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ON THE MOTION OF ROTATION OF A SOLID BODY. 
[6 
Suppose that Ax, Ay, Az, are referred to axes Ax, Ay, Az, by the quantities 
l, m, n, k, analogous to A, y, v, k, these latter axes being referred to Ax,, Ay,, Az,, 
by the quantities l,, vi,, n,, k r 
Let a, b, c; a', b', c'; a", b", c"; a,, b,, c, 
quantities analogous to a, /3, 7; a', /3', 7'; a", 
trigonometry, the formulae 
af, bf, cf\ a", bf, cf, denote the 
/3", 7". Then we have, by spherical 
a = a a, 4- b af + c a ”, 
a' = a' a,+ b'af + c af, 
a" = a"a, + b"af + c"af, 
¡3 = a b, -\-b bf + c bf, 
/3' ^a'^ + b'bf+c'bf, 
¡3" = a"b, + b"bf + c"bf, 
7 = a c, +b cf + c cf ; 
7' = a' c / + b' cf 4- c'c/' ; 
y^a^ + ò'V + cV- 
Then expressing a, b, c; a', b', c ; a", b", c"; a,, b,, c,; af, bf, cf; af, bf', cf, in 
terms of Z, w, n ; Z /( m,, n /( after some reductions we arrive at 
— 4 (1 — ZZ, — mvi, — nnff, = 4H 2 suppose, 
(/3" — 7 ) = 4 (Z + Z / 4- n,vn — nmf) II, 
&&/ (7 — a') = 4 (m 4- m, 4- Z,m — Ini,) IT, 
M', (a' — ¡3") = 4 (n + n, 4- m,n — mnf) II ; 
and hence 
II =1 — 11, — vwi, — nn t , IIX = Z + Z / + n,m — nm,, 
Ilya = vi + m, 4-1,vi — lm,, II c = n 4- n, 4- vn,n — mn,, 
which are formulae of considerable elegance for exhibiting the combined effect of 
successive displacements of the axes. The following analogous ones are readily obtained : 
P = 1 4" AZ 4~ yaw 4~ jhi , 
Pm, = y — vi — An 4- vl, 
and again, 
P, = 1 4- AZ / 4- ym, 4- vn,, 
Pfm = y — m, + An, — vl,, 
These formulae will be found useful in the 
a solid body. 
PI, = A —l — wi 4- yan , 
Pn, = v — n — yaZ 4- A vi: 
If l = A — l, 4- vvi, — fin,, 
Pf n = v — n, 4- ya l, — \vi,. 
integration of the equations of rotation of 
Next it is required to express the quantities p, q, r, in terms of A, y, v, where 
as usual 
,d/3" 
d/3 ,d/3' 
dt +y dt ’ 
'-«3 
Differentiating the values of /3re, (3'k, (3"/c, multiplying by 7, 7', 7", and adding, 
Kp = 2A' (7ya — 4- j") 4- 2ya' (7A — y'y + <y"v) 4- 2i/ ( — 7 — y'v 4- y" y),