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21-3+ &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 + b + c, to the power of n, 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 1 terms; then the nth power of the multinomial will be expressed by (y. + p)" = y* + n. y*-'p + naà
+y* - p2 + &c.; but it follows, à priori, that the sum of all the coefficients in this expansion will be yn it n.yu-tn. na!! y" -9 + &c.; and from the preceding cases, it is manifest that the sum of all the coefficients in
y" is (m – 1)"
yn – is (m - 1)*-, &c. which substituted for y", yu-', yu-, &c. gives (m - 2)" + n (m - 1)"-' + n. no?. (m 1)" - + &c.; but this series arises from the developement of (m - 1 + 1)n = m"; therefore, the sum of all the coefficients in the expansion of a multinomial of m terms to the power of n is m".
Throughout the preceding investigation, the exponent n 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,
Observations on the Presence of Moisture in Modifying 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. Whatever 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 philosophical 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 320. If the principle be true above that degree, it must be equally applicable to those below the
That is, if I be the latent heat, and t the temperature, l + t = c a constant quantity, whatever t may be. differ with Dr. Thomson as to any practical advantage derived from the variable quantity of latent heat at different temperatures either in distillation or in its agency in steam engines. Suppose in the former application that vapour is distilled over at the temperature of 70°, and condensed in a temperature of 50°, a constant succession of liquid will be formed by condensation, which is the practical effect desired, and it must be admitted from the law quoted by Dr. Thomson, that the stock of vapour at 700 constantly passing from the still to the receiver will hold more latent heat than the same quantity at a higher temperature. The difference will consist in having a small excess of latent heat, which is “ in the uncondensed vapour at every period of the process without any disadvantage,” as to the ustimate quantity of liquid condensed. I am inclined to think that if the size of the apparatus be increased so that the same weight of yapour may come over in the same time, the advantage would be in favour of the low temperature, owing to the quantity of heat lost in all processes carried on at high temperatures by radiation and the conducting power of contiguous bodies.
For the same reasons there is no advantage in using steam for engines at a high pressure. Whatever may be the fuel consumed to make a given volume of steam equal to one atmosphere, it will take twice the quantity to give twice that volume, or the same volume of a density to give a pressure equal to two atmospheres. I should think therefore, that the increased temperature of the volume equal to two atmospheres would lose more heat to surrounding bodies than the two volumes of one atmosphere, but the mechanical advantage of the two will be obviously the same. The boasted advantage of the Cornish engines has chiefly arisen from their inventor assuming some erroneous data respecting the power of steam, and many others, even Mr. Herapath, seem to have fallen into the same mistake. In the range of temperature commonly used for high pressure steam, it will be found that from an increase of temperature of every 30° degrees, the density and elasticity of the steam become doubled; that is, at 212°, its elasticity is equal to about 30 inches of mercury, and a cubic foot of such steam would weigh about 253 grains. At 212 + 30 = 242 degrees, the volume remaining the same, it will support 60 inches of mercury, and a cubic foot will contain 253 x 2 = 506 grains. Hence it will appear that the temperature is increasing in arithmetical proportion while the power of the steam increases in geometrical proportion, and hence the apparent advantage by working with high pressure.
The source of this fallacy will be found in the assumption of the quantity of heat being as the temperature; when the fact is, that while the temperature has been advancing by 30°, the real quantity of heat is doubled ; and it will be found that a cubic foot of steam of 60 inches pressure in mercury, although only 30° in temperature above a cubic foot of 30 inches pressure will heat twice the quantity of water to the same temperature, or melt twice the quantity of ice, which is the clearest proof that their respective quantities of heat are as 2 to 1.
* The force of steam has not strictly a geometrical ratio to the temperature. The ratio for 10° below 212° is about 1•23. And this ratio for every ten degrees above will decrease by .01, while steam, for every ten degrees below, has a similar increased
The remaining part of this paper which is applied to calculate the correction for the specific gravity of gases, as affected by the presence of aqueous vapour, is very valuable. If the force of aqucous vapour at different temperatures be correctly taken in order to get the specific gravity of the same, nothing can be more simple than the formula given by Dr. Thomson for finding the allowance to be made for the presence of vapour in any gas. This is the same formula which is contained in my paper sent to your journal for finding the proportions of mixed inflammable gases. I remain, dear Sir, yours very truly,
Extracts from the “ Journal * of a Survey to explore the Sources
of the Rivers Ganges and Jumna.” By Capt. J. A. Hodgson, 10th Reg. Native Infantry.
