167] THE GEOMETRY OF THE CYLINDROID. 167 



The principal screws on the cylindroid both pass through the double 

 point ; the two tangents to the curve at this point must therefore both 

 touch the parabola. 



The leading feature of the central section is expressed by the important 

 property possessed by the chords joining reciprocal screws. If we add any 

 constant to the pitches, then we alter X, and, accordingly, the point P, 

 through which all reciprocal chords pass, moves along the curve. 



The tangent to the cubic at P meets the cubic again in the point 

 reciprocal to P. Two tangents, real or imaginary, can be drawn from P 

 to the cubic touching it in the points 1\, T 2 , respectively: as these must 

 each correspond to a screw reciprocal to itself, it follows that T l and T 2 

 are the screws of zero pitch. We hence see that the two tangents from any 

 point on the cubic touch the cubic in points of equal pitch. 



Let a and ft be two screws, and 7 and 8 another pair of screws, and let 

 the two chords, a/9 and 78, intersect again on the cubic. If d and 6 be 

 the perpendicular distance and angle between the first pair, and d and 

 the corresponding quantities for the second pair, then there must be some 

 quantity ay, which, if added to all the pitches on the cyliridroid, will make 

 a. and /3 reciprocal, and also 7 and 8 reciprocal. We thus have 



(p a + pp + 2&&amp;gt;) cos d sin 6 = 0, 

 (Py + Ps + 2w) cos - d sin 6 = ; 

 whence, p a + pp d tan = p y + p& d tan 6 ; 



in other words, for every pair of screws, a and /8, whose chords belong to a 

 pencil diverging from a common point on the surface, the expression 



Pa + p? d tan 6 



is a constant. The value of this constant is double the pitch of the screw 

 of either of the points of contact of the two tangents from P to the curve. 



167. Section Parallel to the Nodal Line. 



If the node on the cubic be at infinity, the form of equation to the cubic 

 hitherto employed will be illusory. The nature of this section must therefore 

 be studied in another way, as follows : 



Let the plane cut the two perpendicular screws in A and B. Let I be 

 the perpendicular OC from upon G, and let rj be the inclination of this 

 perpendicular to the axis of x. Then, taking OA as the new axis of x, in 

 which case z will be the new y, we have 



x = I tan (t) 0\ 

 y = m sin 20. 



