1836.] Summation of Polynomial Co-efficients. 187 



salt disengaged by beat a sm^ll of sulphurous acid, as an hyposulphate would 

 have done. It appears to m n , therefore, that we should cnnsider artificial tannin 

 as a combination of resin and hyposulphuric acid, which supposition is conform- 

 able enough with the facts that led to the discovery of this modification 

 of sulphuric acid. Pounded resin digested with heat in very dilute sulphuric 

 acid, does not however a;ive rise to the purple matter: it merely takes a brown 

 colour. Gum benjamin and balsam of copahu do produce it : turpentine does 

 not : benzoic acid does not acquire a red colour under similar treatment. 



It follows from the facts contained in this Memoir, that the Chinese varnish is 

 composed, ]st, of benzoic acid ; 2nd, of a resin ; and 3rd, of a peculiar essential 

 oil, and that it is only to the happy proportions of these three, and to the slight 

 differences between their properties and those of analogous resins, that the Chinese 

 varnish owes the superiority which renders it so precious in the arts." 



III. — Summation of Polynomial Co-efficients. By Mr. W. Masters. 



It is stated in most of the treatises on Algebra, that, if a binomial 

 be raised to any power, the sum of the numeral co-efficients of the 

 terms of that power is equal to 2 raised to the same power ; but I 

 have no where met with even a most distant hint of the proposition 

 (which I am about to demonstrate) that, the sum of the numeral co- 

 efficients of any power of a polynomial is equal to the number of terms 

 in that polynomial raised to the same power. This is almost self- 

 evident ; for if a binomial (x-\-a) be raised to any power, it is plain 

 that the numeral co-efficients that appear in the developement originate 

 not from x or a which are heterogeneous, compared with abstract num- 

 bers, but from (1 + 1) the co-efficients of x and a ; for while we deve- 

 lope (x+a) we at the same time develope (1-f.l), and the figures that 

 appear represent a certain power of (1-f.l). 



(1) Let (a + 6-f. c-f &c), («>& ' + c'-|- &c), (a" + b"+ c"+ &c), 

 be m sets consisting of n things each. If one set be taken, and 0113 letter 

 from it at a time, the number of combinations will be n • and as the 

 numeral co-efficient of each combination is 1, n likewise represents the 

 sum of the numeral co-efficients of the combinations. Next, if one 

 letter be taken at a time and two sets be used, the number of com- 

 binations — 2 n ; and since the co-efficient of each combination is 1, 

 2 n also represents the sum of their numeral co-efficients. 



If one letter be taken and three sets used, Cns. = 3 n — S. N. Cts. 



If one letter be taken and m sets used, C. == mn = S. N. C. 



The co-efficients of n form the following progression : 



1,2, 3, 4, m. 



(2) Now take two sets, consisting each of n things ; combine them 

 by taking one letter at a time from each. 



B B 2 



