109 
the result will be of the form y = x (p ± \/q) indicating two 
straight lines through the origin ; unless the radical be 
impossible for all values of A and B, in which case the pro 
posed equation represents a point. 
If the coefficients of the given equation be such that 
D = 0, and the three numerators are not all = 0 at the 
same time, then one at least of the co-ordinates of the 
center will be infinite, which signifies that the surface has 
no center. 
If at the same time that D = 0 the three numerators 
vanish, then the surface admits of an infinite number of 
centers; for in that case the three equations (l) are reduced 
to one, or to two really distinct equations, as is shewn in most 
treatises on Algebra, and may therefore be satisfied by an 
infinite number of values of x, y, z. If they are reduced to 
two, that is, if the values of h and k deduced from the two 
first for instance, satisfy the third whatever l be, then there 
will be an infinite number of centers situated in a straight line 
which is the locus of the two independent equations; the 
surface will therefore be a cylinder on an elliptic or hyperbolic 
base. 
If the three equations (l) are reduced to a single equation, 
that is, if the value of h deduced from the first, for instance, 
satisfies the other two whatever k and l be, there will be an 
infinite number of centers situated in a plane which is the 
locus of the single independent equation, and the proposed 
surface will be a system of two planes parallel and equidistant 
from that plane. In this latter case the proposed equation 
must be capable of being resolved into two rational factors of 
the first degree. 
136. The locus of the middle points of a system of 
parallel chords of any proposed surface is called its dia 
metral surface. This surface will have several sheets, if 
each of the chords has more than two points in common 
with the proposed surface; if, for instance, the proposed 
surface be of the n th order, the points of intersection with