38 
ON A CLASS OF DIFFERENTIAL EQUATIONS, AND ON THE 
[7 
Multiplying by 
— {(vdu — udv) (wdv — vdw) (udw — wdu))- 1 , 
the first of these becomes 
— a?du + —b-dv — &dw , 
(vdu — udv)(udw — wdu) (ivdv — vdw) (vdu — udv) (udw — ivdu) (wdv — vdw) *" ’ 
but writing (17) and its derived equations under the form 
u + (v + w) = 1 (19), 
du + (dv + dw) = 0, 
we deduce — du (v +w) + u (dv + dw) = — du (20), 
i.e. — du = — (vdu — udv) + (udw — wdu) (21), 
and similarly 
— dv = — (wdv - vdw) + (vdu — udv), 
— dw = — (udw — wdu) -f wdv — vdw). 
Substituting, 
ivdv — vdw udw — wdu vdu — udv ^ (^)> 
the integral of which may be written in the form 
/;•.-«= .. 4 ( 23 ), 
WV X — VW X uw x — wu x vu x — v x u 
where, on account of (17), 
u x + v x + Wi = 1 (21); 
and also in the form fu + gv + hw = 0 (25), 
where /, g, h are connected by 
+ (26); 
/99 
this last equation is satisfied identically by 
b 2 - c 2 _ c 2 - a 2 j _ a 2 - b 2 
g CP-A 2 ’ A*-B* { 
Restoring x, y, z, x x , y x , z x for u, v, w, u x , v x , w x , the equations to a line of curvature 
passing through a given point x x , y x , z x , on the ellipsoid, are the equation (14) and 
ffl-c 2 ) (c 2 - a 2 ) (a 2 -b 2 ) _ Q / 28 \ 
a 2 (y x z 2 — y 2 z x ) b~ (z x 2 x 2 — z 2 x x 2 ) c 2 (x x y 2 — x 2 y x 2 ) 
or again, under a known form, they are the equation (14) and 
(b 2 —c 2 ) x 2 t c 2 - a 2 y 2 , a 2 -b 2 * 2 _ A /e)m 
B 2 -C 2 a 2 + C 2 -A 2 b 2 + A 2 -& c 2 { h