630 ON THE RESULTANT OF A NUMBER OF UNIT VIBRATIONS [442 origin, the area is triangular and increases as o-2; afterwards it becomes hexagonal, and after passing through the form of a regular hexagon, when its area is a maximum, returns backwards through the same phases. The calculations by the sequence formula present no difficulty of principle. When n — 4, I find (0«r<a), fa(<r) = <r*IQaa; (a«r< 2a), fa (a-} = {as - 4 (a- - a}3}/6as; when 2a < a- < 4a, the above values are repeated symmetrically. In this case there is no discontinuity either in fa, or fa, or fa'. When <r = 2a, that is in the middle of the range, fa — f 3 fa' = 0- The calculations might be pursued to higher values of n without much trouble. In all cases there is symmetry with respect to the middle of the range. The functions <j>n are algebraic and rise in degree by a unit at each step. At the beginning of the range 0Jl+1 (a) = (ff/a)nfnl, so that the contact at both ends of the representative curves with the line of abscissae becomes of high order. Again, since cr must lie somewhere between 0 and na, we must have (a) da-1 a == 1; .(8) from the above expressions we may test this in the cases of n = 2, 3, 4. A plot of the curves for these cases is given in Fig. 1. The ordinate represents (j> (<r) and the abscissa represents a itself with a taken as unity, so that the area of each curve is unity. Fig. 1. In order to pass from these curves in which a is the sum of the distances from one end to the representative curves for the mean distance, which must lie between 0 and a, we have merely to reduce the scale of the abscissae in the ratio n : 1, and to increase the scale of the ordinates in the same ratio, so that the area is preserved. For instance, when n = 4, the middle ordinate will be increased from f to f.f the line within the square which is drawn perpendicular to the diagonal through the origin, a itself corresponding to the position of the line- as measured along the diagonal *.