284 PROFESSOR AIRY, ON THE DIFFRACTION OF 



to the axis of the telescope) from the focus. Then, the lens being 

 supposed aplanatic, and a plane wave of light being supposed incident, 

 the immediate effect of the lens is to give to this wave a spherical 

 shape, its centre being the focus of the lens. Every small portion of 

 the wave, as limited by the form of the object-glass, must now be 

 supposed to be the origin of a little wave, whose intensity is propor- 

 tional to the surface of that small portion ; and the phases of all these 

 little waves, at the time of leaving the spherical surface above alluded 

 to, must be the same. If then Sx x Sy be the area of a very small part 

 of the object-glass, q the distance of that part from the point defined 

 by the distance b, the displacement of the ether at that point, caused 

 by this small wave, will be represented by 



Sx X. Sy X sin—- {vt — q — A) ; 



A 



and the whole displacement caused by the small waves coming from 

 every part of the spherical wave will be the integral of 



sin — (vt—q — A) 



through the whole surface of the object-glass, q being expressed in 

 terms of the co-ordinates of any point of the spherical surface. 



Now let X be measured from the center of the lens in a direction 

 parallel to i; y perpendicular to x and also perpendicular to the axis 

 of the telescope; and % from the focus parallel to the axis of the 

 telescope. Then 



q=.^{{x- by + y- + x} = -y/ix' +f+x'-2bx) 



omitting squares and superior powers of b. But x^ + y^ + z' —f^^ 

 since the wave is part of a sphere whose centre is the focus ; therefore, 



q = VW^-^bx)=f-j,x nearly; 

 and the quantity to be integrated is 



sm— \vt - f - A + -x). 

 ^ J 



