110 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



of those numbers m, m/2, w/3, ra/4, , mfm which are integers 

 ^ z x and are z's. Then A (m) is derived from the A (m) of the first problem 

 by replacing 5's by -y's, while (y, x) is given by A (y) . Let z x = 2 X ~ 1 and 

 let x increase indefinitely, i. e., use the infinite series 1, 2, 4, 8, . Then 

 y(m) = 2 m 2 m ~ l 1 = 1, (y, x) = 1, so that every integer is a 

 sum of terms 1, 2, 4, 8, in a single way [L. Pisano 11 ]. 



For the number (y, x) of ways y is a sum of x terms of m, m + n, m + 2n, 

 - or x distinct terms, (y, x) = (y nx, x) + (z, x 1), where z = y m 

 or y nx + n m, respectively. Then (y, x) is given by A w for 

 rifj. = y mx or y mx nx(x l)/2, respectively. Hence y is a sum 

 of x distinct terms of the progression as often as y nx(x l)/2 is a sum 

 of x equal or distinct terms. 



Finally (pp. 842-5), to reduce the integration of 



[y, z] = A X \JJ - w00), x~] + B x [y - \fs(x), x - 1] 



to that of (y, x) = A x (y 4>(x), x} + B x (y f(x), x 1), substitute 

 Lu> z] = ({y ~ F(x)}fm, x") into the former and compare the result with the 

 latter. The condition for agreement is F(x) F(x 1) = \l/(x) mf(x}, 

 whence, for a constant c independent of x, 



F(x) = 2{t(x + 1) - mf(x + 1)} + c. 



Thus, in our second problem, [y, ofj = [_y x, x~\ + \_y x, x 1], 

 while in the first problem concerning %i, - , z x = x, 



(y, x) = (y - x, x} + (y, x - 1). 



Hence F(x) = S(z + 1 - 0) = x(x + l)/2 + c and c = 0. Hence 



D/, z] = (y - x(x + l)/2, x), 



so that y is a sum of x distinct parts as often as y x(x + l)/2 is a sum of 

 parts g x. Again, for the equation in our first problem, and 



&, *] = D/ ~ 2z, z] + Q/ - 1, a; - 1] 



of our third problem, we have F(x) = x, [y, x~] = ({y + x}/2, x}, so 

 that y is a sum of x odd parts as often as (y + x)/2 is a sum of x even or odd 

 parts. Finally, for our first and last problems, F(x) = (m ri)x, so that 

 y is a sum of x terms ofm,m + n,m + 2n, - - as often as {?/ (m ri)x}/n 

 is a sum of z positive integers. 



G. F. Malfatti 19 obtained the general term of a recurring series whose 

 scale of relation has a multiple root. In the Appendix, he treated the 

 number of partitions into x distinct terms of the series 1, 2, , extended 

 either to infinity (as by Paoli) or to a given number p. Taking first the 

 former case, he showed how to pass from any of the series 



x = 1: 1 1 1 1 1 1 1 1 1 

 x = 2: 1 12233445 

 z = 3: 11234578 10 - 



19 Memorie di mat. e fis. societd, Italiana, 3, 1786, 571-663. 



