( 822 ) 



the farther reduction to the simples! form (kl) = o(jm), assumed by 

 the equation when the edges kl and jm, the planes ijm and i Id of 

 the simplex of coordinates are conjugated elements of the system* 

 a or «. 



III. If we assign to the elements p, n, a, u vector-nature, expres- 

 sions 2£((jjj>ij, Saijiiy become of importance as virtual coefficients 

 (in Ball's theory of screws) and the disappearing of these coefficients 

 then gives the condition that the force p performs no work at a 

 displacement in consequence of a double rotation a, resp. that the 

 dynam a performs no work at a rotation .-r. 



So in Ball's notation the equations (7), (7)' give the condition oj 

 reciprocity between force and double rotation, resp. between dynam 

 and rotation. 



In like manner the equation 



2a «£ = 0, (8) 



which includes (7) and (7)' and likewise (2), gives the condition of 

 reciprocity between the dynam a and the double rotation «. 



IV. We shall now pass to the general equilibrium of forces and 

 rotations. It will be convenient to understand by p, rr, a, a vectors 

 unity and to indicate the intensity of these vectors by a factor. 



It will be sufficient to limit ourselves to the equilibrium of forces, 

 leaving the treatment of the dual case to the reader. 



In the first place we regard the case of n forces, n > 10 working 

 along lines given arbitrarily. 



It goes without saying that for the equilibrium it is necessary and 

 sufficient that the intensities / v satisfy the ten conditions: 



2fr pM = (9) 



We can therefore in general bring arbitrary intensities along n — 10 

 vectors, those on the other ten then being determined by the above 

 equation (9). 



In particular for n = M the theorem holds: 



To rectors along eieren lines given arbitrarily belongs in general only 

 one distribution of ratios of intensity, so that the system on those 

 lines is in equilibrium. 



The generality of the case is circumscribed by the requirement 

 that no ten lines can satisfy one and the same linear condition in 

 the form 2aijp v .. = 0, where the coefficients (t y do not depend on 



v, in consequence of a well-known property of determinants tending 

 to zero. 



