ON MATHEMATICAL TABLES. 



49 



The numbers of primes in each quarter million in the first four mil- 

 lions are : — 



The following is a list of successions of composite numbers of ninety - 

 nine and upwards occurring in the fourth million : — 



Sequences of 99 and Upwards. 



The meaning of this table is that the 109 numbers between 3,064,751 

 and 3,064,861 are composite, and so on — 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. The above list is not given 

 as being necessarily complete, although it probably includes all, or 

 very nearly all, the sequences of ninety-nine and upwards. 



Similar lists of sequences for the other millions are given in the 

 'Messenger of Mathematics,' vol. vii. pp. 102-106; 171-176 (1877 and 

 1878). 



It may be mentioned that the longest sequence met with in the first 

 million was 113, in the second 131, in the third 147, in the fourth 137, 

 in the seventh 145, in the eighth 147, and in the ninth 151. 



II. — Tables of the Legendrian Functions. 



The tables contain the values of P"(aj) for n=l, 2, 3,... 7 from x = 

 to x = 1, at intervals of 0*01. The functions tabulated are 



P°0)=1, 

 P 1 (x)=x, 



1879. E 



