527 



parts together ; and from which a formula for a greater num- 

 ber of squares might be suggested. Such a principle is found 

 to govern the generation of the four-square results, when these 

 are arrived at by a peculiar process, which the author exhibits. 

 The same process is then extended to the case of eight squares; 

 and it is found that 



(s'^ + f 2 + u'^ + v''^ + w'^ + x'^ -I- y'2 + 2'2) X 

 {s^ + t^ +u' + v'' + w;2 + x"^ + f + z") = 

 {ss' + tt' + uu' + vv' + ww' + xx' + yy' + zz'Y 

 + {st' — ts' + uv' — vu' + wx' — xw' + yz' — zy'Y 

 + {su' — us' + vt' — tv' + yw' — wy' -h xz' — zx'Y 

 + {sv' — vs' + tu' — ut' + wz' — zw' + xy' — yx'Y 

 -f- {sw' — ws' + xt' — tx' + ^ly' — yu' -\- zv' — vz'Y 

 4- {sx' — xs' + tw' — wt' + yv' — vy' + zu' — uz'Y 

 + {sy' — ys' + zt' — tz' + vx' — xv' + wu' — uw'Y 

 + {sz' — zs' + ty' ~ yt' -f- vw' — wv + ux' — xu')'^. 



These results are verified by the actual development of the 

 several squares ; which development, by the mutual cancelling 

 of all the double products, reduces itself to the sixty-four 

 squares furnished by the product of the proposed factors, when 

 multiplied together in the ordinary way. 



The author then enters into a more minute examination of 

 the constitution of the preceding polynomial ; and shows that 

 the cancelling of the aforesaid double products is a necessary 

 consequence of that constitution. 



It is further shown that the product continues to be of the 

 same form as each of the factors, when the coefficients a", a', 

 a^ a'', &c., are introduced in order, in connexion with the 

 squares entering those factors. 



Sir William Rowan Hamilton stated also a theorem respect- 

 ing products of sums of eight squares, which does not essen- 

 tially differ from the foregoing, and was communicated to him 

 by John T. Graves, Esq., about the end of the year 1843. 