As I have had it in my power to explore and survey the course of the Ganges within the Himālaya mountains to a considerable: distance beyond Gangautri, and to the place where its head is, concealed by masses of snow which never melt, I hope, that an account of my journey may be acceptable to the Asiatic Society. I must premise that, as Capt. Raper's account of Capt. Webb’s survey in 1808, has already appeared in the eleventh volume of the Researches, I have nothing to add to that officer's able and fàithful description of the mountainous country, passed through in the route of the survey from the Pūn Valley to Cajani, near Reital, where the survey towards Gañgautri was discontinued iw consequence of the serious obstacles which impeded it. I shall, therefore, only give an account of the course of the river above
* The Editor is favoured with these extraets from almost the only copy of Captain Hodgson's Journal, which has reached England, by Mr. Edmonstone, of Newcastle ; who observes, that in order to shorten the communication, a number of minute and interesting details have been necessarily omitted. This circumstance will serve to explain the breaks which the narrative occasionally assumes,' and we should hope will be received as a sufficient apology for our not doing all the justice that we could wish to the labours of Capt. Hodgson, who has since been appointed to the important situation of Syrveyor-General of India.
the village of Reital, where I halted to make arrangements for my progress through the rugged regions before me, in which I found I had no chance of getting any supplies of grain for my followers: I was consequently obliged to buy grain, and to send it off before me, so as to form little magazines at the places I intended to halt at; and as I learned that several of the sangas or spar bridges over the river had been destroyed by avalanches of snow, I sent a large party of labourers to re-establish them.
Considering Reital as a point of departure, it will be satisfactory to know its geographical position. By a series of observations with the reflecting circle of Troughton, and also by his astronomical circular instrument, I found the latitude to be 30° 48' 28" N.; and having been so fortunate as to get two observations of immersions of the first satellite of Jupiter, and one of the second, I am able to give a good idea of the longitude of the place; and the more satisfactorily, as two of the immersions are compared with those taken at the Madras Observatory on the same night, and with which I have been favoured by Mr. Goldingham, the astronomer there.
The telescope used by me in observing the satellites was a Dollond's 42 inches achromatic refractor, with an aperture of two and three-quarter inches, and power of about 75 applied, having a tall stand, and rack work for slow motion. The watch was a marine chronometer, made by Molineux, of London, and went with the greatest steadiness on its rate, as nightly determined by the passage over the meridian of fixed stars observed with a transit instrument. The time of mean noon when required was always found by equal altitudes.
By a mean of several observations taken at Madras about the time of four emersions of the first satellite, which I observed at Mr. Grindall's house near Seharanpūr. (Mr. Goldingham finds 5 10 24" for the longitude of Seharanpūr.) A snowy peak called Sri Cānta is visible both from Reital and Seharanpūr, its position is determined by means of a series of triangles instituted by me for the purpose of taking the distances and heights of the snowy peaks. I find the angle at the pole or difference of longitude between Seharanpūr station and Sri Cānta to be 1° 14' 47", the peak being east, and at Reital, the difference of longitude of that village, and the peak is found to be 12'6"; the peak being east, consequently the difference of longitude of Seharanpūr and Reital is 1° Ž' 41". On the whole, I think 5k 14' 20:6", or 78° 35' 60-7" may be safely taken for the longitude of Reital, east of Greenwich.
Reital contains about 35 houses, and is esteemed a considerable village ; as is usual in the upper mountains where timber is plentiful, the houses are large, and two or three stories high. When a house has three stories, the lowest serves to shelter the cattle by night; the second is a sort of granary, and in the apper the family dwells ; round it there is generally a strong wooden gallery, or balcony, which is supported by beams that project from the walls. The roofs of the houses are made of boards or slates; they are shelving, and project much beyond the top of the walls, and cover the balcony, which is closed in bad weather by strong wooden shutters or pannels. These houses are very substantial, and have a handsome appearance at a distance, but they are exceedingly filthy within, and full of vermin. The walls are composed of long cedar beams; and stone in alternate courses, the ends of the beams meet at the corners, where they are bolted together by wooden pins. Houses of this construction are said to last for ages ; for the Deodar or Cailon pine, which, I suppose, to be the cedar of Lebanon,* is the largest, most noble and durable, of all trees.
The situation of this village on the east side of a mountain, the summit of which is covered with snow, and the foot washed by the Bhagirathé is very pleasant. It commands a noble view of the Sri Čānta, and other adjoining peaks of the Himālaya, on which the snow for ever rests. Snow also remains until the rains, on all the mountains of the second order, which are visible hence, both up and down the river. Many cascades are formed by the melting of the snows on the foot of the surrounding mountains. One in particular descends in repeated falls of several hundred feet each, from the summit of a mountain across the river, and joins it near Batheri.
In the following account of my progress up the river, I have put down such remarks as occurred at the time, and they were written on the spot, and are here inserted with very little alteration. Though I am aware that such minute descriptions of localities must appear tedious, I hope they will be excused by those who, feeling interested in the subject, may have the patience to read the detail. To give general descriptions of such rude regions is difficult, if not impossible, and I trust that particular ones, though often tedious, will be found more faithful, and to give more precise ideas of those remote recesses of the Himālaya, which I visited.
On the 19th of May, I was joined at Reital by Lieut. Herbert, of the 8th Reg. N. I. who had been appointed my assistant, and from his skill and zeal, the survey has received much benefit.
Mr. Herbert came direct from Calcutta, and brought me a pair of mountain barometers, but the tubes filled in England had been broken before they arrived in Calcutta : there were some spare empty tubes which we filled and used as hereafter mentioned, but we could not succeed in boiling the mercury in the tubes to free it entirely of air. The height of Reital above the sea, as indicated by our barometers, is 7108 feet.
Having received reports that the sanghas were repaired, and that the grain I sent forward was lodged in the places I directed,
• It is the pinus Deodára of Roxburgh the Dévadáru of Sanscrit writers.-H.H,W, New Series, vol. IV,