773] 
ON THE 8-SQUARE IM AGIN ARIES. 
369 
Hence if 0, 1, 2, 3, 4, 5, 6, 7 and O', 1', 2', 3', 4', 5', 6', 7' denote ordinary 
algebraical magnitudes, and we form the product 
(00 +11 + 22 + 33 + 44 + 55 + 66 + 77) (O'O +11 + 2'2 + 3'3 + 4'4 + 5'5 + 6'6 + 77), 
this is at once found to be = 
(00' - 11' - 22' - 33' - 44' - 55' - 66' - 77') 0 
+ (01' + 0'1+e,23 + 6 2 45 + e 3 67 )1 
+ (02' + 0'2 + 6,31 + 6,46 + e 3 57 )2 
+ (03' + 0'3 + ei 12 + e 6 47 + e 7 56 ) 3 
+ (04' + 0'4 + e 1 51 + e 4 62 + 6«73 )4 
+ (05'+ 0'5 + e 2 14 + 6 5 72 + e 7 63 )5 
+ (06' + 0'6 + é 3 7l+6 4 24 + e 7 35 )6 
+ (07' + 0'7 + e 3 16 + 6 s 25 + 6 6 34 ) 7, 
where 12 is written to denote 12 — 12, and so in other cases. 
The sum of the squares of the eight coefficients of 0, 1, 2, 3, 4, 5, 6, 7 respectively 
will, if certain terms destroy each other, be 
= (O 2 + 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + 7 2 ) (O' 2 + l' 2 + 2' 2 + 3' 2 + 4' 2 + 5' 2 + 6' 2 + 7' 2 ) ; 
viz. the sum of the squares contains the several terms 
e,6 2 23.45, 6,6,23.67, 6,6,31.46, f,e g 31.57, 6,6 6 12.47, 6,e 7 12.56, e 2 6 3 45.67, 
6 4 e 7 24.35, e 4 e s 62.73, e 2 e 7 14.63, e 2 e 6 51.73, 6 2 6 5 14.72, e 2 e 4 51.62, 6 4 6 5 46.57, 
e 5 e 6 25.34, e 5 6 7 72.63, e ;i 6 6 16.34, e 3 e 7 7l.35, 6,,6,71.24, e 3 e 5 16.25, 6 6 6 7 47.56, 
and observing that 21 = —12, etc., and that we have identically 
23.45 + 24.53 + 25.34 = zero, etc., 
then the three terms of each column will vanish, provided a proper relation exists 
between the e’s : viz. the conditions which we thus obtain are 
become 
6,e 2 = - e,6 7 = 
e i e 3 = — e 4 e 6 = 
e 5 e 7> 
6,6, = - 6 3 6g = 
f 2 e 7» 
6,65 = e 3 e 7 = 
e,e 6 = e 2 e 5 = 
- *3*4, 
eie? = — e 2 6 4 = 
e 3 e 5 ) 
^2^3 = 6 4 e 5 = 
e<+7- 
generality assume e ; 
+ — — e 4 e 7 — 
e 5 e 6> 
+ = e 4 6 6 = 
<*€7, 
+ = - e 4 e 5 = 
6 S e 7> 
II 
uf 
1 
II 
<¿7 
- e 7 , 
11 
ttT 
11 
<4J 
e 8 , 
e 5 = - e, 
= e. 
C. XI. 
6 7 — — e 4 
47