SURFACES OF THE SECOND DEGREE. 83 



Let the equation of the surface, referred to the principal axes, be 

 Aa;" + A'y" + A"z''+W=0 (30), 



which must be identical with (24) when the values of x', y', «', found in 

 (1) are substituted. We must then have 



a==Aa' +A'a" +A"a"\ 



h = Ali' +A'(i"' +A"(i"\ 



c = Ay^ +A'y" +A"y'\ 



(31), 



a = Al3y +A'fi'y' + A"li"y", 



h=-Aya +A'y'a -{■A"y"a', 



'^ = Aafi+A'a'^ +A"a"li", 



which equations are reduced to those in (2) by substituting unity for 

 A, A', A", a, h, and c; and cos f, cos n, and cos X, for a, h, and c. Thus, 

 whatever equation is deduced from these, we immediately find another, 

 containing a, /3, &c. in the same way, by the last mentioned substitu- 

 tion. Multiplying the first of these by p, the last by q, and the last 

 but one by r; and adding, we obtain by the use of (14), 



pa + qc +rb =Aa\/Vo 



p +qcoS(^ + rcosr]= ay/V^ 



from which, and similar processes, we obtain 



(32), 



p{A — a) + q{A cos ^—c) + r {A cos t} — b) = 0, 



p{A cos^-c) + q(A-h) + r(^cosf-a) = (33), 



p{Acc^ri — b) + q{Acosl^—a) + r{A — c) =0; 



which agree in form with (22), if a, J, and c be struck out, and A — a 

 substituted for a, ^cos^ — a for a, &c. But 1^3 = is the result of (22), 

 with the last terms erased ; that is, if in V^ the substitutions just men- 

 tioned be made for a, a, &;c. the result developed and equated to zero 

 wiU give the equation for determining A, A', and A". That equation is 



r,A'- r^A'+r,A-v,=o (34). 



L 8 



