772 PROFESSOK SYLVESTER ON THE MOTION OF A EIGID BODY 



Let A, B, C be arranged in order of magnitude, and suppose 



1—1 — 1 i-=i— 1 jL— 1 — 1 Ml— M_l 

 A;~A fi' B, B ,1*' C, C lu' L* L* ^' 



and let jm. be so determined as to make one of the quantities A„ B,, C, equal to the sum 

 of the other two. Then 



(1) Any imaginary value of jx must be neglected. 



(2) Any value of (^ which makes A, , B, , C, of different algebraical signs must be 

 neglected. 



(3) If /a-, being real, makes Aj, B,, C, all positive, these quantities will correspond to 

 the moments of a real disk whose representative ellipsoid is confocal to that of the body 

 whose moments of inertia A, B, C are given. 



(4) If /A, being real, makes A,, B,, C, all negative, by taking —A,, — B,, — C,, i.e. 



the reciprocals of — -, -—^, — 7^ as the new moments of inertia, we evidently shall 

 jLtAfiBftL 



have obtained a reduction to a disk of the supplemental or contrafocal kind. 



In case (3) M , and in case (4) M is to be substituted for M, so that the 



fj. ft, 



necessary condition of ^f being intermediate between the greatest and least of the quan- 



titles A, B, C will continue to be fulfilled in the disk by vj- remaining intermediate 



between the greatest and least of the quantities Aj, B,, C,. 

 Suppose A, + B,=C,, then 



A B C_ 



A— /*~'~B— ft C—p.^ 

 or 



(A+B-CV+ABi!i+ABC=0. 



The determinant (^'. e. negative discriminant) of this equation is 

 AB(AB-CA-CB+C^) or AB(A-C)(B-C) ; 



so that if C is the least or greatest moment of inertia, [/j will have real values, but will 

 be unreal if C is the mean moment of inertia. 



Suppose now that A, + B,=Ci for one value of p, to find the values A', B', C corre- 

 sponding to the conjugate disk, we obtain from the above equation in ju,, by substituting 

 A„B,,C, for A, B, C, 



2A,B,^-A,B,C, = 0, or ^=-^, 

 and accordingly 



1 1 i. 2 J_ 2 B.-Ai Ai-B, .J^._J_ 



F • B'" A,~~A, + Bi • B,~A, + B, •• 2A, * 2B, •"a,- B," 



Hence if A„ B, have the same signs, A', B' have opposite signs, and vice versa, if A„ 

 B, have opposite signs A', B', and therefore A', B', C have all the same signs for 

 C'=A'+B'. 



