3i° 
Mr. Robertson's Demonstration 
placed under one another, the products will stand as below ; 
the first two lines immediately following being considered as 
the multiplicand and the multiplier respectively. 
n n • — r 
n-—2r 
71 71 — T 1 1 
— 3 r 
n — r n — 2 r „ 
+» &c. 
+ *z x~T ~~ 
1 1 — r 
+ - X -* x r 
r 2 r 
1 — 2 r 
+ 7 
X X Z 3 X r 
2 r 3 r 
X T + ' ZX r 
+ r X 7 T^ ? 
+ 7 
X— X — — r 
2 r 3 r 
-f > &c. 
” + 1 * + i — r n + i —2 r 
, n .n n — r r n 
X r + -ZX r r +- 
n 4- i — r n-f" 1 — 2r 
~ ZX r +7X ; *** ~ + 7 
Ti + i— 3 r 
7 + , &c> 
„ ” + 1 3 r 
- l zKv r -f, & c# . 
2 r 
_ n + i— 2 r ^ n r, +1 — 3 r 
+ — X r -Z~ X r + “ X ~~~ X — Z 3 X r. &c.. 
” + i — 3 r 
7 x^T 1 * '~JT Z ' X r +» &c. 
Now, in order to establish the laws of arrangement upon 
clear and general principles, it is necessary to observe these 
particulars, ist. The exponents of the terms, both in the 
multiplicand and multiplier, are in arithmetical progression ; 
they have the same denominator r, and r is also the common 
difference in the numerators of each progression. 2d. The 
multiplicand being multiplied by x r , the first term in the 
multiplier, gives the first horizontal line of products ; and 
consequently the exponents in this line are obtained from 
the exponents in the multiplicand by adding 1 to the nu- 
merators.' The numerators, therefore, of the exponents of 
this line are also in arithmetical progression ; and under this 
the other lines of products are to be arranged, so that terms 
which have the same exponents may come under one another. 
3d. The coefficients being neglected, if any term in the mul- 
