6 , REPORT — 1842. 



On substituting these values in the expression Xdx + Ydy-\-Zdz, and placing 

 a X -f- ai Y -f- O; Z = A &c, we find 



X dx + Y dy + Z dz = A dL + ft d M + v d N ; 

 but since the values of d M and d N are quite arbitrary and independent of d L, the 

 expression ft d M + » rf N may always render the second member A d L + ft d M 

 + i> <Z N positive, whilst that for the equilibrium Xdx + Ydy + Zdz must not be 

 positive ; it will consequently be requisite at first, for the equilibrium, that ftdM 

 + v d N = ; and since d M and d N are arbitrary, it is necessary that ft = 0, k = 0, 

 andXdx + Ydy + Zdz = *dL. 



When a displacement makes d L = 0, that is, when the obstacles are expressed 

 by equations, we have Xdx + Ydy-\-Zdz = Ofor the possible displacements, the 

 sign of the quantity A then remains arbitrary ; but for the displacements which give 

 *dL>-0, Xdx-\- Y dy + Zdz must not be positive; it is consequently necessary 

 that A be negative, or carrying the whole to the other side Xdx + Ydy + Zdz 

 + A d L = 0, or A is positive ; that is, it has the same sign as d L for the possible dis- 

 placements. On substituting A.dx-\-Bdy-\-Cdz for d L, and observing that A is 

 independent of the displacements, and always retains the same value, whatever be the 

 displacements under consideration, we shall obtain, since dx, dy, dz are entirely 

 arbitrary, 



X+AA=0 1 



Y + AB=0 [ (3.)* 



Z + AC=0 J 



X Y Z R 



Hence results -_-=.- = --= + . = — A ; but R being positive, 



ABC -VA 2 + B2+C2 



and — A a negative quantity, it is necessary to keep the sign — ; consequently 



X - A Y__B,Z__C 



•=r= » ^5- — , — ' r- — — . — : > i. e. the force R must be op- 



R ^a= + B 2 +C 2R V R V r 



posed to the normal N, and must press the point against the plane ; the magnitude 

 of this force remains arbitrary. 



When there are two conditions, 



Adx + Bdy + Cdz = dL, 

 A 1 dx + B 1 dy + C 1 dz = dM, 

 and the above manner is adopted, it will be shown at first that v = 0, and X d x + 

 Ydy + Zdz = AdL + J c*dM. Since, for one of the possible displacements, d M 

 = 0, and Xdx + Ydy + Zdz = *dL, it is requisite that A be negative, or, carried 

 to the other side, positive; in the same manner it is shown that p. is positive, 

 i. e. that in the expression 



Xdx + Ydy + Zdz + *dL + ftdM = Q, 

 A and ft, have the same signs as d L and d M for the possible displacements. From 

 thi3 equation is obtained X + *A + ^A 1 = 0, 



Y + AB + ^BjrrO, 

 Z + A C + A, Ci = 0, 

 whence it results that the point may be considered perfectly free, if to the given force 

 be added two others, the projections of which on the axes are A A, A B, A C, (*. A, 

 &c. ; the value of these forces remains perfectly arbitrary, but their direction will be 

 determined. 



When a point is subjected to three conditions, and the same course is adopted as 

 above, it will again be found that Xrf* + Yrfy + Zrfz+?irfL + /«(ZL-)-^(/M + 

 v d N = 0, where A, ft, > are indeterminate but positive quantities ; this equation does 

 not afford any equation for the equilibrium, since the three equations which may be 

 derived from it arc identical, when for A, ft, v their values are substituted, but the con- 

 ditions of equilibrium will consist of the three inequalities A > 0, ft > 0, u > 0. 



Suppose, for instance, a solid sphere to be situated within the angle of the posi- 

 tive coordinates, and the conditions that the centre is not affected by any dis- 

 placement be required, then in this case we shall have for the possible displace- 

 ments three conditions, dx>0, dy>0, dz>0; consequently Xdx + Y dy + 



* The point might therefore be considered perfectly free, if to the given forces another X 

 VA 2 + B 2 + C 2 be added, the projections of which on the axes are X. A, X. B, \ C. 



