APPENDIX. oil 



By putting 



x = tan (z q) 



b = tan q 

 a = \j m cos q 

 the roots of the equation correspond to the intersection of the curve 



= 



cos z q) 



with the straight line 



y = a(x + b). [Fig. 8.] 



The straight line cuts the axis of x at a distance equal to tan q, and the axis 

 of u at a distance equal to ^ m sin q, from the origin. 



(d.) The development of the fourth root of the fundamental equation divided 

 by sin (0 q) is, 



3 



^ m sin q (cotan (z q) -f- cotan q) = cosec (s qy 



By putting 



x = cotan (z q} 



1) = cotan 

 a ^ m sin g 1 



the roots of the equation correspond to the intersection of the curve 



with the straight line 



y = a(x-\-l). [Figs. 9 and 9 .] 



The straight line cuts the axis of x at a distance equal to cotan q, and the 

 axis of y at a distance equal to $ m cos q, from the origin. 



(e.) From the reciprocal of the fundamental equation multiplied by m, its 

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



r = cosec 4 s 



with the straight line 



r = m cosec (z q}. [Figs. 10 and 10 .] 



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

 vector, z the angle which the radius vector makes with the polar axis, m the dis 

 tance of the straight line from the origin, and q the inclination of the line to the 

 polar axis. 



