ACTED ON BY NO EXTEENAL FOECES. 773 



Consequently one and one only of each of the two solutions for disks drawn perpen- 

 dicular respectively to the extreme principal axes, makes the three moments of inertia 

 all of the same sign, and consequently each such solution leads either to a direct or 

 supplemental reduction to the disk form. 



Now, suppose that A, B, C being all of the same signs, A has become equal to B+C, 

 so that the equation in (h becomes 



2B(U<''+2AB|«,+ABC=0, 

 or 



^'-A^+^=0. 

 Let /*„ jtt' be the two values of /it from this equation, so that 



JL_1 i 1_L L 1_1 1 



A.-A-^i' B,-B-^,' C -C-|<t; 



and 

 Then 



1_^ 1 1 _ 1 1^ 1_1 1 

 A'-A~;?' F-B~f<,'' C'-C~^'' 



A, + B,= C„ A'+B'=:C'. 



1.1-1 /l.M-?- l-9/-i- 11- A 

 A,"I"A'— A~VfA,"'"]i?/ — A~C— '^|b + C~C/ — ~AC' 



1 1-1 /lj_l\la- ^ 



A,'A'-A* \^,-r^i)A-r^ 



1 

 -A*' 



Hence if A, B, C have the positive sign. A; and A' are both negative, and if A, B, C 

 have the negative sign. A, and A' are both positive ; consequently, on the first supposition, 

 the signs of one of the two systems A„ B,, Cj ; A', B', C will be all negative, and on the 

 second supposition all positive. Hence one of the two reductions falls under case (3), 

 i. e. is proper or direct, and the other under case (4), and is improper or supplemental. 

 As nothing in nature exists in vain, it will presently be seen that the choice which is 

 always possible between these two modes of reduction leads to an important simplifica- 

 tion of the cases which arise in the problem of rotation, and that there need never be 

 any room for doubt as to which of the two sorts of reduction should be employed in 

 any specified problem. 



The case of exception to which allusion has been made in anticipation, arises when 

 two of the moments of inertia are equal ; for then, supposing A, Aj, C to be the original 

 moments of inertia, the new moments of inertia will be A„ A„ C, ; and since Cj cannot 

 be zero, we can only suppose Ci=2A, ; and making 



1—1 i 1—1 1 

 A,-A~/t' C,-C~ft' 



the equation in /// becomes 



2A C 



MDCCCLXVI. 6 N 



