SECTION VI. 
ON THE DISCUSSION OF THE GENERAL EQUATION OF THE 
SECOND ORDER. 
133. To find the position of the center of any surface. 
The center of a surface is a point O (fig. 22) such that 
any chord of the surface PP', drawn through it, is bisected in 
it. (It must be observed, however, that if PP' cut the surface 
in more points than two, it would be sufficient that these 
points combined in a certain order should be, two and two, 
equally distant from O). 
If the surface be referred to any three axes originating 
in O, and PM, P'M' be the ordinates parallel to 0% of the 
extremities of a chord, we see from the equal triangles POM, 
P'OM', that these ordinates are equal and of contrary signs; 
the same thing would be true for the other co-ordinates of P 
and P', as well as for every other chord passing through O. 
If therefore f{cc, y, %) = 0 be the equation to the surface, 
and if it be satisfied by co = a, y = b, % = c, it must also be 
satisfied by a? = - «, y = - b, z = — c; that is, it must be such 
as not to alter when the signs of the three variables are 
changed ; and, conversely, if it have this property, the origin 
is the center of the surface. When/(a?, y, #) = 0 is algebraic, 
it cannot have the above property unless the dimension of 
every term be even in an equation of an even degree, and the 
dimension of every term be odd in an equation of an odd 
degree; for in the former case the equation is not at all 
altered by replacing od, y, % b J — cd, — y, — % ; and in the latter 
case (in which the equation cannot have a constant term) the 
sign of every term will be altered, and therefore the whole 
equation unaltered.