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ellipse or hyperbola according as mM < 0 or > 0. It is concentric 
with the ellipsoid and its axes coincide with those of # and y. 
Let its axes be a and bh’. The equation expressed in these terms is 
a®y? + b*2* = a*h? 
To determine the axes let this equation be differentiated 
dy x _ be 
dia, a*y 
Eliminating y and = by these equations and (a) the result will be 
ea®— bh? = f 
Subject to this condition the axes may be of any magnitude. Since 
Band a’ are the coordinates of the vertices of the angles of the rec- 
tangle circumscribed round the axes of each of the ellipses or hyper- 
bolas, which are the projections of the lines of curvature upon the 
plane zy, it follows that the locus of these points is a curve repre- 
sented by this equation, b and a’ being coordinates measured upon 
the axes of z andy. Hence it appears that the two equations 
a?y? + b® a2 = a* b” 
ea*® —h? =f 
determine all the lines of curvature. The first determines the pro- 
jections of the points of any one line of curvature corresponding to 
any system of values of a® and b® and the second determines the 
various systems of values of these last quantities. 
If a’ = a’, e = 1 and f=0 and the equation (a) becomes 
