PART I. 



THEORETICAL INVESTIGATION. 



Section I. 



PRELIMINAEY ANALYSIS. 



1. In what follows there will frequently be occasion to express a triple integration which 

 has to be performed with respect to all space, or at least to all points of space for which 

 the quantity to be integrated has a value different from zero. The conception of such an 

 integration, regarded as a limiting summation, presents itself clearly and readily to the mind, 

 without the consideration of co-ordinates of any kind. A system of co-ordinates forms merely 

 the machinery by which the integration is to be effected in particular cases ; and when the 

 function to be integrated is arbitrary, and the nature of the problem does not point to one 

 system rather than another, the employment of some particular system, and the analytical 

 expression thereby of the function to be integrated, serves only to distract the attention by the 

 introduction of a foreign element, and to burden the pages with a crowd of unnecessary 

 symbols. Accordingly, in the case mentioned above, I shall merely take dV to represent 

 an element of volume, and write over it the sign fff, to indicate that the integration to 

 be performed is in fact triple. Integral signs will be used in this manner without limits 

 expressed when the integration is to extend to all points of space for which the function to be 

 integrated differs from zero. 



There will frequently be occasion too to represent a double integration which has to 

 be performed with reference to the surface of a sphere, of radius r, described round the point 

 which is regarded as origin, or else a double integration which has to be performed with 

 reference to all angular space. In this case the sign ff will be used, and dS will be taken to 

 represent an element of the surface of the sphere, and da- to represent an elementary solid 

 angle, measured by the corresponding element of the surface of a sphere described about its 

 vertex with radius unity. Hence, if dV, dS, da denote corresponding elements, dS = r*da, 

 dV = drdS = r"drda. When the signs fff and ff, referring to differentials which are denoted 

 by a single symbol, come together, or along with other integral signs, they will be separated 

 by a dot, as for example fff. ffUd Vda. 



d 2 d? d? 



2. As the operation denoted by - — - + — ■ + — - will be perpetually recurring in this 



paper, I shall denote it for shortness by V- This operation admits of having assigned to it a 



geometrical meaning which is independent of co-ordinates. For if P be the point {x, y, z), 



T a small space containing P, which will finally be supposed to vanish, dn an element of 



a normal drawn outwards at the surface of T, U the function which is the subject of the 



d? d* d 2 



operation, and if V be defined as the equivalent of — + — ; + , it is easy to prove that 



da? dy i dz" 



