OlTOBF.R 12, 18.^3.] 



SCIENCE. 



501 



As these series uuiy be broken off at iiiiy point 

 bv tlie vanishing of the arbitrary niiuierical eo- 

 efflcients introduced during integration, these 

 soUitions m.iy l)e in terms of the radius vector 

 of any degree, jjositive or negative. 



It is then found that a most important and 

 simple class of sohitions, called zonal harmonics, 

 is those which are independent of the longi- 

 tude, and consequently contain but two varia- 

 bles. — the radius vector and the latitude. 



If in any iiarmonic we assume some special 

 value of the radius vector for consideration, 

 we evidently confine our attention to a spheri- 

 cal sinface : and tlie ex()ression is then spoken 

 of as a surface Iiarmonic, in distinction from 

 that in which the radius vector is a variable, 

 in whicli case it is called a solid hannonic. 



On the surface of a sphere of given radius, 

 it is possible to sujipose the values of a surface- 

 harmonic to be laid otf grapliically along the 

 radii to each i)oinl, toward or aw.iv from the 

 centre, according to their sign. This will give 

 a picture to the mind of the distribution of the 

 surface-harmonic. 



Now. in a zonal harmonic of the first jiosi- 

 tive degree (which varies as the sine of the 

 latitude) the surface-distribution is all positive 

 on one side of the equator, and all negative 

 on the other. A simple zonal harmonic of the 

 second degree has a distribution like that in- 

 cluded between a nearly spherical ellipsoid of 

 revolution about the polar axis and a sphere 

 when the two intersect along two parallels of 

 latitude. The ellipsoid may be prolate or ob- 

 late. The number of zones depends, in any 

 case, upon the degree of the zonal hannonic. 

 and is such that the number of parallels of lati- 

 tude at which the distribution changes sign is 

 the same as the degree ; and they are symmet- 

 rically situated about the equator, so that in 

 the odd degrees the equator is itself such a 

 parallel. 



There are other .solutions, called sectorial 

 harmonics, in which the surface-distribution 

 changes sign at equidistant meridians, and 

 other solutions still, which are a combination of 

 these two, called tesseral harmonics, in wliich 

 tlie sign of the distribution changes, checker- 

 board fa.shion. at parallels and meridians. The 

 sectorial harmonics are, however, in reality, 

 nothing more than the combination of a num- 

 ber of zonal harmonics of the same degree, 

 whose poles are situated at equal distances 

 along the equator ; and the tesseral liarmonics 

 are combinations of the sectorial with the 

 zonal harmonics. Indeed, the most general 

 harmonic is one by means of which any sur- 

 face-distribution whatever maj- be expressed by 



properly determining the constant coellicieuts, 

 and is merely a combination of zonal harmon- 

 ics superposed one upon another, with poles 

 situated in some irregular manner upon the 

 sinface of the sphere. This lirings ns to the 

 fundamental theorem stated in section r)87, 

 upon which the special importance and useful- 

 ness of these functions rest, — "A spherical 

 harmonic distribution of density (i.e., matter) 

 on a spherical surface produces a similar and 

 similarly placed spherical harmonic distribution 

 of ])otential over every concentric spherical 

 surface through space, external and internal ; 

 and so, also, consequently, of radial component 

 force. . . . Tlie potential is, of course, a solid 

 harmonic throu<£li space, both external and in- 

 ternal ; and is of positive degree in the internal, 

 and of negative degree in the external space," 

 as is evidently necessary, if the series express- 

 ing the potential in these two cases are to con- 

 verge. When we come to treat in the same 

 o(|uation the potentials of a given point due to 

 two ditl'erent bodies, or systems of bodies, a 

 remarkable relation is found to exist between 

 them, called, from its discoverer. Green's theo- 

 rem, which, though somewhat complicated when 

 expressed in rectangular cp-ordinates. has been 

 put by JIaxwclI in a simple form, which may 

 be written 



f Vjdm^ = f V.dm^, (7) 



in which the subscripts refer to the first and 

 second systems respectively, and the integra- 

 tions are to be extended so as to include the 

 total masses w, and m., respectively of the 

 two systems. Laplace's and Poisson's equa- 

 tions are. of course, particular cases of Green's 

 theorem. Thomson has etfected an important 

 extension of Green's theorem, given on pp. 

 1(!7 to 171, part i. Constant references are 

 made to these theorems, not only as to their 

 direct application, as we have presented it, but 

 ill their application to the inverse question of 

 determining what the distribution of matter 

 must be to produce a given distribution of 

 potential. 



The most extended and important applica- 

 tion of the theories of attraction and iiotciitial 

 treated in this cluqiter is tliat of ellipsoids and 

 ellipsoidal shells, — a subject which is closely 

 connected with that of the figure' of the earth, 

 and one which has engaged the prolonged at- 

 tention of many of the most powerful mathe- 

 matical intellects of the past. A full account 

 of the course of discovery in this field is found 

 in Todhunter's History of the theories of at- 

 traction and figure of the earth, 2 vols. 



Ten pages of new matter ( pp. 40-50) have 



