26 Proceedings of the Royal Irkh Academy. 



Values of K,{x\ E^, U^)^ ^^(^) >^ 

 We have seen that, as x increases indefinitely, K^x) approaches 



the limit - , , Ex lo -7=' 



y^TTX V 273-^ 



Substituting in the first expression for Y^x) we find 



••• ^'W^-J- 



Assume now for x large 



1 



Then ^(e) satisfies the differential equation 



whence writing cf>{%) = \ + a,z . . . a„,z-, we have 



{ 2m + ly 

 8 {m + 1 ) 



giving rise to a semiconverging series. 

 Similarly assuming _ 



Y,{x) = -'^^.^^^l{^), 



we find, putting c/)/(s) = I + hz + . . . - 



{2m 



giving another semiconverging series. 



We thus find the series for large values of x 



"^ ^'j2.j^r 8 128 i' 



