208 Ffoceedings of Iloyal Society of Edinburgh. 
On a Certain Locus. By Professor P. H. Schoute, 
University of Groningen, Holland. 
(Read 5tli December 1892.) 
1. If the two ends of a strip of paper are pasted on one another 
after having twisted one of them through 180°, we obtain a surface 
with only one side. On the last meeting of the British Association Pro- 
fessor A. Crum Brown showed 
z a model of a mathematical 
surface closely connected with 
it. This surface, with the 
common name of “ marrow- 
bone,” is the locus of the line 
AB (lig. 1), cutting a given 
circle C orthogonally and 
moving under the condition 
Fig. 1. ^AOD = 2^BAE. 
In the following lines I wish 
to determine the order of the locus and one of its peculiarities in the 
more general case, when the ratio of the two angular velocities 
instead of 2 ; 1 is m : {in and n integer and prime to one another). 
2. Let us seek the section of the locus by the plane a of the 
circle. This section consists of the circle counted a certain number 
of times, and of a certain number of generators. As these two 
numbers are different for m odd and for m even, these two cases are 
to be treated separately. 
First Case : m odd (surface with two sides). 
The surface contains the different positions of the generator cor- 
responding to the values of ^ between 0 and 27 t. 
When m<h increases by 2-7?, the same point A is obtained. This 
proves that A lies on m different generators. For the angle 
i_ AOD + 2/^ admits m different values between 0 and 27 t. In 
m 
other words, the circle is an ^?^-fold curve of the locus, and is to be 
counted m-times as part of the section. 
