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 action (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 (p and ^ the angles 0' and (o are used, 0' 
denoting the angle made by //, with the line connecting the centers 
of the elements, that is, with /; and o) denoting the angle which the 
plane of // and p' makes with the plane of p and /?, the direction 
of the resulting action will be expressed by the following formula: 
U- X XJ ^ = cos 0 cos 0 '— sin d sin 6' cos w 
P P , 
-(- [sin 0 cos 0'-\- cos 0 sin 0' cos oj'] i 
-f COS0 sin 0' sin to .j 
-j- sin 0 sin 0' sin o ). ij. 
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 
—2 
versors) i^j and ij are such that =■ — 1] — 1; and ij (or its 
substitute ^^) =—1, from which it readily appears that^‘=^ = 
— ji; jk —i = — ki; and ki==j = — ik. 
Brief remarks on Mr. Elliott’s paper were made by the Chair¬ 
man and by Mr. Christie. 
Mr. Artemas Martin presented a paper on 
METHODS OF FINDING n'^“-POWER NUMBERS WHOSE SUM IS AN 
n™-power; with examples. 
[Abstract.] 
First method: 
( 1 ) 
Let r + 2" + 3“ + 4" + . . . . .r“ = 
and assume the auxiliary formula 
(i> + 3)”-y‘ = «; . . 
also assume d such that 
whence 
( 2 ) 
