128 History of the Theory of Numbers. [Chap, v 
according as u^ is or is not divisible by w„, while 
fix) =Kx)F{l) +M (2) F{2) +M (f) F{S) + . . . . 
Hence if we multiply the elements of the ith colmnn of A„ by fx{ujui) and 
add the products to the last column for 2 = 1, . . ., n — 1, the new elements of 
the last colunm are zero except the final element, which is/(w„). Thus 
A,=/(ii„)A„_i=/(ui)/(w2) . . .fM- 
[These results are due to Smith,^^ not merely the case Ui = i a.s stated.] 
Cesaro^^ noted that |wy|=/(l) . . ./(n) if 
«,= s/wft0ft.Q, 
where the function h has the property that the determinant with the general 
element h{i/j) is unity, and similarly for hi. 
Cesaro®° gave the last result for the case in which h{x)=hi{x) = l or 
according as x is or is not an integer. P. Mansion (p. 250) stated that he** 
had employed a similar proof. 
Ces^ro®^ duphcated his paper^^ and transformed its final result into 
/(l)/(2) . . .fin) 
F[i,j] 
F\nl) 
where [i, j] = ij/{i, j) is the 1. c. m. of i, j, and F{x) is a function such that 
F{xy)=F{x)F{y). In particular, if F{x) = \/x, then J{x)=4){x)Tr{x)/x^, 
where 7r(n) is the product of the negatives of the distinct prime factors of n. 
Hence 
|Ki]|n=0(l)...0(n)7r(l)...7r(n). 
Ces^ro^^ investigated the r-rowed minors of the n-rowed determinant 
whose general element is F{b)=F{i, j), where 5 is the g. c. d. of i, j. It is 
shown that the {n — v)-Towed determinant whose general element is F{i-\-v, 
j-\-v) is equal to the sum of certain products of /(I), . . ., f{n) taken n — v&i 
a time, the case v = Q being Smith's theorem. Here 
/(x) =^J^F{j), Fix) =S/(d) (d divisor of x). 
Ces^ro^^ stated that the (n — l)-rowed determinant, whose general ele- 
ment Uij equals the number of divisors common to i+1 and j + 1, equals 
the number of integers ^ n deprived of square factors > 1 . 
"Atti. Reale Accad. Lincei, (4), 1, 1884-5, 711-5. 
•"Mathesis, 5, 1885, 248-9. 
"Giornale di Mat., 23, 1885, 182-197. 
"Annales de I'^cole normale sup., (3), 2, 1885, 425-435. 
"Nouv. Ann. Math., (3), 4, 1885, 56. 
