of Edinburgh , Session 1881-82. 529 
forces /3 0 , f$ 0 ' , /? 0 ", &c., p being a versor, the results of the rotation 
being j3, /3\ &c., respectively. 
By the expressions 
(1) P=pf5 0 q, I3' = pfi' 0 q, &c., 
we submit the forces at once to two of Minding’ s conditions ; 
namely, 1° the magnitude of each of the forces remains unchanged 
by the rotation, and 2° the mutual inclinations of the forces, con- 
sidered two by two, remain also unvaried. 
The resultant of the forces being 
2/3 =*>(S/3 0 )<Z , 
remains also unchanged as to its tensor, and we may, without loss 
of generality, assume at once that the unit of force has been chosen 
so as to give 
T2/3=1=T2/V 
We put 
j US/? =S 
t U2/3 0 = 7c' 0 . 
This gives 
( 2 ) h'=pk\q. 
We call a, a', &c., the vectors drawn from a certain origin 
to the points of application of the forces. A third condition will 
be the rigidity of the system of these vectors, so that they remain 
immobile in respect to one another, and to their origin. 
The fourth condition to which Minding' s System of Forces are to 
be submitted demands that the action of the system be reduced to 
that of a single force, and consequently that the moment of the 
resulting couple SV. afi , in respect to the origin, be perpendicular 
to the direction of the resultant of the forces. 
This condition, expressed by 
(3) SAfSVa/3 = 0, 
or in virtue of (1) and (2) by 
%Sap(Vp Q k' 0 )q = 0 , 
constitutes a relation between the data a, a , &c., A)> Po> &c., kf 
on the one hand, and between the three scalar elements of the ver- 
