95 
1888.] Dr G. Plarr on the Boots of - 1. 
the integer N being susceptible of taking the m values 
0_, 1, 2, . . . . , m - 1 . 
When the angle u takes the value of a right angle, say 
then we get 
W = l7T, 
(2) = s , 
so that g becomes the equivalent of what is called a quadrantal 
versor. In this case we have 
n 
S’"" — cos + £ sin , 
n 27t,^ 
^ ^ m m 
N again taking the m values 
0, 1, 2, , (m- 1) . 
n 
The second member of this expression of g”^ represents m versors 
differing from each other, but they all are real quantities (real as 
opposed to imaginary of the form A + B 1), their axis being 
the unit-vector g. 
The first member, that is the expression of the — ^ power of g, 
m 
may be considered principally under two forms, and for stating 
them we will suppose the case : 
n %i 
m 2m' -f - 1 
2 ^^' 
The power ^ ^ of g may be calculated : 
I. Either under the form 
II. or under the form 
1 \2n' 
2m'+l 1 . 
' / ’ 
^g2n' • 
In this second case, applying = ( - 1)**', the result will be 
This is the expression of the roots of the scalar equation: 
