266 WITHDEAWN FKOM THE ACTION OF GRAVITY. 



whence, by replacing c*, sw and c w by tbeir values p, p' and p — p', as well aa cc' 

 by its value found above, we obtain 



^'=-^'^ p^-^p"—pp' ^^^ ^^"—7=^' -^p'+p'^-pp' ; 



the triangles c uj'and. c" v g, on their part, will give 



fc^-^,y/ p^-\-p"^—pp" andgc'=y/—^, y/pm^pii^piipi^ 



There results, then, after substitutions and reductions, and by making, for the 



sake of abbreviation, 



^ • 



'\/p^P^^-\-p'^p'i^+p'^p''^—P^p'p"—PP'^p"—PP'p"^==:?y 

 fi— p p 



-^ {p-p'){p-p'' ' 



and consequently 

 Hence we deduce 



^^ {P-P'){p"-P'f' 



p" 

 fs= " .P, 



•^^ {P-P"){P"-P') 



fd^ p{p"—p') 

 gd . p'{p—p") 



On the other hand, according to the result of § 7, by observing that ./ and ^ o 

 are respectively equal to the radii fu and ^ t; of the two partitions which we are 

 considering, we have 



PP" P"P' . f^ P{P"—P' ) 



fo = -^ — 77, go = -77 r, whence ■ — • = -77 ^. ; 



J p—p'"^ p"—p" go p{p—p) 



the two ratios "^ and — are therefore equal, and consequently the right line 

 gd go 



d o\s> the bisector of the angley o g. 



Knowing, from what precedes, the three sides of the triangle/ g, we thence 

 deduce, after all reductions are made, cos/o g = — ^ ; whence it results that the 

 angle/o g is of 120°, and consequently that the angles/o d and godsxe each of 

 60°. 



Let us seek lastly the length of the bisector d 0. With this view it is to be 



remarked that in every triangle of which one of the angles is 120°, there is a 



very simple relation between the bisector of that angle and the two sides which 



comprise it. Let ab c (Fig. 16) be a triangle in which Tig, 16 



the angle at a is 120° ; let the side Z> a be prolonged by ^^ 



a quantity a d equal to a c, and join d c; this line will 



be parallel to the bisector a h, for the angle d a c will be u ^- , 



60°; and since a di& equal to a c, the triangle dac will "^"---icr 



be equilateral, and the angle dca will be 60° like the angle cali\we shall have 



dc ha ^ , 1 J ^^^ ^^ „.i 



therefore -r = ; — ; — ;. where because dc z^ ad z^ ac, ~~ := z— j- — , and 

 ah ba-\-ad o,c ba-Vac 



a""-- 



