564 



REPORT — 1894. 



and these are the only values of p yielding Lucasians in the range of 500 numhera 

 between 4,490,500 and 4,500.000. An interesting group is given by 



^-2' + 3 = ll;;) = 2'+3 = 131; i) = 2'5 + 3 = 32,771 ; 



and these are the only numbers of form (2^ + 3) yielding Lucasians when 

 x not>26. Higher values go beyond the tables of primes. 



Complete, list of primes p of form {4« + 3), with (2jW + 1) also prime, when 

 p not >2,500 ; these all give composites for N, and {2p + 1) is a factor of N. 



M= 11, 23, 83, 131, 179, 191, 239, 251, 359, 419, 431, 443, 491, 059, 683, 710, 743, 

 911, 1,019, 1,031, 1,103, 1,223, 1,439, 1,451, 1,499, 1,511, 1,559, 1,583, 

 1,811, 1,931, 2,003, 2,039, 2,063, 2,339, 2,351, 2,399, 2,459. 



It seems probable that primes of one of forms p=(2^ + l), (2^ + 3) will, with 

 exception of those yielding Lucasians, generally yield prime vahies of N, and thp.t 

 no others will ; all the known (and conjectured) prime Mersenne's numbers fall 

 under this rule. 



9. End Games at Chess. By Lieut. -Colonel Allan Cunningham, R.F.^ 

 Felloio of Kiny^s College, London. 



Investigation of the number of positions in all the ' end games ' at chess when 

 ibere are only two or three pieces on the board. The results are : — 



P = Total number of positions 

 C = Number of checkmate positions 

 S = Number of stalemate positions 

 I = Number of indifferent positions 



with a given 

 set of pieces. 



Depaktment II. 



10. Experiiiients shoiving the Boiling of Water in an open Tube. 

 By Professor Osborne Reynolds, F.R.S. 



11. Report of the Committee on Earth Tremors. — See Reports, p. 145. 



12. Report of the Committee on Meteorological Photography. — 

 See Reports, p. 143. 



13. Report of the Committee on Solar Radiation. — See Reports, p. 106. 



