388 PROCEEDINGS OF THE AMERICAN ACADEMY § 8 



to Table III. at four ditfcrent temperatures which may 

 divide the whole range of temperature in question into 

 three equal parts. 



We have, for o" and any three temperatures, /,, i ^ and /^ 

 at which the formulae must hold, the simultaneous equa- 

 tions: 



' 1 == I + at, + bt,' + ct,^ + dt,' + et:^ + etc. (2) 



I = I + yl/. + Bt: + a^ (3) 



"^^ \ = I + ^A -f bt^ + c4^ -f dt,' + ^4^ + etc. (4) 



^3 1=1+ at, + ^// + ^Z/ + dt,^ + ^// + etc. (6) 



Subtracting (2) from (i), (4) from (3), and (6) from 

 (5), transposing in each case {A — a) t, and dividing by 

 t, w^e have 



A — a:^—{B — b)t,— {C—c)t,' + dt,^ 



+ et,' + etc. (7) 



yl — rt' = — (^ — (5) /, — (C— c) // + dt,^ 



+ ^4^ + etc. (8) 



A — a = — {B — b) t^— {C—c) t,^ + dt^ 



+ ^^3' + etc. (9) 



Subtracting (8) from (7) and (9) from (8) we have, 



transposing and dividing by 4 — /, and t, — Z^, respectively, 



(i? _ ^) = _ (C— 6-) (A + /,) + ^ it: + t.t. 



+ /'.O + ^ {t^ + ^/A + /./.^ + /.O + etc. (10) 



^B-b)=.-{C-c) {t, + /,) + d (// + //, 



+ 1^) + ^ (/^-^ + t:t. + ^3^3^ + A^) + etc. (11) 



Subtracting (11) from (10), transposing and dividing 

 by {t^ — /,), we have 



C- ^ = r/ (/3 + A + /.) + ^ (^; + t^ + t: + tj, 



+ // + /,/.)+ etc. (12) 



