The question how the other, or triangular, portion of the divided quadrant will next divide leads us to another well-defined problem, which is only a slight extension, making allowance for the circular arcs, of that elementary problem of the triangle we have already considered. We know now that an entire quadrant must divide (so that its bisecting wall shall have the least possible area) by means of an anticlinal partition, but how about any smaller sectors of circles? It is obvious in the case of a small prismatic {364} sector, such as that shewn in Fig. 149, that a _periclinal_ partition is the smallest by which we can possibly bisect the cell; we want, accordingly, to know the limits below which the periclinal partition is always the best, and above which the anticlinal arc, as in the case of the whole quadrant, has the advantage in regard to smallness of surface area. 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 θ, according to our equation _MP_ = _a_θ cot θ; and the angle θ has a definite relation to α, the angle of arc. [Illustration: Fig. 149.] Moreover, in the case of the periclinal boundary, _RS_ (Fig. 147) (or _ab_, Fig. 149), we know that, if it bisect the cell, _RS_ = _a_ ⋅ α/√2. Accordingly, the arc _RS_ will be just equal to the arc _MP_ when θ cot θ = α/√2. When θ cot θ > α/√2 or _MP_ > _RS_, then division will take place as in _RS_. When θ cot θ < α/√2, or _MP_ < _RS_, then division will take place as in _MP_. In the accompanying diagram (Fig. 150), 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 α, which shews on the (left-hand) vertical scale the various possible magnitudes of that angle (viz. the angle {365} of arc of the whole sector which we wish to divide), and on the horizontal scale the corresponding values of θ, or the angle which [Illustration: Fig. 150.] determines the point on the periphery where it is cut by the partition-wall, _MP_. Two limiting cases are to be noticed here: (1) at 90° (point _A_ in diagram), because we are at present only {366} dealing with arcs no greater than a quadrant; and (2), the point (_B_) where the angle θ comes to equal the angle α, for after that point the construction becomes impossible, since an anticlinal bisecting partition-wall would be partly outside the cell. The only partition which, after the point, can possibly exist, is a periclinal one. This point, as our diagram shews us, occurs when the angles (α and θ) are each rather under 52°.