1892 - 93 ,] Professor Schoute on a Certain Locus. 
209 
On the other hand, when ncji is equal to — tt, the generator 
lies in a. This proves that a contains 2n generators. For the 
2A:+ 1 
expression 
-7T admits 2n different values between 0 and 27 t. 
The section contains ??^-times the circle C and 2n generators. 
Therefore the locus is of the order 2{m + ?^). 
Second Case : m even = 2m' (surface with only one side). 
The surface only contains the different positions of the generator 
corresponding to the values of between 0 and tt. 
In the same manner is proved that the section contains ?}/-times 
the circle and n generators. Therefore the locus is of the order 
2m' + n or m + n. 
In the case of the “ marrow-bone ” the order is 3.* 
* If we proceed analytically, the ecjuation of the surface is obtained by 
eliminating p and 0 among the three equations 
x = («-f^sin 0) cos 20) 
y-={n+p sin 0) sin 20 >, 
$ cos 0 ) 
or of 0 between 
X = {a-\- z tan 0) cos 20) 
?/ = (« 4- s tan 0) sin 20) (1). 
Here two pit-falls might be indicated. Firstly, if we deduce the two equa- 
tions 
7/ o o 
“ = tail 20, a;“' + ?/- = (a + s tan 0)- 
tlie elimination of 0 leads to an equation of the sixth order, containing also 
the result of the elimination of 0 among 
x= - (« -f s tan 0) cos 20)_ 
y= -{a + z tan 0) sin 20) 
Secondly, if we go on more cautiously and put tan 0 = we find 
x=-{a + zt) 
y = {a-\-zt) 
l-f- 
l+t 
and by elimination of t 
{x- -f y‘^-){y - zf = {ay- xzf , 
an equation of the fourth degree. Here however is it evident, that the tenus 
x^z^ annihilate one another, and the rest can be divided by y, &c. 
Synthetically the order of the general surface can also be found in seeking 
the section by a plane through OZ. 
VOL. XIX. 21 / 1 / 93 . 0 
