286 PROFESSOR AIRY, ON THE DIFFRACTION OF 



If we make - = w, — — . -2r = n, the expression becomes 

 2a^.sm-—{vt-f~A)J^V^-uf'Cosnw, fromw=— 1 tow=+\, 



A 



or 4«^ sin — - {vt-f~ A) j„\/l — tt;''. cos nw, irom w = to w = l. 



A 



It does not appear, so far as I am aware, that the value of this 

 integral can be exhibited in a finite form either for general or for 

 particular values of w. The definite integral 



J„^/\ — vf . cos nw (from w=-0 to w = \,) 



(which will be a function of 7i only) being expressed by N, it may be 

 shewn that N satisfies the linear differential equation 



n ' dn dv? ' 



which may be depressed to an equation of the first order that does 

 not appear to yield to any known methods of solution. 



If we solve the equation by assuming a series proceeding by powers 

 of n, or if we expand cos nw and integrate each term separately, we 

 arrive (by either method) at this expression for the integral 



TT . rf_ n^ _ _ "" Xr \ 



4 "" ^ 2.4"^2.4^6 ^:^\Q'.S^^^-' 



The table appended to this paper contains the values of the series 

 in the bracket, for every 0,2 from w=0 to w = 12. Each value has 

 been calculated separately, the logarithms used in the calculation have 

 been systematically checked, and the whole process has been carefully 

 examined. The calculations were carried to one place further than the 

 numbers here exhibited. I believe that they will seldom be found in 

 error more than a unit of the last place; except perhaps in some of 

 the last values, where the rapid divergence of the series for the first 

 five or six terms made it difficult to calculate them accurately by 

 logarithms. 



