4-34 Prof. Booth on the Rotation pi 



they generate the moments (Y#—X 3/) d m, (Zy — Y z) dm, 

 (Kz — Zx) dm, in the planes of x y, z y, and ^^respectively ; 

 hence the sum of the moments generated by the forces which 

 act on the whole mass arey*(Y.r — X y) d m,f (Z y — Y z) d m, 

 andy (X z — Zx) dm, in the above-named planes. 



III. Let the axis O z be the instantaneous axis of rotation, 

 u> the angular velocity round this axe, then the effective forces 

 are, X = — coy, Y = w x, Z = ; substituting these values 

 in the above formulae, we find the moments wf (x 2 + y 2 ) d m 

 — w fx x d m, and —cofy z dm in the planes of xy, y z, and 

 x z respectively. 



Now as the impressed and effective moments must by the 

 principle of D'Alembert be equivalent, we shall have, denoting 

 by K the impressed moment, and by « (Z y the angles which 

 its axis makes with the axes of coordinates, the following 

 equations : — 



K cos a = — cafx 1 dm, K cos /3 = — cofy z d m 9 ~\ , , 

 Kcosy = wf(x2+y*)dm J } } 



From these equations it follows that when f x z dm = 0, 

 fy zdm = 0, or when z is a principal axis of the body, that 

 y = 0, or that the plane of the impressed moment must 

 be perpendicular to a principal axe, in order that this axe 

 may be an axis of rotation ; and the angular velocity round 



... K 



this axe is = /^-h- — ^-7— 

 J {x' + y'jdm 



IV. The centrifugal forces generated by the motion act in 

 the direction of the radius vector, and are proportional to the 



square of the velocity divided by the radius ; hence X = w 2 r . — 



= u? x, Y = oo 2 3/, Z = 0; translating these forces to the ori- 

 gin, the moments thence resulting are co 2 fx * dm, —uPfy 9 d m 

 and 0, in the planes of x z, y %, and xy. 



Let G be the centrifugal moment, a!, fi',y' the angles which 

 its axis makes with the axes of coordinates, then 



G cos a! == — vPfy zdm, G cos /3' = w 2 / x z dm, 



y' = 0. 



} (2.) 



V. The plane of the centrifugal moment G passes through 

 the instantaneous axis of rotation, and through the axis of the 

 impressed moment K. 



For as </ is a right angle, the plane of G passes through 

 the axis of z, the instantaneous axis of rotation. 



Again, let V be the angle between the planes of G and K, then 

 cos V = cos a cos a'-Fcos /3 cos /3' + cos y cos </. Putting for 



