446 ME. J. MERGER: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE. 



It follows that 



(**(*,*)<&= 2 -L (m=l,2, ...) 



Ja ii = l "n 



31. In conclusion, it may be pointed, out that the theorem ot 29 holds also when 

 K (a, t) is of negative type. This may be deduced from the theorem mentioned by 

 employing the usual device, or it may be proved directly by commencing with the 

 equation 



Lt [K, (,, *) - R M (X ; a, ,)] = K (*, *) - 2 feifll 



X-^-oo n = l A n 



instead of tluit at the beginning of 27, and proceeding by a method similar to that 

 which we have used above. 



It may also be of interest to remark that by a very slight modification of these 

 proofs we may show that (27) represents , (*, t) when the latter has only a finite 

 number of singular values of one sign, but an unrestricted number of the other. 



