74 Mr. De Morgan on the General Equation of Curves 



The lines determined by these angles are parallel to the 

 principal diameters. The next question is, how can the greater 

 principal diameter be distinguished from the lesser? It must 

 be observed that in the general equation 



u rectangle of segments of the axis of x _ (diameter parallel to axis of xf 

 c ~~ rectangle of segments of the axis of y (diameter parallel to axis of y)- ' 



Since the square of one or both of these diameters may be 

 negative, that which is numerically the greater can be ascer- 

 tained only when we know the sign of c 2 -a 2 . If this be positive 

 the greatest of the two diameters above-mentioned is parallel to 

 the axis of y, &c. 



In (10) one of the values of x must be less than 90 . Take 

 this for the new axis of x, that is, let sin 2<p be positive. From 

 (10) it appears that 



B C-A , , 



sin2( t>=M cos2( P=-jj- ( n ) 



where M= ± ^/B" + (A—Cf, and since sin 2 <p is supposed positive, 

 M must be of the same sign as B. Also sin \^ = cos <p, cos \^ = -sin 9, 

 since ^-9 = 90°. Therefore from (5) 



\a' sin s = A cos 2 <p — B sin <p cos <j> + C sin 4 <p, 



\c' sin 2 9 = A sin <p + B sin 9 cos <p+C cos 2 <p, 



\(c' + a') sin 8 6=A+ C, 



\(c'-a') s\n-0 = {C-A) cos 2<p + B sin 2<p = M from (11). 



Therefore c'- - a" 2 has the same sign as M(A + C) 

 B(A + C), 



since B and M have the same signs; the hypothesis being that 

 the principal diameter which is nearest to the axis of x in the 

 positive direction is parallel to the axis of x. Accordingly there- 

 fore as B(A + C) or sm0 (b- 2 c cos 0) (a + c — b cos 0) is positive or 



