6] ON THE MOTION OF ROTATION OF A SOLID BODY. 29 
then a = cos 2 / + sin 2 /cos0, 
a! = cos / cos g + sin / sin g cos (z — 0), 
cl' = cos/ cos h + sin/ sinh cos (y + 6), 
(3 = cos g cos / + sin g sin / cos (z + 0), 
(3' = cos 2 ¿7 + sin 2 ¿7 cos 0, 
¡3" = cos g cos h + sin g sin h cos (x — 0), 
y = cos h cos / + sin h sin / cos (y — 0), 
7' = cos h cos g + sin h sin g cos (x + 0), 
y" = cos 2 h + sin 2 li cos 6. 
Also sin g sin h cos x = — cos g cos h, 
sin h sin / cos y = — cos h cos f 
sin f sin g cos z = — cos f cos g, 
and sin g sin h sin x = cosf, 
sin h sin/sin y = cos g, 
sin /sin g sin z = cos h. 
Substituting, a — cos 2 / +sin 2 /cos$, 
cl - cos fcosg (1 — cos 6) + cos h sin 6, 
a" = cos /cos li (1 — cos 6) - cos g sin 6, 
/3 = cos g cos/ (1 — cos 0) — cos h sin 0, 
¡3' = cos 2 g + sin 2 g cos 0, 
(3" — cos g cos h (1 — cos 0) + cos /sin 0, 
7 = cos h cos / (1 — cos 0) + cos g sin 0, 
y = cos h cos g (1 — cos 0) — cos/sin 0, 
y" = cos 2 h + sin 2 h cos 0. 
Assume X = tan \0cosf, g — tan \0cosg, v = tan %0cos h, and sec 2 ^ = 1 + + ^ + ^ = 
then kcl = 1 + X 2 — g? — v~, k 0! = 2 (Xg + v), k a" = 2 (vX — g), 
/c/3 = 2 (Xg — v), k/3' = 1 + g 2 — v 2 —X 2 , k[3" = 2 {gv + X), 
«7 = 2 (vX + g), /cy'= 2 (gv-X), icy" — 1 + v 2 - X 2 - g 2 ; 
which are the formulae required, differing only from those in Liouville, by having 
X, g, v, instead of \m, \n y ; and a, a', a"; /3, /3', /3"; 7, y, y", instead of a, b, c ; 
a', b', c'; a", b", c". It is to be remarked, that ¡3', /3", ¡3; y", 7, y, are deduced from 
CL, CL, cl", by writing g, v, X\ v, X, g, for X, g, v. 
Let 1 + a + /3' + 7" = v; then kv = 4, and we have 
Xv = 0"-y', 
X 2 v = 1 + a — /3' — y", 
gv = y-a , 
g 2 v — 1 — a + ¡3' — y 
vv = cl — /3, 
V 2 V = 1 — CL — /3' — y".