528 PROFESSOR DITTMAR AND MR C. A. FAWSITT ON 



should have wished ; but we could uot see our way towards doing better, and 

 consequently left the subject on one side to proceed to calculate interpolation 

 formulae for 



The Relation between the Per-Unitage p of Methyl-Alcohol and the Specific 



Gravity 4 S at 0°. 



As the specific gravity for^> = must be that of water at 0°, which we will 

 call W , we at once adopted the difference 4 W — 4 S as our dependent variable, 

 but found it convenient to take 4 W 4 =1000, and adopted "y" as a symbol for 

 the value which the difference then assumes ; while " x " was substituted as a 

 handier symbol for the per-unitage of CH 4 0. 



A preliminary graphic interpolation showed that there is a change of curva- 

 ture somewhere about # = 0-20 (corresponding to 20 per cent.), showing that if 

 a parabolic formula worked at all it must at least be of the third degree. 

 Warned by Mendelejeff's experience with ethyl-alcohol, we never attempted 

 to obtain one formula for the whole curve, but at once decided upon dividing 

 it into sections. As the part from x = 0-2o upwards exhibited no change of 

 curvature, we tried a variety of functions, including the general equation of the 

 second degree ( Ay 2 + ~Bxy + Cx 2 , &c), for summing up the relation y =/(%) for 

 x = 03 to 1*0 in one formula, but arrived at no satisfactory result. After a 

 deal of pioneering, we ultimately came to divide the curve into the following 

 sections : — 



I. From£ = toa; = 0-4. 



IT. „ £=0-3 bo a;=0-7. 



III. „ a?=0-6 toaj=l'0. 



First Interval 

 We began by bringing the formula y = ax + bx 2 + cx s into the form 



a+bx+cx 2 — — = 0, 

 x 



and then proceeded to calculate the constants a, b, c by means of the method 

 of the least squares. This, as we now see, was an error of judgment, because 

 it is the error in y and not that in y-r-x which must be brought to its minimum 

 value ; yet the result was satisfactory all the same. We found 



a =+185-079; log a = 2-267 357. 

 6= -348-682; log h = 2-542 429. 

 c= + 559-542; log c = 2747 833. 



