OBJECT-GLASS WITH A TRIANGULAR APERTURE. 437 



From the necessity which thus exists of taking in at once the whole 

 of the expression for Z at every step of our examination, we shall be 

 obliged to feign several cases, and effect corresponding expansions and 

 reductions for each : and from these particular results infer, in the best 

 manner we are able, the general appearance and brightness of the image 

 upon the screen. 



1. Let us suppose r and therefore m extremely small. This will be 

 true of parts very near the centre of the screen. In this case we must 

 expand the expression for Z in a series of terms arranged according to 

 the powers of m. This may be effected most readily as follows. 



If f, = sin Q, f, = sin (60° - Q), and f, = - sin (60» + 9), f„ f^, f, wiU 

 be the roots of the equation 



a? - %x + i sin 30 = 0. 



And that part of the expression for Z which is enclosed within the 



brackets is equal to 2 ( — tH^ J . 



.j^ sin- mf _ 1 — cos 2 mf 



... x(S!^) = ....(/,-«..(^).3-|=L,..(^,_... 



By the usual method of finding the sums of the powers of the roots 

 of equations, we easily find 



2(/) =0, 



2(/=)=-i|.sin30, 

 2(/')=-|J-sin30, 



&c. = &c. 



3k2 



