[507 
154 
ON THE MECHANICAL DESCRIPTION OF 
% 
and then 
x = 7 cos <f> — k cos ((f) + X — a) cos (<p — a), 
y — 7 sin <f). 
I will, for greater simplicity, at once write b = 0: the equations thus are 
x = (c — k cos 9 cos \ + k sin 9 sin X) cos 9, 
y = c sin 9 ; 
x = (c + k sin 9 sin X) cos 9 — k cos 2 9 cos X ; 
k cos X , „ ( ky . ^ \ 1 / ——„ 
H (c- — y-) = C + — Sin X - VC“ — y 2 , 
or say the first is 
whence we have 
that is 
c 2 x + k cos X (c 2 — y 2 ) = (c 2 + ky sin X) V c 2 — y 2 , 
an equation which will assume a more simple form if either X = 0 or X = 90°; that is, 
if in the apparatus the nut-sides which guide the bed are either at right angles, or 
parallel, to the sides of the slot. 
Taking the general case, and writing for convenience cos 9 = £, sin 9 = y, the curve 
is given by equations of the form 
® = (£ v, i) 2 > 
2/ = (£, v> i) 2 , 
P + y 2 = i; 
viz. the elimination of f, y from these equations leads to the equation of the curve. 
The points of the curve have thus a (1, 1) correspondence with those of the circle 
^ 2 + if = 1; or, the circle being unicursal, the curve is also unicursal. Moreover, con 
sidering the intersections of the curve with an arbitrary line ax + by + c = 0, the points 
of intersection correspond to the points of intersection of the circle by the quadric 
a (£> y> l) 2 + ^(f> V> l) 2 4- c = 0; viz. there are four points of intersection, or the curve 
is a quartic, and hence it is a binodal quartic. But it is a binodal quartic of a 
special form: to show this more clearly, I introduce for homogeneity the coordinates 
z, so that the foregoing equations become 
* : V : z = (fr, V, D 2 : (I. V, £) 2 : where £ 2 + y 2 - f 2 = 0 ; 
the curve corresponding to these equations is, as just seen, a binodal quartic. But in 
the case in hand the form is the more special one, 
x : y : z = (f, y, £) 2 : : f 2 , where £ 2 + y 2 - ? = 0. 
The intersections by the arbitrary line ax + by + cz = 0 are the points corresponding to 
the intersections of the circle f 2 + ?? 2 — £ 2 = 0 by the quadric a(£, y, ^) 2 + &££+ c^ 2 = 0,