590 



HYGROMETRY. 



Ujrgrome- equal U) 100; and as y is equal to 100 at the same 

 try- point, the curve is symmetrically disposed, with regard 



^"""V"" 1 "' to these two values of x and y. 



79. To render the calculation more simple, the co- 

 ordinates x and y are transformed into x' and y', which 

 are also at right angles to each other, but immediately 

 related to the axis of the hyperbola, and having their 

 origin in some assumed point of it. Hence the new- 

 line of abscissa* will form an angle of 45 with AB ; 

 and if X and Y be taken to represent the primitive co. 

 ordinates of that line, corresponding to the point of new 

 origin, we shall have 



Hygiome. 

 try. 



81. These data are sufficient to determine completely 

 the nature of the hyperbola ; for since its axis coin- 

 cides with the line of the abscissae x', it must necessa- 

 rily have an equation of the form, 



y= Y + (/-*') or Y + Cos. 45 (y'~ *') 



Before reducing these equations, it will be convenient, 

 for the sake of operating with small numbers, to repre- 

 sent by unity, the abscissae x corresponding to the ten- 

 sion 100 : then, from the inclination of the axis of the 

 hyperbola, the equation for the axis, in terms of x and 

 y, will be y = 1 x ; and since the primitive co-ordi- 

 nates X and Y must be similarly related, Y = 1 X. 

 The general expressions for x and y, thus restricted, 

 become 



in which a, b, and c. are three constant coefficients, the 

 values of which may be determined by the several va- 

 lues of x' and y', given above. The solution of the re- 

 sulting equations gives the following values : 



a = .0000605 

 b = 1.149338 

 c = 4.08683 



82. If the value of (x .3815) ^/2 be represented 

 by*, then x' s y', andy' = t x'. This value 

 of y being substituted for that quantity in the general 

 equation of the hyperbola, we obtain, 



(s x') 2 = a + 2 b x' + c x'* 

 The solution of this quadratic gives, 



- 



c 1 



By adding together both sides of these equations, the 



quantity X is exterminated, and the following values of And y' = 



y and y' are obtained, 



C ^^ L 



80. If the values of x and y be substituted for these 

 quantities, as determined by experiment, (taking for 

 example the muriate of lime, whose specific gravity is 

 1397;) then * = . 376; # = .61 3; 



83. The value of y being determined by the last for- 

 mula, and substituted for it in the equation, 



will give the value of y in terms of x. By means of 

 this formula, Biot calculated the following Table, which 

 he found to accord almost exactly with actual observa- 

 tion: 



The value of y thus determined is so small, that the 

 point in the curve to which its extremity refers, nearly 

 coincides with the axis, and might be taken, without 

 any great error, for the vertex of the hyperbola; but to 

 avoid the introduction of any unnecessary inaccuracy, 

 it will be better to assume the origin of the abscissae 

 of x', at that point of the axis where the latter is in- 

 tersected by y', and then X will be determined by ad- 

 ding to .876 the projection of y' on the axis, along 

 which the abscissas of x are reckoned, that is, Cos. 

 45X .0077718 or .0055. We thus obtain X = .3815, 

 and x'= (x .3815) /2y'. When xandy are given, 



we obtain the value of y by the equation y' x ^ 





t 



and that value being substituted for y', in the equation 

 x"= (x .3815) ^--y, the value of x' is also deter- 

 mined. In this manner were found tl.e following va- 

 lues of x 1 and y 1 ', from the corresponding values of x and 

 y, as ascertained by observation. 



Table of 

 the tension 

 of vapour 

 for the de- 

 grees f 

 baussure's 

 hygrometer 

 at the 

 ttmp. SO". 



