Heat by the Method of Mixtures. 269 



From (4$) we see that to diminish the effect of an error in 

 T we must, as for t, decrease M+W, and increase T — 0, as 

 much as possible ; but that contrary to the previous case of t. 

 an increase in 6 — t will act injuriously. Likewise by reference 

 to (46), we see that an increase of T — 6 would really be bene- 

 ficial, although from (46) it would seem to be in conflict with 

 the condition that 6 — t should be a minimum. 



Since, however, an increase of T — 6 involves an increase in 

 M+¥, and this, as we have seen in case of t, to be highly 

 undesirable, it is best to decrease the value of T— 6, even at 

 the expense of an increase in the involved error of s, which is 

 much less in this case than in the preceding. 



If, as before, we let the maximum error of s be 1 in 1,000, 

 or 0*1 per cent, we have 



When T— = 50°, At = 0-05° 

 « T — = 80°, At = 0-08° 



Or if we read T to ^g-° = 0*04° the resulting error in s will be 

 less than that involved in reading t to -^L- or g4 ¥ °. 



(C) We have last to consider the effect of an error in deter- 

 mining 0. This is the most important of the errors in the 

 temperature readings, both because of its effect upon s, and the 

 numerous causes which go to produce it, combined with the 

 difficulty of guarding against them. 



1st. The error produced in s will be found as before, by 

 differentiating (2). 



^_a _(M + W)c(T-0) + (M + W)c(0-O ^ 



— -a 6 s-— w{T -ey ~ 



c (M + W) c (M + W) (0-0 



y (5a) 



w ' (T-0) + io " (T-0) 2 J 



thus involving both corrections already found. Here, since 

 the first correction is generally larger than the second, the most 

 desirable thing to do will be to increase 6 — t (by decreasing 

 M+W), rather than to increase T — 6 % except by increasing T, 

 which in most forms of heating apparatus has its limit at 

 100° c. 



If as before, we take the maximum allowable error as 0*1 

 per cent, we have 



For 6-t = 10° and T-0 = 50°, A0 = 0-0083° Z T h° 

 « 0-t= 5° " T-0 =80°,&6= 0-0041° Z-gbt° 

 Conversely, if we read 6 to only ^° we get 



AqS = 0*0024 = i of one per cent for the 1st case. 

 A d s= 0-00425 =f+ " " " " " 2d case. 



■jy 



