THE INTERNAL CONSTITUTION OF BODIES. 465 



is done by substituting the volume v for CCfdi d -q d l, the term 



which stands under the sign S in the first of the equations (II) will 



be represented by tzr v — ». 



If we now write in their places all the expressions just found for 

 the integrals which constitute the first of the equations (II), we shall 

 have 



7 da di „dr 



ax dx dx 



By similar substitutions the second and the third equation will give 

 respectively 



, da d ^ „ dr, 



A q V — J = s; v -— — OT V S — - 

 rfy dy dj 



, do di ^dTy 



« q V -p = ra-v-- zsrvS — - . 



MZ dz dz 



These three equations must hold good for the particular values x, y, 

 z; Xj, y„ z, ; X , y , z^, &c., which answer to the centre of the 



molecules in their state of equilibrium ; and as each molecule furnishes 

 three similar equations, the whole collectively will be sufficient to enable 

 us to determine the unknown quantities. 



If from the formulae marked (III)", (IV)', (V)' we derive, by means 



of the changes already indicated, the expressions for— J - — » 



dx' dx^ d X ' 

 we find 



— ar 



l3= _lsv(at^ +/q)(L±.^£!)£ \ ^^IZLi^ 



dx 



^=g.sv,(^^,+/q,){ <^ + °^^:>- "^^ -A-j — 



dx f L r»- r.* j r» 



dT> /-_L„„\J^ xv 



= -v» (y^v + ^rq,) — . 



dx ^ r»'^ r, 



and shall obtain —9 r— , -, — . • -r^, &*•> ^Y changing in these formulae 

 rfyayay'az 



X into y and into z. 



If we introduce these expressions into the foregoing equations, re- 

 collecting that, according to the hypothesis of Franklin and i^piims, 

 we must makc/= <7, and take y a little less than ff, the result will be 



