112 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ill 



ni* } in place of A (i) . Taking x = 1 in the general integral, we see that 

 (t, 1 + X) - (t - 1, 1 + X) = 0(0, which is shown to be A (t ~ of Paoli's 

 case. Hence 0(0 = n^~ l \ and 



(t, x + X) = ni'- 1} + A'wi'- I} + A"ni'- 3) + - 

 He gave the f ollowing results f or x ^ 4 : 



,. 6^ 2 + 24* + 17 (-1)' a 



I, 6) - 



4) = 



72 8 9 



68 t 



288 32 27 





27 16 



where a and i are the imaginary cube roots of unity, and /3 = i, /3i = i. 

 He also gave the general term of the first subtraction series : 



(* ' 4 ) = 144 ie~ 



\(X ~T~ tti jL\d i QJi J ./ r, 



_: '_ : _ t' / __ fi _J < 



27 27 ' ~ P + 



Earlier in the paper (pp. 618-26), he gave (t, 5) and the general terms of 

 the addition and subtraction series; these and various other results given 

 above occur in his earlier two articles in Prodromo dell' Enciclopedia 

 Italiana and (in more detail) in Antologia Romana, 11, 1784. 



V. Brunacci 20 reproduced Paoli's 18 treatment of his first problem. 



S. Vince 21 proved by induction that every positive integer is a sum of 

 distinct terms of 1, 2, 4, 8, . For, if true for numbers up to s = 1 -f 2 + 

 . . . _j_ 2 n-1 = 2 n 1, it will be true for the remaining numbers up to 

 s -f 2 n . The proof for d= 1, =t 3, 3 2 , is longer. 



S. F. Lacroix 22 reproduced part of the discussion by Euler. 9 



Fre"gier 22a proved that a m equals a sum of a terms of the arithmetical 

 progession whose first term is unity and common difference is 



2 + 2a + + 2a m ~ 2 . 

 Cf. Volpicelli, 37 Lemome, 76 Mansion, 87 and Candido. 213 



20 Corso di Matematica Sublime, Firenze, 1, 1804; 108-9, pp. 237-248. Cf. Compendium 



del Calc. Subl., 1811, 114. 



21 Trans. Roy. Irish Acad., 12, 1815, 34-38. Euler. 13 



22 Traite" du Calcul Diff. Int., 3, 1819, 461-6. 



224 Annales de math, (ed., Gergonne), 9, 1818-9, 211-2. 



