182 UNDULATOEY THEORY Of LIGHT. 



The distances from P to tlie corrcspoiuliiig parts of the obstructing- bars will 

 difier from the distances to the adjacent openings by half an undulation; and, 

 accordingly, if the bars were removed, the wave -which would proceed from those 

 points Avould neutralize the efiects of the former : but being obstructed, P remains 

 illuminated by the resultant effect of all the first set of waves. 



Furthermore, since the position of P is determined by the condition that ch 

 sliall be the length of an undulation, it will be necessary to take P further from 

 B for the longer undulations and nearer for the shorter. The different colors 

 will thus be separated, and a perfect spectrum will be formed on the screen. 



Should the point P be taken so that ch is equal to two undulations, there will 

 be no spectrum: for in this case cd wilLbe equal to one undulation, and as in 

 the cases we have considered of a single aperture, one-half of cacli opening, a, 

 will hold in check the other half. If we find still another point -where be is 

 equal to three iindiilations, then cd will equal one undulation and a half; two- 

 tliirds of each ojoening will then be neutralized, but the remaining third will lie 

 effective ; and there will be another spectrum, but less brilliant than the first. 

 If he z= four undulations, the spectrum will again fail. If he z=i oX it will rt'turn, 

 and so on. 



If the bars are broader than the open spaces, there will, be a spectrum for 

 hc^ii^nl, 11 being any integral number; until the light is too feeble, or until 

 cd =: r^X, n' being also any integral numbci-. If the spaces arc broader than the 

 bars, there will be a spectrum for every integral value of « in Zr.—^^A until 



n^:^-——, [a and h standing for the breadths of the spaces and bars severally.) 

 If, however, -"!"— is not integral, take ^ =: the greatest common measure of a 



and h. Then n z=: —^ will give the number of the first spectrum which Avill fail . 



Put this value of n equal to ?«, and we may say generally that the mth spectrum 

 will fail, and also the mnih, n being, as before, any integral number. If a and 

 h are incommensurable, there could be theoretically no perfect spectra, or spectra 

 of maximum brilliancy; nor would any spectrum absolutely faiJ: but a near 

 approach to failure would occur for approximate values of q. All these propo- 

 sitions result so obviously from the construction above given, that they require no 

 demonstration. 



The same construction indicates a simple expression for the deviation of each 

 spectrum from the directioii'* JlS, of the radius of the original wave. For re}i- 

 resenting this deviation by (">, we have — 



sni',= [17. 



Putting ??:rz 1, A =:: one fifty-thousandth of an inch, the length of the mean 

 undulation, and supposing one thousand opaque lines to the iuch, the formula 

 gives us, by substitution, sin'j=i0.02=zsin I^ 9'. As the sines of small angles 

 are very nearly proportional to the angles themselves, the deviations of the suc- 

 ceeding spectra will be nearly the tlouble, triple, 6:c., of this. And as the' de- 

 nominator, a-\-h, is the reciprocal of the number of lines to the unit of ineasure- 

 nient to which X has been referred — in this case to the inch — it is evident that 

 the sines of the deviations Avill increase directly as this number. With five 

 thousand lines to the inch, the fifth spectrum will have a deviation of thirty 

 degrees. The force of the derivative v/aves from minute apertures tlius appears 

 to be great even at large obliquities, when the obstructing effects of interference 

 are removed. 



In the above expression for siu'?, if « be put equal to 1, and a\-h equal to X, 



