( 794 ) 
piece is transferred by this transformation into we shall 
express by the formula 
S^S a =Sy, 
which operation is associative and commutative. 
Let us finally choose, in order to make the resemblance to ordinary 
ciphering as complete as possible, all ?n„’s equal to 10, let us take 
for each system of w th indices the digits 0,1, 2, 3, 4, 5, 6, 7, 8, 9 in 
this order, and let us give to the piece zero only indices 0. 
The different pieces of p we can then represent biuniformly by 
the different infinite decimal fractions lying between 0 and 1, in 
such a way, however, that finite decimal fractions do not appear 
and that 30 is not equal to -29, whilst each group y n consists of 
the different ways in which one can add the same number to all 
w th decimals, modulo 10. 
Now according to the above we understand by *5473...+ *9566... 
the decimal fraction, into which *5473... is transferred by the 
transformation which transfers *6 into *9566..., or, what comes 
to the same, the decimal fraction, into which *9566... is transferred 
by the transformation which transfers *6 into *5473... 
We shall call the operation furnishing this result, on the ground 
of its associativity and commutativity, the “sham-addition” of *9566 . .. 
to *5473_; it takes place just as ordinary addition, with 
this difference that in each decimal position the surplus beyond 10 
is neglected, thus that different decimal positions do not influence 
each other. So we have *. 
•5473 .... + *9566 .... = *4939 .... 
Let us understand analogously by *5473 .. .. — *9566 .... the 
decimal fraction, into which *5473-is transferred by the trans¬ 
formation which transfers *9566 .... into *6, and let us call the 
operation furnishing this decimal fraction the “sham-subtraction” of 
•9566_from *5473_; then this sham-subtraction is performed 
in the same way as ordinary subtraction with this difference, 
that “borrowing” does not take place at the cost of the preceding 
decimal positions, so that here again different decimal positions do 
not influence each other. So we have: 
*5473 . .. . *9566 .. . . = 6917 .... 
By operating only with a finite number, great enough, of conse¬ 
cutive figures directly behind the decimal sign, sham-addition and 
sham-subtraction furnish in the type of order £ a result agreeing 
with the exact one up to any desired degree of accuracy. In this 
too they behave like ordinary addition and subtraction of real numbers* 
