476 PROCEEDINGS OF THE AMERICAN ACADEMY. 



/i = (ei"xi — ei'x2) . . . (V'^i — ^A ^2), 

 /2 = (^A+i"^i - «A+i'^2) . • • (ep,+pj'xi - ep^+pjz^), 

 the field of force is 



A'[-2 {p.^ e/'xj-ejxr^' 2 e/'aX-e/x^] 

 *- ^ i-l ' '^ i=p,+i • -= / -I 



5A 3/i 



•* = ^. (^' 



Accordingly 



which reduces, when we apply Euler's theorem for homogeneous func- 

 tions, to the Jacobian of /i and/a- Heuce 



The vanishing of the Jacohian of two binary forms fi and i^ of degrees 

 Pi and P2 respectively determines the points of equilibrium in the field 

 of force due to pi particles of 7nass pj situated at the roots of fx, and pj 

 particles of mass — pi situated at the roots of i^.^ 



It is easy now to express the covariant ^ which we have in the general 

 case integrally and rationally in terras of the ground-forms and their 

 Jacobians. For this purpose let us write as the field of force : 



. , Sf . ^ ^ 9f 9f, 



t:=k TIT 1 I— l—k—l 



4-S'"'j]-4s2"<-t--|-)] 



t=A--l 



i=l 



where ^t denotes the Jacobian of/, and fi.. Accordingly : 



i=k—l 



^ ~ ^ \^ '^''^ -f^ ' ' ' -^-i-^+i • • • '^*-i )' 



* If one of the two ground-forms is linear, the theorem may be stated tlius : 

 If (p is the first polar of the point (y-^ , y.,) with regard to a binary form / of de- 

 gree p, tiie vanishing of (p determines the points of equilibrium in the field of force 

 due to p unit particles situated at the roots of /i and one particle of mass —p at, 



(^l.!/2)- 



The special case of this where yo = ^ leads us back to Gauss's theorem referred 

 to near the beginning of § 1. 



