48 Statics. 



sine of the same angle Z)GL, multiplied by the sum of the mo- 

 ments with respect to the axis TS. 



Now it is manifest, that with regard to any other body n, we 

 should arrive at a similar result, with the exception only of the 

 signs according to which the bodies are on the same or on diffe- 

 rent sides of LH. Consequently, if we take the sum of all the 

 moments with respect to the axis LH, we shall find that it is 

 equal to the cosine of the angle DGL, multiplied by the sum of 

 the moments with respect to DE, minus the sine of the angle 

 JDGL, multiplied by the sum of the moments with respect to TS. 

 But each of these two last sums is by supposition equal to zero ; 

 consequently their products by the cosine and sine respectively 

 of the angle DGL, will be each equal to zero ; therefore, also, the 

 sum of the moments with respect to any axis whatever LH, which 

 passes through the centre of gravity G, is equal to zero. 



81. Hence we infer that the resultant action of all the partic- 

 ular actions of gravity, which are exerted upon the several parts of 

 a system of bodies, passes always through the same point of this 

 system, whatever be its position ; for it is not with respect to 

 the direction of the resultant that the sum of the moments of the 



68. several parallel forces may be equal to zero. 



Moreover, although the inquiry hitherto has been only res- 

 pecting bodies whose centres of gravity are in the same plane, 

 the method is not the less applicable to the case where the parts 

 of the system are in different planes. 



82. If the bodies, still regarded as points, are not in the same 

 Fig. 23. plane, let us imagine a horizontal plane XZ, and from each of 



the gravitating points/*, 9, r, let the vertical lines Ap, B q, Cr, 

 be supposed to be drawn ; and in order to determine the point 

 JE, through which passes the resultant p ?, in the direction of 

 which must be the centre of gravity, we take the moments with 

 respect to two fixed lines FX^ jPZ, assumed in the horizontal 

 plane, perpendicular to each other ; we take, I say, the sum of 

 the moments, as if the bodies were all in this horizontal plane ; 

 and having divided each of the two sums of moments by the 

 sum of the masses or forces jo, ^, r, we shall have the two distan- 

 ces E'E, E"E. It will only remain, therefore, to find at what 

 distance .EG, below the horizontal plane, this centre is situated. 



