BOCHKR. — A PROBLEM IN STATICS. 473 



equal angles with corresponding conaponents, are the stereographic pro- 

 jections of each other; and the intensities of the resultants are in the same 

 ratio as the intensities of tlie components. Thus theorems I and II are 

 proved. 



It has long been a familiar fact that one and the same plane vector 

 field gives the force due to n particles pi,p2, • • • />„ in a plane, of 

 masses mi,m2i . . . >«„; and the steady flow in a plane conducting 

 lamina due to electrical sources at /?i , . . . p„ of intensities mj, . . . m„. 

 On the other hand it is well known that the vector field on a spherical sur- 

 face which i-epresents the steady flow of electricity on the surface due to n 

 sources Pi, ... P„ with intensities rui, . . . m,^ (rui + . . . + m„ = 0) 

 may be obtained by projecting stereographically onto a diametral plane 

 and considering the flow due to sources at pi, . . . p„ (the projections 

 of Pj , . . . P„) of intensities /Wi , . . . m„. If the length of each vector 

 in this last mentioned plane field be multiplied by ^ (1 + r^), — the 

 radius of the sphere being taken as unit of length, — and the field thus 

 modified be projected back onto the sphere, we obtain the desired flow 

 on the sphere. Comparing these facts with theorems I and II we see 

 that one and the same vector field on the spherical surface represents the 

 jiow in the electrical problem just mentioned and the force in the me- 

 chanical problem considered before. 



§ 2. Relation to Algebraic Invariants. 



Let us now, in the discussion of the field of force, introduce homo- 

 geneous variables by the formulae : 



a: = — , e, = - 



The plane field (1) then becomes; 



^\^'2^eJ'^,-eJx,\ 



In this formula we may, without real loss of generality, assume that 

 "^1 + "^2 + • • • + w„ = 0. For if this were not the case we could 

 regard our formula as the special case of the corresponding formula 

 where n is replaced by n + I, e„+/' = 0, and 7w„+i is so chosen that 

 »ii + . . . + m„4.i = 0. This expression for the field of force, when 

 reduced to a common denominator, becomes, when we take account of 

 the relation /«i -\- ni2 + . . . -f m„ = which we now assume to hold, 



