1866.] 



alluded to by Fermat, 



127 



Diagram No. 7. 



There is an interval of w —3^ + 3 

 squares from the 1st term. 



*2n— 6p-f 4 

 R 



2n— 6p + 5 



n— 2^ + 2, n—22)+i 



2n— 62^+ 1 



- 3i?+l, p-f 1 

 w — 2p+ 1, n— 2p 



There is here an interval of 

 (p — 1) squares from S to «w. 



2n—4jp + 3 

 2n— 4p+ 1 



"2 29+ 1, 1 

 n — 2p + l, n 



There is here an interval of 

 (p— 1) squares from an to T. 



2n--2p + 3 

 j^ + 1, j^ + 1 



n — 1 , ?z. 



2m— 2p— 1 

 p, ^, n4- 1 



This Diagram (No. 7) 

 shows in terms of n and 2? the 

 roots of R, S, an (which 

 comes down diagonally from 

 AN), T, and V, and the 

 number of squares between 

 them ; the relative positions 

 of these terms depend entirely 

 on 29, and are always the 

 same for the same value of 



Terms similar to these in- 

 crease indefinitely as n in- 

 creases. The value of cer- 

 tain roots is independent of 

 w, and therefore is the same 

 for everv value of w. 



* [The numbers above the letters are the indices of the vertical series.] 



