90 Mr DE morgan ON THE GENERAL EQUATION OF 



First, with regard to the coefficients K^, Vi, V^, V3 in equation 

 (34) it appears from spherical trigonometry, that V^ is always positive 

 when '(;, J/, and ^ are the sides of a spherical triangle; while from the 

 possibility of the roots, as well as from the quantities themselves, we 

 infer that if V3 is finite, Fj and Vi can never vanish at the same 

 time, while if ^i = 0, and ^ = 0, Fj, must be negative. 



If we suppose TV finite, and the order of signs in (34) to be 



H (-- or + + + +, in which case all its roots are of one sign ; 



that is, if K2 be positive, and Vi and V3 of the same sign, the equa- 

 tion (30) shews that the surface is impossible or an ellipsoid, according 

 as W and F'a have the same or different signs. From (36) it appears 

 that in this case, a^^, b,^, and c„ must be positive, whence a, h, and c have 

 the same sign ; which conditions, together with that of V^ having the 

 same sign as a, are equivalent to those given in the Table for the 

 impossible case or the ellipsoid. If we examine independently into 

 the conditions under which the aggregate of the first six terms of 

 (24) always has the same sign, we shall find them to be that a^, b„, 

 and c„ must be positive, and V3 must have the common sign of a, h, 

 and c. And it is evident that the first three terms of (30) are the first 

 six terms of (24) in a different form. It may be worth noticing, that 



these conditions are equivalent to supposing ,-- , -- — , —7=5= to be 



's/ he \/ca y/ab 



the cosines of the sides of a spherical triangle. When any other order 



of signs except the two already noticed, is found in (34), we shall have 



one positive root only, or one negative root only, according as V3 is 



positive or negative ; that is to say, one possible axis, or a double 



hyperboloid, when V^ and W have contrary signs ; and one impossible 



axis or a single hyperboloid, when they have the same signs. 



When W—0, V^ being finite, equation (30) represents a point, or 

 a cone; the first when all the roots of (34) have the same sign, the 

 second in any other case. When V3 = 0, Vi being finite, or W infinite, 

 the center is at an infinite distance, and the equation belongs to an 

 elliptic or hyperbolic paraboloid, according as V^ is positive or negative. 

 Since when V3 = 0, «,,, 5„, and c,, have the same sign, (10), which is 



