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Cylinderj r rI t'21 r2 Fig. 17. Inverted cylindrical coordinates The terms located at cylinder 2 can be transposed to cylinder 1 by means of Grains addition theorem, Eq. 79) in Ref. [1]. Using the inverted cylindrical coordinates shown in Fig. +,(krzOI,(kQOexp(inq~l). 56) -oo Applying Eq. 52) at the surface of cylinder l, we obtain +m a(m, 1)+A(m, 1) ~ n = -oo a(n,2)K,,+,(kr20=O. 52) at the surface of cylinder 2 yields +co a(n, 2)+A(n, 2) ~ a(m, 1)Km+,(kr2i)=O. 55) can be further reduced by using the fact that the modified cylindrical Bessel functions Im(k ~), Km(k O) are even functions of order m, yielding a(-m,j) = +_a(m,j) ; j = 1, 2.

2n+l (m-p)[(n-#)! ~ (m+n)! [ri-rl] m (m + #) ! (n + #) ! 26) where d(m,j) = 4z~jXjm/[(2m + 1) + (4~/3) ojXj(m - 1)]. 27) We obtain Sik in terms of a double multipole expansion with respect to r i - r 1 around axis r 2 - r 1 and with respect to r k - r 2 around the inverted axis rl - r2. -. over sphere 1. 28) b) r2 r1 Fig. 15. Addition theorem rk r2 52 Macroscopic Particles where V(r~, rk) is an arbitrary potential function, to obtain tr{~ gk V(gi, rk)}i= k =1Q1 iel I dri" gf V(ri, ri). 24) and Fig.

19) ~" " J~j " T jk 9 Xk " " Tki I" The second term in the braces vanishes if the poles of Xi(co), Xj(cn), Xk((D ) lie infinitesimally close to the imaginary frequency axis. 20) Turning to quadruplet interactions, it is obvious from Eq. 21) 9 Xk'Tk l' XI'Tli }. The summation in Eq. 21) includes the possibility that the tertiary dipole p~,O and the quarternary dipole piznd are located at the primary molecule i and the secondary molecule j, respectively. In order to avoid double counting of these contributions, we have to multiply the terms k = i, l = j in the sum of Eq.

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