NEWTON. 



1327. Two triangles CAB, cAb, have a common angle A and 

 nn of the sides containing tnat angle the same in each; BC, 

 l<- intersect in D prove that ultimately, when b moves up to B y 

 CD : DB :: AB : AC. 



1028. Two equal parabolas have the same axis, and the 



focus of the outer is the vertex of the inner one, J//V'> ^'Qv are 



common ordi nates; prove that the area of the surface generated 



by the revolution of the arc PQ about the axis bears to the area 



A' a constant ratio. 



1329. Common ordinates from the major axis are drawn to 

 two ellipses, which have a common minor axis, and the outer of 

 which touches the directrices of the inner ; prove that the area of 

 the -urface generated by the intercepted arc of the inner ellipse 

 revolving about the major axis will bear a constant ratio to the 

 intercepted area of the outer. 



50. A /! is a diameter of a circle. /' j.oint on the circle 



J, and the tangent at P meets />'.( produced in T\ prove 



that ultimately the difference of BA, ///' bears to AT the ratio 



! L'. 



1 .">:'>!. If 1'Q be an arc of continued curvature, and 7? the 

 between 7' an. 1 V :<t ^hieh the tangent is parallel to /\> ; 

 then the ultimate ratio 7*7? : RQ, when /'V is indefinitely dimin- 

 i.-he.l, is one of equality. 



'<-. If J be a point on a curve, and 7\ <} two neighbour- 

 ing points, the ultimate ntio of the triangle formed by the tan- 

 - at these points to that formed by the normals is 



1 : d 



