(See the second sum in Equation [S] . ) Here a- and a-' are functions of ^ 



th 9** 

 To calculate the associated force on the i spar, we must evaluate 



9t 

 for R. = a.(z.). 



For small values of (kR^), the Bessel function in $* can be approx- 

 imated by the beginning of its Taylor series, that is. 



jQ(kR.) = 1- i(kR.r + .. 



Similarly, 



— Jn(kR.) = - ik^ R. cos X. + . . 



dx:, ^ ^ "i 1 1 



— Jn(kR.) = - ik^R. sin X. + 



ay. 0^ l' 2 1 1 



Keeping a one-term approximation in each case, we find 



kz4 



= |wk [zq + si^ia sin 9^ - p cos 9.)] [5(9) - k Tq] e i 

 •i~^i 



- icok^a.(z.)e^^i |[(xq - y a^ sin 9.)Tq + pT^] cos X. 

 + [(yQ + ^^aQ cos 9.)Tq - aT^] sin X.| 



The pressure due to this potential is, when evaluated on R- = a-, 



- p $'. 



R.=a. 



R.=a. 



- I pook [zq + &Q(a sin 9. - (3 cos 9. )] [ S( 9) - k Tq] e ^ 

 + ipcok^a.(z.)e^^i ^[{i^ - y a^ sin e.)TQ + p T J cos X. 

 + [(Yq + ys^Q cos 9.)Tq - aT^] sin X. | 



23 



