110 
its chords, real or imaginary, will be in number n, and 
their combination on the same indefinite line will form 
\n{n — l) different chords, and as many middle points; 
and therefore the diametral surface, since it may be met 
by an indefinite line in points, will have an 
equation of the degree ^n(n — l). For surfaces of the 
second order where n — 2, the diametral surfaces can only 
be planes. 
When any surface admits of a diametral plane, if we 
make it the plane of scy, and take the axis of % parallel 
to the chords which it bisects, the equation to the surface, 
for every pair of values so = a, y = b, must furnish for z 
values which, taken two and two, are equal and of contrary 
signs; and therefore the equation, supposed algebraic, can 
only involve even powers of z. And, conversely, whenever 
an equation contains only even powers of one of the variables, 
z for instance, the plane of soy is a diametral plane, and 
is said to be conjugate to the chords parallel to the axis 
of z. Also if a diametral plane be perpendicular to the 
chords which it bisects, it is called a 'principal plane, and the 
chords principal chords. Moreover the intersection of any 
two diametral planes is called a diameter of the surface. 
137. To find the equation to a diametral plane of a 
surface of the second order. 
Let 00 = mz, y = nz, be the given equations to a line 
through the origin to which the proposed system of chords 
is parallel, and f{oc, y, z) = 0, the general equation of the 
second degree, the equation to the surface. Let A, k, l, be 
the co-ordinates of the middle point of any chord, and let 
,the surface be referred to axes parallel to the former ones 
passing through it; then the equation will become 
/(»' + h, y’ + k, % + l) =0, 
and the equations to the chord itself will be so = m.z\ 
y = nz'; therefore the values of % belonging to the points