474 PROCEEDINGS OF THE A3IERICAN ACADEMY. 



where <^ is a binary form of degree n — 2. We will now prove that 

 <^ is a covariant of weight 1 of the system of linear forms which stand 

 in the denominator of (2). 



For this purpose let us notice that in the case n = 2, <^ reduces at 

 once to mi (e-i' e^" — e./ c-^") which is an invariant of weight 1. If then 

 we regard the particle f„ of mass m„ as consisting of « — 1 coincident 

 particles of masses — mi, — m^, . ■ . — »?„_i, we are led to write (2) in 

 the form 



K [x ^^ Me.'eJ'-eJe.") 1 



[_ * -^ (e/' xi — e/ x^) (e„" x^ — ej x^jj • 



Each term under the sign of summation being a covariant of weight 1, 

 the same is true of their aggregate, which is uierely the fraction in (2) of 

 which (f> is the numerator. The denominator of this fraction being a 

 covariant of weight zero, it follows that its numerator is a covariant of 

 weight 1, as was to be proved. 



The form <^ obviously vanishes at every point in the field where the 

 intensity of the force is zero. Besides these points of equilibrium <^ 

 cannot vanish anywhere except perhaps at the points e^ where the de- 

 nominator of (2) vanishes. If a particle e, whose mass is different from 

 zero does not coincide with any of the other particles, c}> will not vanish 

 there; for if it did, formula (2) would yield a force whose intensity 

 remains finite as we approach e,. If, however, k of the particles whose 

 total mass is not zero coincide at the point e„ the form <^ must contain 

 the linear factor ef Xi — ej Xo exactly k — 1 times in order that (2) 

 should yield a force which becomes infinite to the first order at e,. Such 

 a point, in spite of the vanishing of cfj, is, of course, not a point of equi- 

 librium. We shall, however, speak of it as a point of pseudo-equilibrium, 

 in justification of which term we may notice that if we let k distinct 

 particles whose total mass is not zero coincide at e„ k — 1 points of true 

 equilibrium fall together at this point. 



It remains then only to consider a point at which k of the particles 

 whose total mass is zero coincide. At such a point we have a force of 

 finite intensity and, accordingly, (f> must have at least a ^vfold root there. 

 If the multiplicity is exactly k, we do not have equilibrium, but such a 

 point we shall again call a point of pseudo-equilibrium. If the multi- 

 plicity of the root of 4> is greater than k we have true equilibrium. 



