LYMAN. — THE FALSE SPECTRA FROM DIFFRACTION GRATINGS. 41 

 of the liglit given by the whole spectrum for the wave-length X will be 



.,,, , , 7r27r (« — l)7r,., 



appreciable for those values a = — , — , -^ — which come 



^^ n n n 



near to the values a = -, a = -^. Here n may be taken at pleasure. 



o o 



For the sake of a definite case, let n = 70. Then /„ will have an appre- 

 ciable value at a = ?§ tt, f * tt, f *5 ir, f 5_ tt; for f§ and f ^ are the nearest 

 values to ^ : |^§ and 4 J are the nearest values to §. Moreover, since 

 ^ of 70 is 23.33, the 23d spectrum will be nearer this position than the 

 24th, and therefore the stronger of the two. Similarly, as § of 70 is 

 46.66, the 47th spectrum will be stronger than the 46th. Therefore, 

 taking the number 70, the two spectra which are the stronger will be the 

 lower of the first pair and the higher of the second. 



If, then, we let n = 70, and consider the error to occur every third 

 line in the grating, for each line in the spectrum there will be four repe- 

 titions between its normal position and the direct image of the slit. 

 These repetitions correspond to the 23d, 24th, 46th, and 47th order 

 of the grating of in groups, the 70th order corresponding to the normal 

 position. 



This simple treatment of false spectra was suggested to the author by 

 Professor Runge. The values used are those which seem most nearly 

 to fit the case of the false lines obtained by the author from the grating 

 called No. I, and illustrated in a former article. 



Before proceeding further to the numerical verification of this theory 

 it may serve to illuminate the matter if we place in contrast with it the 

 results which may be expected from the theory of Rowland.* The 

 theory of Rowland is divisible into two parts, one dealing with the pro- 

 duction of ordinary ghosts, the other part dealing with the production of 

 lines at a considerable distance from the parent. It is this second part 

 which could alone be expected to fit the case in hand. It does not do so, 

 however, for it demands lines whose apparent wave-lengths bear simple 

 ratios to the parent line. Professor Runge's theory shows the possibility 

 of the total absence of lines at these positions indicated by Rowland, 

 and it shows the probability of the formation of lines on either side, 

 of these positions at different distances and of an indicated relative in- 

 tensity. Thus, in the present case, the theory of Rowland might be made 

 to call for lines at positions corresponding to § and ^ the wave-length of 

 the parent line. The lines actually observed do not fulfil this condition, 

 but occur in flanking positions. For example, in the spark spectrum of 



Rowland's Physical Papers, p. 535. 



