§7 



OF ARTS AND SCIENCES. 



383 



Summing up our results, we have 



dt ' 2 dr 



f^w/'s I /•* d'e . 



6 ^/-^ ' 24 rt^2^ 



^' = 



d^ 



I /^^ 



r/^e 





/■^ d'e , . 



— . + &c. 



6 r//+ ' 



d' 



dt' 



I / <f ^e 

 '6 



24 





— /- 



- — / 



^'e 



+ 



/^ ^/l-: 





— &c. 



> ix.-:^ 



&c 



+ &L^\ 



equations by which, at any temperature, /, the values of ^', 

 b' , c, d' and e' may be calculated in terms of e and its 

 derivatives, the relations between which have been already 

 determined (I. to V.) The temperature specially adapt- 

 ed to this calculation is, however, the freezing tempera- 

 ture; for, putting t^o in equation IX. all but the first 

 terms disappear. 



It remains to be determined whether the successive 

 terms in which e is expressed, 



e=a' -{- b't + c't' + d't^ + e't' + etc. 



form a convergent series. 



Referring to equations I. — V. we see that the ;^th deriva- 

 tive of e may be expressed 



^"^ — yl£" + ' 4- Be" + ' TA 1-^V" + ' T\ 



dT" ' ^ ' ' 



* It will be noticed that equations IX. are the expression of the converse of Maclaurin's the- 

 orem, from which they might have been derived by considering « as a constant, while a', b' , etc. 

 are variables. Not being able at once to find a proof of the proposition in this form, it was 

 thought advisable to give the calculation in full. The theorem is known as Bernouilli's. See 

 Williamson's Differential Calculus, § 64. 



