274 Dr. G. J. Stoney on the Resolution of Light 



sheaf of u fw's, it is often convenient to substitute a single 

 undulation for the whole of the little sheaf. That this is 

 legitimate when the macula of the sheaf is small enough, 

 may be proved as follows. The guide-cone of the sheaf is, 

 by supposition, a very acute one. Draw a line from its 

 vertex, within the cone. This we may call its axis. Let J 

 be the corresponding point in its little macula. Let u be any 

 one of the undulations of the little sheaf, and let I x be its 

 index-point. Along the axis imagine two 

 undulations w 1 and 17/ to be sent, with the -"&• ' ) - 



same wave-length, state of polarization, and ( s ^ J 



intensity as « l3 but of which t\ reaches k in 



H 

 the same phase as u u and i\ f in the opposite ^ 



phase. Under these circumstances v 1 and \~J 



r/ cancel one another, so that their addition The small macula, 

 to the system makes no change. Now u^ (Part of indicator- 

 and Vi produce a luminous ruling on plane diagram.) 



K of which one of the lines of cipher illu- 

 mination passes through l\ and of which the spacing is 



where d is the very short distance from 1 Y to J. If this 

 distance is short enough the nearest maximum of brightness 

 on either side will be so far from k, that within a limited 

 field of view, such as is seen on looking into an optical 

 instrument, there will be no appreciable light from this 

 ruling. And if so, we may omit u ± and 1;/, and r : alone 

 remains. Similar substitutions of v 2 , r 3 , &c, all of them 

 advancing along the axis, may be made for u 2 m$, &c, the 

 other undulations of the little sheaf. When this has been 

 done, all these v's may be combined into a single resultant 

 v travelling along the axis. This establishes Theorem X., 

 which is as follows : 



Theorem X. 



When dealing ivith a limited field of view we are at liberty 

 to substitute the resultant v for any very small sheaf of undu- 

 lations, and this substitution is legitimate, however unrelated 

 the phases, states of polarization, and intensities of the undu- 

 lations in the little sheaf may have been. 



This is the first of the two substitutions spoken of above in 

 § 4, which are of the kind that must be legitimate to justify 

 our applying to nature conclusions obtained by mathematics 

 from data which are of necessity almost immeasurablv 

 simpler than the complex details that nature everywhere 



