150 THE THEORY OF SCREWS. [159 



from the same principles. If a and yS are screws of zero pitch, then any 

 reciprocal transversal 6 will be also of zero pitch ; and as p must be reciprocal 

 to 6, it will follow that the pitch of p must be equal to the product of the 

 shortest perpendicular distance between p and 6, and the tangent of the 

 angle between the two lines. In short, the pitch of p must simply be equal 

 to what is sometimes called the moment between p and 6. 



We are also led to the following construction for the cylindroid. 



Draw a plane pencil of rays and another ray L, situated anyivhere. Then 

 the common perpendiculars to L and the several rays of the pencil trace out 

 the cylindroid. 



I have already mentioned (p. 20) that the first model of the cylindroid 

 was made by Plucker in illustration of his Neue Geometric des Raumes. The 

 model of the surface which is represented in the Frontispiece was made from 

 my design by Sir Howard Grubb, the cost being defrayed by a grant from 

 the Scientific Fund of the Royal Irish Academy. A hollow cylinder was 

 mounted on a dividing engine and holes were drilled at the calculated points. 

 Silver wires were then stretched across in the positions of the generators 

 and a beautiful model is the result. 



The equation to the tangent cone drawn from the point x, y , z to 

 the surface, 



z (x 2 4- y 2 ) - 2mxy = 0, 



is of the fourth degree and is given by equating to zero the discriminant 

 of the following function in &&amp;gt;, 



to 3 (xz- zx) - &amp;lt;u 2 [yz -zy + 2m (x - x }} + o&amp;gt; {xz - zx + 2m (y - y )} + zy - yz. 



This cone has three cuspidal edges, and accordingly the model exhibits 

 in every aspect a remarkable tricuspid arrangement. 



I here give the details of the construction of the much simpler model of 

 the cylindroid figured in Plate II.* A boxwood cylinder, O m lo long and 

 O m&amp;gt; 05 in diameter, is chucked to the mandril of a lathe furnished with a 

 dividing plate. A drill is mounted on the slide rest, and driven by overhead 

 gear. The parameter p a -p ft (in the present case O ^OGG) is divided into 

 one hundred parts. By the screw, which moves the slide rest parallel to the 

 bed of the lathe, the drill can be moved to any number z of these parts 

 from its original position at the centre of the length of the cylinder. Four 

 holes are to be drilled for each value of z. These consist of two pairs 

 of diametrically opposite holes. The directions of the holes intersect the 



* See Transactions of Royal Irish Academy, Vol. xxv. p. 216 (1871) ; and also Phil. Mag. Vol. 

 XLII. p. 181 (1871); also B. A. Report, Edinburgh, 1871. 



