BOCHER. — A PROBLEM IN STATICS. 479 



to locate the roots of cf). Let us, for simplicity, consider again merely 

 tlie case of two ground-forms fi and fo, and let us su[)[)Ose that these 

 forms are real, so that the roots of each, so far as they are not real, will 

 be conjugate imaginary in pairs. The force at any point on the axis of 

 reals will here be in the direction of this axis, and we have the theorem : 



If (f) is the Jacohian of two real binary forms fi and i^, then in an 

 interval of the axis of reals hounded by roots of one of these ground-forms 

 and containing no root of either form there lies at least one root of<^. 



The interval in question may, it should be noticed, be infinite, extend- 

 ing through the point at infinity if this is not a root oi f orf^, otherwise 

 extending up to this point. Immediate consequences of the last theoiem 

 are these : 



If all the roots of f^ and fg o?-e real, their Jacobian has a number of 

 real roots at least as great as the difference between the degrees of f^ and fj. 



If all the roots of i^ and fg are real and distinct, and if all the roots of 

 one of these forms lie in one of the intervals into which the roots of the 

 other form divide the axis of reals, then all the roots of the Jacobian of 

 fi and fj are real and distinct, and just one of these roots lies in each of 

 the intervals into which the roots of ii and fj divide the axis of reals, ex- 

 cept that the two intervals which are bounded at one end by a root of f^, 

 at the other by a root of fg, contain no root of the Jacobian. 



These theorems also can easily be extended to the case of more than 

 two ground-forms. 



The proofs of the theorems which we have here deduced from mechan- 

 ical intuition can readily be thrown, without essentially modifying their 

 character, into purely algebraic form. The mechanical problem must 

 nevertheless be regarded as valuable, for it suggests not only the theorems 

 but also the method of proof. 



§ 3. Stieltjes's Generalization. 



Up to this point we have been considering the problem of determining 

 the positions in which a single particle free to move in a certain field of 

 force can rest in equilibrium. This problem may be generalized in a 

 fruitful manner by considering with Stieltjes* a number of particles of 



* Acta Math., 6 (1885), 323, where, however, only the case in which all fixed and 

 movable particles lie on the axis of reals is considered. Tlie case in which all the 

 particles are free to lie anywhere in the complex plane was taken up for the first 

 time in the book of the present writer entitled, Ueber die Reihencntwickelungen 

 der Potentialtheorie, Leipzig, 1894, p. 215. Cf. also Bull. Amer Math. Soc, March, 

 1898, p. 256. 



