168 PROFESSOR KELLAND, ON THE 



Now the circular functions in this expression are 



/> x IT 27T ff + f 



(COS aj — COS ap) COS ap, Calling — j~ > «• 



But by Laplace's Formula, 



and the integral between our limits is merely the double of this ; 

 .•. the integral of the term 



ilfiV cos op = A {cos afe' a ^~ l - i - £e~ 2aA/rT ! 



= A {cos af cos af— <J — 1 sin af — ± — ^ (cos 2af- •/ — 1 sin 2a/*) I 



= A {COS 2 a/* - i — 1 COS 2af— <s/ — 1 (sin a/cOS a/ — ^ sill 2a,/)} 



= 

 a very remarkable result. 



If M be not in the plane COP, there is a factor — in M and JV, 



which amounts to the factor f I — j dq or -g- in the final result. 



/^sin <7A\ 2 7 ^^ 



* 1 ri n r\w 



q 



We find then that the whole intensity is the sum of the intensities 

 due to each of the mirrors separately. Should the form of the function 

 M* as integrated for the whole space be objected to, the only reply is, 

 that one or other of two things must be supposed ; either 1° that 

 the integration for spaces perpendicular to the plane of the paper would 

 take away X, or 2° that the intensity is a function of the length of the 

 wave. In either case our conclusion is correct. There is evidently some 

 factor required to render this result of the same dimensions as that with 

 which we set out. Perhaps I am not warranted in assuming, from the 

 coincidence of my results with the principle of vis viva, and their con- 

 sequent probability, that this factor is not variable from point to point. 

 When the question first arose in my mind respecting this matter, I thought 

 to answer it at once by an appeal to the transformations effected by the 

 " Differential Calculus to any indices." Although the result of this appeal 

 is very far from satisfactory, I do not think it will be deemed an un- 

 pardonable digression to take it here. 



