542 BESSEL ON BAROMETRICAL 



in the transition of the temperature from t to t', from the rule 

 prescribed by the equation. It remains, for instance, quite 

 doubtful whether between the two elevations the true tempera- 

 ture may not differ from the value which would follow from the 

 rule by a quantity amounting to one-tenth part of x — t'. It is 

 not superfluous, therefore, to investigate further the influence of 

 such possible deviations. 



I will suppose that the true value of 



1 l—4ak i 



T T' 



E 



I + kt 



where t denotes the height of the thermometer at the elevation x, 

 corresponding to equation (23.), and a is a constant coeflBcient, 

 greater or less according to the amount of the deviation from the 



law. This expression of -^ is so chosen, that it agrees with the 



previous one at the two limits, and that the deviation of the 

 temperature which it supposes attains its maximum = « (t — t') 

 somewhere about t = ^ (t + t') or o? = i (/* + h'). We obtain 

 thence 



\a + x/ lEi t \ T — T J' 



and the integral taken from h to h', 



lie C ^ ~\ 



= ^{r-,')|l-y«i(T-T')| 



It does not seem probable that in any case which is likely to 

 occur the value of a would be comprehended within any very 



narrow limits, as for instance + — ; if it should reach either of 



these limits, the consequent correction in the resulting difference 

 of elevation, according to formula (23.), w^ould be 



_ H'-H T -i' 



~ - 1 + AT ' 4000" 



So, for example, for a difference of elevation of 1000 toises, for 

 which T — t' is usually 12°, the correction would be about 

 + 3 toises. We should be the less inclined to assume that « 

 must necessarily be very small, as it should not be overlooked 



