70 



NEWTON S PKINCIPIA. 



be taken equal to three times A S, and B being drawn 

 perpendicular to A B, between B O, B A asymptotes, a 



S M 



rectangular hyperbola is drawn, H P, whose semi-axis or 

 semi-parameter is to D in the proportion of 6 to m; 

 it will cut the parabolic trajectory in the point P, 

 required. For calling A M = x and P M = y and A S 

 = a ; then A B = 3 a and y x (x + 3 a) = half the 

 square of the hyperbola s semi-axis, which axis being 



36 D 2 18 D 2 fx \ 



= -3-^2- = tf-&amp;gt; or y ( 3 + a ) 



3D 2 /2 1 1 



6D 



equal to ,3/0 



m 



X X - 



3 -D 2 m, , 2 1 , , 3D 2 

 r . Therefore - x y ^ (x a) y 



2 



2 211 



and - , A M x P M = ^ x y ; and - (x - a) y = - 



SM.PM=SMP; therefore the sector A S P = 



3 D 2 



so that the radius from the focus S cuts off the given area, 

 and therefore P is the point where the comet or other 



o 



body will be found in ^ parts of the time. 



If the point is to be found by computation, we can 

 easily find the value of y by a cubic equation, ?/ 3 + 3 a 2 



