2. The improper integral defined as the limit of a sum. We first consider the case where ak = 1 (k =1, 2, 3, * * * ). A fundamental difficulty not encountered in the previous paper is apparent from the definition of .fof(x)dx. Bromwich and Hardy have considered this problem [2]. As they point out, the ordinary definition of this improper integral is by means of a double (repeated) limit; and in order to replace this limit by the single limit,