350 PHILOSOPHICAL TRANSACTIONS. [ANNO I798. 



M ' 



X (q — p) X ACD + q. as X AG 



&±«* , multiplying both by 0" + ») * (g -j) 



m + " n 



we have ^—^ X (q — />) X acd + q . an X ag = ap X (mp X (/> + 



( (M+«)x(g-g) )+ Q)> 



From these equals take ^ . ag X an, and there remains — - — x {q — p) x 

 acd equal to ap X pm X ( (W> + ,"* + Jlf """ ^ r P) + ?.ag X (ap - an) ; or, 

 dividing by q - p, ^ X acd = ap X (-^±^) + -E-) X pm + ^4- X 

 ag X (ap — an). Now, m " X ap X pm is equal to the area apm ; there- 

 fore the area apm together with ~ X ap . pm, and — ~- X ag X (ap ~ an), or 



apm with — - — X ap . pm — X ag X (an — ap), or apm -\ ^— x ap . 



q — p q-p v 9-P 



pm £— x rect. pt, is equal to — — — X acd. Now ic' is an hyperbola of 



the order/) + q; therefore its area is ■ X rect. gh . mh. But q is greater 

 than p; therefore — — is neerative, and *LlH?L j s the area mhkc'; and the 



r p -q & q-p 



area ntkc' is equal to — - — X gt X tn ; therefore mnth is equal to (mhkc' — 

 ntkc'), or to -—— X (gh . mh — gt . tn). From these equals take the com- 

 mon rectangle at, and there remains the area mpn, equal to -— — X ap x mp — 

 —2— X pt ; which was before demonstrated to be, together with apm, equal to 



q -. p 



— — — . acd. Therefore mpn, together with apm, that is, the area amn, is equal 



to — — - . acd ; consequently amn : acd : : m : m + N J ana< (dividendo) amn : 

 nmdc :: m : n. An area has therefore been found, which the hyperbola ic' always 

 cuts in a given ratio. Therefore, a conic hyperbola being given, &c. a. e. d. 



Scholium. This proposition points out, in a very striking manner, the connection 

 between all parabolas and hyperbolas, and their common connection with the conic 

 hyperbola. The demonstration here given is much abridged ; and, to avoid cir- 

 cumlocution, algebraic symbols, and even ideas, have been introduced : but by at- 

 tending to the several steps, any one will easily perceive that it may be translated 

 into geometrical language, and conducted on purely geometrical principles, if any 

 numbers be substituted for m, n, p, and q ; or if these letters be made representa- 

 tives of lines, and if conciseness be less rigidly studied. 



Prop. 14. Theorem. — A common logarithmic being given; if from a given point, 

 as origin, a parabola, or hyperbola, of any order whatever be described, cutting in 



