to the Products of the Sums of Squares. 125 



whether or not the theorem admits of generalization ; and in 

 the Philosophical Magazinefor April 1845, Mr. J. T. Graves 

 announced that he had arrived at the truth that " the product 

 of two sums of eight squares is a sum of eight squares ;" but 

 adds, that " the full statement and proof of the theorem must 

 be reserved for another time." I have anxiously inquired at 

 every likely source of information to which I have access, for 

 the publication of this proof; and am disposed to conclude, 

 from the result, that such publication has not yet been fur- 

 nished. 



I have been thus led to enter into an independent investi- 

 gation of the subject; and find that the theorem holds not 

 only for four and eight, but also for sixteen, and indeed for 

 any number of squares expressed by an integral power of 2. 

 This investigation I propose to forward to the British Asso- 

 ciation, at its meeting in June, provided I be assured that I 

 have not been anticipated in the generalization here announced. 

 By thus alluding to the results at which I have arrived, in 

 the pages of this Journal, I shall afford timely opportunity 

 for the information being communicated to me, if the above- 

 mentioned extension of Euler's theorem has ever as yet been 

 published. 



The notation in which the subordinate theorems and the 

 general theorem is announced is this, viz. 



S2(D)x22(a')=22(n"), 24(n)xS4(D')=24(n"), 



and generally 



where n is any integral positive power of 2. 



The theorem holds too when certain coefficients are intro- 

 duced : thus, taking eight squares, it is true that 



X (s'2 + bi'-^ + cu'-^ + bcv" + 6cto'2 + c.v'-' + bij'- + z'-) 

 = s"2 ^ bi"^ + cu"'- + ba/'^^ + bcw"-^ + cx"-^ + bif'^ + z"\ 

 And also that 



(s2 .^ ^^2 ^ «2„2 ^ ^3„2 _^. ^4.^2 ^ ^5^y2 ^ „6^2 ^ ^7_j2) 

 = 5"2 -}- «<"2 + «2,//2 + ^.V/2 ^ a4a,H2 + ^h^\n ^ ^GyUi j^ ^7JI2^ 



And the same has place for any number of squares expressed 

 by a power of 2. 

 Belfast, May 8, 1847. 



