600 



SCIENCE. 



[Vol. II., No. 3(5. 



follow pi'ecisel3' the same laws as the velocities 

 aud flow of incompressible fluids ; for, consider 

 for the moment the lines of force starting from 

 the surface of some attracting body (a magnet, 

 for example) . Thej' gradually diverge as the 

 distance increases, and curve away into space. 

 Each one of these lines may be taken as the 

 representative of a definite amount of attraction, 

 which is the same at all points along it ; and if 

 a tubular surface be supposed to exist, includ- 

 ing everywhere certain of these lines which lie 

 beside each other, and no others, the total 

 amount of force acting across everj' cross-sec- 

 tion of the tube is the same : hence equations 

 (1) and (2) apply as well to forces of attrac- 

 tion as to velocities of an incompressible fluid, 

 provided F, k, v, w, be taken to be the compo- 

 nent forces along the normal and along a;, y, z, 

 respectively, and provided that none of the ut- 

 tractiug matter be contained within the closed 

 surface considered in equation (1), or at the 

 point considered in equation (2) . In order to 

 the farther development of these equations, let 

 us compute the work which would be obtained 

 in carrying a unit of attracted material from 

 one given position to another. The work is 

 found from the usual expression 



r =^ - /{udx + vd7j + wds:), (3) 



in which it^v,w, being component forces, the 

 limits of the integration are the co-ordinates of 

 the two given points ; but what path is taken 

 between these points is of no consequence, 

 hecause the amount of work depends alone 

 upon their difference of level : 



(4) 



dx' ■ dy' dz 



in which the right-hand numbers are partial 

 differential coefficients. Fis evidentlj- a func- 

 tion of the co-ordinates such that its value de- 

 pends upon position, and not upon the kind of 

 co-ordinates employed. The point which fixes 

 the lower limit of the integral in (3) is usuallj^ 

 taken at infinity ; and the value of Ftaken be- 

 tween it and the point fixing the upper limit 

 is called the potential of the latter point. 



By help of (3), we may put equation (1) in 

 the form 



rdV ^ 

 du 



J\ 



!S = 0, 



(5) 



in which d u is the element of the normal to 

 the closed surface considered. 



And b3' substituting in (2) the values given 

 in (4), we have, 



d^ V d^V 



dTF 

 dz- 



0, 



(6) 



djf' ' dy'^ 

 which is Laplace's equation, and is often 



written in the abbreviated form, V- V = 0. 

 Poisson showed, that, when the point at which 

 the potential is to be computed is within the 

 mass of the attracting matter, the right-hand 

 member of (6) should no longer be 7iil, but 

 47r/j instead, in which i> is the density of the 

 matter at that point. Similarly, the right-hand 

 member of (5) becomes iwrn when an amount 

 of matter m is included within the closed sur- 

 face considered. 



Equation (6) states that T^must be such a 

 function of the co-ordinates, that, if we take 

 its three partial second diflerential coefficients 

 and add them, their sum is iiil. What possible 

 algebraic forms are there which fulfil this con- 

 dition? Thej' are, of course, to be found bj' 

 attempting to solve the differential equation 

 (6). But it is to be seen beforehand, from 

 the manner in which that equation was es- 

 tablished, that it must have an infinite num- 

 ber of solutions ; for Fmust be such a function 

 as to be capable of expressing the work to be 

 obtained from a unit of attracted matter when 

 brought from infinitj- into the presence of 

 attracting matter, whatever its distribution in 

 space. The function T^ must therefore, in 

 general, be difl5erent for every different dis- 

 tribution of attracting matter. 



The integration of equation (G), and the 

 discussion of its various solutions, constitute 

 the branch of mathematics called spherical 

 harmonic analysis ; and to it the authors have 

 devoted pp. 171 to 219, in part i. The for- 

 mulae there obtained are employed, whenever 

 required in the present chapter, to express 

 the potential, or the attraction of matter dis- 

 tiibuted according to laws not conveniently to 

 be treated bj' less elementary methods. 



As the study of spherical harmonics has 

 been comparati\-ely neglected in this country, 

 a short digression, explaining some of their 

 properties, ma^- be useful. 



From the nature of attraction, it being to- 

 ward fixed centres, it appears that polar co- 

 ordinates would be more suitable to express its 

 relations than rectangular co-ordinates ; and, 

 in fact, equation (G) is usually tr.ansformed to 

 polar co-ordinates in space before integration, 

 which co-ordinates maj' be taken to be the 

 radius vector, the latitude, and the longitude 

 of the i)oint at which the potential is com- 

 puted. 



It may be shown that there are two general 

 forms of solution of this polar differential equa- 

 tion, — one in ascending powers of the radius 

 vector ; and the other in ascending powers of 

 its reciprocal, with coefficients depending upon 

 sines or cosines of the angular co-ordinates. 



