Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/138

 118 velocity upon that phærical body, in the directions of right lines parallel to AC; and let FB be one of thoe right lines. In FB take LB equal to the emi-diameter CB, and draw BD touching the phere in B. Upon KC and BD let fall the perpendiculars BE, LD, and the force with which a particle of the medium, impinging on the globe obliquely in the direction FB, would trike the globe in B, will be to the force with which the ame particle, meeting the cylinder ONGQ decribed about the globe with the axis ACI, would trike it perpendicularly in b, as LD to LB or BE to BC. Again, the efficacy of this force to move the globe according to the direction of its incidence FB or AC, is to the efficacy of the ame to move the globe according to the direction of its determination, that is, in the direction of the right line BC in which it impels the globe directly, as BE to BC. And joining thee ratio's the efficacy of a particle, falling upon the globe obliquely in the direction of the right line FB, to move he globe in the direction of its incidence, is to the efficacy of the ame particle falling in the ame line perpendicularly on the cylinder, to move it in the ame direction, as $$\scriptstyle BE^2$$ to $$\scriptstyle BC^2$$. Therefore if in bE; which is perpendicular to the circular bae of the cylinder NAO, and equal to the radius AC, we take bH equal to $$\textstyle \frac {BE^2}{BC}$$ then bH will be to bE as the effect of the particle upon the globe to the effect of the particle upon the cylinder. And therefore the olid which is formed by all the right lines bH will be to the olid formed by all the right lines bE as the effect of all the particles upon the globe to the effect of all the particles upon the cylinder. But the former of thee olids is a paraboloid whoe vertex is C, its axi CA and latus rectum CA; and the latter olid is a cylinder circumcribing the paraboloid