680 
DR. J. CASEY 0 N CYCLIDES AND SPHERO-QTJARTICS. 
the second follows from the fact that the bisectors of the angles ABD and DBN pass 
respectively through O and H. 
244. The points O, P, Q are the centres of J, J', J". Let their distances from H be 
denoted by l, e>', l" respectively, and the preceding equations may be written 
tan 2 r — tan o' tan h",' 
tan 2 r 1 = tan o" tan o , > 
tan 2 r"=tan o tan V . 
(151) 
Hence we get the three following equations : 
tan 6 =tan r 1 tan r" : tan r ,' 
tan o' =tan r" tan r : tan r 1 , > . 
tan h" — tan r tan r' : tan r". 
Hence also we get 
tan c$ tan V tan h" — tan r tan r 1 tan r". 
(152) 
(153) 
245. If we denote the radii of the circles of inversion J, J', J" by g, g 1 , g", we easily 
get the system of three equations, 
tan 2 ^ =tan (b — ^)tan(!$ — <F),j 
tan 2 §' = tan (b 1 — b") tan (b 1 — h ), > 
tan 2 g" = tan (l" — l ) tan (b" — b 1 ), 
(154) 
with this other system of equations, 
tan(e> — b' )=tan g tan g 1 : tang'\/ — 1 , 
tan (b 1 — ^ ,, ) = tan g 1 tan g" : tan g’ s/ — 1, ' 
tan (b" — ei )=tan§"tang : tan g "\/ — 1.. 
Hence also 
tan (b — b 1 ) tan (b' — b") tan (b" — B') = tan g tan g' tan g"\/ — 1. 
(155) 
(156) 
From either system it follows, as we know otherwise, that one of the circles of inversion 
is imaginary. 
246. From combining the equations of the two preceding articles, we easily get the 
relations between the radii of the F’s and the J’s ; thus 
tan 2 g — tan (b — b') tan (<$ — b") 
tan 2 8 — tan 8 (tan S' + tan S") + tan S' tan S" 
( 1 + tan 8 tan S') (1 + tan 8 tan 8") 
(tan 2 r — tan 2 r 1 ) (tan 2 r — tan 2 r") 
tan 2 r sec 2 r 1 sec 2 r" 
4 (sin 2 r — sin 2 r') (sin 2 r — sin 2 r") 
sin 2 2 r 
