424 
MR. SYLVESTER ON THE RESIDUES OF ^ EXP.\NDED 
when 11=4, it beeoiiies 
( 0 , 1 ) 
( 0 , 2 ) 
( 0 , 3 ) 
( 0 , 4 ) 
when n=&, it becomes 
( 0 , 1 ) 
( 0 , 2 ) 
( 0 , 3 ) 
( 0 , 4 ) 
( 0 , 5 ) 
( 2 , 5 ) 
( 3 , 5 ) 
( 4 , 5 ), 
and so forth. Every such square it is apparent may be conceive^ as 
pyramid, formed by the successive superposition of square layers, w net . 
sess not merely a simple symmetry about a diagonal (such as is piopei o a 
cation table), but the higher symmetry (such as exists in an addition ta e), evi ■ 
all the terms in any line of terms parallel to the diagonal transverse to the ax. o 
symmetry being alike. Thus for u=5, the three layers or stages m question will 
be seen to be, the first— 
( 0 , 1 ) 
2 ) 
(0 
the second- 
and the third- 
( 0 , 3 ) 
( 0 , 4 ) 
( 0 , 5 ) 
( 0 , 2 ) 
( 0 , 3 ) 
( 0 , 4 ) 
( 0 , 5 ) 
( 1 , 5 ) 
( 1 , 2 ) 
( 1 . 3 ) 
( 1 . 4 ) 
( 0 , 3 ) 
( 0 , 4 ) 
( 0 , 5 ) 
( 1 . 5 ) 
( 2 , 5 ) 
( 1 , 3 ) 
( 1 , 4 ) 
( 2 , 4 ) 
( 2 , 3 ). 
( 0 , 4 ) 
( 0 , 6 ) 
( 1 , 6 ) 
( 2 , 5 ) 
( 3 , 5 ) 
( 1 , 4 ) 
( 2 , 4 ) 
( 3 , 4 ) ; 
( 0 , 5 ) 
( 1 , 5 ) 
( 2 , 5 ) 
( 3 , 5 ) 
( 4 , 5 ) ; 
In general, when (») is odd, say2;i+l, the pyramid wilt end with a single term 
. A squMS arrangement haring this kind of symmetry, viz, such as obtains in 
addition table as distinguished from that which obtains in the multiplication ta e, may 
Persy mmetric. 
