30 Lord Rayleigh on the Incidence of Aerial 



but the nine coefficients are not independent. By the law of 

 reciprocity the coefficient of the #-part due to v must be the 

 same as that of the ?/-part due to u, and so on*. Thus 

 B 1 = A 2 , &c. ; and we may write (4) in the form 



, . dF dF dF ... 



r + =u d^ +v dti + w -T3> ' ' ' ' (5) 



where 



F = i V' 2 + \B 2 if + iC 3 z* + B v vy + G 2 yz + G x zx. . (6) 



In the case of a body, like an ellipsoid, symmetrical with 

 respect to three planes chosen as coordinate planes, 



B l =C f =C 1 =0, 



and (4) reduces to 



r^ = Aynx + B 2 vy + C z wz (7) 



It will now be shown that by a suitable choice of coordi- 

 nates this reduction may be effected in any case. Let u } v, to 

 originate in a source at distance R, whose coordinates are 

 x' , y' , z', so that u = x f /R 3 , &c. Then (5) becomes 



.,-,-., , ,dF ,dF ,dF . , t, , ,, . 



r s W^=x f ^ + y'-^ +z' Tz = A 1 xx'-\-B 2 yy' + C 3 zz' 



+ Bj [x'y \y'x) + C 2 {y'z + z'y) + 0, {z'x + x'z) 



= F{x + x',y+y> ) z + z>)-F(x,y,z)-F{x',y>,z'). 



Now by a suitable transformation of coordinates F(x,y, z), 

 and therefore ¥(%', y' , z') and F(x+x', y+y f , z + z f ), may be 

 reduced to the form A 1 <z ,2 -f-B 2 3/ 2 + C 3 2 2 ; &c. If this be done, 



. r'li'f = A X W + B 2 yy f + C z zz' , 



or reverting to u, v, ic, reckoned parallel to the new axes, 



7 s y{r = A l ux + B 2 vy + GsWZ, .... (8) 



as in (7) for the ellipsoid. It should be observed that this 

 reduction of the potential at a distance from the obstacle to 

 the form (8) is independent of the question whether the 

 material composing the obstacle is uniform. 



For the case of the ellipsoid (a, l>, c) of uniform quality 

 the solution may be completely carried out. Thus f, if T be 



* ' Theory of Sound/ § 109. u and v may be supposed to be due to 

 point-sources situated at a great distance E, along the axes of x and y 

 respectively. 



f The magnetic prob] era is considered in Maxwell's 'Electricity and 

 Magnetism,' 1873, § 437, and in Mascart's Lemons, 1896, §§ 52, 53, 270. 



