64 
Proceedings of the Royal Society 
on a circular cylinder. Let a be the radius of the circle thus 
formed by the axis of the closed helix ; let r denote the radius of 
the cross section of the ideal toroid on the surface of which the 
helix lies, supposed small in comparison with a; and let 0 denote 
the inclination of the helix to the normal section of the toroid. 
We have 
~ 27 ra a 
taD 6 = NlSn 7 = Nr ’ 
because is as it were the step of the screw, and 2ttt is the cir- 
cumference of the cylindrical core on which any short part of it may 
be approximately supposed to be wound. 
Let k be the cyclic constant, I the given force resultant of the 
impulse, and /x the given rotational moment. We have (§ 28) 
approximately 
Hence 
I — - K7ra 2 , /x = KN7rr 2 a . 
\J K7r’ \f Nk^ttHs ’ 
= /i. 
\/ N/XK 5 7T5 
tan 6 
11. Suppose, now, instead of a single thread wound spirally 
round a toroidal core, we have two separate threads forming as it 
were a “ two-threaded screw,” and let each thread make a whole 
ring, or any solid with a single hole through it, may be called a toroid ; but 
to deserve this appellation it had better be not very unlike a tore. 
The endless closed axis of a toroid is a line through its substance passing 
somewhat approximately through the centres of gravity of all its cross sec- 
tions. An apertural circumference of a toroid is any closed line in its surface 
once round its aperture. An apertural section of a toroid is any section by 
a plane or curved surface which would cut the toroid into two separate toroids. 
It must cut the surface of the toroid in just two simple closed curves, one of 
them completely surrounding the other on the sectional surface : of course, it 
is the space between these curves which is the actual section of the toroidal 
substance, and the area of the inner one of the two is a section of the 
aperture. 
A section by any surface cutting every apertural circumference, each once 
and only once, is called a cross section of the toroid. It consists essentially 
of a simple closed curve. 
