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1919]         ON  THE  RESULTANT OF  A  NUMBER  OF  UNIT  VIBRATIONS           629
We start from fa (cr).    This is zero, unless 0 < a- < a, and then is unity. Hence between 0 and a
0a (0-)=     0! (cr) cta/ft = cr/a. J o
If <r lies between a and 2a,
02 (o-) = I      0j (cr) dcr/a = --- -
J cr  a                                                         M
cr  a
02(o-) = 0,    (o-<0);   02(cr) = cr/a,   (0<or<a);             ]
02(<r) = (2a-o-)/a, "(ar<2a); 02(<r) = 0, (2ar)J' by which 02 is completely determined ; and it will be seen that there is no breach of continuity in the values of 02 itself at the critical places. These values are symmetrical on the two sides of cr = a, and can be represented on a diagram by two straight lines passing through cr = 0 and a-  2a, and meeting at <r = a. (See Fig. 1.)
In like manner we can deduce 03 from 02.    If o-<0, 03 = 0, and indeed generally 0n = 0.    If 0 < cr < a,
do-     <72
If a < cr < 2o,,
a                    -a~-(r da- = -
From the symmetry it follows that when 2a < cr < 3a, 03 (cr) = (3a - <7)2/2a2.
When                             cr > 13 a,    03 (o-) = 0.
It may be remarked that in this case not only is 03 continuous, but also the first derivative 03'. The representative curves for all three portions are parabolic. The maximum of 0a, occurring at <r = 3a/2, is f.
These problems might also be attacked in another and perhaps more direct manner by expressing the probabilities as multiple definite integrals. Thus in the case of two points the chance of distances x and y from the chosen end is docdyfa?, and what we require is the integral of this taken between the proper limits. If we treat us and y as rectangular coordinates of a point lying within the square whose side is a, the probability we seek is represented by the length of 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 *.
For three points we have to consider a cube of side a, when the chance is represented in like manner by the area within the cube of a plane drawn perpendicularly to the diagonal through the origin. At first, that is near the
[* o- = \/2 x shortest distance of the line from the origin. For the cube (next paragraph of text) o-=\/3 x shortest distance of the area from the origin.    W. P. S.]nd we commence with a sequence formula connecting <f>n+1 with </>. If for the moment we suppose <n known and consider the inclusion of an additional point, we see that