166 THE THEORY OF SCREWS. [165- 



There are two critical cases in which this expression vanishes. It does so if 



h tan ft + m sin 2a = ; 

 where the plane of section is tangential to the cylindroid. 



But we also note that the discriminant will vanish if 



h = 0, 

 i.e. if the plane of section passes through the centre of the cylindroid. 



We might have foreseen this from the results of the last article ; for the 

 plane of Y is a central section, and the hyperbola has evidently degenerated, 

 for all the reciprocal chords, instead of touching an hyperbola, merely pass 

 through the common apex P. The case of the central section is therefore 

 of special interest. 



166. The Central Section of the Cylindroid. 



By this we mean a section of the surface, special in no other sense, save 

 that it passes through the centre of the surface. The equation to the 

 central section is ( 160) 



y 3 sin ft cos 2 /3 + yx z sin (3 mx z sin 2a + 2mxy cos ft cos 2a + my- sin 2a cos 2 /3 = 0. 

 The chord joining points of equal pitch + 6 is 



x (cos 20 + cos 2a) + y cos ft sin 2a + m cot ft sin 2 20 = 0. 

 The apex P through which all reciprocal chords pass is 



_ m (\ 2 - 1) cot ft (\ + cos 2) 

 1 + 2X cos 2a 4- A, 2 



_ m (A 2 1) cosec ft . sin 2a 

 V ~ 1 + 2X cos 2a + X 2 



and in general the co-ordinates of a point on the cubic are 



x = y cos ft tan (6 a), 

 y = m cosec ft sin 26. 



One of these curves may be conveniently drawn to scale, from the 

 equations 



as = % tan (0 - 25), 

 y = - 66 sin 20. 



The parabola, which is the envelope of equal pitch-chords, would in this 

 case have as its equation 



