37^ PROCEEDINGS OF THE AMERICAN ACADEMY § 6 



we substitute the value of P' in terms of P^ derived 

 from equation I. § 2, namely, 



p: 7; A h I -I T, p{i-vy 



since the density varies inversely as the cube of any 

 length, and if, moreover, we substitute the value of i^" in 

 terms of /^o", 



p» J6 



~pr ~~ T^' 



as required by our hypothesis for bodies suffering no 

 change of state, we have 



^"f"^° 7; i'{i—r) "I" -^^ /" 



Differentiating with respect to log V^ and remembering 

 that, since V \ V^=^ l^ : /^, 



log V — log F^ = 3 log / — 3 log 4, whence differ- 

 entiatmg, _- — = -^1^^ — , we have 



.. dP , I dP' , / dP^' ^, ^ . 



K — -— + — - ,- -4- — =: o, that IS, 



dV ^ 3 dl ^ ^ dl 



y dP _ ±p, T^ n(L-n (3/^—2/0 



dV 3 ° ^o /' (^—ry 



_i_Pl^K {h-n ydT_ cLr^JlR^^ 



3 "7; /^ (/-/') dV 3 " ^'^ 



Simplifying, and substituting P' and jP" in place of 

 their values, also putting Z-*-]- ^' i" place of — P," we 

 have 



