SURFACES OF THE SECOND DEGREE. 81 



The equations of the center, central line, or central plane, as the 

 case may be, of the surface expressed by (3) are 



aX+'cT+bZ+a = 0, 



cX+bY+aZ+% = (22), 



bX+aY+cZ+c = 0, 



and in the two following sets of quantities, it will be found that the 

 sum of the products made by taking a term from each in the same 

 horizontal line is = F^ ; while if the terms be taken from different horizontal 

 lines, it will be = 0. 



Thus 



aa„-\-cn„-\-bm„—Vz, an,, + cb^^+bl,, = 0, &c. 



Hence, if the three equations in (22) be independent of one another, 

 the co-ordinates of the center are 



j^^ _ a,,a + n,J + m,,c ^ y^ _ n„a^bj)^l„c ^^ _ m^^a^-l^-\-c,fi 



r- 



The equation of the surface, referred to this center, and to axes 

 parallel to the primitive axes, becomes, calling ^ {x, y, %) the first side 

 of equation (3), 



aa? ^-bf +cz^ + 2ay% + 2bzx ■{■^'cxy+^{X, Y, Z) = ...;:.. (24), 



and by multiplying the three equations in (22) by X, Y, and Z respec- 

 tively, and adding, we get 



0(X, F, Z) = aX+bY+7z+/=JV (25). 



When only two of the equations (22) are independent, there is a 

 central line. The conditions of this case are, that the numerators and 

 Vol. V. Paut I. L 



