144 The Rev. Samuel Haughton's Account of 



will have for components, round the axes of .r and y I'espectively, the quantities 

 w-\yzdm, w-\.vzdm taken with their 2:>roper sign ; i. e. the components are ± u?L' , 

 ± o?M'; L'M, being coefficients in the equation of the ellipsoid 



A'3? + B'y"- + Cz^ - 2L'yz - 'iM'xz - 2N'.Ty = /x. 



The tangent plane to this ellipsoid, applied at the point (a; y, z) will be 



{A'x- Mz - N'y ) x' +{B'y- N'x - L'z) y' + ( C'z - L'y - M'x) z' = ^. 



At the point R' the tangent plane will be perpendicular to the plane XOZ, and 

 will be found by making x = 0, y = 0, and destroying the coefficient of y' in 

 the preceding equation. These conditions give usi'= 0, which proves that 

 the statical couple produced by centrifugal force lies altogether in the plane 

 XOZ. The equation of the tangent plane is the same as the equation of the 

 line R'P', and is 



C'z'-M'x'=-^ . 



Hence we obtain 



M' 

 tan = - ^ . 



The value of the centrifugal couple is w'M', which is fomid from the pre- 

 ceding equation by replacing C" and tan by their values fxP\ and -p ; Q being 



the line RP. 



We thus obtain finally the centrifugal couple lying in the plane XOZ, and 



expressed by the equation 



ur\xzdm = — fiorPQ. (8) 



It thus appears that the centrifugal couple lies in the plane of radius vector 

 and perpendicular, is proportional to the area of the triangle ROP, and has a 

 direction opposite to the direction of rotation. 



V. — To find the Relation between tJte Plane of principal 3foments and the Axis of 



Rotation at any Instant. 



The motion of the body at any instant consists of a rotation of a certain 

 magnitude round a certain axis; this rotation might be produced by an im- 



