312 APPENDIX. 



(/). From the reciprocal of the fourth root of the fundamental equation, its 

 roots may be seen to correspond to the intersection of the curve 



r = cosec^ 9 

 with the straiht line 





in which 



y = 2 q. [Fig. 11.] 



Both these equations are referred to polar coordinates, of which (p is the 

 an ale which the radius vector r makes with the polar axis, \1 - the distance of the 



T m 



straight line from the origin, and q the inclination of the line to the polar axis. 



3. The third method of representation is by a curve and a circle. 



(.) The roots of the fundamental equation correspond to the intersection 

 of the curve 



sin 4 z 



with the circle 



r = - sin (z z}. [Fig. 12.1 



m V. 1 L O J 



Both these equations are referred to polar coordinates, of which r is the radius 

 vector, the angle which the radius vector makes with the polar axis, the 



in 



radius of the circle which passes through the origin, and 90 -(- q is the ano-le 

 which the diameter drawn to the origin makes with the polar axis. 



(b.) From the fourth root of the fundamental equation it appears that its 

 roots correspond to the intersection of the equation 



r ^ sin &amp;lt;f 

 with the circle 



[Fig. 13], 



in which 9 (2 q) is the inclination of the radius vector to the polar axis, 

 ^ m is the diameter of the circle which passes through the origin, and 90 q 

 is the inclination of the diameter drawn through the origin of the polar axis. 



In these last two delineations the curve I K I K I&quot; incloses a space, within 

 which the centre of the circle must be contained, in order that there should be 

 four real roots, and therefore that there should be a possible orbit. The curve 



