Lags of Thermometers. 61 



Any number of successive inversions of the type con- 

 sidered can be combined, and the exact solution, analogous 

 to (19) and (29), is easily obtained. Unless the inversions 

 are of very short duration, the approximate lag can be 

 deduced from the steady lags on the various gradients 

 without much trouble. 



5. Effect of Varying the Bore of the Capillary Tube. 



The steady lag of a thermometer is given by (1), where M 

 has the dimensions of c 2 , being equal to c 2 /15 for the sphere, 

 and c 2 /8 for the cylinder ; and N has the dimensions of c, 

 being equal to c/3 for the sphere and c/2 for the cylinder. 

 The four examples of liquid thermometers we considered in 

 para. 4, all gave the same accuracy of reading (same open- 

 ness of scale) when used with the same thermometer stem. 

 If the area of section of the capillary tube is multiplied by a 

 factor fju, the volume of the bulb must be multiplied by the 

 same factor to retain the previous accuracy of reading. 

 Hence for the new spherical bulb the radius is c'—fi*c, while 

 for the new cylinder the radius is (fjul/l'^c, V being the new 

 length, and I the previous length of the cylinder. The lags 

 of the new thermometers, all of which have the same accuracy 

 when used with the new stem, are therefore 



for the new sphere of radius cjj,i ; and 



for the new cylinder of radius c \//jlIJI'. Now the lag of the 

 cylinder is not affected appreciably, so long as the cross- 

 section is small in comparison with the length, by varying I. 

 Hence if we choose a new length for the cylinder such that 

 V=fd, the lag (44) has still the previous value. If V has a 

 greater value than fil the lag will be diminished because the 

 new radius c must be less than c. For the sphere, however, 

 the lag will always be less when fi<l, and vice versa. 



Commencing anew, let us suppose that the bulbs con- 

 sidered in para. 4 are all used in conjunction with a stem, 

 which has the smallest practicable capillary tube that may be 

 used with alcohol. For mercury, a much smaller bore is still 

 possible. Let us now designate by p the ratio of areas of 

 the sections of the smallest capillary tubes practicable for 

 mercury and alcohol. Then, owing to this further reduction 



