1881.] On Harmonic Ratios in the Spectra of Gases. 345 



If there is only one quantity, in the first compartment, b is given 

 by the equation — 



b 

 a 



b = aa + ha. 



Hence, if there is only one b which can range between - and 1, and 



must lie between a + and a {a. — B), if there is a coincidence with 

 the given fraction, the probability of such a coincidence is 



If there is more than one quantity, a, in the first compartment, we 

 observe that these quantities may lie so near together that one and 

 the same b can have, within the limits within which we count coin- 

 cidences, the required ratio with more than one of the quantities, a. If 

 these quantities, however, are not sufficiently close together to admit 

 of any such double coincidence, the probability that one b should have 

 the required ratio with one a is 



fi(a 1 + a 2 + . . . a t ). 



I — <x 



Call the sum in brackets St. 



If we drop the limitation that b should not possibly have at the same 

 time the required ratio with more than one a, the expression just 

 found will not any more represent the probability of a single coinci- 

 dence, but it will represent the expectancy for the coincidences. For 

 in the most general case there is a certain range, A, within which b 

 may lie in order to have the required ratio with one of the quantities, 

 a; there is a range, A 2 , within which a double coincidence will 

 happen, and so on : hence the expectancy for the coincidences is 



(A 1 + 2A 3 + 3A 3 + ...)---!, 

 but the expression in brackets is always equal to 



and hence the expression which we have found will represent the 

 expectancy if there is only one quantity, b ; for r quantities it is 



rsp 



I — a. 



We have hitherto supposed that the quantities, a, are at given fixed 

 places, or that s t has a certain given value. Let ptdst, be the proba- 



2 c 2 



