Coincidences between the Lines of Different Spectra. 463 



equal to 



d 1 -\-d 2 + . . . + <fr + 2(ft— v)x 



A» — \) 



From this expression one may deduce a simpler one for the 

 probability that the difference between the arbitrary number 

 A and the nearest of the wave-lengths A , A x . . . X^lies between 



-q and . It is 

 z z 



d } +d 2 + . . . + d v +(n—v)d v+1 _ d 1 + d 2 + ... + d v - 1 + (n — v + l)d t 



or 



( n — v)(d v+1 — d v ) 



X w — \q 



Let us take -^, -^-. , . 7? as abscissas, and let us draw a rectangle 



z z z 



over the interval -£ to -^ with v+1 — - as base and 2 - 



'Z Z Z A, n — A 



as height. Then the area of this rectangle will represent the 

 probability, that the difference of \ from the nearest wave- 

 length lies between -^ and -~- (for v = write d = 0). 



Thus we get what we may call a curve of difference analogous 

 to the curve of error. The curve of difference forms the pro- 

 file of a staircase, the steps of which may be of different 

 height and depth. If n is a large number and the scale of 

 the drawing not too large, the staircase will resemble a 

 smooth curve ; and it may be seen from the following applica- 

 tion that there are cases where it closely resembles the curve 



2c 

 y= —p=e~~° 2x2 , if one is allowed to choose the constant c 



accordingly. 



I have taken as an example the first five ultra-violet bands 

 of the water-spectrum as observed by Liveing and Dewar*. 

 They consist in 598 lines extending from 2268 to 3203*5. 



* Phil. Trans, of the Roy. Soc. 1888. 



