SERIES AND WEIGHTS PROBLEMS 173

In general, it is found that, if the total number of pounds
we wish to take as the upper limit is expressed in the form
1(3™—1), then the weights required to weigh any amount up
to that limit will be:

i NS
If 40 be expressed in this form, then:
3" =81 =3¢
and the solution is therefore:
By e G R
or 1 31 32 33

This solution depends upon the facts that every positive
integral number can be expressed in terms of positive powers
of the number g (if we employ minus signs). Similarly, the
solution: ;

A S D
depends upon the fact that all numbers can be expressed in
terms of the number 2 without the employment of minus
signs.

There is no other number which satisfies the conditions
that all other numbers may be expressed in terms of its posi-
tive integral powers, and there cannot, therefore, be any
further solutions to the weights problems, provided all the
weights must be diferent.