310 APPENDIX. 



the roots of the equation will correspond to the points of intersection of the 



curve 



_ sin 4 z _ x* 



with the straight line 



y = a(x b}. [Figs. 6 and 6 .] 



It will be perceived that the curve line, in this as in all the following cases 

 under this form, is not affected by any change in the values of m and q, and that 

 the position of the straight line is determined by its cutting the axis of x at 

 the distance tan q from the origin, and the axis of y at the distance ^^ 



m 



from the origin. The tangent of its inclination to the axis is obviously equal to 

 , which may in some cases answer more conveniently for determining its 

 position than its intersection with the axis of y. 



(b.) The development of the fundamental equation divided by m sin z, is 



sin 8 z = - (cotan q cotan z) ; 



and by .putting 



x = cotan 2 



b = cotan q 



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



y = sin 8 * = (1 -|- *)&quot;&quot;&quot; t 

 with the straight line 



y = a (b x}. [Fig. 7.] 



The position of the straight line is determined by its cutting the axis of x at a 

 distance equal to cotan q from the origin, and the axis of y at a distance equal to ^-^ 

 from the origin. This form of construction is identical with that given by M. 

 Binet in the Journal de FEcok Poll/technique, 20 Cahier, Tome XIII. p. 285. His 

 method of fixing the position of the straight line is not strictly accurate. This 

 mode of representation is not surpassed by either of the others under this form. 



(c.) The fourth root of the fundamental equation developed, and divided by 

 cos (z q\ assumes the form 



cos ? (tan (z q\ 4- tan?) = n 



cos (z 



q) 



