1866.] 



alluded to hy Fermat, 



121 



The first of these going up gives the roots of N, the second gives those of 

 M, and these roots are identical with those obtained from AM and AN. 



The index of A is 2 n — 2^+1, therefore when n in moving down- 

 wards becomes ^i^- 2, n+ 3, ....... 



2 n—'6 p + 2, 



2 /I— 3 p + 1 



^ + 1, p being two of the roots of each term in this series ; by adding 

 to the first-mentioned root and p to the other, these two terms will 

 be found to be terms in two horizontal series, of which the first terms 

 are in the first vertical series, and these terms both of them make the 

 diagonal index, and therefore are terms in a diagonal series which, rising 

 towards the right, give the roots of W. 



A descends diagonally to the left, and on each side of the line which 

 leads to W changes to cross diagonals which lead to M and N. W leads 

 diagonally to the left to R and S, and where it crosses AT changes and 

 goes up to A. M goes down into R, and then diagonally to where it meets 

 the horizontal series from T ; its roots there correspond with the series 

 from T ; it returns to T and up to A. In like manner N goes down 

 through S to the line from V, and so to V and up to A. E. and S go each 

 of them across to the vertical line from A, and so up to A. : every term 

 through which these lines pass has the four roots indicated whose squares 

 would make the term. 



The number of terms in the whole square, whose four roots may be ex- 

 pressed in term of p and is very considerable ; and it may be well now 

 to present some skeleton diagrams of the many ways in which certain 

 members of the square are invariably connected. 



I propose to exhibit several (to avoid confusion, which would arise from 

 putting all in one diagram) ; these do not by any means include all the 

 connexions tliat exist ; but whatever may be the value of w or ^, a number 

 of the form {^p-\-2). n-\-l, whether it be 2n-\-\, 6?z+l, 10^+1, &c., 

 and whatever be the value of n, gives the following results. See dia- 

 gram No. 2. At the {n—p)i\\ term there will be A, which descends 

 vertically till the index is 2n — \, then 2^+1, and from these rise up in 

 diagonals AN and AM, as already mentioned. A then further descends 

 till the index is equal to 2/z + 2^+ 1, when it rises in a diagonal to W. 



Diagram No. 3 shows certain connexions between AM and AN and 

 other terms in the square. AM is always — 2^-f 1, 0, ?^ — 2j3-|-2, 

 AN is always — 2p+l, 0, n — 2p, n. 



If the series in which AM is a term be carried down diagonally till 

 becomes 2/)+l (that is 2^9+1 places), it becomes a term in a diagonal 

 series that intesects it and rises to the top, then goes down to the hori- 

 zontal series from R, where it becomes a term in that series and passes to 

 where it is below M and rises vertically up to it. AjST does the same with 

 respect to the series from S, and rises up to N. . 



