1822.] in the Expansion of a Multinomial. 299 
Q"-3 + &c. which series is the developement of the binomial 
(2+1)" = 3"; therefore, the sum of all the coefficients in the 
expansion of a trinomial a + 6 + ¢, to the power of 7, is 3", 
4, To find the sum of all the coefficients in the expansion of 
a muitinomial of m terms, to the power of n. 
Let p = the last term, and y = the sum of all the remaining 
terms; that is, of the m — | terms; then the wth power of the 
multinomial will be expressed by (y + p)" = y"+ 2. y""p + 
n. uP + &c.; but it follows, a priori, that the sum. of 
all the coefficients in this expansion will be y* + n.y"-' +n. 
= .y"—* + &c.; and from the preceding cases, it is manifest 
that the sum of all the coeflicients in 
. y” is. (m,— 1)” 
Of 8 (a LPs 
yy"? 1s (m — 1)"~*, &e. 
which substituted for y", y"~', y*~*, &c. gives (m — 1)" + n 
(m —1y- +0 “ge .(m — 1)? + &e.; but this. series 
arises from the developement of (m— 1 + 1), = m"; therefore, 
the sum of all the coefficients in the expansion of a multinomial 
of m terms to the power of m is m”. 
_ Throughout the preceding investigation, the exponent » has 
been taken arbitrarily, it may, therefore, be expounded by any 
number whatever, either positive or negative, whole or fracted. 
I am, Sir, yours truly, 
S. Jongs. 
ARTICLE X. 
Observations on the Presence of Moisture in Medifying the Spe- 
cific Gravity of Gases. By C. Sylvester, Esq. 
(To the Editor of the Annals of Philosophy.) 
DEAR SIR, 60, Great Russell-street, June 5, 1822. 
Wuarever Mr. Herapath may say of Dr. Thomson’s paper, 
its foundation is good, and the principal facts from which his 
conclusions are drawn have been long known to the philosophi- 
cal world, and confirmed by experience ; I allude particularly to 
the fact of the same weight of steam at all temperatures 
containing the same quantity of heat; and that the sum 
of the degrees expressive of the latent and sensible heat is 
a constant quantity. He is doubtless wrong in making these 
sums commence at 32°. If the principle be true above. that 
degree, it must be equally applicable to those below the 
