338 report— 1865. 



£-*-l)T ^ 



2^ " v ' 1-2-' 



which is the formula (2). "We shall now show how the equation (4) can be 

 obtained by squaring this formula. For this purpose we represent by a and 

 /3 any two unequal positive uneven numbers congruous to one another for the 

 modulus 4, and by a' and /3' any two positive uneven numbers not congruous 

 to one another for the modulus 4. "We then have 



x-l^-v r iv v, g* (a+w ,v fl , ^"'^ 



■K 



( i- r f ■ "cwoa-*) p (i-^)(j-20' 



= P-fQ_R, for brevity. 

 Here 



P = V 9* _y vS' /ym_V n ?" . 



(1— q v f L 1— j2« 



again in Q, if we double each term we may suppose /3>a; let /3=a + 4?i; 

 observing that a may be any positive uneven number, and n any positive 

 number whatever, we find 



0=22,^- i 



2«+" 



(1— 2")(1— 9 4 "+") 

 =2v y V 2 " r ?" 9 4 "+" 1 



*•-! 2« + >* 



— 9v v V 



» (l- f /»)(l-^) 



Lastly, in R let a'+/>' = 4» ; so that 



■11 2tn ^v — ^r 



i (1— 3")(1— q in - v ) 



qin 



[JL.+ 9*-" , +l1 



'Ll — '/' l-? 4 "-" J 



Al •"!' 



1 



4,1-1 nSB+X 9 »0 2 " 



Consequcntlv 



*rK 



^_=P + Q-R 





l l_ 5 2« "l — cfn \ — cf"' 



which is the formula (4). 



Thus by a purely analytical process we deduce from an equation which ex- 

 hibits the number of compositions of the double of an uneven number by the 

 addition of two uneven squares, an equation exhibiting the number of com- 

 positions of the quadruple of an uneven number by the addition of four 

 uneven squares. This analysis Jacobi has expressed arithmetically as follows. 

 Representing by N an uneven number, by [4N] the number of compositions 

 of 4N by the addition of four uneven squares, we resolve 4N in every possible 

 way into two unevenly even numbers 2^ and 2N 2 , and each of these in every 

 possible way into two uneven squares ; we thus obtain the equation 



