of Edinhimjh, Session 1883-84. 
561 
4. On an Equation in Quaternion Differences. By Professor 
Tait. 
{Ahsf7^act.) 
When the sides of a closed polygon are bisected, and the points 
of bisection joined in order, a new polygon is formed. It has the 
same number of sides, and the same meaii point of its corners, as 
the original polygon. In what cases is it similar to the original 
polygon ? In what cases will two, three, or more successive opera- 
tions of this kind produce (for the first time) a polygon similar to 
the original one ? 
Take the mean point as origin, and let q^a, . . . be the 
71 corners. Here a. is any vector, which, if the polygon be plane, 
may be taken in that plane ; and q^, . . . q^ are quaternions, which 
in the special case just mentioned are powers of one quaternion in 
the same plane. We obviously have, if ^qr = 9^r-\-i for the plane 
polygon, two conditions: — the first, 
(1+D + D2 + . . . . + D”~^)g,.a = 0 , 
depending on our choice of origin; and the second, 
^(1 +D)'”g,a = QD*$,a, 
depending on the similarity of the derived polygon to the 
original. In this last equation, Q is a scalar multiple of an unknown 
power of the quaternion of which the qs, are powers, expressing how 
the original polygon must be turned in its own plane, and how its 
linear dimensions must be altered, so that it may be superposed on the 
derived polygon. Also 5 is an unknown integer, but it has (like 
Q) a definite value or values when the problem admits of solution. 
r has any value from \ to n inclusive, as may be seen at once by 
operating by any integral power of D, and remembering that we 
have necessarily 
The solution of this case is easily effected, and gives the well-known 
results : — the general solution involving all equilateral and equi- 
angular polygons, where m may have any integral value. Besides 
this, there are special solutions for the triangle, and quadrilateral 
reduced at one operation to a parallelogram. In the former of 
these m may have any value ; in the latter (unless the figure be a 
square) m must be even. 
