of determining Stream Lines and JEquipotentials. 249 
out o£ the equation \/ 2 Y = 0, so that all three coordinates 
must still be present in the integrals. Clearly then, our 
deductions, though necessary, are not sufficient. I have little 
doubt that the omission lies in this : that to leave us unre- 
stricted as to the direction o£ the orthogonal traces of a and 
/3 upon the surfaces y, it is not sufficient that the surfaces 7 
should be spheres. For the curves normal to 7 which we 
have called the " guiding lines '' must be such that they form 
one set of lines of curvature of any surface whatever passing 
through them. To satisfy this condition it seems likely that 
except when the radius of the spheres 7 is infinite, the 
guiding lines will have to be straight and the spheres con- 
centric. This is the symmetry when V is independent of the 
radius in spherical coordinates, but may vary anyhow with the 
latitude and longitude. 
The only case remaining uninvestigated is that in which 
the surfaces 7 are spheres with centres which do not lie on a 
straight line. 
By this application of Lame's formula?, aided by those due 
to Hicks, we have discovered no new type of symmetry 
which allows two coordinates to be used instead of three. 
We have proved that within the stated limits the well- 
known types are the only possible ones. A summary of 
these may be useful. 
Summary of Types of Symmetry ivhen the guiding lines are 
orthogonal to a family of surfaces. 
Guiding lines, j Chequer ratio. 
i 
If v 2 V" is made equal 
to/(V, a, (3) over one 
surface y its value 
on the others will be 
Analytical 
methods. 
Parallel straights. 
Constant. 
V 2 V=/(V, a, fi). 
Conjugate 
functions. 
Circles with their 
centres on a com- 
mon axis and 
their planes nor- 
mal thereto. 
Proportional to 
distance 
from axis. 
V *V=f(Y,a,p). 
Zonal harmonics 
of the 
cylindrical, 
spherical, 
spheroidal, 
and toroidal 
systems. 
Kadii from a com- 
mon point. 
Constant. 
r' 2 
where r is the distance 
from the radiant point. 
