into Uniform Undulations of Flat Wavelets. 273 



14z. To quite obliterate these glimmers requires, in fact, 

 more light to be thrown upon plane K than is furnished by 

 the undulations of the sheaf, where these are all alike, as we 

 have supposed them to be. The requisite additional light, 

 when resolved into its component u f w's, would furnish on 

 the indicator-diagram certain out-lying and on-lying appen- 

 dages * to the rectangular macula of the sheaf, of the kind 

 described in a paper " On the Cause of Spurious Double Lines 

 and of Slender Appendages," read at the Oxford Meeting of 

 the British Association (see B. A. Report for 1894, p. 583). 

 But usually the light that is indicated by these appendages 

 in the indicator-diagram is a very small proportion of the 

 total light, and in many practical cases need not be taken 

 into account. Whenever it is legitimate to exclude them 

 from consideration, we may regard the sheaf of undulations 

 as producing illumination upon plane K, only within the 

 central patch of fig. 8, which is shut in by dotted lines. 



15. When, as more frequently happens, the macula of a 

 sheaf of undulations is a circular disk or oval instead of a 

 rectangle, the modification of the above analysis is obvious. 

 Airy's investigation of the image of a star in a telescope is a 

 particular case of the above more general theorem, viz. the 

 case where the macula is a circular patch situated exactly in 

 the middle of the indicator-diagram. 



10'. From Theorem VIII. the following one may with ease 

 be deduced. 



Theorem IX. 



When light reaches the point k in the same phase fyc, from 

 tivo or more sheafs of u f w's, represented on the indicator- 

 diagram by separate maculce, then, as before, the illumination 

 of plane K is chiefly ivithin a central patch surrounding k 

 such as that represented in fig. 8 ; but this central bright 

 patch now consists of a luminous ruling of parallel bands if 

 there are two sheafs, this ruling being finer the more distant 

 the two maculae are on the indicator-diagram. Similarly, if 

 there are three maculce not in a straight line, the bright patch 

 of fig. 8 is resolved into rows of bright specks ; and so on. 



All of these effects can be shown experimentally. 



17. When we have occasion to deal with a very small 



* The demand made by nature for the presence of these appendages 

 is a requirement analogous to the mathematical necessity of substituting 

 a series of sines and cosines of which the ares are not commensurable 

 for the ordinary Fourier's series, when the series is to represent an 

 isolated event without repetitions: i.e when beyond that event the 

 sum of the series is to be everywhere absolutely cipher. 



Phil. Mag. S. t>. Vol. 5. No. 26. Feb. 1903. T 



