1888.] in a Liquid when the impulse reduces to a couple. 275 
and taking L=np, y=ng, z=nr, 
we have 
ni {0° /A’ + f' |B’ + r?/C?} — {o?|A? + y |B’ + 27/0", 
so that n’ =e/E, 
and p=cAr|no, g=eBp/no, r=eC*v[na......... (11). 
6. To determine X, », v as functions of the time we have 
only to express that this direction is fixed in space. This gives 
A+ pr —vq =0 
b+up—rAr=0 | 
b+ dg — pp =0 
If M be a factor of homogeneity, and we write 
«A? —o’ =MaP 
eB -—-a= MsQ | 
eC? -o’ = Mak 
the first of equations (12) becomes 
Ns TE Eee 
dt a ie (B al, ) PY, 
or ee (Q — ft) py, 
ndu no 
so that these equations are 
dr 
= (Q— RB) wv 
djs 15 
qq he Pe AAD BU DALOD DOODIOROL OOOO ( 5): 
dv 
du ~ Q) | 
7. The integration of this system of differential equations 
has been considered by Halphen in his Memoir “Sur le mouve- 
ment d’un solide dans un liquide” *; the following is an outline of 
his process (artt. 7—9): 
* Liouville’s ‘‘Journal,” 1888; see also the author’s ‘‘Traité des Fonctions 
Elliptiques,”’ Part II. 
