36 
[7 
7. 
ON A CLASS OF DIFFERENTIAL EQUATIONS, AND ON THE 
LINES OF CURVATURE OF AN ELLIPSOID. 
[From the Cambridge Mathematical Journal, vol. III. (1843), pp. 264—267.] 
Consider the primitive equation 
fa + gy + hz+ =0 
(1). 
between n variables x, y, z, the constants /, g, h being connected by the equation 
H (/, g, h ) = 0 (2), 
H denoting a homogeneous function. Suppose that f, g, h are determined by the 
conditions 
Then writing 
M + gyx 
+ hz 1 
... =0 
fan—%+gg n —2 
*4“ —2 • • 
... =0 
x = 
y » 
2/i 
z .... 
z \ > • • • 
y n-2 ) 
z n—2 > • •• 
with analogous expressions for y, z ; the equations (3) give f g, h, proportional 
to x, y, z, or eliminating f g, h by the equation (2), 
H{X, Y, Z ) = 0 (5). 
Conversely the equation (5), which contains, in appearance, n(n — 2) arbitrary 
constants, is equivalent to the system (1), (2). And if H be a rational integral 
function of the order r, the first side of the equation (5) is the product of r factors, 
each of them of the form given by the system (1), (2).