302 APPENDIX. 



There will, therefore, be a contact of the curves when we have 



m sin 4 z = sin (z q) 



and 



4 m sin 8 s cos z = cos (z q) 



or when 



4 sin (z &amp;lt;?) cos s = cos (s q) sin 



which may be more simply written 



sin (20 q] = sin q. 



When the value of z deduced from this equation satisfies 



m sin 4 z = sin (z q) 



then there is a contact of the curves, or the equation has two equal roots. These 

 equal roots constitute the limits of possibility of intersection of the curves, or the 

 limits of the real roots of the equation. 



For the delineation of both curves it is only necessary to regard values of 

 a q between and 180, since for values between 180 and 360 the solution 

 is impossible ; and beyond 360 these periods are repeated. 



The curve 



/ = sin (z q) 



is the simple sine-curve, always on the positive side of /, and concave to the axis of 

 abscissas, and has a maximum for 



q 90. 



The curve 



y =. sin 4 z 



is of the fourth order, and since it gives 



-^ = 4 m sin 3 z cos z = m sin 2 z J m sin 4 z 

 dz 



-r- 

 it lias a maximum for 



-r-j- = 12 m sin 2 z cos 2 z 4 m sin 4 z 



dz 



= 4 m sin 2 z (1 -f- 2 cos 2 z] = 2 m (cos 2 z cos 4 z} 

 ^J. = _ 4ra(sin2z 2sin4z) 



dz* 



-r- = 8 m (cos 2z 4 cos 4 z) 



