318 Proceedings of Ptoyal Society of Edinburgh. [sess. 
where x, y\ are the coordinates in the circular cylinder correspond- 
ing to x, y, in the elliptic cylinder. If y' = xt&nO\ we get 
k = - tan<£sin2 0 
Jv 
= 2 ^tan<£^ = 2 Jtan 
sh r' 2 Ji V 2 
-where r " 2 = x 2 + y 2 = x 2 + y 2 g/^ • Hence 
2 
z(fx 2 + rjy 2 ) = -- tan cf>xy . 
v t 
Thus the ruled surface is the elliptic cylindroid , for which 2tan<£ 
= J€(pf~2hv)- 
A very simple model of the elliptic cylindroid may he made by 
means of two elliptic rings, hinged together at one extremity of 
their major axes. Points at the same distance from the hinge on 
each are joined by a pair of elastic cords crossing each other. As 
the rings are opened out, the cords lie on different elliptic cylin- 
droids, one for each angle between the rings. 
5. If TV/ be the magnitude of the free screw, while TVristhat 
•of the constrained screw, we have as the components of these 
TVr'cosi// , TVr'sin ' and tcosxf/ , rjsimf/ respectively, while TV?* = 
icos 2 if/ + r]sm 2 i[/. If we write x = pcosif/, y = psim]/ we see that 
TVr-p 2 — 1, provided that £x 2 + yy 2 = l . Thus the magnitude of 
the rotation Vr is the reciprocal of the square of the parallel radius 
vector in the ellipse £x 2 + gif — 1 regarded as having its x and y 
axes respectively parallel to Vr, and Vr 2 . 
But this ellipse is the ellipse, when referred to its centre, which 
is the transverse section of the elliptic cylinder above considered 
by the plane z = 0. It may be termed the amplitude conic. 
The pitch is given by pfx 2 + pf) =pfx 2 +p 2 yy 2 . 
6. If we eliminate 0 between the equations 
p{£c,os 2 0 + ys\xi 2 6) ^pfco^O -\- p 2 g^m 2 6 
2 (£cos 2 0 + ^sin 2 0) = (pf - p. 2 g)sin0cos0 
wve get 
(p - 2h)(p - P 2 )(p f - Ihvf + Pi = 0 
