402 
Proceedings of the Royal Society 
Every positive integer can be expressed, in one way only, by the 
sum of a finite number of terms of one of the infinite set of series 
1 + 2+ 4 + 8+16 + 
2 + 3 + 6 + 12 + 24 + 
4 + 5 + 10 + 20 + 40 + 
6 + 7 + 14 + 28 + 56 + 
8 + 9 + 18 + 36 + 72 + 
&e., &c., 
the partial sums for each being the several places occupied in the 
above series by each particular integer. This, however, is obvious 
when we consider that the sum of (n + 1) terms of any one of these 
series is of the form 
( 2 r + 1 ) 2 * - 1 , 
and that this expression can be made to equal any given positive 
integer by one definite pair (and one only) of values of r and n. 
Thus we see that we may write 
u x — + x + 1 )? 
where the bar under x + 1 means that it is to be divided by the 
highest power of 2 that it contains. 
The numbers of operations, for classes of different numbers of 
boys from 2 to 25 inclusive, are in order as follows : — 
2, 4, 9, 20, 30, 36, 28, 72, 36, 280, 110, 108, 182, 168, 75, 1120, 
306, 432, 190, 140, 4410, 2772, 2530, 1440. 
The calculation of the numerical value of any particular term is 
easy, but I have not attempted to express the general law of this 
very curious series. It seems, however, to be well worthy of 
attention, especially from the point of view of the expressions for 
numbers in the binary scale. 
(b.) On a Graphical Solution of the Equation Vp<£p = 0. 
This equation has been exhaustively treated in our Transactions 
by M. G. Plarr. The present note is a mere sketch of a graphical 
