equations of all orders , by continuous approximation . 315 
15. This theorem employs exactly the same total number 
of addends as Theorem II, but with the important improve- 
ment, that the number of addends to each derivee is inversely 
as their magnitude, contrary to what happened before. Figu- 
rate multipliers are also excluded. And it is easy to convince 
ourselves that no embarrassment will arise from the newly 
introduced functions. For if we expand any of the addends 
in the general formula equivalent to M , and analyze 
it by means of the third property of figurate series, we shall 
find 
M / — + P k rr ‘ 
And since we take the scale in our Theorem in a reverse or 
ascending order, this formula merely instructs us to multiply 
an addend already determined by r, and to add the product 
to another known addend ; and if we trace its effect through 
all the descending scale, to the first operations, we observe 
that the addends to the last derivee, from which the work 
begins, are simply r repeated n — 1 times. 
16. Because N q = N, the addend exterior to the paren- 
thesis, might for the sake of uniformity be written N^'r 7 . 
The harmony of the whole scheme would then be more com- 
pletely displayed. To render the simplicity of it equally per- 
fect, we may reflect that as the factors r, r ', See. are engaged 
in no other manner than has just been stated, viz. in effecting 
the subordinate derivations, their appearance among the 
principal ones is superfluous, and tends to create embarrass** 
ment. Assume therefore 
*N = N \r. 
and we have 
