CASE XIV. 123 



When the integration has been carried so far back that P—D is small 

 enough to make the second term in the right member of (129) the dominant 

 one, it is convenient to change from P to D as the independent variable 

 of integration. We may now write (129) in the form 



^^ =f{P,D) Sm,{P-DyPi 



dD 



C.[(P_D), + (a.)\p._D).(^y] (134) 



For P—D = this equation does not have a singular point when D is the 

 independent variable, for at this point/ (P, D) vanishes. 



Suppose corresponding values of P and D have been found by (132) 

 and (130) until Pq and Dq are obtained, and that their difference is small. 

 Then P may be expressed in terms of D by an expression of the form 



P-Po = lAi (D-D.y (135) 



1=1 



provided the modulus of D — D^ is sufficiently small. Substituting (135) 

 in (134), expanding the right member, and equating coeflScients of corre- 

 sponding powers of D — Dg, we find 



where 



dPo dD, 



(136) 



^ 3m,iP,-D,yP,i 



a/(Po,Do) _ 2m,(Po-J)o)(4Po-Do) 



^^° Po'D,^ [{P,-D,y + {^)\p'-D,y (^«)] 



67n,(P,-D,yP,^ [p^_Z)„ + 3(^^)'(P'-Z)„)^^]] 



df (Pq, Dp) ^ 6m,(Po-Z)o)Po§ 



Qm,iP,-D,y Po? [p„-Do + (^)' (P'-D„) (^^)'] 



