METHODS EMPLOYED IN CALIBRATION OF MERCURIAL THERMOMETERS. 191 



105°, while 33 determinations of the corrections of equal value are made 

 between 122"^ and 129°, without including the additional observations. 



Such an arrangement of the experiments is evidently unsatisfactory. 

 It will be hereafter shown that von Oettingen has probably under- 

 estimated the value of the corrections of the principal points, but making 

 all allowance for the much higher value which should probably be 

 assigned to them, there can be no doubt that the correction is either 

 determined unnecessarily often in the middle of the scale, or at too great 

 intervals in the lower part. 



It appears, then, that the extremities of the scale are the portions 

 upon the corrections of which least reliance can be placed. This is 

 especially unfortunate. Delicate thermometers are now commonly 

 graduated for a part only of the range between 0° and 100° C. As, there- 

 fore, such instruments can have at the most but one fixed point, the 

 temperature value of a scale division must be determined by comparing 

 them at two temperatures with another instrument upon which the other 

 fixed point can be read. If two thermometers do not cover the whole 

 range between the freezing and boiling points, the comparison may have 

 to be made by several intermediate steps. Each instrument in a set of 

 such thermometers is, therefore, constructed so as to overlap those which 

 read above and below it. To avoid the necessity for having too many 

 thermometers, this overlap rarely exceeds one-fourth of the scale and is 

 often less. Hence the errors in the absolute measurements made by a 

 given thermometer will be several times greater than the errors made in 

 the measurement of the temperature- difference necessary to determine the 

 value of the overlap. The extremities of such thermometers should, 

 therefore, be the most carefully corrected, and yet it is here that the 

 method under discussion is weakest. 



Objection may also be taken to the method of grafting the initial 

 point correction curve with those portions of the correction curves for 

 the upper points which affect the same part of the scale. 



Von Oettingen, referring to an example in which eight threads were 

 used, says (he. cit., p. 14) : ' Fiir die Curve der arithmetischen Mittel . . . 

 sind ausser den 64 Verbesserungen der 8 mal 8 oberen Fadenenden, noch 

 die enimal 8 Correktionen der unteren Enden hinzugenommen. In der 

 That haben diese Verbesserungen dasselbe Grewicht wie der oberen Enden.' 



This view appears open to objection. It is quite true, as von Oettingen 

 adds, that the expressions obtained by him for the corrections of an upper 

 and a lower point both contain the same assumptions, which are only 

 approximately true — viz., that the sum of the errors of any row and of any 

 column are separately = 0. 



But he determines the ei'ror of the topmost initial point (116) from 

 the means of the curves of the upper points. To this, therefore, the full 

 weight, 10 (if 10 threads are used), must be assigned. 



In using the formula (p (i'g) — ([> ('i,o) = /ig — 7i,o, the only quantity 

 neglected is the difference of the means of the corrections of the upper 

 points in the ninth and tenth rows. Now, since the upper points in these 

 rows extend over nearly the same part of the scale, the difference of these 

 quantities — i.e. //'t, — /i',o — must be small. In the example given it 

 amounts to 0°'007. Hence (/> (uj) must be known with considerable 

 accuracy. The value of f (ig) will be more doubtful, and so on. On 

 the other hand, the weight 10 is perforce assigned to corrections given 

 by the initial point curve at the lower end of the scale, inasmuch as no 



