BOGHER. — A PROBLEM IN STATICS. 477 



an expression from which the lack of symmetry could be removed by 

 well known and obvious methods.* 



The mechanical significance we have thus attached to the vanishing 

 of certain covariants makes it sometimes possible by direct mechanical 

 intuition to obtain information concerning the position of the roots of 

 these covariants. The following theorem serves as a starting point : 



A point (in the plane or on the sphere) cannot he a position of true 

 equilibrium if it is possible to draw a circle through it upon which not 

 all the particles lie, and which completely separates the attractive particles 

 which do not lie on it from the repulsive particles which do not lie on it. 



This is at once obvious when we consider the spherical field of force, 

 for there will clearly be in this case a component at the point in question 

 perpendicular to the circle. 



This principle enables us in many cases, when we know the positions 

 of the roots of the ground-forms, to find regions in the plane in which no 

 root of the covariant ^ lies. 



Let us look at the case of two ground-forms f and f ; and letv us 

 suppose that there are two regions Si and S^ on the sphere, bounded by 

 circles Cx and Cj respectively, which do not overlap, and such that all 

 the roots of /i lie within or on the boundary of S-^ and all the roots oif 

 lie within or on the boundary of S2, ; provided, however, that if C^ and 

 Cz have a point or points in common, not all the roots of both f and f 

 shall lie at such common points. In this case the principle just stated 

 shows that the Jacobian of yi and/g can have no root lying in the region 

 between the circles Cx and O^. In other words all the roots of this 

 Jacobian must lie within or on the boundary of Sy and S^. 



We can in general go further by constructing in Sx and S2, curvilinear 

 polygons Tx and T^ bounded by arcs of circles such that each side of 

 T^ passes through at least two roots oi f and, when extended, through at 

 least one root of the other ground-form, and is so situated that the circle 

 of which it forms a part separates from one another all the roots of ^i 



* We thus get as the field of force : 



r„2 |=* >=* TO .7. 



-miM' 



and the covariant becomes : 



where we must remember that J — 0, / = — J 



f>-kZ2^-^. f.f. -^ij' 



