31 4 Mr. Robertson's Demonstration 
multiplied by 1 r are equal to the whole of the next per- 
pendicular line of products towards the right hand. 
To do this in a manner applicable to any part of the series 
concerned, and to avoid numeral coefficients, which would ob- 
scure and encumber the general reasoning, it is necessary to 
find the value of the numerator of -- - j = -- I - ~ Z-LL j n terms of A, B, 
J + ixr 
C, D, &c. and of a , b, c , d, &c. and to ascertain the relative 
values of «, /3, y , S, &c. and that we may do this with due pre- 
cision and perspicuity, it is proper to fix upon two contiguous 
perpendicular lines of products. 
17. Let the lines be those which have in their first terms 
G F 
F and G respectively, and then n-\- 1 — £ r = -f 1 = -^ + 1 
e , d . c . b , 
— r = D + 1 — 2r= C+ 1_ S' - = T+ 1_ * r = A + 
1 — 5 r = + 1 — 6 r; and therefore, according to the sub- 
stitution in the last article, n 1 — £r = y+^==-|' + ‘7 := '5 
+ T=r+T = §+T=X+T=T+ 7- N ° W the firSt 
of the two contiguous perpendicular lines fixed upon being 
multiplied by these values, viz. the first term being multiplied 
by the first value, the second term by the second value, &c. 
and the denominators f r 6 , » s r 6 , &e. in the first line, and 5j r T 
a £r 7 , &c. in the other being omitted, the two lines will be as 
represented in the following columns. 
