1866.] 



alluded to bij Fermat. 



125 



values of p and n, these connexions occur, but tbe form of them varies 

 with the values of p and n, 



I have pointed out some of the results of having as the first term in the 

 square an odd number of the form {Ap + 2), w + 1 ; the forms will be 2?i + 1 , 

 6/«+l, lOw + 1, 14?^-f 1, &c., in which p may be 0, 1, 2, 3, &c. But 

 there is another form which gives similar results, viz. Apn-\- {2p-\- 1), which 

 is in many respects the " converse " of the other ; the forms will be An -\- 3, 

 8;^^-5, 12w+7, 16?i + 9, &c. These are the first terms to which this 



system gives rise. The (^n—p — \)i\i term is always j p,py n, n+ 1 



In 



the former system n produced the equal roots and p the unequal ; here, p 

 produces the equal and n the unequal roots, and 1 call the (n—p — l)th 

 term A as in the other. This system has also W and M and N on each side 

 of it, and other squares similar to the other, but the roots of vyhich they 

 are composed are differently formed. M and N come from below. W, 

 instead of being 



-p + 1, —p, n—4p H- 2, n, 



wiUbe -3p,p, (n-2p), (.^ + 1-2;^), 



and other terms are similarly altered, but the general result is the same. 

 A specimen of a square of this form is given in Diagram No. 8, but which 

 cannot be reduced to the size of the Proceedings, wherej3 = 3 and n=l 6 ; 



the first number is {199^, and the square is completed so far as to show 

 that the roots of the 4 squares whose sum is (199) = 



5, 



13, 1, 



2 



-9, 



-3, 3, 



10 



1, 



10, 7, 



7 



but neither in this Diagram nor in Diagram No. 5 is the whole square com- 

 pleted (to avoid confusion) ; but if the series be traced in succession, the 

 entire Diagram 6 would be filled up, and every terra would disclose the 

 roots whose squares compose it. In this manner every odd number in all the 

 series is divided into the squares that compose it (not exceeding 4), the 

 squares being indicated by their roots. 



T"he two systems of (4p -{- 2) X ^^ + 1 and 4 p.n + 2p-\-l include every 

 possible odd number; 4n-f 3 includes every alternate odd number from 

 3; 8?i + 5 every fourth number from 5, and so on ; 2?i+l includes every 

 odd number; 6>i4-l every third odd number. Many odd numbers belong 

 to both systems, and to more than one in each. 151 is an example ; it is 

 either 10)1+1 (}i= 1 5), or itis l2n + 7 {n=12). The paper contains a Dia- 

 gram (No. 9) exhibiting the odd number 151 as belonging to both systems ; 

 but the Diagram cannot be reduced. The roots of the squares that com- 

 pose 151 are (10. 1. 5. 5), or (3. 9. 5. 6). 



