219 
‘‘Hyom the peculiar nature of the symbols it is evident 
that the expressions last stated may be written in the forms, 
aie Uo (ée;, e", e*) Eee (e, é", é*) 
and 
p-P(n-S, ¢-&, E-n)+¥(n-¢, c-&, E-n), 
where &, n, Z, are the co-ordinates of the point 2, y, Z, in mag- 
nitude and direction, or, if I may presume to invent the phrase, 
the components of the points «, y, 2, and p the vector of this 
same point. 
‘‘ There is a peculiarity about this expression to which it 
may be well to solicit attention. It is known to all physicists 
that, in the lunar theory, and in that of the perturbed motion 
of pendulums, there occur equations of the form, 
w= sin (n+) + Acos (n0+ B)+ &., 
implying that the value of u is not simply periodic, but admits 
of indefinite increase. 
‘‘ Similarly, in the above expression, we observe in the 
first term the vector p outside the arbitrary function, a circum- 
stance likely to add considerably to the interest of the physical 
interpretation of the solution. 
‘‘ That the modified form of this expression will satisfy 
the requirements of physical research, appears probable from 
the considerations, that it must be perfectly symmetrical, that 
it must be spatial, and that even the notation exhibits a semi- 
physical character. 
“‘ The exact nature of the requisite modification I am not 
at present prepared to state to the Academy, but with the 
existence of such I am strongly impressed, and as the subject 
has recently attracted much attention, these remarks have been 
submitted in the hope of contributing to the production ofa 
result which possesses much interest both for the mathemati- 
cian and the physicist. 
“Tt may be well to add, that the two other forms in which 
