1868.] On the Mysteries of Numbers alluded to hij Fermat. 251 



And he submits for consideration whether it may not be desirable to try- 

 two shorter wires, the two ends of each wire making connexion with the 

 earth on opposite sides of the Observatory, and the register of each being 

 made, at the Observatory, near the middle of its length. 



February 13, 1868. 

 Lieut. -General SABINE, President, in the Chair. 

 The following communications were read : — 



I. On the Mysteries of Numbers alluded to by Fermat." — Second 

 Communication. By the Right Hon. Sir Fredertck Pollock, 

 Bart., F.R.S. Received January 14, 1868. 



(Abstract.) 



This paper is not adapted to be read in extenso ; so much of it is con- 

 nected with mere calculation, so much more of it requires continual re- 

 ference to diagrams, that no adequate knowledge of its contents would be 

 acquired by merely hearing it read aloud ; but a statement has been pre- 

 pared of what it contains which will give a general view of the result. 



The properties ascribed to all odd numbers, in addition to those con- 

 tained in Fermat's theorem, are these: — 1st. The algebraic sum of the 

 roots in some form of the 4 squares which compose the number will equal 

 1, 3, 5, 7, &c. (every odd number which it is large enough to produce) ; 

 2ndly, the difference between some 2 of the roots will be any odd or even 

 number whatever, subject to the same limitation. 

 1 3 5 7 9 



The series 1, 3, 7, 13, &c. (n, n, n-\-l) will give 1, 3, 5, &c. as the sum 



2 4 6 8 



of the roots of its terms ; and each term is the smallest that will give that 



13 5 7 



amount. So 1, 5, 13, 25, &c. is the series whose terms are the smallest that 



4 8 12 16 



give the odd numbers as a difference of the roots, and 1, 3, 9, 19, &c. that 



2 6 10 



give the even numbers. And these are the three series that compose The 

 Square (the subject of the last paper) when the 1st term is 1 ; and they 

 are the cause of its properties. A portion of the paper is devoted to an 

 investigation of the change effected in the sum of the squares, by a change 

 in the roots. If 2 roots differ by 7i, they may be represented by a and 

 « + and if the smaller be diminished by 1, and the larger increased by 

 1, the sum of the squares is increased by 2w + 2 ; if ?z = 0, the difference is 

 2; and it becomes 4, 6, 8, &o. as n becomes 1, 2, 3, 4, &c. On the other 

 hand, if the smaller root be increased and the larger diminished by 1, the 

 sum of the squares becomes less by 2n—2, 



2 A 2 



