530 
Proceedings of the Royal Society 
sor p on the other : by this relation (3) the number of the arbitrary 
scalar elements of the question becomes reduced to two. Obviously 
the scalar elements on which the unit-vector k' depends are also two 
in number ; but as the system may, in virtue of the variations of 
p( )q, possibly turn round the very direction of k\ it will be 
more convenient to consider the elements of p ( ) q as the true in- 
dependent parameters, at least two of them. 
The position of the single resultant of the system will be that of 
the straight line 
(4) SY(a-p)/3 = 0, 
where p designates the vector of a point of that line, the origin ,of p 
being the saipe with that of the vectors a, a , . . . 
Having 
2VpP^Yp$P = Ypk', 
and putting generally, with Professor Tait, 
(5) 
j cfi o) = 2iaS/?(o , 
(</>' a> = 2i/3Sa.(o , 
we get the solution of (4) by 
p = <h'k' — <f)k' + ulfi 
where u is an arbitrary scalar on which the position of the extremity 
of p on the line depends. 
We remark now that f>k’ , owing to pq = 1 , becomes 
cf>h' = '%a$p/3 0 qpk' 0 g = 5aS /3 Q k Q ; 
namely, cf>k' will be constant in direction and value, whatever be the 
state of rotation of the system. 
This result permits of a change of origin of the vectors a, a', &c., 
and p, by which the new value of cf>k' will vanish. The new origin 
will be at the extremity of the former ( - <£&'). This change of 
origin will not affect k' — 'Sp, but it will change the vector moment. 
Hot to change notations, we assume that the origin 0 was from 
the beginning at the point for which 
( 6 ) 
o = <j>k' = !§aS j3 Q k ' Q . 
