330 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xi 



Set 



f(x, y) = cos xt - cos yz, 



Mm) = Z sm ^-cos dz, co(w) = Z cos d- sin 8z (d8 = m). 



d d 



Then (c) yields the result 



-Lt(m')t(m") - 2co(m')w(w") = 2 a - 1 2d{sin 2 2 a ~ l dt - sin 2 2 a ~ l dz\. 

 We now include the case in which a = and set 



2 a m = 2 a W + 2 a "m" (a f ^ 0, a" ^ 0, m', m" odd). 

 Let m = dd, etc., as before. Liouville 2 stated the formula [a case of (e) 3 ] 



Z { d Z[/(2 a 'd' - 2""d") -/(2*'d' + 2 a "d")]} 

 (G) "''"" = Z(5-2d){/(2*d-/(0)}, 



d 



where d, d', d" range over all the divisors of m, m', w", respectively, and the 

 first summation extends over all the pairs of even or odd integers 2 a W, 

 2 a "ra" whose sum is 2 a m. Consider the case a = 0; then a' or a." is zero; 

 but, by introducing the factor 2 before the first member of (G), we may 

 restrict attention to the case m = m' + 2 a "m". Since 25 = 2d, we get 



(F) 2 Z { Z [/W - 2'"d") - /(d' + 2*"d")]} = Z (5 - d)/(d), 



d', d" d 



a case of (d). For example, if f(x) x*, 



2 {2 a "f i(m') Mm" 

 For /(z) = x 2 or z 4 in (G) we get 



= f 



Again using the notation m = m' + 2 a "m", Liouville stated the follow- 

 ing two cases of (d) : 



/(0)ri(m) = Z {/(O) + 2/(2) + 2/(4) + . . . + 2/(d - 1)} 



(D) * + 2 Z ( d Z [/(^ - d") - /(d' + d")]}, 



Z { ,[W - d " + 1) - ^W - d"' - 1) - F(d' + d" + 1) 



(E) "'"' + F(d' + d" - 1)]} = F(l}Urri) - Z F(d), 



<i 



where F is an odd function: F(- a) = - F(x). For /(re) = (- 1)* /2 , 

 (D) gives 



II Mm) - P (m)\ = 2p(mOp(m"), p(m) = 2(- l)w- /2 . 

 The first expression is therefore the number of decompositions of 2m into 

 (1) 2/ 2 + z 2 + 2> 2 + y 2 ), 



with y, z, u, v odd positive integers and a > 0; it is also the number of 

 decompositions of m into the form (1) with y and z positive odd numbers, 



* Jour, de Math., (2), 3, 1858, 201-8, 241-250. Third and fourth articles. 



