346 M. R. Ttiihlmann on the Alteration produced by Heat 

 But 2S 2 should be a minimum, i. e. 



d2S 2 dS3 2 _ 



^ 7 =0 and -^ — = : 



ob oc ■ ' 



from these two differential equations we get 



Sro^sKS* 4 + c2* e and 2 l mt* = b'2tP.+ c2 l t*, 

 and hence 



Accordingly the following formulae were found for the altera- 

 tion of the index of refraction of water with the temperature : — 



(1) For the lithium-line, 

 ?z=l-33154-0-000003072£ 2 + 0-0000000001123/ 4 . 



(2) For the sodium-line, 

 ^=l-33374-0-000003147^ 2 + 0-0000000001205/ 4 . 



(3) For the thallium -line, 

 ^=1-33568-0-000003267* 2 + Q-Q000Q00001476* j4 . 



The agreement of these formulae with the observed values, 

 which can be seen in Tables I., II., III., and which may easily 

 be judged of from the graphic representation (Plate V.), may be 

 considered completely satisfactory. For it is only in the highest 

 temperatures, in observations of the lithium- and thallium-lines, 

 that divergences of three units in the fourth decimal place occur, 

 one in each ; and these may be fairly attributed to errors of ob- 

 servation. 



Further, a Table has been formed from the three formulae 

 containing the indices of refraction for our three spectral lines 

 for every 2°. 



