262 



Mr Mayall, On the Diffraction Pattern [Feb. 22, 



quantity, the series above are absolutely convergent. The last 

 line is what we should obtain by integrating by parts the integral 



[•CO 



iire^ z I e~ iiLZ J (z) dz. 



Hence we have 



/ e iz COS * 9__^ = ^gi^Z I Q - i^z J / g \ fa 



Jo /i-COS0 J z 



fjb — COS tfi 

 1 + K 2 



and putting — - — = /j, in (1) it becomes 



V + iV 1 = ^J (z) + i l ^e^jj-^J (z)dz (2), 



in which k, and therefore \i, are nearly equal to unity, while z is 

 large. 



To get an approximation to the value of the integral on the 

 right-hand side of (2) we may use the approximate expression 

 for J 



«*.(*) = a Az: 



irz 



7T \ 1 . /7T 



with which value the integral becomes 



'2 



Q—lf-Z 



IT 



ens I -— z 



1 . (IT 



— sm T " — z 



8z \4 



dz 



*Jz 



(3). 



This may be expressed in terms of Fresnel's integrals 



/"" 1 f" 1 



P = J cos g 7ry%, Q = J sin ^ ^2/%, 



the values of which have been tabulated by Gilbert* from v = 01 

 to v = 5'5. For we have 





I \dz If . I i(j-) -<(l— )) & 



i 



e-^+D* — +- e l 



\]z 2 



e -i(lj.-l)z ...(4). 



\] Z 



If we neglect terms in z the first of the integrals in (4) is 



* I/em. couronnes de VAcad. de Brnxelles vol. xxxi. 



