PROCEEDINGS OF THE SOCIETY. 125 



ing amplitude becomes zero, which is manifestly absurd in the case of 

 two unequal amplitudes. 



The correct substitute for these two formulas is, adopting Mr. Gordon's 

 symbols, 



/ 2 IT 



A (i + 2) = V Al2 + A 2 2 + 2 Al A2 C0S X ^ " ^ 



an equation which possesses the necessary symmetry with regard to the 

 quantities applying to the two combining waves, which satisfies common 

 sense inasmuch as it shows that both waves invariably contribute to 

 the result, and which fully explains every possible case in accordance 

 with experience. 



The phase angle -r- <£ of the resulting wave, which Mr. Gordon 



A 



does not even attempt to deduce, is obtained quite definitely from thu 

 two equations — 



A(i + 2 ) cos — ^ <f> = A x cos ^>i + A 2 cos ~'.<f> 2 . 



AAA 



•> _ •> _ 2 7T 



A(i + 2 ) sin ^— <£ = A x sin l_ ^ -f A 2 sin — <f> 2 . 



Seeing that these three equations contain the complete and only 

 possible solution of this simplest problem in the study of interferences, 

 and that they are, therefore, to be found in the earliest chapters of 

 any book on the mathematical treatment of such problems, it is highly 

 significant that they should be unknown to Mr. Gordon, and that he 

 should have found it necessary to invent those absurd and incomplete 

 substitutes. It seems hardly worth while to examine any further attempts 

 at mathematical proofs from one so ill-prepared for such tasks, but 1 will 

 follow him a little further. 



Having, in the face of Airy's results, derived conclusions to his liking 

 from the assumption of a " polyphasal antipoint," he attempts to prove 

 the existence of such " antipoints " in an appended note. And here he 

 •entirely brushes aside the well-established principles of the undulatory 

 theory, and calmly suggests that Airy should have integrated not over 

 the aperture which passes the light, but over a small portion of the cone 

 which encloses the wave-train converging towards the focus. Surely a 

 more startling proposition has never been made. We have energy in 

 the form of light being transmitted in converging waves towards a focus, 

 and we are to disregard nearly the whole of that energy and of those 

 waves, and are to confine our attention to a narrow strip down the side 

 of a cone — which is indeed " polyphasal " with a vengeance, but which 

 conveys only an infinitesimal fraction of the total energy. 



It need hardly be stated that even a correctly worked out result based 

 on such assumptions would be entirely worthless, but the mathematical 

 expressions which follow have not even that merit, for, from what has 

 just been pointed out, it is clear that the correct result of the proposed 

 integration must necessarily be an infinitesimally small amplitude, whilst 



