OF THE MOTIONS OF A SOLID BODY. 243 



the particles Dm, tending to turn the body about z", since 

 this force is simply as the distance from x" (261) or as V 

 (y'2_^2,"% but when reduced to the direction ofy, as y" 

 only, acting on the lever x" ; and the same sum will ob- 

 viously be the rotatory pressure with regard to the same 

 axis, if y instead of x' be the axis of rotation. 



Scholium 2. The rotatory inertia, with respect to 

 these three axes, \sS{x"^+f'') Dm= C, S (j:"^-\-z"^) Dmzz 

 By and S{y"^-{-z"^)Tim:=iA respectively. 



Scholium 3. The evanescence of Sx"i/'Dm and Sx" 

 ^'nm determines only the position of the axis x" ; but when 

 that of ^y"z"iim is added, it obviously gives us the two 

 necessary conditions with respect both to y" and to z'', 

 since we have for the former SyVomziO and Sy'z'nm 

 = 0, and for the latter Sz"x"Bm—Q, and Sz'y'DmziO.] 



345. Theorem, li x \ y\ and z\ parallel 

 to the principal axes of rotation of a solid, be 

 the coordinates of the particle d//z, A^ B, and 

 C, the rotatory inertia with respect to these 

 axes, 6 the angle made by the plane of a:' and 

 y" with that of s and ?/, <p the distance of x" 

 from the intersection, and ^ [that of a?, which 

 is also] the complement of the angle made 

 with <r by the projection of z'' on the same 

 plane ; putting d?-— d4' cos 6=:pdt, d>^ sin q sin 

 <P—dQ cos (p:^qdt, and dvj' sin 6 cos <p+d9 sin <p= 

 rdt, we shall have 



Aq sin Q sin <pi-Br sin q cos <P—Cp cos &=—N 

 {Aq cos fi sin <p + Br cos o cos (p-^-Cp sin 9) cos'^ 



R 2 



