Lkathem — On Doublet Distributions in Potential Theory. 35 



If the relevant region is externally unbounded in all directions the form 



of (j) at great distance B from a definite origin may be arranged in powers 



of B, and it is known 1 that the only terms which can occur are spherical 



solid harmonics of positive or negative integral orders. The following terms 



might be present, 



vR cos + C + M/iwR - M' cos d'/iirB 2 , 



where v, C, M, M' are constants, and 6 and 0' are angles measured from fixed 

 directions. Terms of a greater order of magnitude would correspond to cases 

 of no practical interest. Of the above terms the first corresponds to what is 

 usually called " uniform motion at infinity," the second has no physical 

 significance, the third represents a source of strength M at infinity, and the 

 fourth is introduced in order to show that it and terms of lower order do not 

 affect the final result. 



In applying Green's Theorem as above we have to bound the region of 

 integration externally by a sphere with centre and radius R. And we 

 must add to the right-hand side of equations (2) and (3) the limit (if any), 

 for B ->- oo , of 



i 1 d S AVf t> a n M M'cosB'} 

 |r a« - a« Irjj r C ° S e + ° + 4^R - -4.BT \ d& ' 



taken over the sphere B ; here it is allowable to substitute R- 1 + OP cos A B' 2 

 for r'\ where A is the angle between OP and B. 



It is to be noted that the combination, in one, of the two integrals of the 

 types $r- l d<l>ldnd8 and jtydr'/dndS gets over the difficulty of non-convergence 

 or semi-convergence which might seem to be unfairly evaded by the choice of 

 a specially simple form for the outer boundary. For if we take an alternative 

 outer boundary S' of any shape, lying completely outside the B sphere, and 

 apply Green's Theorem to the functions r' 1 and 4> in the space between the 

 two, we see that 



\(r- x dip/dn - 4>dr- l /dn) dS 



has the same value for both surfaces provided A^ = in the region of inte- 

 gration. And each term in the second factor under the integral which we 

 are studying is a legitimate value of ip, so the spherical boundary gives results 

 which are not special, but general. 



Let us consider each term separately. 



The term in v yields SOPvj cos X cos# dw, which = 47r OPv cos 6 

 where is the angle between OP and the direction of the stream at infinity. 



The term in C yields 4irC. 



1 Thomson and Tait, Natural Philosophy, edition of 1890, vol. i, p. 181. 



