Motion of Solid and Fluid Bodies. 185 
(B — Nn)’. tan‘a + [2(a +N) (B-+N) — 44B 4+ 12N*]tan’a + (A — nN)? = 0; 
or, assuming 
w= (A—N) (B —N) — 4n’, 
(Bs — Nn). tanta — 2(w — 4n*) . tan’a + (A —n)* = 0. 
Now, in order that this equation should give real values for tana, it must give 
real and positive values for tan?a; but it can be shown that the roots of this qua- 
dratic equation, if real, are negative, and, consequently, that no real value exists 
for tana. 
The equation solved for tan’a may be put under the form 
tan’a.(B — N)? = (w — 4n®) + V(w — 4y*)*— (w $ 4°) 
or 
tan’a.(B — N)* = (w — 4y*) + V —16Nio. 
These equations show that the condition for real values of tan*a is, that # must 
be negative or zero; and that in either case tan’a must be negative. Hence we 
have proved, in general, that the two branches of the curve cannot have a real 
point of intersection. The same thing is true of the curves of the fourth degree 
in the other principal planes. 
The geometrical meaning of the condition w = 0 is, that the curve of the 
fourth degree, in each plane, should be the product of an ellipse and circle whose 
equations are 
av’ + By —1=0, na’ + ny? —1=0, in plane (2, 7). 
By + cz? —1=0, Ly? + 12° —1=0, in plane (y, 2). 
c2” + az’— 1=0, mz’ + mz’ —1=0, in plane (2, z). 
The curve of the fourth degree consists of two branches, each of which cuts 
at right angles, the axes of coordinates, in four points, and the semiaxes of one 
branch are , and the semiaxes of the other branch are equal, and each 
Le 
Va VB 
1 j . 
a and similarly for the other coordinate planes. 
N 
The equation of the ellipse in the plane (a, y) being 
Ly’ + mz” —1= 0, 
2c2 
