Retardations, and on the Theory of FoucauW s Test. 171 



spectra. Referring to (5) we see that if positive values of f 

 be alone regarded, the vibration in the place of the second 

 aperture, represented by £ _1 sin (0f), is the same in respect 

 of phase as would be due to a theoretically simple grating- 

 receiving a parallel beam perpendicularly, and the directions 

 (f)—±d are those of the resulting lateral spectra of the first 

 order. On account, however, of the factor f -1 , the case 

 differs somewhat from that of the simple grating, but not 

 enough to prevent the illumination becoming logarithmically 

 infinite with infinite aperture. But the approximate resem- 

 blance to a simple grating fails when we include negative as 

 well as positive values of £, since there is then a reversal 

 of phase in passing zero. Compare fig. 2, where positive 



Fig. 2. 



values are represented by full lines and negative by dotted 

 lines. If the aperture is symmetrically bounded, the parts 

 at a distance from the centre tend to compensate one another, 

 and the intensity at </>=+# does not become infinite with 

 the aperture. 



We now proceed to consider the actual calculation of 

 I = (19) 2 + (21) 2 for various values of (p/0, which we may 

 suppose to be always positive, since I is independent of the 

 sign of d>. When %6 is very great and <£/# is not nearly 

 equal to unity, $i{ (# + <£)£} in (19) may be replaced by ^tt 

 and Si{(# — $)£} by +2 7r -> according as <f>j0 is less or greater 

 than unity. Under the same conditions the Ci's in (21) 

 may be omitted, so that 



1=^(1, or 0)+^log||±| 



But if we wish to avoid the infinity when </> = #, we must 

 make some supposition as to the actual value of 0f, or 

 rather of 27r0£/\. In some observations to be described 

 later a = l inch, f=i inch, 1/X = 40,000, and ti = \a\f. Also 

 /was about 10 feet— 120 inches. For simplicity we may 

 suppose /=407r, so th.it 2ir6^\ = o00, or in our usual 



(25) 



