IX THE HIGHEB GEOMETRY. 13 



Canstr. Let m + n be the order of the parabolas, and 

 p + q that of the hyperbolas. Find <j> a 4th proportional to 

 m + n? q _ p and m + 2/i ; divide G B in A, so that A R : A G : : 

 2 : j9 -f. ; then find v a 4th proportional to M + N, <j> + Pi an( i 

 q p, and y a 4th proportional to q, A G, and q p ; and, 

 lastly, 6 a 4th proportional to the parameter* of L M, TT and M. 



7/1 I W M 1 ^ 



If, with a parameter equal to x Q of the 



m + 2 M 



rectangle r . A x, and between the asymptotes A B, A c, 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 : 

 A G : : q : p -+- <j>, and that : m + n : : q p : m + 2n, A R is 

 AG X q 



(m + 11) x (q p} ^ , 



equal to n + *-* ; and, because L M is a conic 



TO -4- 2n 



hyperbola, the rectangle M s . R s, or M s . A p, or A p . 

 (M r + A R) is equal to the parameter, or constant space, 



therefore this parameter is equal to A p x ( M P + p 4- 

 AG . q 



(m + M) . (g - 



m + 2/t 



Again, the space A c D is equal to r- of the rectangle 



A c . c D, since A D is a parabola of the order m + n; but by 



m -4- n f M + N \ 



construction A c . c D is equal to - ot -- . r . AN ; 



m+2;i \ M ) 



therefore, A c D = Q -- - . r . A N, of which 6 : parameter 

 M 



of L M : : TT : M, and TT : *i + ~s :: <j> + p : q p ; therefore 6 = 

 Par.LM X (M + x) /(m+jOx (7^) \. ^ o T 



M (q p) 



S 

 * i. e., the constant rectangle or space to which A p . s M is equal. 



