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) ^(0,0) + S 2 (0,0) + 
is the following type of series of Hertzian functions of different orders 
G = 15 + div + div div (^) + div div div + . 
Here go, a 0 , ai, bo, bi, bs . . . . 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. 
1 As each shell of electricity moves outwards it induces a secondary separation of elec- 
tricity so that electricity flows back to a new position of the primary singularity (£, 77, f) 
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 ^r/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 concen- 
trated charge, is directly responsible for the field. 
2 See for instance Wilson, E. B., Washington Acad. Sri., 6, 1916, (665-669). 
3 Larmor, J., London, Proc. Mathe. Soc, 13, 1913, p. 51. 
4 Whitehead, A. N., The Anatomy of some Scientific Ideas, The Organization of Thought, 
London, 1917, p. 182. 
INVARIANTS WHICH ARE FUNCTIONS OF PARAMETERS OF THE 
TRANSFORMATION 
By Oliver E. Glenn 
Department of Mathematics, University of Pennsylvania 
Communicated by H. H. Donaldson. Read before the Academy, November 21, 1917 
A systematic theory and interpretation of invariantive functions which 
contain the parameters of the linear transformations to which a quantic 
