gia Mr. Robertson's Demonstration 
zzzii 
n ~-'x Vj= ~- z 3 x r > the term °f the multiplicand directly over 
i 
it, into x~\ the, first term in the multiplier. The second'term 
in the perpendicular line of products is obtained by multiplying 
that term of the multiplicand in the next perpendicular line 
towards the left, by the second term of the multiplier. Thus 
" + 1 — 3 r n — 2 r 
I x - x n —z 3 x T , is the product of - x ? — z'x r into 
r r zr r zr 
i — r 
~zx r . And in general, if p be put for a number denoting 
the place of a term in. the perpendicular line of products, and 
if the terms in the multiplicand be supposed to be numbered, 
beginning with that directly above the perpendicular line of 
products under consideration, and reckoning towards the left 
hand ; and if the terms in the line of the multiplier be num- 
bered, beginning with x r , and reckoning towards the right, 
then the product whose place is p will arise from the multipli- 
cation of that term in the multiplicand whose place is denoted 
by p into that term in the multiplier whose place is also de- 
noted by p. The observations in this ’and the last article are 
evidently general ; being applicable to any extent to which 
the series in the multiplicand and multiplier may be carried. 
16. The laws of arrangement being thus established by the 
exponents, the summation of the coefficients, in any perpen- 
dicular line of products, is next to be attended to. And in or- 
der to do this, with as little embarrassment as possible, put 
A = n> R — n x. n — r, C — nxn — r xn — 2 r, D = nxn — i 
x n — 2 r x /I ~ — 3 r> See. and put a — i, 6 =ixt — r, i 
