Ill 
of the surface where the chord meets it, are given by the 
equation 
f(ni% + h, nz + k, z + l) = 0 (l), 
which is of the form 
Rz* + S% + T = 0 ; 
and since the values of % are equal and of opposite signs, 
S = 0. But S is the coefficient of % in equation (1) ; 
mdf(h, k, l) df(h, k, 1) df(h, k, l) 
dh + n dk~ + di °’ 
or m (ah + h'l + ck + a") + n {bk + a'l + c h + b") 
+ cl + ak + b'h + c" = 0 (2), 
or (am + c' n + b') h + (bn + cm + a ) k + (c + b'm + an) l 
+ am+bn + c =0, 
the relation among the co-ordinates of the middle point of 
any chord, or the equation to a diametral plane. 
Cor. As the coefficients of A, k, l are possible, there will 
be a diametral plane for all values of the constants m and n, 
unless the three coefficients should all become nothing at the 
same time, when the diametral plane will be situated at an 
infinite distance. But the equations 
am + c n + 6' = 0, bn + cm + a = 0, c + b'm + an = 0, 
containing only two unknown quantities, have an equation 
of condition which is the same as D = 0 (Art. 134) ; in this 
case therefore the surface has not a center. 
When the surface has a center, every diametral plane 
passes through it, or through the locus of the centers ; for 
equation (2) is visibly satisfied by the co-ordinates of the 
center furnished by equations (l), (Art. 134). This also 
follows from the definition.