147, 148] DEVELOPMENT OF POTENTIAL. 403 



If the body is homogeneous, and is symmetrical about an axis 

 of revolution, since V is independent of #, y, it is evident that all 

 the harmonics are zonal, and we have 



128) F=^^ + ^^ + ^i + . 



r r 2 r s 



where every P n is the zonal harmonic in cos #. 



If we know the value of V for every point on the axis of 

 revolution, so that we can develop it in powers of as 



129) F, =0 = ^(r)==|* + i + *.+... ' 



then putting cos#=l in 128) and comparing with 129), we find 

 4* = B n so that V is completely determined as 



130) 7= + + + .... ; ~ : 



If in addition the body has an equatorial plane of symmetry, so 

 that F(cos#) = F( cosfl-), evidently the development will contain 

 only harmonics of even order. As a case of this we shall develop 

 the potential of a homogeneous ellipsoid of revolution in 161. 



Whether the body is homogeneous or not, we may easily obtain 

 the physical significance of the first few terms in 126). For making 



use of the values in 111) since n = ax+ + c * we have 

 131) r'Pii (a 



There occur in the first three terms the volume integrals 

 J J J ^ dad ^ dc==M y I J I Qadadbdc = Ma, 



jjJQbdadbdc = Mb, jffycdadbdc = Me, 



///< 



A + B-C 



gabdadbdc = F, 



26* 



