316 Proceedings of Royal Society of Edinburgh. [ sess . 
conical rotation. In the present Note I employ the quaternions 
which directly turn one side of the polygon to lie along the next. 
The successive sides are expressed as ratios of one of these qua- 
ternions to the next. 
Let p v p 2 , &c., p n be (unit) vectors drawn from the centre of the 
sphere to the corners of the polygon ; a v a 2 , . . . a n , the points 
through which the successive sides are to pass. Then (by Euclid) 
we have 
( p 2 ~ a i)(pi " a x ) = 1 + aj = Aj , suppose. 
(pS - a 2)(p2 - a 2 ) = 1 + a 2 = A 2 > 
&c. = &c. 
(pn+ 1 - a-n)(p n ~ a n ) = 1 + = A n . 
These equations ensure that if the tensor of any one of the ps be 
unit, those of all the others shall also be units. Thus we have 
merely to eliminate p 2 , . . ., p n ; and then remark that (for the 
closure of the polygon) we must have 
Pn+l — Pi • 
That this elimination is possible we see from the fact already 
mentioned, which shows that the unknowns are virtually mere 
unit-vectors ; while each separate equation contains coplanar vectors 
only. In other words, when p 1 and cq are given, p 2 is determi- 
nate without ambiguity. 
We may now write the first of the equations thus : — 
(p 2 ~ a 2)(pi ~ a i) = + ( a i - a 2)(Pi “ a i) = 9.1 > suppose. 
Thus the angle of q x is the angle of the polygon itself, and in the 
same plane. By the help of the second of the above equations this 
becomes 
^(Pi — a i) = (Pb _ a 2 ) 9 i 3 
whence 
9.2 ~ ^(Pi “ a i) + (2 — a s)9i = (pb — a B)9i • 
By the third, this becomes 
(p 4 - a 3)^ = Ag^i y 
whence 
(P4 — a i)92 = A-b9i "b ( a 3 — a 4)92 = 9s * 
The law of formation is now obvious ; and, if we write 
9q — Pi~ a ] , = a i - ®2 ’ P 2 = a 2 “ a 3 > ^ c, » 
