514 On the Resultant of a Number oj Unit Vibrations. 



\cl— cos -1 (-£?'), so that the chance of r lying between r and 

 r + dr is 



4dr\±a— cos -1 (h*) j 



a 2 */(4-r 2 ) 



(37) 



Expressions (36) and (37). obtained on the supposition 

 that is positive, are to be doubled when we allow for tlie 

 equally admissible negative values of 0. 



When (36), (37), as they stand, are integrated over the 

 ranges of r to which they apply and added, the sum is J, as 

 it should be under the suppositions made. 



It still remains to consider the case where 37r/2<«< 27r. 

 From = to = \[a — 7r), r (as before) ranges from to 2. 

 From = ±(ct — 7r)to6 = \ (3w — a) , r ranges from 2 cos {\u — ) 

 to 2. At this point (fig. 6) a second range enters for *. 

 From = ^(37T — a) to = ^x, the first range is, as before, 

 from 2cos(.]a — 0) to 2, and the second range is from to 

 2cos(2«7r — \ol — 6i. Lastly, from — \a to — ir^ the first 

 range of r disappears, while the second continues to be from 

 to 2cos(2tt— \*— 0). 



The probabilities of various #'s being positive and lying 

 within specified ranges can be obtained as before. For the 

 range from = to fl= 2 (a — ir) we get the expression (33), 

 and from = ±(a. — it) to = } 2 ('6ir — oc) we get (34). For the 

 third range from = ^{ott — a) to = } 2 x, we get 



U0VC 2 pcos^-ia-en dr 



« 2 Lj2co8(!a-0) Jo J \/(4— r 2 ) 



±d$( 3tt\ 



and from #=!* to = 7r, 



-?J 7 (4 -r») = ^H> + *~~ x; • (39) 



If the integrations with respect to are effected over the 

 appropriate ranges and the results added, we get \, as was 

 to be expected. 



Finally, lor the probabilities of various r's when is left 

 open, we get for r between and 2 cos (77- — ia) two ranges 

 for 0, viz. from to \ol— cos _1 (^r), and again, from 

 # = 2tt— \a>— cos _1 (y ? ') t° = 7r, making altogether {oc—tt). 

 Thus for these values of r the probability is that expressed 

 in (36). 



When r lies between 2cos(7r — 2 -a) and 2, we recover in 

 like manner (37). And as before we may verify the results 



