TRANSACTIONS OF SECTION A. 



321 



3. Report of the Committee to aid in Establishing a Solar Observatory in 

 Australia. — See Keports, p. 25. 



4. Report on Magnetic Observations at Falmouth Observatory. 

 See Reports, p. 78. 



FRIDAY, SEPTEMBER 1. 



Discussion on the Principle of Relativity. Opened by E. Cunningham. 



See p. 236. 



Department of Mathematics. 



The following Papers and Eeport were read : — 



1. On Mersenne's Numbers. By Lieut.-Col. Allan Cunningham, R.E. 



These are of form M,= (2« — 1), with q prime. 



A pretty full account of all that was then known about these numbers was given 

 by Mr. W. W. Rouse Ball in a paper in ' Messenger of Mathematics,' vol. xxi., 1892 ; 

 three omissions occur in that paper (about numbers proved by Ed. Lucas), viz., 



M 67 , M 89 , known to be composite ; M U7 known to be prime. Since 1892 the progress 

 has been as follows : — 



M 67 =1937077121-761838257287; due to F. N. Cole 

 M 7l =228479-10334355636337793* ) due to the ( 

 M I63 =0 (mod. 150287) . . I present J 

 M 197 = (mod. 7487) . . .J writer I 

 M 181 = (mod. 43441) ; due to H. J. Woodall . 



1903 

 1909 

 1908 

 1895 

 1911 



This leaves still unverified (as composite) only 15 out of the 44 numbers (with q < 

 257) originally affirmed by Mersenne to be composite, viz., when 



g=101, 103, 107, 109, 137, 139, 149, 157, 167, 173, 193, 199, 227, 229, 241. 

 A complete list of all the possible prime divisors < 1 million of these 15 still unveri- 

 fied numbers has been prepared by Mr. A. Gerardin (of Nancy, France) and the 

 author jointly (but working independently). These ' trial divisors ' have been 

 tested by the author up to 500,000 without success (every ' trial divisor ' was 

 tried twice). 



1911. 



The composition of this large divisor is not known. 



