§7 



OF ARTS AND SCIENCES. 



381 



then the absolute coefficient of expansion will be ex- 

 pressed b}' the series, 



so that the voluminal expansion, e, which is the quotient 

 ot' the absolute expansion by the volume, will be 



_ rt- -f- 2^/ + 3^/^ + 4rt^/3 -(- 5^/4 _|_ etc. 

 ^ ~ I -f- ^7^ + bt' 4- cV^ -|- dt' + et^ -f etc. 



whence, by actual division, we find 



e = ,7 -|- (2^ — a'~) t^ (3c — -^ab + a^) f 

 {-{-^d — 4^c — 2^^ + ^cfb — a') fi 

 + (5^ — S'^d — 5^6- + 5«V + 5^^^ — 5«3^ _j_ ^5) p 

 + etc. " VI. 



Putting a, b\ c\ d' and e in place of the coefficients of 

 the successive powers of t, so that 

 a! =. a 



b' ^ 2b — a^ 

 <:' = 3r — T^ab -\- a} 

 d' =^ 4</ — ^ac — 2(5^ -f~ 4^*^ — (^^ 

 e' =z ^e — s^^d — 5(^(f + ^a^c -\- ^ab^' — ^a^b 

 + a^ 



we have 



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



r 



VIL 



d^e 



~d¥ 



d' 



^ // ^ 2rV + 3^7^ + 46'73 + etc. 

 = 2t:' + 6dt + 1 26-7^ + etc. 

 = 6^' -|- 24t'7 -j- etc. 



. VIII. 



dt 

 d^e 



dt 



- = 24^' -|- etc. 

 := o -|- etc. 



