302 
Proceedings of the Royal Soeiety 
The roots of this equation are actually 
c r 
257-27 = 286-06= 4-4934 
442-37 = 491-80= 7-7252 
624-46 = 694-17 = 10-9041 
805-56 = 895-48 = 14-0662 
For the crossings of the curve belonging to the function jFa;, we 
get the condition 
x= -timx, 
and, in this case, the points p and q must move away from A in 
opposite directions, and the first coincidence will take place when 70 
is somewhat beyond the extremity of the quadrant AB ; the phases 
are so closely analogous to those of the preceding case that it is 
unnecessary to detail them. 
The functions jf>x and give the conditions 
x = cotx and x = - toix . 
In these cases, when q begins its motion from A along the 
tangent, the travelling point p must leave B, in the one case travel- 
Fig. 7. 
ling backwards towards A, in the other case forwards towards C. 
Thus the same artifice serves for the detection of the roots in all the 
four equations. 
As an example of the other cases, we may take the function 
_3Fa? = ircosit;“ 3 sina?; this, when equated to zero, becomes 
X — 3tan^i?. 
