278 M. R. Ruhlmann on the Alteration produced by Heat 



increase of temperature, and also that a slight touching of the 

 glass plates might cause such to take place, because the plates 

 are connected to the tube with scarcely any friction. Accord- 

 ingly, along with the measurements of the angle of least deviation, 

 continuous measurements must also be made of the refracting 

 angle of the prism. For this reason the arrangement of the in- 

 strument shown in fig. 2, PL V. was chosen. 



The hollow prism itself showed by the most careful testing 

 absolutely no proper refraction, so that it was not necessary to 

 apply any correction on this account*. 



(5) The thermometer used for measuring the temperature in 

 the inside of the prism was a little one by Greiner (Reaumur, 

 with paper scale) divided from —55° to 80°. It was possible 

 to judge to one-tenth of a degree, without, however, being sure 

 of the accuracy to two-tenths of a degree, since the thermometer 

 had often to be read very rapidly, and the degrees were not quite 

 1 millim. apart. Only the lower part of the thermometer dip- 

 ped into the Jiquid. From —55° it was below the sheet of 

 cardboard, by which it was protected from the heat radiated or 

 convected from the prism or the lamp beneath. It was neces- 

 sary, therefore, to correct for the projecting portion of the scale"; 

 and for this purpose the formula given by Kopp was employed, 



T = t + v.*(t— t), 



where t is the temperature read off, t the temperature of the scale, 

 a the apparent coefficient of expansion of mercury in glass 

 (0-000154), and v the number of degrees in the projecting 

 partf. 



* A formula of correction for this purpose is found in Biot, Precis Ele- 

 mentaire de Physique Expenmentale, 1842, vol. ii. p. 113. 



f As I have nowhere found a derivation of this formula, I may be per- 

 mitted to give it shortly here. Let t be the temperature of the air, T that of 

 the liquid, r the temperature read off, v the number of degrees not immersed, 

 7 the volume of mercury in the thermometer at zero, the space between 

 two degrees being taken as unity, V the volume of the immersed part re- 

 duced to zero, v the number of degrees representing the volume not im- 

 mersed. Then 



l + «. 100-7=100, . . (2) 

 7=~; (4) 



H-V(l-MT)=F(l+*r), 



because a reading r corresponds to a state as if 7 had become 7(1 + ««"). 

 Hence from (3) and (4), putting in V and 7, 

 T=T+vu(T-t). 

 For a first approximation T= r, and 1\ is thus found. This is substituted 

 for T, and thus T 2 is got, and so on. In most cases a first approximation is 

 sufficient. 



I^F W ' • 



■ (1) 



V=i-V(l-*0, 



. (3) 



further, we have ,,._ 



i-vri-j- 



