of a Number of Unit Vibrations, 501 



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 [joints 

 the chance of distances x and y from the chosen end is 

 dxdyjd 2 , and what we require is the integral of this taken 

 between the proper limits. If we treat x and y as rect- 

 angular 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 perpen- 

 dicular to the diagonal through the origin, a itself corre- 

 sponding 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 

 origin, the area is triangular and increases as a 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<tt), (f>Jcr) = (T ?J i iGa* ; 



(a<o-<2a), <p±(o-)= { <r 3 -r- 4 («x -— a) 3 }/6a 3 ; 



when 2a<s<4:a, the above values are repeated sym- 

 metrically. In this case there is no discontinuity either 

 in </> 4 , or <j> ', or cf> 4 " . When v = 2a, that is in the middle 

 of the range, 



^=2/3, >/ = 0. 



The calculations might be pursued to higher values of h 

 without much trouble. In all cases there is symmetry with 

 respect to the middle of the range. The functions </> n are 

 algebraic and rise in degree by a unit at each step. At the 

 beginning of the range <j> n + 1 [o~) = (o-/a)*/nl, so that the 

 contact at both ends of the representative curves with 

 the line of abscissae becomes of high order. 



Again, since a must lie' somewhere' between and na, we 

 must have 



t 



cj> n (a)da/a=ly .... . (8) 



from the above expressions we may test this in the cases of 

 n = 2, 3, 4. 



