Astronomical and Nautical Collections, 437 



more slowly than the angles themselves, while they remain 

 inconsiderable. 



If, in fact, we consider rays sensibly inclined to each other, 

 such as EP, FP, IP, meeting in the point P, which we may 

 suppose at the distance of a great number of breadths from 

 the undulation EA : and if we take two arcs, EF and FI, of 

 such a length that the differences EP — FP and FP — IP may 

 be equal to half an undulation : on account of the marked 

 obliquity of the rays, and of the smallness of a semiundula- 

 tion, in proportion to their length, these two arcs will be 

 almost equal, and the rays which come from them to the 

 point P will be nearly parallel ; so that on account of the 

 difference of a semiundulation between the corresponding 

 rays of the two arcs, their effects will mutually destroy each 

 other. 



We may therefore suppose all the rays sent by the differ- 

 ent }:>arts of the undulation at AE, to the point P, to be of 

 equal intensity, since the only rays, with respect to which 

 this hypothesis would be incorrect, are such as have no sen- 

 sible influence on the quantity of light which it receives. 

 For the same reason, in order to simplify the calculation of 

 the result of all these elementary undulations, we may consi- 

 der their constituent motions as performed in the same direc- 

 tion, the angles which they form with each other being 

 inconsiderable.. The problem is thus reduced to that which 

 has been solved in the Memoir on Diffraction, already 

 quoted : To find the result of any number of systems of 

 parallel undulations of lights of the same frequency , when 

 their mtensities and relative situations are given. — The 

 intensities are here proportional to the length of the small 

 illuminating arcs, and the relative situations are given from 

 the differences of the paths described. 



We have considered, correctly speaking, only the section 

 of the undulation made by a plane perpendicular to the mar- 

 gin of the screen represented by A. We may now take into 

 account the whole extent of the undulation, and suppose it 

 to be divided, by equidistant meridians perpendicular to the 

 plane of the figure, into infinitely thin wedges or strata; 

 and we may apply to all of these the reasoning which has 



