416 
Miscellanea 
The whole arithmetic is shown here. 
Example. In P 3 : The first coefficient is C 3 = B 2 .b; look in P 2 , we see "2 [3...," hence 
the required coefficient is 6. Place this in front of its [ ] in P 3 . 
The second coefficient is £> 3 = (B 2 + C 2 ) c. Look in P 2 we see " 2 " and get (2 + 3) 5 = 25. 
3 [5... 
Place this in front of its [ ] in P 3 . 
2 
The third coefficient is E 3 = (P 2 + C 2 + D 2 ) d. Look in P 2 . we see "3 " and get 
5 [7... 
(2 + 3 + 5) 7 = 70. Place this in front of its [ ] in P 3 . 
2 
The fourth coefficient is F 3 = (B 2 +C 2 + B 2 + E 2 ) e. Look in P 2 we see " j? " and get 
7 [11... 
(2 + 3 + 5 + 7) 11 = 187. Place this in front of its [ ] in P 3 . 
6 
Similarly 217 in P i comes from "25 [7...," which gives 31x7 = 217: and so on. 
Now note that as placed the [ ] factor continues always the same in the same horizontal line, 
and therefore needs only to be calculated once for all in step (II). 
Perform each calculation in step (II) directly under its place in step (I), 
First, do all the parts in ordinary type, j 
Second, insert the parts in Italics, 
Third, make the final additions. 
Note. The [ ]'s are all calculated in P 2 , and keeping the calculation lined as above, the value 
of each [ ] has merely to be repeated for each successive P, until it runs out. Thus 31 is the 
third [ ] in P 2 , and it reappears automatically in P 3 and P 4 and then runs out. 
The process as detailed above, may be shortly described thus : — 
Of the series of letters a, b, c, d, e, f, g, k, i, I; I... select any one, say /, and place it so as 
to have free positions in front of and after it : then place after it as a multiplier, the [ ] con- 
taining the sum of all the letters that follow it (that is, formally, the sum of the products of all 
the letters that follow it taken one at a time). This letter / and the square bracket [g + h+...] 
constitute a sum of products of two letters each, hence if we are calculating say P 7 , we need 
each multiplier of f[g + h + ...~\ to be the product of 5 letters: and this multiplier is "the sum 
of the products 5 at a time " of all the letters which precede /. Thus one term in Pi is 
/ abcde + abcdf+abcdg + abcef+abceg + ...\ 
I +bcdef+bcdeg+... j A [>' + £ + ...]. 
\ +cdefg-\-... 21 terms in all / 
The algorithm presents a systematic method of calculating the whole series of values of 
Pi, P 2 , P 3 ...P n , up to any required value of n. so that each calculation not only serves for its 
own place, but is a step in the calculation of future terms. In short the process is an 
economical method for the calculation of P r , where r has all the values r = 1, 2, ...n, and 
where P r is the sum of the products r at a time of a series of n quantities. 
The calculation of the problem of the Amritsar District is exhibited On the opposite page. 
It will be noted in this calculation, in which the whole arithmetic is shown, that in step (I) 
10 products, and in step (II) 14 products of two terms each were evaluated by logarithms, 
and that to obtain the final values of the P's, 5 quotients of two terms each were evaluated. 
In all 29 logarithmic calculations of two terms each were made. The remainder of the calculation 
is simple addition. A complete check may be made by comparing the value of F 0 derived as 
above from all the P's with the value of F 0 obtained from F 0 = (l— a) (1 —b) ... (1 — n). 
