JEolotropic Potential. A'21 



B y (74) ^-S^C,** 



which is quadratic in xyz\ the isotropic solution suggests 

 that ir may he made proportional to it, in which case 



I «W = 2^) ! + S^| g-4A4. . . (70) 



The second and third members have the coefficients of each 

 power and product (as regards .r y z) alike.it' six conditions 

 are fulfilled, belonging to the types 



Ai = p« 2 + 77 '- + rff 2 + 2///3' 7 ' + -V«£' + 2^7', 



Ai' =/>/3 V + ?«'/3 + r*' 7 +y (£7 + a'- ) + ,/ («'£' + 77') + >'' (V*' + £#) . 



Differentiate the second and third members of (76) with 

 regard to X only, as it appears in the coefficients; if A is 

 taken to he constant, the result is 



These terms then are cancelled in (75), which becomes 



iV^.+2S( j p* + 2^ / <0=0 (77) 



The effect of making A constant is that we secure the para- 

 meter X commonly used in the isotropic case, instead of a 

 function of that parameter ; the constant A determines the 

 scale of the parameter, and the value A= — 1 gives the 

 precise usage in the isotropic case. The relation (7(3) is 

 then 



+ ^f+gf+ P f)+M q >f+p'f+A . (78) 

 d//\ dx *dy L dz) oz\ dx l dy dzj 



Then (77) and the first part of (78) make the equation in <f> 



"g/S+^+V^- • • < 79 > 



Thus is to be found in terms of X through the medium 

 of a. ..of ..., which are connected with X by six equations, 

 the types of which are here rewritten with A= — 1, 



k +x {pa + r'i + ,/#) 4- y'(r'x + y7 ' +/,'£') + /S'( 7 '« +7/7' + r/?) = J 



a + 7 '( />£' + /V + / 7 ) + Wff + '/*' +PV) + *' (#£' + A' + T) =• >. ( 80) 



lora + 3' / / y + //3 + 7 V) + a '(r' 7 ' + 7^ + / /a')+7(/V + //3+/^j srOJ 



2 F 2 



