OF THE MOTION OF A SYSTEM. 221 



mains undetermined, it follows that, whatever may be its 

 value, the projections of the areas on the planes perpen- 

 dicular to this plane will vanish. Hence we may at any 

 time find the situation of this plane, in the same way as 

 the centre of gravity may at any time be found, notwith- 

 standing- any mutual actions of the system, and for this 

 reason, it is as natural to suppose the coordinates x and 

 y to be situated in this plane, as to make them begin at the 

 centre of gravity." 



[This proposition may be much more simply and intel- 

 ligibly demonstrated by means of Lemma D ; for if c^^^, 

 c',^^, and c^\^^ be the sums of the products of the masses by 

 the projections of the areas described by the revolving 

 radii of the different bodies of the system on the three 

 planes belonging to the system of ordinates x^^^y y^,^, 

 and z^^^y the sum of the squares of c^^,, c',,^, and c",^^ will be 

 equal to the sum of the squares of c, c, and c" : and 

 since this sum is a constant quantity for all systems of 

 planes, it is obvious that when the portion belonging to 

 any one plane is equal to the whole, there can be none 

 left for any plane or planes perpendicular to it. We 

 have, therefore, only to determine the inclination of the 

 section of the parallelepiped to either of its faces, and 

 we shall have the angle 0, the cosine of which will be ex- 



pressed by ^ ,^ ,,^. for the plane of J7 and y (329) : 



and since v^ is the distance of x from the intersection of 

 the planes (324), that is, the angle of the face c adjoining to 



X, its tangent will be — =—r, since the areas of the faces 

 c' and c" are to each other as the y\ 



sides X and y to which they are ad- cX ^J^^" 



jacent, the side z being common to /^^^^Hk 

 both triangles.] ^^^ ""^^ 



