40 Prof. Miller on the employment of the Gnomonic 



6. Let the gi'eat circle EF 

 meet the gi'eat circles BC, C A, 

 AB in the points D, E, F. Let 

 uvtv be the symbol of any point 

 P in E F. Draw the great 

 circles A P, B P, C P meeting 

 B C, C A, A B in U, V, W. 



By (7), sinBD sin CU 

 sin CD sin BU 



_ sin BPD sin CPU _ sin VPE sin CPA _ sin VE sin CA 

 ~ sin CPD sin BPU "" sin CPE sin APV ~ sm CE sin AV " 



But sin VE sin C A = sin CE sin AV - sin AE sin C V. 



Therefore 



sinBD sin CU _ sin AE sin CV 

 sin CD sin BU ~ sin CE sin AV 



The symbol of P is uvw, therefore by (f ), 



V sin BU w sin CU w sin CV u sin AV 



Hence 



6sinAB csinCA' c sin BC wsinAB' 

 sin AE sin BC u sin BD sin CA v w ^ 



sin CE sin AB a sin CD sin AB b c 



Let hkl, pqr be the symbols of any two points in EF, not 

 being opposite extremities of a diameter of the circle. Substi- 

 tute h, k, I, and p, q, r successively for u, v, w in the preceding 

 equation, and two other equations are obtained. The three 

 equations give 



{kr — lq)u +{lp — hr)v + {hq — kp) 10 = Q. 

 If 



\\ = kr — lq, v=:lp — hr, w — hq — kp, . . (t)) 



uvw may be taken for the symbol of the great circle passing 

 through the points hkl, pqr. 

 The equation 



uu + vv + \\w = {6) 



expresses the condition that the point uviv lies in the great 

 circle uvw. 



7. Let uviv be the symbol of the point in which the great 

 circles hkl, pqr intersect. Then by {$), since uvw is a point in 

 each of the great circles, 



hu + kv + \w = 0, 



pM-|-qw + rw = 0. 

 Whence 



M=kr — Iq, v = lp — hr, H; = hq — kp. . . (t) 



