528 
One form of the theorem is the following: 
(00’ — 11’ — 22’ — 33’ — 44 — 55’ — 66’ — 77’)? 
+ (10’ + Ol’ — 32’ 4 23’ — 54’ + 45’ — 76’ 4 67’)? 
+ (20 + 31’ + 02’ — 13’ — 74’ — 65! 4 56’ 4 471 
+ (30/— 21’ + 12/ + 03’ — 64’ 4 75’ + 46’ — 57’)? 
+ (40 + 51! + 72’ + 63’ + 04’ — 15’ — 36’ — 27’)? 
+ (50 — 41’ + 62’ — 73’ + 14’ + 05’ — 26’ + 37’)? 
+ (60’ + 71’ — 52’ — 43’ + 34’ 4 25’ + 06’ — 17/7 
+ (70 — 61’ — 42’ + 53’ + 24’ — 35’ 4 16’ + 07/2 
= (0? + P+ 2? + 3? + 4? + 5? + 6 + 7?) (0? + 
IE te a oe ae wet ie Sr) 
In a letter dated January 18th, 1844, Mr. Graves commu- 
nicated to Sir William Hamilton his theorem respecting sums 
of eight squares under the form: 
(P+P+O+P+ Ps fP+g +h’) x | 
(a? +b? +0°4+d? +e? +f? 4+9° +h?) = . 
GPP + Pa deter frag P4 pr; 
where a”. . h’ denoted the following expressions: 
a” =aa' — bb' — ce’ — dd’ — ee' — ff’ —gg' —hh'; 
b’ = ba’ +-ab’ — de’ +.ed' — fe’ + ef’ + hg’ —gh’; 
ec” =ca' + db’ + ac’ — bd’ — ge’ —hf’ + eg’ + fh’; 
d” = da’ — cb’ + be’ + ad’ — he'+ gf’ — fy’ + eh’; 
e” =ea' +fb' +ge' + hd' + ae'—bf' —cg’— dh’; 
SJ’ = fa — eb! +he' —gd' + be’ +af’ + dg’ — ch’; 
g” =ga' — hb’ — ec’ + fd' + ce’—df' +ag'+bh’; 
h" =ha' +gb’ — fe’ — ed’ + de’ + cf’ — bg’ +al’. 
In a letter of somewhat earlier date, but evidently written 
in haste, upon a journey, and dated December 26th, 1843, 
analogous expressions had been given, containing, however, 
some errors in the signs, which were soon afterwards corrected 
as above. ‘That earlier letter also indicated an expectation 
that a theory of octaves, including a new and extended system 
of imaginaries, which had thus been suggested to the writer 
(J. T. Graves, Esq.) by Sir William R. Hamilton’s theory of 
ieee eclt aie. 
nd 
