28 



Let the coefficients for the expansion of the potential due to the singularity about x Jl^t) 

 be a^, P. This potential may also be expressed as an expansion about r^(i^) which will con- 

 verge for all lR| > |R {bt)\. Let the coefficients of this expansion be o^ , &^ • The latter 

 expansion is precisely of the form due to a singularity fixed at r.{t^). If we find a^ , 6^^ in 

 terms of a^, P, we may insert the values directly into the formulas for the force and moment. 



It is evident that 



Therefore, the formulas for the Lagally force and moment, which depend only upon the instan- 

 taneous values of the coefficients, remain unchanged. Further, it is only necessary to deter- 

 mine a° and A' , the source and doublet strength of the equivalent singularity, since the time 

 derivatives of these quantities appear in the expression for F2('iJ but no higher order terms 

 appear. 



The potential about r.(t)ma.y be written 



+ terms or higher order 



I'^-Ro^S^)! |R-R(§«) 



3 



-1/2 /. „ A n \ / „n . n ,,9 \~3/2 



</_2R^R^^\-^ (A-R-A-t^) / 2R-R^-Rl \ ^ 



R \ p2 / p3 \ P2 / 



Expanding by the binomial theorem and collecting terms, we have 



_^+J_(A+a°R )'R + terms of higher order 

 R R^ 



Therefore 



and 



a°'{t)^a"{t) [70] 



A'=A + a°R„ m] 



