400 
Proceedings of the Royal Society 
Many of the above results were previously obtained quater- 
nionically by Professor Tait. The interest of the present com- 
munication is less in the results obtained than in the methods 
employed to treat a particular problem in Pliicker’s “ Line Geometry.” 
In the development of the results Pliicker’s “ Neue Geometrie ” has 
been followed as far as possible. Any interested reader may be 
referred to that work for farther information on this and like 
matters. 
3. Mathematical Notes. By Professor Tait. 
(a.) On a Problem in Arrangements. 
While making some algebraic problems last summer for an 
examination, I devised the following question : — 
“A schoolmaster went mad, and amused himself by arranging 
the boys. He turned the dux boy down one place, the new dux 
two places, the next three, and so on till every boy’s place had 
been altered at least once. Then he began again, and so on ; till, 
after 306 turnings down, all the boys got back to their original 
places. This disgusted him, and he kicked one boy out. Then he 
was amazed to find that he had to operate 1120 times before all 
got back to their original places. How many boys were in the 
class ? ” 
It is clear that one of the factors of the number of turnings 
down is (n- 1), where n is the number of boys in the class. The 
factors of 306 are 18 and 17, and those of 1120 are 7, 10, and 16. 
If we try 1 7 as the original value of n — 1 , 16 will be the value for 
one boy less : from which it appears by a tentative process that the 
class consisted of 18 boys. But it is interesting to examine the 
nature of the question more closely. It is intimately connected 
with one of the problems suggested in my paper on “ Knots ” (Trans. 
R.S.E., 1876-77, § 5). If we know the arrangement of the boys — 
after one of them has for the first time been turned to the foot of 
the class, the processes given in that paper lead easily to the 
complete solution. 
Now it is easy to see that the particular arrangement just men- 
tioned can be found diagrammatically as follows : — 
