CHAP. XI] LIOUVILLE'S SERIES OF EIGHTEEN ARTICLES. 335 



The second member of () equals S/(i), the summation extended over all 

 the decompositions 



(4) 2m = i 2 + i\ + p* (i, i\ odd and > 0, p even). 



For /(re) = (- 1) ( *- 1 >%, the first member of () is S(- l) m 'fi(w - 2m' 2 ) 

 and equals %E, where E is the excess of the number of cases in which m' 

 is even over the odd cases in 



m = 2m' 2 + m 2 + + ml (m', m/ any integers), 



since 8ri(m) is the number of representations of m as a sum of 4 squares for 

 m odd. 



Let &li be the number of sets of solutions of 



m = 2m' 2 + m? + + mi 



in which m' is odd, d7 z the number in which m' is even. Then a discussion 

 of () for the case f(x) = x 2 leads to the result, relating to (4), 



i = 2i 2 - Zp 2 if m = 1 (mod 4), 

 = 2i 2 - Sp 2 if m = 3 (mod 4). 

 If M is the number of solutions of (4), 



2mM = 2Zx 2 + 2p 2 . 



Let f(x, y) be a function even with respect to x and to y. Then 

 M 2JZ(- IJ^-^'VCS" - 2m', 2d" + 4m') = SS(- l) (a2 - 1)/2 /(m!, d z + 5,), 

 where the summation on the left relates to (3) and that on the right to 



2m = m 2 + m 2 , m 2 = d 2 8 2 (mi, m 2 , d 2 odd and > 0). 

 If /reduces to/(z), (TT) becomes (). Also, 



(p) 4S2(-l) ro ' + ( 8 "-" /2 /(2 a V / + m') - SS(-l)*/(sO = 2(-l)" l ~ 1 /( Vm~) or 0, 

 according as m is a square or not, where m is any integer and 



m = m' 2 + m", m" = 2 a "d"S", m = s 2 + s' 2 + s" 2 , 



m", d", 5" being positive and the last two odd. 



A discussion of the case m = Sv + 7,f(x) = x 2 , shows that Ni/N 2 = 17/20, 

 where NI is the number of representations of m as a sum of 7 squares hi 

 which the first square is odd, and N 2 the number in which the first square 

 is even, including zero. 



For m odd and f(x) any even function, 



if m- square, 



4 " ifro* square, 



m = 4m' 2 + d"B" (d" } 8" odd and > 0). 



For m = 4v + 1, this formula holds for any f unction /(x). 



Liouville 8 stated that for F(x, y, z) odd with respect to x, y, and z, 



(u) Z2F(2 a "d" + m', 6" - 2m', 2 tt//+1 d" + 2m' - 5") = or 2F( Vm", j, j), 



8 Jour, de Math., (2), 5, 1860, 1-8. Twelfth article. 



