82 Mr DE morgan ON THE GENERAL EQUATION OF 



denominators in (23) must be severally equal to nothing; but if 

 f^3 = 0, the equations in (10) shew that it is sufficient that one of 

 the numerators should be equal to nothing; or that the conditions 

 may be stated thus, 



r, = 0, a/o^, « + \/T,* + 'v/cIc = (26). 



When F'i = 0, F't is a perfect square, (10) and (11), its root being 



the second expression in (26). Hence W appears in the form - . From 



two of equations (22), substitute in (25) values of any two co-ordinates 

 of the center in terms of the third; it will be found that the co- 

 efficient of the third disappears under the conditions in (26), and that 

 the resulting value of W, which we denote by W, may be expressed 

 in either of the following ways: 



„^, b(^ — 2cab + a¥ , „ cb^ — 2acb + b<f ^ 

 ab — & bc — tt 



^ _ ad'-^bac + ca' .^^. 



ac — V 



When no two of the equations (22) are independent, there is a 

 central plane. The conditions of this case are, as appears from the 

 equations, that a„, 6,,, c^,, /„, «»,,, «,,, must be severally = ; of which how- 

 ever it is sufficient that any three should exist. We have moreover 



a 



a : c '. b (28). 



From all which it appears that W is now in the form -. From 



one of the equations (22) substitute in (25) the value of one of the 

 co-ordinates in terms of the other two; the coefficients of the last two 

 will disappear, as before, and the different forms of the value of W, 

 which we call W", will be 



W"^ - I +/= - J +/= - 7 +/• (29). 



By substituting W or W", when necessary, for W or <p {X, Y, Z) 

 in (24) the equation of the surface will be obtained, referred to any 

 point in its central line or plane. 



