H. Skey. — An Astronomical Telescope on a Nevi Construction. I2l 



the surface in the direction PC. Let PC represent this force ; but P is also 

 subject to another force, namely, its own weight acting vertically down- 

 wards, which we may suppose represented by PQ ; the resultant of these, 

 therefore, PR, is the whole force acting on P, and so must be perpendicular to 

 the surface, and therefore to the curve. To prove that this curve is parabolic — 

 NM : MP : : PQ : QR (=PC). 

 KM : MP : : Weight P : Centrifugal force. 

 But the dynamical measure of the force of gravity at this latitude is 32*17, 



4: Ti T 



expressed in feet every second, and of the latter force — (see note), n repre- 

 senting 3-1416, or the semi-circumference of a circle whose radius is 1, ^ being 

 the number of seconds in one revolution, and r the radius = MP. 



.-. NM : r :: 32-17 : ^-^ 



t^ 



consequently 32-17r -- —^ = 32-17 x-p-= 8-04 -V = KM. 



C ^ 7i Kir' 



The line NM thus determined is called the sub-normal to the curve at the 

 point P, and when the angular velocity of rotation is constant then the sub- 

 normal is also constant in length, no matter in what part of the curve the 

 point P is situated. This property belongs exclusively to the parabola. 

 Hence the surface of a fluid rotating on an axis perpendicular to the horizon 

 is a paraboloid. 



To determine then the length of NM for different times of rotation — 



Let ^ = 1 second then NM = 8-04-h = 0-814 feet. 



n^ 



^ = 4 „ „ =13-037 „ 



Now that part of a paraboloid where a ray of light parallel to the axis will be 

 reflected along a line forming a right angle to the axis must itself be inclined 

 at an angle of 45°, consequently such reflected ray will, when it meets the 

 axis, have traversed a distance equal to the length of the subnormal, 

 therefore at that part of the curve the two forces, namely gravity and 

 centrifugal force, have the same measure, for they are represented in magni- 

 tude and direction by difi'erent sides of the same square. 



Moreover this particular ray is the only one which 

 will be reflected in a horizontal direction along the 

 parameter of the paraboloid until it meets the axis 

 in the focus of the curve. And since the distance 

 FY equals the half FP' it also equals half NM, by 

 which we can obtain the focal length of the telescope 

 for any velocity. 



Within the range of our acquaintance with nature 

 1^'ig- 3. "we have one remarkable and brilliant metal which 



