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VOL. IV.] PHILOSOPHICAL TRANSACTIONS. 353 



The Generation of an Hyperbolical Cylindroid; and a Hint of its Ap^ 

 plication for grinding Hyperbolical Glasses. By Dr. Christ. 

 IVRENy LL.D. Translated from the Latin. N"4S,p.96l. 



Let (fig. 5. pi. 9.) D B, E C, be two opposite hyperbolas, having the transverse 

 axis B C, their centre A, and one of the asymptotes GP; and through the 

 centre draw O M perpendicular to B C. Then if the two hyperbolas revolve 

 round their axis O M, it is plain that a body, called an hyperbolic cylindroid, 

 will be generated by that rotation, whose bases, and all the sections parallel to 

 them, are circles. — I say also, that if the body be cut through the asymptote 

 G P, the section will be a parallelogram. 



Let it be cut through the transverse axis by a circular section B N C, and also 

 through O and M in equal circles, and at equal distances from the centre ; and 

 also through the axis, into the generating figure, whose half, is B D E C, and 

 in the plane of which will be the asymptote G P, through which let the plane 

 B D E be cut at right angles in the plane F H P ; and join O H. 



Because the triangle O G H is rectangular, therefore the square of O H, or 

 O D, minus the square of O G, is equal to the square of G H ; and because 

 D O is parallel to B A, and cuts the asymptote in G, it will be (from the pro- 

 perties of the hyperbola, which are demonstrated in conic sections) the square 

 of O G, together with the square of A B, equal to the square of O D ; that is, 

 the square of O D minus the square of O G, equal the square of A B, or of 

 A N ; therefore the square of G H, is equal to the square of A N ; therefore 

 GH and AN are equal, and at right angles to G A. And the same also is 

 demonstrated of all other sections parallel to the base ; consequently an hyper- 

 bolical cylindroid being cut through an asymptote, the section is a parallelogram 

 Q. E. D. 



CoroL Hence it appears, that on the surface of a cylindroid, though con- 

 sisting of a double flexure, innumerable right lines may be drawn. It appears 

 also, that this body may be otherwise generated, viz. by the revolution of a 

 parallelogram about the axis, the angle at the axis G A O remaining the same, 

 or the generating line H R continuing immoveable, and either generating or 

 cutting the body. , 



And if a sharp and straight edge-tool have the same situation to the axis 

 with the generating line, while the mandrel turns round ; it is plain, that hy- 

 perbolas may be as accurately wrought by the lathe as circles ; since nothing 

 more is required for the formation of a cylindroid, than for that of a cylinder, 



VOL. I Y T 



