53 



which has presented itself as a sort of construction of the law 

 of living force of the system ; and a common properly of these 

 three rectangular directions, which in general belongs exclu- 

 sively to them, and to their respectively opposite directions, 

 may be expressed by the rules of this calculus under the very 



simple form, 



= FS . 'm{iaf ; {&5) 



or under the following, which is equivalent thereto, 



With respect to the geometrical and physical significations 



of the three values of the positive scalar s, the equation (49) 



gives 



si^+S.Lai = 0; (67) 



and consequently by (36), and by the general rules of this 



pQ I pill lie 



s='2..m{S.aUif=^.mx\ (68) 



if a; denote the perpendicular distance of the mass m from the 

 plane drawn through the fixed point of the body, in a direction 

 perpendicular to the axis i. We may therefore write the fol- 

 lowing expressions for the three roots of the cubic (56) : 



Si=S.?na;^; S2 = S.m/; s^^-^.mz^; (69) 



if xyz denote (as usual) three rectangular coordinates, of which 

 the axes here coincide respectively with the directions of 

 ii, t2^ ta; and we see that the three principal moments of in- 

 ertia, or those relative to these three axes, are the three sums, 



of pairs of roots of the cubic equation which has been em- 

 ployed in the present method. At the same time, the condi- 

 tions above assigned for the directions of those three axes take 

 easily the well-known forms, 



= S . mxy = S . myz = S . inzx, (71) 



if (for the sake of comparison with known results) we change 

 the vectors a, a, . . of the masses m, m, . . to the expressions 



