GRAVITATIONAL STABILITY OF THE EARTH. 187 



This choice of the q's amounts to subjecting the functions F. and Q'. to the 

 equation 



^r f ........ (55) 



dr 



To see that this equation is compatible with the differential equations (48) and (50) 

 for P', and Q',, we observe that 



_ 

 dr ~ r**+ l dr \r dr 



and that from the equation (48) we can form the equation 



d 2(n + l)1 /- .rfF,\_. n+1 P,d [ ..+,/! dE.\1 

 J\ dr]-2n+lh> dr\ \rdr)\' 



r dr 

 while from (50) we can form the equation 



d" 2(n+l)d 2(n+l)1 f <* /^ip/i] . * . d/,/1 dE. 

 P r dr r> )\dr ( ~ 2^1 h^ dr 



12. We have still to satisfy the condition 



If we form the expression in the left-hand member from the expressions of the 

 type (44) for ,, v lt ?,, we have 



a*, at, ^ = jn dp. _ IHO [rf 



3x 3y 3z lr rfr r*"^ J rfr 



By means of the formulae (47) and (49) for P, and Q,, the coefficient of eu, in the 

 right-hand member of this equation is transformed into 



1 f ^ , n+l d_ / >, dE.\-\ n+1 f n 

 2n+ 1 [r dr r" +8 rfr \ </r /J r*" + ' L+ 1 



The first term is equal to 



[ [/. + (+ 1)/.] or /. 



and the second term vanishes identically in virtue of equation (55). It follows that, 

 with our choice of p t and q a , the equation (56) is satisfied identically. 



13. We have now completed the determination of the forms of u, v, w in terms of 

 the spherical solid harmonics & F,, <f n , and of certain functions of r, viz. : f., O n , E,, 



2 B 2 



