210 
Proceedings of Boycd Soeiety of Edinhurgh. [sess. 
3. As the projections of the generators on a pass through the 
centre O of the circle, two generators can meet only in two different 
ways. If they lie in different planes through OZ, they can only 
meet in OZ ; if their projections on a coincide, they can meet else- 
where. This remark will enable us to determine the order of the 
double curve on the surface. 
The intersection of the generator c/) with the axis OZ is given by 
the relation 2 = 
a cot 7 icf). As cfi + BAE-f/iTT 2^2 different 
values between 0 and 27 t, n different values between 0 and tt, 
the line OZ is an 2w.-fold line for m odd and an ?^-fold line for m 
even. 
The two sets of m {in') generators passing through two dia- 
metrically opposite points of the circle procure (??z'^) points of the 
double curve. As the expressions 
— (mcf) 2 Il7t) and ~(m^ -f tt - 1 - 2k7r) 
m m 
never are equal, it never happens that one of the rd' {m^) points 
lies on the line OZ. Therefore the locus of the {m'^) points is a 
double curve of the order {m'^). 
For m odd the double curve of the surface consists of a circle 
counted 7n-tinies, a right line counted 2?z-times, and a curve of the 
order This corresponds to a single double curve of the order 
m{m-\) 2n{2n-\) 
_ _j ^ _ 
+ or 2{iid‘ -l- n^) - {m -1- n) 
For m even the corresponding numbers are successively ^ , n , 
A 
111 ^ n m + n 
and^ 
111 the case of the marrow-bone is duly found the single double 
OG — 
line, the line 
y 
::)• 
Two generators with coincident projections on the plane a are represented 
by the equations 
aj = (a-l-^zsin0)cos 20 'j x= - cos 0) cos 20 1 
y = {a sin 0) sin 20 > , ~ + O' cos 0'' sin 20 f 
c = 2 ^ cos 0 J s = - g sin 0 J 
