73 
Such planes may be found relatively to every system of axes of 
coordinates. It therefore follows that an algebraic surface has an 
infinite number of diametral planes, 
- A . 
The sum of every two values of z is and the number of 
. nn—l 
these products is —~—>—+ Hence a surface represented by the 
equation 
oo Aas + Bao =O 
ce n.n—l. Cc 
will have the same diametral plane with the given surface, and also 
the rectangle under the coincident values of z, for this surface will 
- 5 | th 
be equal to the —a part of the sum of rectangles under every 
pair of corresponding values of z for the proposed surface. It fol- 
lows therefore that the sum of the posilive rectangles under the 
intercepts of the values of 2 between this surface and the given 
surface is equal to the sum of the negative rectangles. Since a; is 
a formula of the second degree between x and y, it follows that 
the surface possessing this property is one of the second degree. 
It follows in general that a series of surfaces may be thus de- 
termined of degrees inferior to that of the given surface, and hay- 
ing properties with respect to it similar to those of curvilinear 
diameters with respect to plane curves. Such surfaces may be 
called diametral surfaces. (Geometry, 607.) 
An absolute diametral plane is one which bisects its ordinates. 
It is evident all diametral planes of surfaces of the second degree 
must be absolute. 
In order that a surface should admit an absolute diametral 
