TRANSACTIONS OP THE SECTIONS. 5 



In the first place, I demonstrate the parallelogram of forces*, and then pass on to the 

 composition and decomposition of forces, and to the equilibrium of a free point. 



To find the conditions of a point, or of a system of points, which is not free, it is 

 necessary, and at the same time sufficient, that the forces cannot effect any displace- 

 ment which the obstacles allow of, or that they can only produce impossible displace- 

 ments. It is this condition which we will endeavour to express analytically. 



I show that a force acting on a point subjected to a certain obstacle can never give 

 rise to a displacement, forming either a right or obtuse angle with the direction of 

 the force, but that it can cause a displacement which forms an acute angle with the 

 direction of this force. As a point can sometimes. only be displaced in a straight 

 line by an infinitely small quantity, I shall only consider infinitely small displace- 

 ments, but perfectly arbitrary. I will designate one of these displacements by ds, 

 and its projections on the rectangular axes x, y, z, by dx, dy, dz, then the condition* 

 that a force R cannot cause a displacement d s will be expressed by cos (R d s) _ ; 



n , t •,• ..•*• j ,t> j \ Xdx , Ydy . Zdz 



or since R and ds are positive quantities, and cos (K, as) = + ■,.,/ + „ . ■• 



r l Rds Rds Rds 



the analytical expression of the condition that a force does not tend to produce the 



displacement d s will be 



Xdx + Ydy + Zdz<0 (1.) 



where X, Y, Z are the projections of R on the three axes x, y, z. Let us now see how 

 we can express the condition that the displacement of a point subjected to obstacles 

 is possible, independently of the means which may occasion this displacement. 

 For this purpose I observe, that whatever may be the condition which hinders some 

 displacements, it may always be represented by one or by several fixed planes. I sup- 

 pose, at first, that there is but one sole plane of dimensions infinitely small which 

 prevents the displacement of the point. I imagine a normal prolonged from the 

 r point in space where the displacement is possible, and I de- 



j-N . signate the angles of this normal with the axes x, y, z, by 



—, 1 >/y *> P> V ; it is then evident that only such displacements are 



/ | / / possible as form a right or acute angle with the normal, i. e. 



/ O^ / that cos (N, d s), or cos a. — + cos /3 ^1 + cos y d — > ex- 



/ . / ds ds ds = 



presses the condition that one displacement is possible ; cos 

 «, cos H, cos y may be functions of the coordinates of the point, which renders 

 this expression an exact differential or not ; in the first case the point will be found 

 on a curved surface, and a., J3, y will be the angles of the normal of the surface with 

 x,y, z ; in the second case it will not be so ; consequently, making generally cos « 

 dx-\- cos 0dy + cos ydz = Adx + Bdy + C (Z^, where this expression is or is not 

 an exact differential, the condition will be expressed that a displacement is possible by 



Adx + Bdy + Cdz^O (2.) 



and the whole of these conditions (1.) and (2.) will be the analytical expression that 

 a force cannot effect a possible displacement, the projections of which are dx, dy, dz. 



Let us see what conclusions we should draw from these two inequalities for the 

 equilibrium of a point. 



For the sake of shortness, I make Adx+Bdy + Cdz=dL, where d L ex- 

 presses an arbitrary quantity infinitely small, which is the complete differential of a 

 function of the three variables x, y, ss, or simply an infinitely small one, which does 

 not possess this property ; I add to this equation two others, perfectly arbitrary, 



A 1 dx + B l dy + ddz = dM, 



A 2 dx + B 2 dy + C 3 dz = dN, 

 where A,, B u C u A 2 . . . are the arbitrary functions. I find the expressions of these 

 three equations in the following manner : — 



dx = adh+bdM + cd'N, 



dy = a l dh + b l dM+c 1 dl<i, 



dz = a 2 dL + b. 2 dM + c 2 dN. 



* It seems to have escaped attention, that in order to demonstrate that the resultant of two 

 forces falls within the angle of these forces, it is necessary to admit that the dependence be- 

 tween the resultant and the components must be given by a continuous function. 



