Centre of Gravity. 45 



75. By the centre of gravity of a body, or system of bodies (that 

 is, any assemblage whatever of bodies), we mean that point 

 through which passes the resultant of all the particular forces ex- 

 erted by the gravity of the several parts of the body, or system 

 of bodies, in whatever position the body or system is placed. 



If, for example, in the actual position of the triangle ABC, the Fi S- 26 - 

 resultant force of all the actions of gravity, upon the several parts of 

 this triangle, passed through a certain point G of its surface, and in 

 another position a b c, it should pass through the same point G, this 

 point is what we call the centre of gravity. We shall see here- 

 after that the resultant in question passes through the same point 

 in all possible positions of the given body. 



76. The centre of gravity is easily determined by means of 

 what we have said upon the use of moments in finding the resul- 

 tant of several parallel forces. . 61. 



Let there be any number of bodies m, n, o, whose masses we Fi S- 27 

 will consider for the present as concentrated in the points A, B, 

 C, situated in the same plane. Let u be the velocity which 

 gravity tends to give to each in an instant ; 



u X m, u X n, u X o, or u m, u w, u o, 



will be the quantities of motion, or forces, with which the bodies 

 tend to move according to the parallel directions A" A, B"B, C"C. 

 In order, therefore, to find the position of the resultant, we take the 

 sum of the moments with regard to any point F, assumed at pleas- 

 ure, in a line perpendicular to the directions of the forces, and 

 divide this sum by the sum of the forces ; we have therefore for 

 the value of the distance FG", at which this resultant passes, 66. 

 FG , f _ umx FA" + unx FB" + uo x FC" 

 urn -f un -f- u o ' 



or, by suppressing the common factor w, 



FG = m X FA" + n x FB' + o x FC" 

 m + n + o 



In like manner, if we draw the lines AA\ BB, CO, parallel to 

 FG\ and terminating in the vertical FC ; and suppose moreover, 

 that the point G, taken in the direction of the resultant, is the 

 centre of gravity sought, by drawing G'G also parallel to FG\ 

 we shall have 



