408 Lord Rayleigh's Investigations in Optics. 



whence by separation of real and imaginary parts, and putting 

 x equal to unity, 



+ [5 5.9.13 + 5.9.13.17.21 " ' j> 



[5 5. 9.13 + 5. 9. 13. 17. 21 " " " J" 

 Calculating from these series I find 



f 1 fl 4 . . 1-36704 f l . ,, 4N _ 



(7) 



(8) 



•21352 



ll cos(i7r^ 4 )^l + r(sin(iTO 4 )^l =-9576. 

 L Jo J LJ J 



Again, 



I cos (!7r# 4 )da?= -87704, I sin (i7r^ 4 )^ = '26812, 



Jo J© 



[I 'cos^tt^ 4 )^] + [" j W(j9T^*)d^l =-84109. 

 Jo Jo 



Again, 



I cos(tt^ 4 )^=-64357, I sin (va^)daf= "33363, 



J Jo 



Fl cos(7r^ 4 )^J + ("l sin (t^ 4 )^! =-52549. 



Thus an extreme aberration of one eighth of a period reduces 

 the intensity at the central point from unity, corresponding to 

 no aberration, to -9576, With an aberration of one quarter of 

 a period the intensity is '84109 ; and with an aberration of half 

 a period the intensity is reduced to '52549. We must remem- 

 ber, however, that these numbers will be sensibly raised if a 

 readjustment of focus be admitted. 



In most optical instruments other than spectroscopes the 

 section of the beam is circular, and there is symmetry about 

 an axis. The calculation of the intensity- curves as affected 

 by aberration could be performed by quadratures from tables 

 of Bessel's functions ; but, as in the case last considered, the 

 results are liable to a modification in practice from readjust- 

 ment of focus. For the central point we may obtain what we 

 require from a series. 



