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



where e is the number of partitions of w not containing the element 1, and 

 4> is the number of partitions of w 1 not containing 0. 



N. Trudi 98 gave an account of the early history of partitions, made 

 extensive applications to isobaric functions, and finally enumerated the 

 combinations of n letters into a sets each of p letters, |8 sets of q letters, etc., 

 first when the n letters are distinct and second for repeated letters. 



C. M. Piuma" treated the following problem: From an urn containing B 

 balls marked 1, -, B, three are drawn and the three numbers written on 

 them are added; find the number of times the sum is ^ C. To find the 

 number S H of sets of solutions of<j> + \l/ + x = H with 0<<j><$<x=B- 

 First, let C < B + 4. Then every solution satisfies the inequalities. Of 

 the six cases H = Qh + j (j = 0, , 5), let H = 6h + 4 and set ^ - < = x, 

 x - = y. Then x + y = Qh - 30 + 4, < x < y. If <j> is even, 

 <f> = 2a, there are evidently 3h 3a + 1 sets of solutions x, y, and h is 

 shown to be the largest a giving a solution. Thus there are 



2Ui(3ft - 30 + 1) = h(3h - l)/2 



sets (f>, $, x- For < odd, we get h(3h + 3)/2 sets. Adding, we get 

 $6^+4 = h(3h + 1). Then T c = 2g =6 # is found by treating six cases; 

 for example, T 6c = c(12c 2 15c + 5)/2. Finally, there is treated the case 

 C ^ B + 4. 



P. Boschi 100 treated partitions into s parts from 1, , n. Let 



Si. r = x r + x r+l ++", 



Si. r = of Si. r+i + x* l Si. r+2 + + x^S lt n , 



S Z , T = tfS*. r+l + X r+1 S 2l r+2 + + X^S*. n-1, 



Expand and collect the terms of S u , r ; the coefficient of x p is the number 

 of ways P is a sum of distinct numbers chosen from r, r + 1, , n. It is 

 proved by induction that 



H . . .n-r+lW1 . _ v n-r\ . .(] . - r n-r-u+2\ 

 g _ a .(2r+-l)/27T JT s ii _ ? _ Ik 1 _ ? _ J _ ^ - ? - A. 



w>r (1 -a;)(l -x 2 )---(l -a;") 



Thus the coefficient of x p in /S Ml i = a: ( "+ 1)w/2 7 7 u , i is the number of ways P 

 is a sum of s different terms of 1, , n. For u = 2, 



where A 8 is the number of ways s + 3 is a sum of two numbers of 1, -, n. 

 Then A r = A 2n - r -4, 



A r = i(2r + 3 + (- \Y] if 2 < r =i w - 2; 



A r = w -- 2 + r + J{2r + 3 + (- l) r } if n - - 2 < r ^ 2n - - 4. 



98 Atti R. Accad. Sc. Fis. Mat. Napoli, 8, 1879, No. 1, 88 pp. 



90 Giornale di Mat., 17, 1879, 360-372. 



100 Memorie Accad. Sc. 1st. Bologna, 1, 1880, 555-571. 



