892 



PROCEEDINGS OF THE AMERICAN ACADEMY 



and, obtaining corresponding values for siu bc, &c., we have 

 sin AD sin nc sin aod sin boc 



sin AB SUl CD 



sin Aou Sin cod 



80 that the ratio depends only on the angles at o, and is constant for 

 any given pencil. 



31. To prove that an arc drawn from a conic pole O is harmoni- 

 cally divided by the point o, the conic, and the polar of O, as defined 

 in Art. 7. 



I shall deduce the proof from the corresponding property of plane 

 conies. 



Let o be a pole, and pr its polar ; then, from any point of the polar 

 as p, there will radiate a spherical pencil. 



Project the spherical 

 conic upon a tangent 

 plane at p; and suppose 

 O, A, c, and B to be 

 projected into o', a', c', 

 and b'. It is evident, 

 then, that o' is the pole 

 of PC', and 



o'b'.a'c' 

 o'a'.c'b' 



= 1. 



This ratio, however, de- 

 pends upon the sines of 

 o'pa', &c. ; hence, since 

 all lines on the plane are 



perpendicular to the radius drawn to P, 



sin OPB sin apc 



sin OB sin AC 



isin OPA sin cpb sin oa sin cb 

 32. Let OB = Pi, OA =z Q^,oc ^ q; then 



sin (p — p.,) sin p.2 



sin (pi — p) sin pj' 



tan p — tan p^ tan pj 



= 1. 



tan Pi — tan p tan pi 

 tan p 2 \tan Pi ' tan Pj/ 



