of the Second Degree. 75 



negative, the less or greater principal diameter is nearer to the 

 axis of 1. 



When B(A + C) = o there are two cases to be distinguished. 

 If A + C = o the curve is equilateral, but can only be an hy- 

 perbola. If 5 = o and C-A is not =0 from (11), one of the 

 principal diameters is parallel to the axis of x. Which it is may 

 be determined from the sign of c" - a". 



When B=o and C-A = o, or which is the same, 6 = 2c cos 9 

 and a = c, the position of the principal diameters is indefinite; 

 that is, the curve is a circle. 



To determine the magnitude of the principal diameters, the 

 equation must be reduced to the form 



\dy- + Xc'x- -t- X/" = 0, where ff = go". 



In this case the equations (6), (7) and (9) become 



b" -4ac 



-4Wc' = 



shr'0 



. , , _ , a + c — b cos 



\a +Xc = r— r 



surfl 



\f— cd' + ae 2 - bde . 

 X/ b"—4ac + J- 



Whence the squares of the principal semidiameters or - 1 , 



a, 



f 

 and - -r are contained in the formula 



c 



cd z + ae" — bde 



+/ 



- 2 gin'* b *~ 4ac 



a + c-bcos0±*/(a-cy + b*-2cose{a + c)b-2accosd) 



In this way the curve might be referred to the conjugate 

 diameters which make a given angle ff, and the limits of tin- 

 value of ff might be determined. 



k2 



