10 Transactions of the Society. 



In like manner the aperture A may be expressed in terms of 

 the divergence angle u thus 



A = 2 tan u (c . . . rj) . . . . (4) 



and dividing (3) by (4) we obtain for a parallel beam of light the 

 following equation between the diameter of the aperture and the 

 diameter of the unfocussed antipoint — 



V • • • Vi t an /"">\ 



— — . a . . . V-*/ 



A 2 tan u 



V. Helmholtz deduces for the case of the spherical wave-front 

 focussed upon rj a law from which it may be inferred that in that 

 case the expression (5) becomes 



p_ = jin0 (6) 



A 2 sin u 



where p is written for the radius of the antipoint, or rather of 



that zone in the antipoint which is formed by the focussing of those 



diffracted beams which have the retardation = <f>, and is, therefore, 



a general expression for (77 . . . rj x ). 



This last expression can be further simplified. For in the case 



of an aperture with straight parallel edges, the value of 6 in a 



plane perpendicular to the edge is known to be, as above stated, 



such that , 



sin - ?., 

 A 



whence <b s*r\ 



p = h— £ — » CO 



2 sin u 



and this in the case of a circular aperture becomes 



P-l"2 5-*- (8)* 



2 sin u 



When p is the radius of what is commonly called the false disc 

 the phase value <f> is equal to one complete cycle of phase change, 

 and may therefore be expressed by one wave length. Thus we 

 obtain the well-known expressions 



P — s— • — or p = 1*2 . -, 

 2 sin u 2 sin u 



according as the aperture is limited by a rectilinear or by a circular 

 boundary. 



1 - 2 \ 



* The expression - is very approximately correct, if we adopt Sir Geo. 



2 sin u 

 Airy's equation for the antipoint light curve, for the inner zone3 (say those 

 within the boundary of the false disc) of the circular antipoint— with which alone 

 we shall be concerned in the present paper. In the outer zones the circular anti- 

 poirt tends to conform to the dimensions of the rectilinear antipoint, that is to say,. 

 A 

 2 sin u 



