APPENDIX. 303 



and a point of contrary flexure for 



z = 60, and 2 = 120. 



From s to s 60, it is convex to the axis of abscissas, from 60 to 

 120 it is concave, and convex from 120 to 180. 

 For oscillation, the three equations, 



m sin 4 3 = sin (z q) 

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

 4 m sin 2 2(1 -f-2 cos 2 2)= sin (2 q) 

 must coexist, or 



m sin 4 z = sin (z q) 

 sin (2 3 q] = f sin q 



cos 22 = f . 

 In this case we should have 



sin (2s q) = | cos ^ -f- I snl ?&amp;gt; 

 consequently, 



tan = f 

 and 



mn? = , 

 or 



* = 46H-isin- 1 f. 



From these considerations we infer that for the equation 



m sin 4 z = sin (2 q) 

 or even when it is in the form 



nf sin 8 z 2 m cos ^ sin 5 2 -)- sin 2 2 sin 2 q = 



of the eighth degree, there can only be four real roots ; because, in the whole 

 period from z &amp;lt;^=0to z q = 360,only four intersections of the two curves 

 are possible on the positive side of the axis of ordinates. 



Of these, three are between 2=0 and z = 180, and one between 180 

 and 180 -\-q; or, inversely, one between and 180, and three between 180 

 and 180 -\-q; consequently, there are three positive and one negative roots, or 

 three negative and one positive roots for sin 2. 



