164 THE THEORY OF SCREWS. [165 



The envelope of this chord is found to be 

 = +# 2 ra 2 sin 2 2a, 



+ y 2 | ra 2 cos 2 & cos 2 2a - hm sin cos sin 2a - A 2 sin 2 /3 



+ |m 2 X cos 2a cos 2 /3 + |ra 2 X 2 cos 2 /3, 

 w 2 



+ x y \ a s i n 2 cos 2 cos ~ m ^ cos ^ a s ^ n & 







| m 2 X sin 2a cos /3 ra&X sin /3, 

 + ac + m/t 2 cos 2a tan fS + hm 2 X sin 2a + mA 2 X tan /3, 

 + y | - w 2 A cos /3 + w/i 2 sin y3 sin 2a + 2/i 3 sin tan /3 - w 2 AX cos cos 2a, 

 + | m 2 A 2 - A 4 tan 2 0. 



Using the data already assumed in 160, and, with the addition now 

 made of taking X to be , the equation reduces to 



122a; 2 - 2(% 2 - 16 Ley - 2417^ - 5003y + 245436 = ; 

 which, for convenience of calculation, I change into 



x = 1-6 + 21 6 sec &amp;lt;J&amp;gt; 47 5 tan &amp;lt;/&amp;gt;, 

 y = - 12-8 + 32-8 sec &amp;lt;/&amp;gt;. 



This hyperbola has been plotted down in Fig. 36. It obviously touches the 

 cubic at three points. I had not anticipated this until the curves were care 

 fully drawn ; but, when the theorem was suggested in this manner, it was 

 easy to provide the following demonstration : 



The cone of reciprocal chords drawn through any point P breaks up 

 into a pair of planes when P lies on the cylindroid. (I use the expression, 

 reciprocal chord, to signify the transversal drawn across a pair of reciprocal 

 screws on the cylindroid. This is very different from a screw reciprocal to 

 the cylindroid.) For, take the screw reciprocal to that which passes 

 through P. Then the plane X, through P and this screw, is obviously one 

 part of the locus. Draw through P any transversal across a pair of reci 

 procals on the cylindroid, then the plane Y, through the centre and this 

 transversal, will be the other part of the locus. This pair of planes, X 

 and Y, intersect in a ray which we shall call 8. 



A plane of section through P will, of course, usually cut the two planes 

 in two rays, and these will be the two reciprocal chords through P. But 

 suppose the plane of section happened to pass through 8, then there will be 

 only one reciprocal chord through P, and this will, of course, be 8. Now, 

 8 must be a tangent to the cylindroid at P. Every chord through P, in 

 the plane of Y, must cut the surface again in a pair of reciprocal points. To 

 this 8 must be no exception, and as it lies in X, it intersects the screw 



