54.2 



Dr. Booth on a New Class of Properties J'\p 



curve in P, the other a minor directrix in R ; the right line 

 P R envelopes a circle whose diameter in the case of the el- 

 lipse is the minor axis, and in that of the hyperbola the trans- 

 verse axe. 



V. Let a right line be drawn (fig. 2) cutting the conic sec- 

 tion in the points P P', and the minor directrices in the points 

 Q Q', the segn|e^^, V <^,f[j^ m^m^ equal angles ^|j t^(|. 



centre. 'lo ./t/ruj. cj ?i,n()it,'i'i;, '■ [yittno-j'ja 



VI. Through th6 pomts P and P' let two tangents to the 

 curve be drawn, one meeting the minor directrices in the 

 points T T', the other meeting the same directrices in the 

 points T t' ; the sum of the angles subtended at the centre by 

 the segments of the tangents T T and t t' intercepted between 

 the directrices is double of the angle subteiMied by /Q Q'^ thel 

 cord of contact, at the same centre. bnaqiDq aaoihlo oihn mli 



VII. From the points Q, Q', let perpericnculai^s"be let fall 

 on O T, O T, and O T', O t' respectively ; these four perpen- 

 diculars are equal. 



VIII. Calling the length of this perp^dicular _p, we sh^lfc 



sin POP' p "'' ^^^ 3""^ eaoiiJDsiib lonim 

 have the. relation - — 7\-rrr\i — 4^*^* ' ■ ^ ■.''•.-• - >< >^ f ' /^ 

 r , sin QOQ' b 



IX. Let a right line touch a conic section in P (fig. 3), and" 

 meet the mijior directrices in Q Q', the distances O Q, O Q' of 

 the points Q, Q' from the centre are rational functions of the 



Fig. 3. 



^1 



iO 



mbM^i o>ir>ni amluolbnoqioq aaarlj il^dw elj^nfi srii sd A isJ 



«a'>iiJ39'iib lonim 

 joctnoo lo bio3 9 

 ^b^^}d^J8 H O g/ 



anxsUj adi ni O Jnioq 



o yno-jjniiagun li oi ii 

 ■■■■■■■■! lo fnoo 9itj ,w bric w ?)n 

 11 O 3fiil ofli fH ifiioq edi in y.i 



ioq inlt ni ^.yoi 

 vuh nmm yd: 





rl .IIIX 



9(|j J99fn 



Jdj^i'i £ 

 X 



iJoa'irixioiivn 

 iifiij3jin 



O Jni6q 9f{j 6) anil «. wsib aw ^cuxjI -lonirn v,nibnoq?.9'nor» 

 co-ordinates of the point of contact P; and the rectangle under 

 these distances is to the square of the central radius vector 

 passing through the point of contact, as the square oi the di- 



