CASE X. 



Ill 



X. CASE i = i^ = 1*2 = a2 = e^O 5 = 0.' 

 In this case equations (9) and (10) become 



TV PS "^D' 



Eliminating D and P in turn between these equations, we get 



(98) 



K 



+ 



m, 



(MD-w,)=^ ' D» 



(99) 



Solving'for'e we obtain 



Vp»-[M±V"^H-|i]' e=Vl-©-f)^^V^ COO) 



Pi 



The conditions for a maximum or a minimum of the first of (99) are 



M 



27t de 



^1-i 



Pi e=0 



1 



^p§-||=p!(l_e2)_ /l_e2MP + mi=0 



(101) 



The first equation may be satisfied by the vanishing of any of its three 

 factors. P» = is a physical impossibility and makes E infinite. Setting 

 the second factor equal to zero and substituting in the second equation 

 we have mi = 0, which is impossible. The remaining possibility is e = 0, 

 and then equations (101) reduce to 



e=0 



PJ-MP+mi=0 



(102) 



the second equation being precisely the same as (28). 



Similarly the second equation of (99) gives as the conditions for a 

 maximum or a minimum 



e=0 -D*(MD-mi) + MD^-m,{MD-miy=0 (103) 



The latter of these equations gives 



D^-MZ)+?Wi=0 (104) 



which, together with the second of (102), shows that for either a maximum 



or a minimum 



e=0 P = D 



* See 3, parts V and VI, and the appendix, pp. 886-891. 



