76 Mr. De Morgan on the General Equation of Curves 



When the asymptotes are the axes of co-ordinates, the equa- 

 tion of the hyperbola is Xb'x'y' + \f'= o. 



The position of the asymptotes is then determined from the 



equation 



^tan 2 x + B tan x + C=o, 



_sin0 b -2ccos6± *Jb 2 — 4ac 

 or tan x =— . a _b cosd + ccos ,2 9 • 



where the two values of x are those of (f> and ^. 

 The equations (6), (7) and (9) become 



\ 2 6' s = b 2 -4ac 

 sin 2 ^ sin 2 9 



Xb'cosO' _a + c—bcos9 

 sin 2 0' sin 2 



. n , cd 2 +ae t — bde „ 

 a «d */ - b °- 4ae +/ 



. ., sin 6 tJb* — 4ac 



whence tan 6 = + — v , - b - 



a + c — b cos 



and the equation is 



V-4ac , , cd- + ae--bde 



± ^/( a -cY + b"-2cos6{a + c)b-2aecose) Xy ' b 2 -4ac + *~ 



In referring to the expression for the principal semidiameters, it 



, * . cd" + ae 2 — bde 



appears that if a + c -b cos 6 has the same sign as — ^ e _ — +J, 



the greater diameter is possible, or the curve lies in- the acute 

 angles of the asymptotes, and the contrary. 



If the equation of the parabola be reduced to the form 



dy' % + e'x' ss o, where = go , the equations (7) and (8) take the 

 following form 



a + c — b cos a-2«/«ccos^ + c 



\a' = 



sin*0 sin'fl 



