MATHEMATICAL SECTION. 107 
denotes action (transference) in the primary plane, but perpendic- 
ular to the direction of the connecting line; the third line denotes 
the actgn (transference) perpendicular to the connecting line in a 
plane perpendicular to the primary plane; the fourth line denotes 
action (torsion) in a plane perpendicular to the primary plane. 
If instead of the angles ¢ and ¢ the angles # and w are used, 
denoting the angle made by »’, with the line connecting the centers 
of the elements, that is, with p’; and w denoting the angle which the 
plane of y’ and p’ makes with the plane of » and p, the direction 
of the resulting action will be expressed by the following formula: 
See us = cos cos ’— sin @ sin @ cos w 
+ [sin @ cos &’+ cos @ sin @ cos w]i 
+ cos@ sin sinw .j 
+ sin @ sin sinw . i. 
The values of the four parts of this formula are identical, respect- 
ively, with the corresponding parts of the former one. 
The mutual action of the unit-right- quotients (or quadrantal 
versors) 2,7 and 77 are such that ? = —1; 7?= —1; and ri (or its 
substitute k*) = —1, from which it readily appears that 7=h= 
— ji; jk=i=—ki; and ki=j = — ik. 
Brief remarks on Mr. Etuiorr’s paper were made by the Chair- 
man and by Mr. CuristTIE. 
Mr. ArremMAs Martin presented a paper on 
METHODS OF FINDING N™-POWER NUMBERS WHOSE SUM IS AN 
N"™-POWER; WITH EXAMPLES. 
[ Abstract. ] 
First method: 
Let 17+ 27+ 374+ 474+ 1. Le = 8 
and assume the auxiliary formula 
GD. SS Orie ts e1 va Ey 
also assume d such that 
whence 
Seep (pF gee oh 2) 
