2fi Dcnnonstrations of cert am Points in 



B= -fm((^r +yt«^J/r), G = - ^mht^r, 



being quantities depending only on the nature of the medium, 

 and not at all on the quantities p, a, jS, y, though in the same 

 medium they may vary with the position of m. Hence if R 

 be the whole force on w, developed by its displacement p, and 

 X, fj., V the Its its direction makes with the axes, we have 

 R =z p k, where 



K« = ( A^ + G^ + H2) cos"- « + (B2 + F^ + H*) cos« ^, 

 4- (C2+ F^ +■ G^) cos^y + 2 H (A + B) cos a cos A 

 H- 2 G ( A + C) cos a cos y + 2 F ( B + C) cos /3 cos y, 



and K cos X = A cos a + H cos /3 + G cos y, 

 K cos jx = H cos a 4- B cos /3 + F cos y^ 

 K cos V = G cos a + F cos |3 + C cos y. 



Generally, therefore, the whole force on m a displacement, 

 but does not act in the direction of the displacement. More- 

 over, from the linear form of the equations (A) it appears that 

 we must obtain the same values for the component initial 

 forces on tw, and therefore the same whole initial force, both 

 in magnitude and direction, whether we suppose the whole 

 displacement p communicated at once to m, or the component 

 displacements w, v, w separately communicated, and take the 

 sums of the separate forces which would be thus produced. 



Suppose now there be a direction of displacement, such that 

 the whole force developed on m acts in the direction of the 

 displacement, i. e. that A = a, [jI' = ^9 " = y? then our three 

 preceding equations assume the well-known form of the equa- 

 tions which occur in the investigation of the principal axes of 

 a body, and the cubic resulting from the elimination of cos X, 

 cos [X,, cos V, will be 



(K-A) (K-B)(K-C) -F2(K- A) - G2(K~B) 

 - H^(K - C) = 2FGH. 



The results, therefore, will be similar in the two cases, i,e, 

 there will always be three, and generally only three, such di- 

 rections. 



Suppose we have determined one of these, take it for 

 axis of z; then, since a displacement in direction of this axis 

 produces a force also in that direction, it follows from (A) that 

 F = G = 0, and that for this axis K = C. Suppose also that 

 the plane of x z passes through another of the axes, then to 

 determine its position we have the equations 



