215 



If 2an is summable of order p -\- ^ and its mean-values of order 

 p are limited, if Sh„ is summable of order q -\- \. and its mean- 

 values of order q are limited, then tlie product-series is summable 

 of order p -\- q -\- "i, . 



This theorem is an extension of Cauchy's theorem that the product 

 of two absolutely convergent series is convergent, analogous to the 

 extension of Mektens' theorem given in theorem 3, and, like Mertk^s' 

 theorem implies that of Oauchy, so theorem 3 implies that of 

 Rosenblatt. 



