12 



GENERAL THEOREMS, CHIEFLY PORISMS, 



curve with all the parabolas, or hyperbolas, of any given 

 order whatever, lying between straight lines to be found, the 

 moving line shall cut, in a given ratio, the areas of the curves 

 described, contained by the peripheries and co-ordinates to 

 another conic hyperbola, that may be found, at the points 

 where this cuts the curves described. 



PROP. 12. Porism. A conic hypei'bola being given, a 

 straight line may be found, along which if another move in a 

 given angle, and pass through any point whatever of the 

 hyperbola, and if this point of section be joined with another 

 that may be found, the moving line shall cut, in a given 

 ratio, the triangles whose bases and altitudes are the co- 

 ordinates to a conic hyperbola, which may be found, at the 

 points where it meets the lines drawn from the point found. 



Scholium. I proceed to give one or two examples, wherein 

 areas are cut in a given ratio, not by straight lines, but by 

 curves. 



PROP. 13. Porism. Fig. 14. A conic hyperbola being 

 given, if through any of its innumerable intersections with 

 all the parabolas of any order, lying 

 between straight lines, of which one is 

 an asymptote, and the other may be 

 found ; an hyperbola of any order be 

 described, except the conic, from a given 

 origin in the given asymptote perpen- 



K dicular to the axis of the parabolas, the 



hyperbola thus described shall cut, in a given ratio, an area, 

 of the parabolas, which may be always found. 



If from G, as origin, in A B, one of L M'S asymptotes, there 

 be described an hyperbola i c', of any order whatever, except 

 the first, and passing through M, a point where L M cuts any 

 of the parabolas A M, of any order whatever, drawn from A a 

 point to be found, and lying between A B and AC, an area A CD 

 may he always found (that is, for every case of A M and i c'), 

 which shall be constantly cut by I c', in the given ratio of M : x ; 

 that is, the area AMN:NMDC::M:N. I omit the analysis, 

 which leads to the following construction and composition. 



