204 PROCEEDINGS OF THE AMERICAN ACADEMY 



There is a very simple geometrical construction of the roots of the 

 equation in a. Making cos o- = x> and sm v = V' tne values of x and 

 y are the co-ordinates of the intersections of the curves whose equa- 

 tions are 



x 2 + f=h 



(x~a)(j-b) = ab 



Consequently, if we construct the equilateral hyperbola whose equa- 

 tion is 



x y=±l, 



and from a point on it, whose co-ordinates are 



.1 a 





y l = - 



V ±ab' 



• iab' 



as centre, we describe a circle, whose radius is , and then draw 



V/±ab 



radii to the points of intersection of the curves, the angles made by these 

 radii with the x axis °f co-ordinates are the values of or. Since the 

 centre of the circle is on the hyperbola, there are at least two intersec- 

 tions, and thus the equation in o- has at least two real roots. The geo- 

 metrical construction readily affords the condition which a and b must 

 satisfy in order that there may be four real roots. The condition is, 

 that the length of the straight line drawn from the point a, b, on the hy- 

 perbola whose equation is 



xy=ab 



normal to the opposite branch, shall be less than unity. The equation 

 to the normal which passes through the point %", y" on this curve, is 



x"(x-x")-y"(y-f) = o. 



The condition that it passes through the point a, b, gives 



X (X - a) - y" (y" - b) = 0, 



X V" = a b. 



If we multiply the first of these by x"% we get 



X"*(X" — a) -ab(ab — b *") = 0, 



