92 Mr DE morgan ON THE GENERAL EQUATION OF 



The equations (42) and (43) take the forms 



Ax"+jr" = (44), 



F,A - F, =0 (45). 



The first of which is impossible if W and Fl have the same sign, 

 that is, if W" and a have the same sign ; for when a„ = b,^ = c„ = 0, 

 T^i takes the same form with respect to a, b, and c which Vs took with 

 respect to a„, b^,, and c„ in the last case. When a and W" have 

 different signs (44) belongs to two parallel planes, which coincide in 

 one where W" = 0. That is (29) the surface is impossible, two parallel 

 planes, or one plane, according as af—c^ is positive, negative, or nothing. 

 When W becomes infinite, or a = 0, in which case b, c, a, b, and c are 

 severally = 0, the proposed equation (3) is in fact of the first degree. 



Though oblique co-ordinates have hitherto been used, yet they 

 might have been dispensed with so far as the criteria of distinction 

 between the different classes of surfaces are concerned. It would take 

 some space, and complicated algebraical operations, to prove this in 

 all the individual cases, but the following general consideration is equally 

 conclusive. So long as we only consider those distinctions which are 

 implied in calling the surface bounded or unbounded, of one sheet or 

 of two sheets, &c. in which no numerical relations of lines, &c. appear, 

 it is evident that any equation will preserve the same character, how- 

 ever the axes on which its results are measured are inclined to one 

 another. That is, when the sign of a quantity is alleged to be a cri- 

 terion of distinction, it cannot stand as such, if by any alteration of 

 I, >/, or ^, consistent with V^ remaining positive, the sign of that quantity 

 can be changed. Again, if the signs of two out of the three, a, b, and c 

 be changed, as well as that of the third letter in a, b, and c, (those of 

 a, b, and c, for example) it is evident that the surface remains the same 

 in form and magnitude, those parts which were below one of the co- 

 ordinate planes being transferred above it, and vice versa. That is, 

 it is impossible that any aggregate of terms of an odd degree, with respect 

 to a, b, and c, b, c, and a, or c, a, and b, can affect the sign of any 



