74 
plane, it is necessary that a transformation of coordinates, which 
would make all the terms involving odd powers of the variables 
disappear, should be possible ; and as this is not always the case, sur- 
faces of degrees exceeding the second may not have an absolute 
diametral plane. 
(7) The centre of a surface is a point which bisects all right 
lines passing through it and terminated in the surface. 
Prop. 
(8) To determine when an algebraic surface has a centre. 
If the origin be at the centre, the value of z corresponding to 
4+ 2 and +y ought to be equal and opposite to that corresponding 
to —a and —y ; or which is the same, it is necessary that the equa- 
tion should not be affected by changing the signs of the three co- 
ordinates. In order that this should take place, it is necessary that 
the sum of the exponents of the variables should be either even or 
odd in all the terms of the equation. If a change of origin which 
will produce this effect be possible, the surface has a centre ; 
otherwise not. 
