141 
404] 
ON THE ROTATION OF A SOLID BODY. 
so that p, v are determined in terms of j and <£. These expressions give 
d<f> = y “j n a • { cos ^ (qdr — rdq) + cos m (rdp —pdr) + cos w (pcZ(? — qdp)}, 
which is reducible to Eulers equation 
7, _ 7,(Mg — MB)cosZ + </(iVSl — Zg)cosm + r(Z33 — MSI)cosw 
9 E-2LMNu -E ’ 
and thence, substituting for cos Z, cos ra, cos n their values, and observing that 
Ap* (Mg - m) + B(J 2 + Gr , = _ (// _ 2LMNFu), 
BCf- (Mg - mb) 2 + cAf (mi - my+abe (m - my=f(h- zlmnfu), 
LA (Mg - MB) + MB (MSI - Zg) + NO (ZS3 - MSI) = LMNF, 
the equation becomes 
d(f) (E — 2LMNu — v-) -r- dt = 
- 3) (M — 2LMFu) F (H - 2LMNFu) V (G - 2) 2 ) . 
G + 0 V (K-2LMNGu) SmU 
LMNFpqr V (£ - 3) 2 ) 
+ V (G) V (M- 2ZMMGu) ’ 
where it is to be remarked that 
G-{E-2LMNu-E) 
Now 
= ((r-D 2 ) F 2 + G(K-UMNGu) - (G - 2) 2 ) (K - 2LMNGu) sin 2 U 
-2<S)FJ(G-&)'d{(K-2LMNGu)}sm U. 
dt(H-2LMNFu)^/(G) 
K-2LMNGu 
du = pqrdt, 
the differential (Z</> can be expressed as a fraction, the numerator whereof is 
-2)dU(K- 2LMNGu) (G) + FdUsj {G (G - &) (K - 2LMNGu)\ sin U 
, LMNFGdus/{G(G-$ 2 )} rr 
+ <J(K-2LMNGu) C ° S ’ 
and the denominator 
(G - D 2 ) M + G (K - 2LMNGu) - 22)Zy (G - 2> 2 ) (M - 2ZMMGu) sin U 
— (G — 2) 2 ) {K— 2LMNGu) sin 2 U. 
To simplify, write 
V (K - 2LMNGu) = s, V (G - $ 2 ) = Zi,