IN GEOMETRY, MECHANICS, AND ASTRONOMY. 425 



the equations (38) become 



da dfi dy 



- = 2)..c, -^ = D..(i, ~'=Z)a,.v...(40), 



and therefore (37) becomes 



du ^ 



~ = Dw.u (4.1). 



Now, referring to Art. 5, Dw.u is a line drawn perpendicular to u and w, whose numerical 



magnitude is wr sin Q, where n and r are the numerical magnitudes of ui and ?< respectively, and d 



du 



the angle made by at and u. Hence, the equation (41) indicates that the velocity is due to 



dt 



the rotation of the rigid body about the axis w with the angular velocity n. In other words, the 



symbol a, represents completely the motion of the rigid body ; for w represents, in direction, the 



axis of instantaneous rotation, and, in numerical magnitude, the angular velocity of the body about 



that axis. 



Returning to equation (36), we find by (37), observing the properties of principal axes, 



2Z)m .-— ^m = 'S.hnD{xa + y/3 + zy) . f.r -— + « — + ^r — -^ 

 dt \ dt dt dt. 



dt dt dt 



Now, by Art. 8, and by equations (38), we have 



Z)a . -— = (D37 + a)i./3, Z>/3. -— = (D.a + 0)37, Dy .—- = w^Q -^ Wia. 

 dt at dt 



Hence we find, 



'S.Du .-^Sm = tola's, (y^ + x^) Sm + a,,/32 («= + *') Sm + (0372 {x'' + y') Sm 

 — Au>ia + Bw^jj + Cw^y. 

 Hence the equation (36), cleared of the sign 2, becomes 



— \Aw,a + /3e02/3 + Cw^y} = 2Z)m . Uim* ... (42). 



28. We shall now apply this equation to the problem of Precession and Nutation. 



To effect the integration 2 in the second member of (42) when the force U arises from the 

 attraction of a very distant body, which may be supposed to be collected into its centre of gravity. 



Let u be the symbol of the centre of gravity of the distant body, and let m denote its absolute 

 attractive force; then since u — u denotes the line drawn from Sm to to', the attraction of m' on 

 im is 



to' {u- u) 



U = 



{A(m'-m).(m'- m)}S' 



• If we perform the operation -— in the first member of this, by 

 dt 

 means of equations (38), it becomes 



Whence it follows that the first three of Eulcr's six equations 

 follow immediately from (42). 



The last three of Euler's equations follow immediately from 

 the equations (IJH), in the same manner as I have shown in my 

 Mathematical Tracts. 



31 2 



