u 



llEPORT — 1876. 



+ 1, + 1, 0, 0, - 1, 0, ~ 1, 0, 0, 0, 0, 4- 1, 0, 0, + 1, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 

 - 1, 0, 0, 0, 0, 0, 0, 0, 0, + 1, . . . . 



viz. between two unities of the same sign the numhers of zeros follow the law 

 0, 1, 2, 3, 4 .... , while the numbers of zeros between two unities of opposite sign 

 follow the law 2, 4, 6, 8 .... ; the second column is the same as the first column, 

 but lowered one line, and witli — 1 as the top constituent ; the third column is the 

 same as the second line, but lowered one line, and with zero as the top constituent ; 

 the fourth column is the same as the third column, but lowered one line, and with 

 zero as the top constituent ; and so on. 

 For example : — 



+ 1,-1, • 

 +1, +1, -1 



• . +1, +1 



=3 



which is right, since 3 = 1 + 1+1=1 + 2. 



II. If i/'(h) be the sum of the divisors of m, then 



(« rows) 



where all the constituents remain as before, except those in the first column, in 

 which the »-th constituent appears multiplied by r. Thus the non-zero constituents 

 in the first column are the pentagonal numbers, of the form |«(3«+1). 

 For example ; — 



+ 1, 

 + 1, 



^(.3)= 



+1, 

 +2, 



-1 

 +1 



=4 



which is right, since i|r(3) = l + 3: 

 III. Consider the determinant 



v= 



=4. 



(«— 1 rows) 



wherein the top lice contains the natural numbers beginning with 2 and ending 

 with n ; the second Hue is obtained by dividing the constituents of tlie top line by 

 2, and entering the quotient or zero according as each constituent is or is not divi- 

 sible by 2 ; the third line is obtained by dividing the constituents of the top line by 3, 

 and entering the quotient or zero according as each constituent is or is not divisible 

 by 3 ; and so on. 

 " Then V = Oif M is divisible by any square factor, and V = (— )"+'^» if m is asimple 

 product of prime factors; viz. a/3y". . . . |, where a, Q, y ._. . . ^ are different primes, 

 and N denotes the number of prime divisors of «, unity included. 

 For example : — 



•1. 

 1, 



= (-)''+»6 



which is easily verified. 



