VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 340, 



If from g, as origin, in ab, one of lm's asymptotes, there be described an hyper- 

 bola ic', of any order whatever, except the first, and passing through m, a point 

 where lm cuts any of the parabolas am, of any order whatever, drawn from a a 

 point to be found, and lying between ab and ac, an area acd may be always found, 

 (that is, for every case of am and ic',) which shall be constantly cut by ic', in the 

 given ratio of m : n ; that is, the area amn : nmdc :: m : n. I omit the analysis* 

 which leads to the following construction and composition. 



Constr. Let m 4- n be the order of the parabolas, and p -f q that of the hyper- 

 bolas. Find <p a 4th proportional to m -f- n, q — p and m + 2n; divide gb in A, 

 so that ar : ag :: q :p + p ; then find t a 4th proportional to m + n, <p + p 3 and 

 q —• pt and y a 4th proportional to q, ag, and q — p ; and, lastly, 9 a 4th pro- 

 portional to the parameter* of lm, «■ and m. If, with a parameter equal to 



m ■ X of the rectangle r . an, - and between the asymptotes ab, 



ac, a conic hyperbola be described, it shall cut the parabola in a point, the co- 

 ordinates to which contain an area that shall be cut by ic' in the ratio of m : n. 



Demonstration. Because ag is divided in r, so that ar : ag :: q : p -f~ <p, 

 and that <p : m -\- n : : q — p : m + In, ar is equal to p + 

 ag x g 



(m + n x g — p . an ^ Decause LM .\ s a CO nic hyperbola, the rectangle ms . rs, 



or ms . ap, or ap . (mp + AE ) is equal to the parameter, or constant space, 



ag . q 



therefore this parameter is equal to ap X (mp -j- p + \Z% — )• 



Again, the space acd is equal to — of the rectangle ac . cd, since ad 

 is a parabola of the order m -+- n ; but by construction AC . cd is equal to 

 of (0 — !t — * . r . an) ; therefore, acd = 6 — - - . t . an, of which 



m + 2n v m ' m 



G : parameter of lm :: tt : m, and ?r : m + ** :: p -{• p i q — p ; therefore = 



Par. lm X (m + n) Am + n) y.(q— p) . . . 



i tf-p) X P m + 5 * P ; ) also t : ? :: ag : ^ - p ; consequently 



Par. lm x (m + n) ... r . , , t m + n x q — p . * x j j- • • u j u 



acd = ; — — - multiplied by ( — - ~ — - + p) and diminished by 



M (? — P) m + 2ra 



^ X ak X ^; therefore, transposing^^* X £±£ X *£ 

 -{- jb) is equal to acd -f- X an x ? ' A ° ; and par. lm will be equal to 



(ACD H X AN X ) X : X (q - P) X ACD + q . AN X A& 



V « ■"-"' '-' , that is, JL+i 



Now it was before demonstrated, that the parameter of lm is equal to ap X 



q . AG 



( MP + p 4. (P+ t n n ) +l~ p) ). This is therefore equal to 



* i. e. The constant rectanglft or space to which ap . sm is equal.— Orig. 



