OP THE SECOND DEGBEE. 4? 



the parallelogram contained by any two conjugate semi-dia- 

 meters of an ellipse or hyperbola is invariable ; being equal to 

 V sin y (AC — B*)~* when the curve is an ellipse, and to 

 F" sin y (B* — AC)~~* when it is a hyperbola. 



(d.) From equations (cj and (d) we readily obtain 



{m^+2 m cos y+ 1) (A m"^-\-2 B m'+C) 

 a'^-^b"^ +(?7t'H2w^'cosy+l)(Aw''+2Bm+C) .« 



— F" "A m^+'-Z B w+C) (AW2+2 B w'+C) ^ '' 



Now we have just seen that the denominator of the fraction 

 in the second member is equal to — (B'* — AC) {m'-mf, and by 

 subtracting the first member of equation (21) multiplied by 

 2+2 w m' from the numerator, it becomes 



(A+ C) {m'—mf+'2, cos y {(A m m'+ C) (m'+ w)+ 4 B w' w] , 

 which in virtue of equation (21) easily reduces to 



(A+C— 2 B cos y) {m'—mf. 

 Hence, by substituting these reduced expressions for the nu- 

 merator and denominator in equation (f), we obtain 



a'2+6'2=F'' (A+C— 2 B cos y) (B^— AC)-i (23). 



When the curve (1) is an ellipse a'* and b'^ are both positive, 

 but when it is a hyperbola one of them is positive and the 

 other negative. Hence equation (23) shows that, in an ellipse, 

 the sum, and in a hyperbola, the difference of the squares of 

 any tioo conjugate diameters is invariable. 



(e.) We are now prepared to determine the lengths a' and b' 

 of two conjugate semi-diameters which shall contain a given 

 angle 8. For, since by equations (22) and (23) we have , 

 a^+5^^ _ A+C— 2Bcosy 



F' - B^— AC 

 a'^b'^ — sin2 y 



p„a - ^B*— AC) sin* S ' 

 it follows that if zi and Z2 denote the roots of the quadratic 

 equation, 



(B^— AC) s"— (A+C— 2 B cos y) «— sin» y. sin-^ 8... (24), 

 we shall have 



a'»=F''2, , 6'«=F''a2 (25). 



