into Uniform Undulations of Flat Wavelets. 269 



to project the guide-points into index-points on to the base- 

 plane, whereas in experiments the projection is usually 

 exhibited on some parallel plane. The positions of the index- 

 points — /. e. o£ the projections o£ the guide-points — are the 

 same in the resultino- diagram, whether it is formed on the 

 base-plane or on any parallel plane. 



10. Accordingly, if u u u 2 , &c. are the u f w's we have to 

 deal with, then the directions in which they are advancing 

 and the orientations of their wave-fronts, are adequately 

 represented by the index-points I l5 I 2 , &c. got by the above 

 construction ; and whether the projection by which we arrive 

 at the index-points is a projection on to the base-plane or on 

 to some parallel plane, they occupy the same positions within 

 a circle in that plane which is the projection upon it of the 

 flat side of the reference-hemisphere, and which, therefore, 

 has R for its radius. This circle and its contents may be 

 called the Indicator-diagram. 



11. By applying simple geometrical considerations to the 

 foregoing construction, we get the following propositions : — 



Fig- 2. 



Indicator-diagram. 



Let Ui and u 2 be two in ward -bound u f w's, consisting of 

 waves similarly polarized and of wave-length X : and let I 1 

 and I 2 be their index-points upon the indicator-diagram. 

 Then a simple geometrical proof furnishes the following 

 elegant theorem. 



© v 



Theorem VII * 



The light which these two ufw's throio <>» the base-plane, 

 or on any parallel plane, is a luminous riding perpendicular 



* The propositions in this paper are numbered in succession to those 

 in the paper at p. 570 of the B. A. Report for L901. 



