Operation Groups of Order ftp, p being any prime number. 195 



integration with respect to x. The result may be written 



2ir*§da:di/ (75) 



upon the understanding that, while the integration for y 

 ranges over the whole vertical aperture, that for x is limited 

 to such values of x as bring x + s (as well as x itself) within 

 the range of the horizontal aperture. The coefficient of the 

 Fourier component of the intensity involving cos sp, or 

 cos (2r7rp/p 1 ) , is thus proportional to a certain part of the 

 area of the aperture. Other parts of the area are inefficient, 

 and might be stopped off without influencing the result. 



The limit to resolution, corresponding to r= 1, depends only 

 on the width qf the aperture, and is therefore for all forms of 

 aperture the same as for the case of the rectangular aperture 

 already fully investigated. 



If the object be a row of points instead of a row of lines, 

 <? = 0, and there is no integration with respect to it. The 

 process is nearly the same as above, and the result for the 

 coefficient of the rth term in the Fourier expansion is pro- 

 portional to \y*d.v, instead of yy dx, the integration with 

 respect to x being over the same parts of the aperture as 

 when the object was a grating. The application to a circular 

 aperture would lead to an evaluation of 



I 



+ ° Jl 2 M C0S S1( 7 



XVI. The Operation Groups of order 8p, p being any prime 

 number*. By GL A. Miller, PLB.-f 



ACCORDING to Sylow's theorem these groups contain 

 kp + 1 conjugate subgroups of order p and 6>1 in the 

 equation 



-§p -h P . 



kp + 1 



Hence they must contain a self-conjugate subgroup of this 

 order when jo>3 and p^7. We shall first consider all the 

 possible groups that contain such a self-conjugate subgroup, 



* M. Levavasseur gives an enumeration of these groups, without ex- 

 plaining how they were obtained, in Comptes Rendus, March 2, 1896. 

 His enumeration is not quite correct. He states that there are three 

 groups which exist only whenp — 1 is a multiple of 4 without being also 

 a multiple of 8. We shall prove that there are only two such groups. 



"f Communicated by the Author. 



