The figure 9, multiplied by itself, or by one of the other digits , always gives a number whose two digits, when added together, give 9 for the sum. The digits composing the sum of the series of nine digits (that is 45), added together, give 9. The sum of all the products of 9, multiplied by the series of digits (that is, 405) and divided by 9, gives for a quotient 45, and the digits forming the dividend or quotient, added,together give 9. If a row of any digits be multiplied, either by or by any one of the products of 9 multiplied by one of the digits of the series, . such as by 18, 27, 36, 45, 54, 63, 72, or 81, the sum of the digits of the product will be divisible by 9. If these nine digits ot the series are multiplied in the following order: 1, 2, 3 , 4, 5, 6, 7, 8, 9 by 9, or one of the other products mentioned above, the 'product obtained will contain only similar digits except at the tens, where there will be a 0 ; that is to say, if the series is multiplied by 9, there will be all ones, if by 18, all twos, if by 27, all threes, and so on,.exceptat the tens, where the” will be always 0 j this 0, coming always under the I digit of the multiplicand that destroys the uniformity of the digits of the product. But if the 8 in the multiplicand is taken out, the 0 will likewise disappear from the product, in which there will be found only ones, twos and threes, 'c., according to the multiplier made use of