214 Proceedings of the Royal Irish Academy. 



Art. 6. Eciucdions for a range of p from r to h corre&ponding to those of 



Arts. 3-5. 



If the left-hand member of (16) be altered by changing the lower Ihnit 

 of p to r, and the upper to h, where r < h, its value becomes 



' y <^ (''■ + ; 



the proof is similar to that given of (16), but the second form of the left-hand 

 member is to be employed instead of the first. 



And, if the same change of limits of p be made in the left-hand member 

 of (27), the value of the new integral is seen to be zero also, the argument 

 which was applied to (27) being absolutely unaltered. 



By combining these results, we obtain the two equations — 



c J - 



XdX 



XdX 



J,fXr)J,f\p)p^{p){dp) = 1(1 -,=«-■) ^(r + £), (31) 



J.n{Xr)J.nQ<p)p<f>{p)dp = |(l-g-^"-Xr + £): (32) 



r being supposed < h. Of course, if r = h, each integral is zero. 



And, of course, by adding (29) and (31), (30) and (32), w^e obtain (14) for 

 + n and for - n respectively. 



When, in (14), r is made > h, or < a, we have only to extend the limits of p 

 so as to include /■ ; but, outside the range from a to &, replace ^ [p) in the 

 integral by zero, and the required result is included. 



Art. 7. Forms assumed hj preceding Uquations when Orders of Functions are 



Integral. 



In the preceding investigations, it has been supposed that n is not an 



integer. In this excluded case the limiting forms of the above results are 



to be taken. 



"We mav write -r^ ,. . , .,„ tt ,,, , , • -r / m ,^,^s 



- Kn {ix) = (- if - [ Yn (x) + iJ„ (x) } , (33) 



- K„ {- ix) = iir \ [Yn (a:) - iJn ix)], (34) 



where ttY,, (re) = 2 • Y„ (x) + {y - log 2) J„ (x) ] .* (35) 



The limiting forms obtained directly from (16), (27), of the original 

 equation (14) for + n and - n give, when oi is an integer, in addition to 



* The notation is tliat ot Gray and Mathews, " Bcssel Functions." 



