530 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON 



values x, using each formula for the full range which it had been calculated for. 

 We thus found that the best agreement between experiment and calculation 

 was obtained by using formula II. for the interval ,2? = 0*4 to ^ = 0*6, and 

 formula I. and III. for their entire intended ranges. But the actual utilisation 

 of the formulas in this manner would have produced unpleasant, though by no 

 means alarming, discontinuities in the final table at x — 40 and x = 60 ; we 

 therefore ultimately decided upon exhausting each formula, so as to obtain 

 duplicate values for the ?/'s corresponding to ^ = 0*3 to 04, and to# = 0'6to 

 07, and for each such x took the mean of the two competing values. 



This is the history of the entries for 4 S in the following tables. 



From these values 4 S and the values a = (S — S t )-^£ as calculated by means 

 of the interpolation formula, pp. 526 and 527, we calculated the specific gravities 

 4Si5- 5 fi for 60° F. for all the percentages from 31 to 100. 



The corresponding values 4 S 15 . 56 for the alcohols from 30 per cent, down- 

 wards might have been calculated similarly from the values a and b in formula 

 S — S t = at + bf as obtained graphically from the directly calculated values (see 

 p. 527), but we had no perfect faith in these interpolations, and therefore 

 preferred to calculate the specific gravity at 60° F. of each of the alcohols 

 experimented on from the results obtained at 0°, 14°7, and 19° 7, and, from 

 the set of values 4 Si 5 . 5G thus obtained, to calculate the coefficients of a formula 

 I) = ax + bx 2 + cx % by the method of the least squares. 



In this case j/ = 4 W 15 . 5G — 4 Si 5 . 56 , where the first symbol stands for the specific 

 gravity of water at 60° F. ; water at 4° being taken as = 1000. Calculating 



on the basis of a + bx + cx 2 — y =0, we found 



x 



a= +180-522; log « = 2-256 530. 

 b= -191-450; log b = 2-282 055. 

 c = +318-220; log c = 2-502 727. 



To test the equation, Ave calculated the y's for 20 per cent, and 40 per cent, 

 and found 



Value of y by 

 x. Formula. Experiment. 



0-20033 31-039 30-92 



0-40029 61-995 61-98 



The values given in the following table for 4 S in . M U P to 30 per cent., are 

 calculated from this formula ; those calculated in the same way for 31, 32, . . . 

 40 per cent, agreed very well with those deduced from the values 4 S and the 

 • I nation Si 5 . 5C = S — 15'56 a ; but we preferred to let the latter stand uncorrected. 

 As some of our readers may consider this an error of judgment, we give the 

 1 wo values in the following table : — 



