— an 
‘i | 
: 
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? + 4 ul? + vy + w? 4+ a’? + y’ + 2")x 
(s? +P ev ¢r4¢uw 42 + y? + 2%)= 
(ss’ + tt! + uu! + vv’ + ww’ + aa’ + yy! + 22’)? 
+ (st? — ts’ + uv! — vu’ + wa’ — aw’ + y2' — 2) 
+ (su’ — us’ + vl! — be! + yw! — wy + 22’ — zr’) 
+ (sv’ — vs’ + tw! — ut! + we! — zw’ + xy’ — yw’) 
+ (sw’ — ws’ + at! — ta’ + uy’ — yu’ + 20’ — v2’)? 
+ (su! — xs! + tw’ — wt’ + yo — vy’ + zu’ — uz’)? 
+ (sy — ys’ + 2t! — tz’ + va! — xv! + wi’— ww’)? 
+ (sz2’ — zs’ + ty’ — yt’ + vw! — we’ + uz’ — wu’)? 
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, 
