TRANSACTIONS OF TIIE SECTIONS. 



23 



viz. the numbers in the first two columns are primes, and the numbers intermediate 

 to the lower limit and the upper limit are all composite. It is to be noted that the 

 above list is not complete, as the method pursued does not, of course, give all the 

 sequences. The two long sequences in the first million seem to me very remarkable. 



Lists of sequences of 79 and upwards in the first million, and of 99 and upwards 

 for the other five millions, are given in the ' Messenger of Mathematics/ vol. vii. 

 pp. 102-106 and 171-176 (November 1877 and March 1878), where also are to be 

 found several other tables of a similar kind, including a complete list of sequences 

 exceeding 50 between 1 and 100,000. 



Tho following sequences that occur very early in the series of natural numbers 

 deserve to be specially noted, viz. — 



The next sequence, which is so large as 33, after that between 1327 and 1361, 

 occurs between 8467 and 8501, where there is another sequence of 33. 



It is known that any sequence of composite numbers, however long, must occur 

 at a certain definite pface in the series of natural numbers ; for p and q being any 

 two consecutive primes, the q— 2 numbers immediately following (2 . 3 . 5 . 7 . 11 

 , _ ,p)-\-\ must be all composite ; but the number (2.3.5.7... 109) +1 (immedi- 

 ately following which there must be 111 composite numbers) contains 45 figures, 

 whereas a sequence of 111 actually occurs at 370,261 ; so that long sequences are 

 met with far earlier than this theorem requires them to happen. 



On the Variation of the Modulus in Elliptic Integrals. 

 By Dr. D. Bebrens de Haan. 



1. The following results can be easily verified : — 



these give the symbolic equations 



\l-2p-^\v( P , X )=F(p,x), 



r, a , J I-. , 1 [~tv >, l + (l-?r)sin 2 x 2 . "I 



Hence, denoting the operation 



by P, and 



