VIII] THE BISECTION OF A QUADRANT 587 



as our diagram shews us, occurs when the two angles (a and 9) are 

 both rather under 52°. 



Next I have plotted on the same diagram, and in relation to 

 the same scale of angles, the corresponding lengths of the two 

 partitions, viz. RS and MP, their lengths being expressed (on the 

 right-hand side of the diagram) in terms of the radius of the circle 

 (a), that is to say the side wall, OA, of our cell. 



The hmiting values here are (1), C, C[, where the angle of arc 

 is 90°, and where, as we have already seen, the two partition-walls 

 have the relative magnitudes of MP : J?*S = 0-875 : Mil: (2) the 

 point D, where RS equals unity, that is to say where the periclinal 

 partition has the same length as a radial one; this occurs when 

 a is rather under 82° (cf. the points D, D'): (3) the point E, where 

 RS and MP intersect, that is to say the point at which the two 

 partitions, periclinal and anticlinal, are of the same magnitude; 

 this is the case, according to our diagram, when the angle of arc 

 is just over 62 J°. We see from this that what we have called an 

 antichnal partition, as MP, is only Ukely to occur in a triangular 

 or prismatic cell whose angle of arc hes between 90° and 62|°; in 

 all narrower or more tapering cells the periclinal partition will be 

 of less area, and will therefore be more and more likely to occur. 



The case (F) where the angle a is just 60° is of some interest. 

 Here, owing to the curvature of the peripheral border, and the 

 consequent fact that the peripheral angles are somewhat greater 

 than the apical angle a, the periclinal partition has a very slight 

 and almost imperceptible advantage over the anticlinal, the relative 

 proportions being about as MP\ RSwO'lZ.O-n. But if the 

 triangle be a plane equiangular triangle, bounded by circular arcs, 

 then we see that there is no longer any distinction at all between 

 our two partitions; MP and RS are now identical. 



On the same diagram, I have inserted the curve for values of 

 cosec ^ — cot ^ = OM, that is to say the distances from the centre, 

 along the side of the cell, of the starting-point (M) of the anticlinal 

 partition. The point C" represents its position in the case of a 

 quadrant, and shews it to be (as we have already said) about 3/10 

 of the length of the radius from the centre. If on the other hand 

 our cell be an equilateral triangle, then we have to read oif the 

 point on this curve corresponding to a -- 60°; and we find it at 



