VOL. LXXXVII.] PHILOSOPHICAL TRANSACTIONS. 103 



found by the experiments, are represented by ordinates or lines perpendicular to the 

 line ab, at the points where the measures of the densities end. The curve line ad, 

 drawn through the extremities of these ordinates, which must necessarily be regular, 

 will, by bare inspection, give a considerable degree of insight into the nature of 

 the equation which must be formed to express the relation of the densities to the 

 elasticities ; one principal object of these experimental inquiries. Now, putting the 

 density = x, and the elasticity = y, the line ad will be the locus of the equation 

 expressing the relation of x to y; and had Mr. Robins's supposition, that the elas- 

 ticity is as the density, been true, x would have been found to be to y in a constant 

 (simple) ratio, and ad would have been a straight line. But ad is a curve, which 

 shows that the ratio of x to y is variable; and it is a curve convex towards the line 

 ab, on which x is taken; and this circumstance proves that the ratio oiy to x is 

 continually increasing. 



Though these experiments all tend to show that the ratio of y to x increases as 

 x is increased, yet when we consider the subject with attention, we shall I think 

 find reason to conclude that the exponent of that ratio can never be less than unity; 

 and further, that it must of necessity have that value precisely, when, the density 

 being taken infinitely small, or = O, x and y vanish together. Supposing this to 

 be the case, namely, that the exponent of the ultimate ratio of y to x is = ] , let 

 the densities or successive values of x be expressed by the series of natural numbers, 

 0, 1,2, 3, 4, &c. to 1000, the last term = 1000 answering to the greatest density; 

 or when the powder completely fills the space in which it is confined; then, by 

 putting z = the variable part of the exponent of the ratio of y to x, 



To each of the successive values of x = O, 1, 2, 3, 4, &c. 



The corresponding value of y will be accuO 



ju 1 .- >O l + * 1' + *, 2 l + * 3 l +* 4' + * & c 



rately expressed by the equations J ; ' 



For, as the variable part z, of this exponent, may be taken of any dimensions, it 

 may be so taken at each given term of the series, or for each particular value of x, 

 that the equation x 1 + ■ = y, may always correspond with the result of the experi- 

 ments; and when this is done,* the value of z, and the law of its increase as x 

 increases, will be known ; and this will show the relation of x to y, or of the elas- 

 ticities of the generated fluid to their corresponding densities, in a clear and satis- 

 factory manner. Without increasing the length of this paper still more, by giving 

 an account in detail of all the various computations I made, in order, from the results 

 of the experiments in the foregoing table, to ascertain the real value of z, and the 

 rate at which it increases as x is increased, I shall content myself with merely giving 

 the general results of these investigations, and referring for further information to 

 the following table 2, where the agreement of the law founded on them, with the 

 results of the foregoing experiments, may be seen. 



Having, from the results of the experiments in table 1, computed the different 

 values of z, corresponding to all the different densities, or different charges of 



* True, when this is done for every term or value of x : but the law of progression will not be known 

 farther than the terms which are actually compared. 



Y2 



