and (2) give the same value for cos y, and that (3) and (4) give the same value for sin y; and we must also shew that these values satisfy the condition cos γ + sin3 y = 1. and it will be found that (2) reduces to the same; so that (1) and (2) give the same value for cos y. In like manner it will be found that (3) and (4) agree in It only remains to shew that the condition cos3y + sin3y = 1 is satisfied. put X for cot r{1 − k cos (s − a)}, and Y for cot r,{1-k cos s}. Then (1) and (2) may be written respectively thus: 2 A 2 therefore 4 sin2 ß cos y = (X + Y + 2XY) sin cos p... (12). But from (10) and (11) by subtraction therefore (x2 + Y2) cos2 p = 4 sin2 p + 2XY cos3 p. Substitute in (12) and we obtain A sin2 ß cos" y = (sin2 p + XY cos' p) sin' 4 Again, put ...... .(13). X, for cot r,{1-k cos (8—c)}, and Y1 for cot r,{1-k cos (8 — b)}. · 1 Then (3) and (4) may be written respectively thus: and from (14) and (15) by addition, whence 1 (X1+ Y1) cos p = 2 sin p, A 2 sin3 ß sin3 y = (sin3p - X,Y, cos3 p) cos2 ......(16). 1 Hence from (13) and (16) it follows that we have to establish the relation But sin3ß = 1-cos2 ß = sin3 p + cos p-k3 cos' p, so that the relation reduces to cotrcotr, {1-k cos s} {1 − k cos (s — a)} sin (s — b) sin (s—c) sin b sin c {1-k cos (s—b)}{1 – k cos (s — sin b sin c Subtract the latter from the former; then we obtain 143. Thus the existence of a circle which touches the inscribed and escribed circles of any spherical triangle has been established. The distance of the pole of this touching circle from the angles B and C of the triangle will of course be determined by formulæ corresponding to (9); and thus it follows that 144. Since the circle which has been determined touches the inscribed circle internally and touches the escribed circles externally, it is obvious that it must meet all the sides of the spherical triangle. We will now determine the position of the points of meeting. Suppose the touching circle intersects the side AB at points distant λ and μ respectively from A. μ From (2), when we substitute the value of tan tan tan given λ 2 |