VIII] THE BISECTION OF A QUADRANT 585 



best, and above which the anticHnal arc has the advantage, as in 

 the case of the whole quadrant. 



This may be easily determined; for the preceding investigation 

 is a perfectly general one, and the results hold good for sectors of 

 any other arc, as well as for the quadrant, or arc of 90°. That is 

 to* say, the length of the partition- wall MP is always determined 

 by the angle 9, according to our equation MP -^ ad cot 6; and the 

 angle 6 has a definite relation to a, the angle of arc. 



Fig. 235. 



Moreover, in the case of the periclinal boundary, RS (Fig. 233) 

 (or ab, Fig. 235), we know that, if it bisects the cell, 



RS = a. a/V2. 



Accordingly, the arc RS will be just equal to the arc MP when 



^ cot ^ = a/v/2. 



When ^ cot ^ > a/ V2, or MP < RS, 



then division will take place as in RS, or perichnally. 



When 6* cot ^ < a/ V2, or MP > RS, 



then division will take place as in MP, or anticlinally. 



In the accompanying diagram (Fig. 236), I have plotted the 

 various magnitudes with which we are concerned, in order to 

 exhibit the several limiting values. Here we see, in the first place, 

 the curve marked a, which shews on the (left-hand) vertical scale 

 the various possible magnitudes of that angle (viz. the angle of arc 

 of the whole sector which we wish to divide), and on the horizontal 

 scale the corresponding values of 6, or the angle which determines 

 the point on the periphery where it is cut by the partition-wall. 



