46 



ON THE EQUATION OF CUKVES 



chords parallel to the first. On account of this remarkable 

 property the diameters (a) and (b) are said to be conjugate 

 to each other when their direction-indices satisfy the con- 

 dition (21). 



(a.) If 2a' and 26' denote the lengths of the two conjugate 

 diameters whose equations are (b) and (a), we shall have 

 (IV. 12) 



'2 — — F''(^'+2mcosy + l) / X 



" ~ Am2 + 2Bm-hC ^'^^' 



,,, _ — F"(m'2+2m'cosy + l ,.. 



^ ~ Am2 + 2Bw'+C ^°^' 



where m and w' are subject to the condition (21). 



0.) By taking the product of eqns. (c) and (dj we get 



'2 y2— F'"^ (m^+^m cosy + 1) (m'^+ 2m' cos y -f 1) . . 



and by subtracting the square of the first member of equation 

 (21) from the denominator of the fraction in its second mem- 

 ber this equation becomes 



,^ 7/a_ F"g(m^ + 2m cos y-|-l) (m"--h2m' cos y+\) 

 ** {AC—B^)[m'—mf 



Again, if S denote the inclination of the conjugate diameters 

 Caf &nd (bj, we obtain, by the theory of the straight line, 

 . 5 » {m' — mf sin^ y 



sin o:zr - - ' • 



[m'^+2m cos y-f- 1) [m'^+2m' cos y+ 1) ' 



hence, by taking the product of the last two equations, we 



have 



a'2 b'^ sin'»8=F".2 sin" y : (AC— B'^), 



or a' b' sin S=F"sin y (AC— B'*)"* (22). 



The second member of this equation is real or imaginary ac- 

 cording as B* is less or greater than AC ; hence when the 

 curve {\)is an ellipse any two conjugate semidiameters are both 

 real, but when it is a hyperbola, of any two conjugate semi- 

 diameters one is always real and the other imaginary. 



(y.) Since a\ b'. sin 8 denotes the area of a parallelogram 

 having two adjacent sides equal to o', b\ and the contained 

 angle equal to S, it follows from equation (22) that the area of 



