Theory of Molecular Action according to Newton's Law. 205 



T \8 r 5 12 r 5 



+ !<* a * (« -/) 3 + ffl (y-g)* + y 4 (*-A) 4 + 6a»j3« (x -/)« (y-g)2+&c. 1 



105 6 fr -/)« (y - g )« («» |3* + «* y* + /3 2 y 2 ) & T 



Now the hypothesis of symmetry, from which we have re- 

 duced the results by making 



v m (x -ff 1 v r 2 fi 



imposes further the condition that V — V is a function of 8 

 only, independent of a, /3 and y. Consequently, 



2 205 / (x -/)« ( g 4 +i8 4 + y 4) + 6 (x-/)» Q/-g) 2 i(« 2 /3 2 + a 2 ? 2 + /3 2 y 2 \ 

 24 \ r» / 



24 r 9 



Hence we obtain the equation 



This equation is of considerable importance. The method 

 by which we have obtained it appears to be totally different 

 from the ordinary methods, such as that employed by Cauchy, 

 Exercises, 3. 201. 



By substitution 



The coefficient of S 4 depends on the value of 



But 



r 4 = (* -/J 4 + ■ (* - g) 4 + (z-h) 4 + 2 (x -/)« (y - £) 2 

 + 2 (* -/)» (z - hf + 2 (y - jtf {* - hf 

 ... 3 £ **(*—/)* _ ^ ^ __ 6 j m{x~ff{y-gY 



= S^-2S?l^l 4 (b y A.) ) 

 Hence the coefficient of 8 4 is zero. 



