88 Mr DE morgan ON THE GENERAL EQUATION OF 



Taking the first line of this table, and the signs of W, V^, V^, and 

 V^, (on which, as will presently be shewn, the variety of the equation 

 depends,) being such as to denote that the equation is impossible, a 

 change of sign in W only will indicate the ellipsoid, the elliptic cylinder, 

 or parallel planes, according as the centre is a point, a line, or a plane. 

 When the sign changes, if W be then = 0, the variety of the equation 

 belongs to a point, a right line, or a plane ; while if W be infinite, 

 we have an elliptic paraboloid, a parabolic cylinder, or a plane. In 

 using W, we mean its real value, W or W", when the primitive form 



of W becomes - . 



The following table, from which the preceding may be deduced, and 

 which I proceed to establish, gives the signs of W, &c., and also of V^, 

 &c., for the different cases. When p alone, or p and n occur on the 

 same line, p may signify either sign, provided n stands for the other. 

 Also when a sign is enclosed in brackets, it is a necessary consequence 

 of what precedes it, and not an independent assumption. The num- 

 bers over the headings are references to the equations. 



The last part of the table, including all the varieties under W= - , 



forms a similar synoptical table for the curves of the second degree. 

 The following are the values of W, W", V^ and Fi, expressed in the 

 notation of my former paper, the equation of the curve being 



ay* + hxy + ca^ + dy->rex +f= ; 



and the angle made by the axes being Q, 



.^, _ cd^ + ae^ — hde „ 



™,„ _ _ dr-^a£_ &-^^cf 

 id ~~ 4c ' 



V, =- (b'-iac), 



Vi = a + c — h cos Q. 



