( 220 j 
p from this point. If the locus passes through 0, a and a’ are the 
pair of axes we looked for in our problem. 
In general there is but 
one circle through O and 
P having its centre in n; 
then the reduction is in 
general possible only in 
one way. 
However, the case may 
be, that O and P coincide. 
Then the numberof solu- 
tions is oo?. 
If in this case @ and 
Fig. 2. @' are the components 
about the planes Oa and Qa', @ and a! the rotations about a and 
a’, then 
a—oasn OAA' = a 27 
si AA ( ) 
== OD S O : = Ye SE 
Now by the theory of the rotation in 8, : 
aX CAS pt, 
a XI A= DEN 
from which ensues: 
Mr NOEM 
ge O'A (2) 
whilst from (1) follows : 
OF) of OA? 
al? ©? OA? (3) 
OM SOAs). 
As BER Enon? it follows from (2) and (3) that @? =” . (4) 
So we see that the indefimteness of the reduction coincides with 
the circumstance, that the components of the double rotation are 
equal to one another. 
