102 PROFESSOR AIRY ON THE CALCULATION OF 



of course in any particular case, when by very tedious summation nume- 

 rical results might be obtained). The only thing which can be attempted 

 is, to shew that the integrals are precisely the same as those that occur 

 in a very different instance where Fresnel's method of experimenting 

 is adopted. Even thus far however I have not succeeded except in one 

 case, namely, where the hole is a rectangular parallelogram of any length, 

 and where the diffracting aperture is also a rectangular parallelogram 

 in a similar position ; including in this general case the particular instance 

 in which one or both parallelograms have no boundary on one side. 



To consider, in the first place, a case similar to Newton's. A plane 

 wave is supposed to enter an external parallelogram and then to pass 

 through a slit with sides parallel to those of the parallelogram ; and the 

 intensity of the light which falls upon a screen at a given distance is to 

 be found. First, it is to be observed, that in estimating the comparative 

 intensity of light in a direction parallel to one side of the parallelograms 

 (suppose for instance the shorter) there is no necessity to take into ac- 

 count the length of the parallelograms in the other direction ; as it will 

 easily be seen, upon attempting an integration, that the intensity of light 

 is expressed by the product of two quantities, of which one depends only 

 on the lengths of the parallelograms and the position of the point of 

 the screen in one dimension, and the other depends only on the breadth 

 of the parallelograms and the position of the point of the screen in the 

 other dimension. The intensity of light along a given line parallel 

 to one side of the parallelogram will therefore, so far as it depends on 

 the other side, be affected only with a constant multiplier. Neglecting 

 therefore the lengths (by which term I designate that dimension of the 

 parallelograms which is perpendicular to the line on which the comparative 

 brightness is to be ascertained), suppose a normal to the front of the 

 wave to be di-awn, and suppose the limits of the breadth of the external 

 aperture measured from this line to be a, fi, (the distance of any point 

 of the aperture being v), and suppose the limits of the breadth of the 

 slit to be 7, 5, (the distance of any point of the slit being w)'. and 

 suppose the distance of the point on the screen, whose illumination we 

 wish to ascertain, to be x. Let the distance of the external aperture 

 from the slit be a, and the distance of the slit from the screen h. Suppose 



