366 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[December, 



tion as before, we get, in this case, for the value of the overcoming- 

 force, 



ifAy 6in (2c'-{-S-,p) 

 ~ 2 sill (5 — 2e'-j-()!) 



M'lieie y is equal the perpendicular from ( F) on the face, or face 

 jiroduceil. 



If we put Bi=2c' + e, and Sj^S— 2c'', the above equation, 

 after a few reductions, becomes 



_ why cosB, tan B,— tan o 



H ^Z X X — 



2 cosSj tan Sj+tau j, 



^^'hen this is a minimum, 



jj^^ tang, V(lan fl tan SJ 



" "" -/(tan 9 tan 5 J - -/r(tan S-tan fijx (tan;8j-(-tau 5,)] 



(15) 

 r (16) 



K = 



ujA^ tan e sin /S, tan /3, 

 2 cos 5j 



e-tane,)]) 3 



,/(17) 



1 = 0, 



R = — sin / 

 2 



\ ^/[tane (tanfij + tanSJ] - V[tan 8, (tan S- 



iu %vhich the usual changes of signs are to be made for the negative 

 values of 5j, and for arcs greater than 90^ 



When the direction of the force makes an angle equal to c with 

 the face, then S^ = o, and, 



(18) 



(19) 



If the force exceed the value of R here found, it will slide along 

 tlie face, and wlien the face is vertical this value is equal to the 

 maxbnum maximorum value of the resistance, in the same case, 

 already found; or, 



R = — sin c 

 2 



When 6 = 90°, the general equations become 



tan (B = tang,^(tan5.) 



^ V(tar.Sj- ^/[(tan/3,+tan8J] '- "-^ 



u-h'' sin/S, taniS, / 1 \2 



~ W(tanB,+tan5j)-^((an87)/ ^^^^ 



R - 



2 cos 5, \V(tanB,+tan 



If the force in this case be supposed to act in an horizontal direc- 

 tion (5j -j- /3j := 90^)j these equations may be reduced to 



tan ^ = cot (<' — - ) (22) 



Rz. 2 cot 



■ (-9 



(23) 



If the face be vertical, then S = c, and the equations may be 

 further reduced to 



tan ip = cot Je (24) 



R = -^ cot= ic 



(25) 



TEMPERATURE AND ELASTICITY OF VAPOURS. 



On an Equation between the Temperature and the Maximum Elas- 

 ticity of Steam and other Vapours. By \Vu.i,\au John Macquorn 

 Rankine, C.E. — [From tlie Edinburgh New Philosophical Journal 

 for July 1849.] 



In tlie course of a series of investigations founded on a peculiar 

 hypotliesis respecting the molecular constitution of matter, I liave 

 obtained, among other results, an equation giving a very close 

 a])pr(iximation to the maximum elasticity of vapour in contact 

 with its liquid at all temperatures that usually occur. 



As this equation is easy and expeditious in calculation, gives 

 accurate numerical results, and is likely to be practically useful, I 

 liroceed at once to make it known, without waiting until I have 

 reduced the theoretical researclies, of which it is a consequence, to 

 a form fit for publication. 



The equation is as follows: — 



(1-) Log. P = a---^ 



t t- 



Where P represents the maximum pressure of a vapour in contact 

 with its liquid ; t, tlie temperature, measured on the air-thermo- 



meter, from a point which may be called the absolute zero, and 



which is — 



274°-6 of the centigrade scale below the freezing point of water. 



462°'28 of Fahrenheit's scale below the ordinary zero of that scale, 

 supposing the boiling point to have been adjusted under a pres- 

 sure of 29'992 inches of mercury, so that 180° of Fahrenheit may 

 be exactly equal to 100 centigrade degrees. 



4Gl°-93 below the ordinary zero of Fahrenreit's scale, when the 

 boiling point has been adjusted under a pressure of 30 inches of 

 mercury, 180° of Fahrenheit being then equal to 100°'0735 of the 

 centigrade scale. 

 The form of the equation has been given by theory; but three 



constauts, represented by o, /3, and 7, have to be determined for 



each fluid by experiment. 



The inverse formula, for finding the temperature from the pres- 

 sure, is of course 



(2.) 



W 



o-Log i^ e- 



2y 



7 ^ 472 



It is obvious that for the determination of the three constants, 

 it is sufficient to know accurately the pressures corresponding to 

 three temperatures; and that the calculation will be facilitated if 

 the reciprocals of those temperatures, as measured from the abso- 

 lute zero, are in arithmetical progression. 



In order to calculate the values of the three constants, for the 

 vapour of water, tlie following data have been taken from M. Reg- 

 nault's experiments : — 



These data give the following results for the vapour of water, 

 the pressures being expressed in millimetres of mercury, and the 

 temperatures in centigrade degrees of the air-tliermouieter : — 



Log7 = 5-0827176 Log /3 = 3-1851091 a = 7-831247 



Table I. exhibits a comparison between the results of the for- 

 mula and those of M. Regnault's experiments, for every tenth 

 degree of the centigrade air-thermometer, from 30° below the 

 freezing to 230° above it, being within one or two degrees of the 

 whole range of the experiments. 



M. Regnault's values are given, as measured by himself, on the 

 curves representing the mean results of his experiments, with the 

 excejition of the pressures at 26°'86, one of the data already men- 

 tioned, and that at — 30°, which I have calculated by interpolation 

 from his Table, series h. 



Each of the three data used in determining the constants is 

 marked with an asterisk*. 



In the columns of differences between the results of the formula 

 and those of experiment, the sign -{■ indicates that the former 

 exceed the latter, and the sign — the reverse. 



Beside each such column of differences is placed a column of the 

 corresponding differences of temperature, which would result in 

 calculating the temperature from the pressure by the inverse for- 

 mula. These are found by multiplying each number in the pre- 

 ceding columns by — -r=, or by -t-j =, as the case may require. 



In comparing the results of the formula with those of experi- 

 ment, as exhibited in Table I., the following cii-cumstances are to 

 be taken into consideration : — 



First, That the uncertainty of barometric observations amounts 

 in general to at least one-tenth of a millemetre. 



Secondly, That the uncertainty of thermometric observations is 

 from one-twentieth to one-tentli of a degree, under ordinary cir- 

 cumstances, and at high temperatures amounts to more. 



Thirdly, That, in experiments of the kind referred to in the 

 T able, those two sorts of uncertainty are combined. 



