528 



One form of the theorem is the following: 



(00' - 11' _ 22' - 33' — 44' - 55' — 66' - ITf 

 + (10' + 01' - 32' + 23' — 54' + 45' - 76' + 67'f 

 + (20' + 31' + 02' - 13' - 74' - 65' + 56' + 47')^ 

 + (30' - 21' + 12' + 03' - 64' + 75' + 46' - 57')^ 

 + (40' + 51' + 72' + 63' + 04' - 15' - 36' - 27')^ 

 + (50' _ 41' + 62' - 73' + 14' + 05' — 26' + 37')^ 

 + (60' + 71' - 52' - 43' + 34' + 25' + Ob' - 17')^ 

 + (70' _ 61' - 42' + 53' + 24' — 35' + 16' + 07')^ 

 = (02 + 12 + 2^ + 32 + 42 + 52 + 62 + 7') (0'2 + 

 1/2 + 2'2 + 3'2 + 4'2 + 5'2 + 6'2 + 7'2). 



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 : 



(aH6' + c^ + rf' + e'+/'+^'+/i')x 



(a'2 + i'2 ^ c/2 + ^/2 + g/2 ^yV2 + ^/2 ^ ^/2) _ 

 a"2 + 6"2 ^. C//2 + ^//2 ^ g//2 ^y /2 ^ ^//2 ^. y^'/2 . 



where a". . A" denoted the following expressions: 



a" = aa' - bb' - cc' - dd' - ee' -ff-gg'-hh'; 

 b" zzba'+ab'- dc' + cd' ~fe' +ef + hg' - gh'; 

 c" = ca' + db' + ac' - bd' - ge' -hf+ eg' +fh'; 

 d" = da' - cb' + be' + ad'-he'+ gf -fg'+ eh'; 

 e" = ea' +fb' + gc' + hd' + ae' - bf - eg' - dh'; 

 /" =/«' - eb' + he' - gd' + be' + af + dg' - ch'; 

 g" - gd - hb' - ec' +fd' + ce' -d/' + ag' + bh'; 

 h" = ha'+gb' -fc' - ed' + de' + ef - bg' + ah'. 



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 II. Ilamiltou's theory of 



I 



