622 RECORD OF CURRENT RESEARCHES RELATING TO 



vertex ; from the termination of tlie perpendicular take a line c inter- 

 secting the half-diameter of the base line at d ; another line is set 

 off from this point at right angles to c ; the distance at the inter- 

 section of the axis beyond the base line will be the required focal 

 distance of the parabola. 



To those not possessing the requisite tools, this method of cutting 

 out a temj^late, of course, cannot be available. The plan of drawing 

 with a square and piece of string, described in all elementary works 

 on geometry, is so irregular in its action, as to be iiseless for small 

 parabolas; bisection must therefore be resorted to, which by careful 

 manipulation gives a very true figure. This operation depends upon 

 the following property of a parabola : that any point taken on the axis 

 at a distance beyond the vertex equal to the distance of the focus 

 within it, to any transverse line on the axis, will be equidistant from 

 the same line to the focus. Proceed as follows : — 



Provide a thin brass plate (Fig. 3) perfectly square and flat, of 

 sufficient size to enclose the required parabola for which it is to serve 

 as a template. In a centre line quite parallel with the sides, prick off 

 two equidistances, the directrix a without, and the focus h within 

 the vertex of the parabola. Draw a number of parallel lines at right 

 angles across the centre line ; these lines need not be set at any par- 

 ticular distance, but may be ruled at sight, taking the precaution of 

 setting them close together towards the vertex, and progressively in- 

 creasing the distance between tliem towards the base. With fine-pointed 

 dividers, take the distances of these lines on the axis from the directrix 

 or outside mark a in succession. For each measurement, shift the 

 point to the focus b within the vertex, and bisect the line from which 

 the distance was taken on both sides ; the intersection of all the lines 

 by the arcs from the focus will give the outline of a true parabola. 

 The crossing points are now to be dotted in with a thin, sharp-pointed 

 centre punch applied at the right spot under a hand magnifier. The 

 surplus brass is cut out with a fine saw, and the template carefully 

 filed up till the punch marks appear as sunk in the metal. 



The block of glass if intended for a flat-topped or immersion para- 

 boloid, should have both its base and apex polished off to the right 

 thickness before it is cemented to the lathe-chuck with black sealing- 

 wax. By means of the rough edges of an old saw-file, ground on one 

 side and used with plenty of turpentine, the glass is turned away at a 

 very slow speed till it is seen apj^roximately to fit the temj^hite. The 

 edges of this are then slightly smeared with reddle and oil, and the 

 paraboloid fine turned with a keen edge, until the template marks it 

 evenly all over. In order to take out the rings left from the turning, 

 a block of brass, not larger than half an inch square, is traversed over 

 the revolving glass with coarse and then smoothing emery, till all 

 scratches disappear. The glass is then polished with a buff-stick, and 

 crocus and water, and finally a piece of hard beeswax is held against 

 it with finer crocus, in order to obtain the last degree of polish. 



If the paraboloid is to be a non-immersion one with a cupped top, 

 it may be turned flat on the end till the required thickness is arrived 

 at and the hemispherical cavity roughly turned out to a half-circle 



