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MATHEMATICS: O. E. GLENN
145
The process of differentiation may be carried out any number of times with respect to different parameters using formulae of differentiation analogous to the above. When the various derivatives are added together the result indicates that the natural generalisation of a series of spherical harmonics of form
So (0, 0) +S (o,) +S2 (0,) r
is the
G
7r2
+
ra
+
following type of series of Hertzian functions of different orders =
)div ° d bbb2+ div div div c0cl div -+ \(v ()\
v
) +.
Here go, ao, al, bo, bl, b2 .... are arbitrary vector functions of a. It should be remarked that the vector with suffix n is treated as the vector in forming the n divergence while the other vectors are treated for the moment as scalar quantities. The product of k vectors which occurs in the (k + l)th term is to be regarded as a tensor of the kth order with k components each of which is a product of components of the separate vectors; there appear to be enough arbitrary functions in a sum of products of this type with k = 0, 1, 2, ...., K for the representation of the sum of a number of Hertzian functions up to order K. 1As each shell of electricity moves outwards it induces a secondary separation of electricity so that electricity flows back to a new position of the primary singularity (t, ), and tends to maintain the electric separation. The volume density of the compensating electricity created at the primary singularity isthus not p but is proportional to k/r, it is this electricity which is regarded as forming the elementary aether associated with the primary singularity and it is this electricity which, on account of its displacement from the concentrated charge, is directly responsible for the field. 2 See for instance Wilson, E. B., Washington Acad. Sci., 6, 1916, (