1866.] 



alluded to by Fermaf. 



123 



of the following roots, either 3, 12, 2, 2, or 10, 0, 5, 6 ; but to set forth 

 the roots of the numbers which are in Diagram No. 1, would require a 

 diagram larger than could conveniently be put into a publication of the 

 size of the Proceedings. The roots 10, 0, 5, 6, if arranged thus, —10, 0, 

 5, 6, have 1 for their sum ; but it was proved in the former paper (see 

 p. 410, Trans. Koy. Soc. for 1861) that if the algebraic sum of the roots 

 be 1, then the number is the double +1 of a number composed of 3 tri- 

 angular numbers, 161 = 80x2+1, and 80 is composed of the 3 triangular 

 numbers 55, 15, 10. If therefore any number be doubled, and 1 be added, 

 an odd number will be obtained, to which the same process may be applied 

 as is here applied to 161. 



I have stated that the cause of the success of " the method'^ (though it 

 does not at present amount to a demonstration) may be easily shown. It 

 arises, first, from " the method" requiring every odd number that is a term 

 in any of the series to be represented by the roots of the square numbers 

 that compose it ; and secondly, and more particularly, from every series 

 being connected with at least six others of a different kind which intersect 

 it, each of which is again connected with at least five others, so that when 

 the whole network has been pursued, and the roots which in succession 

 form every term have been recorded, it will be found that many different 

 modes of dividing each term into four squares or less will be discovered, 

 i. e. if the numbers be large. I propose to show the manner in which the 

 series are apparently interwoven by an example from each kind. 



Let the first term in a horizontal series be 22 3, 7, 13, 14 with 22 as 



the index in the margin, and 21 and 23 being the indices of the diagonal 

 series which pass through this square ; for, except at the top line, two 

 diagonal series pass through every square. 13 and 14 are the variable 

 roots, which become 12. 15, | 11, 16, | 10, 17, | 9, 18, | &c. | in the suc- 

 cessive terms; when in the second term 14 becomes 15 (15 + 7 = 22), 

 and the roots are 3, 7, 12, 15, a term in the series which would come down 

 from the second square ; the roots in that square will therefore be 

 3, 12, 4, 4; when 15 becomes 19 the roots will be 3, 7, 8, 19, and the 

 roots at the top will be 7, 8, 8, 8 ; so when 15 becomes 25 at the tenth 

 term the roots are 3, 7, 2, 25 ; and as —3, 25 = 22, another series rises 

 up with roots of the first term, 7, 2, 14, 14. 



13 diminishes to 0, and then increases, giving 2 more when it becomes 

 15 or 19 ; but the series is also crossed by diagonal series ; and when the 



index in the two rows from the first term j 23 25 27 29 31* &c" } ^'^ 

 (the sum of all the roots), or 31 (the sum of the variable plus the dif- 

 ference of the constant roots), or 17 (the sum of the variable roots minus 

 the sum of the constant roots), or 23 (the sum of the constant roots minus 

 the difference of the variable), a diagonal series arises. Here are no less 



