362 
Proceedings of the Royal Society 
the problem arises, “What must be the degree of flattening in order 
that the points N x and N 2 may coincide?” in which case no more 
than one normal can be drawn to the curve from any point in its 
axis. 
Again, by augmenting the ordinates, as in figure 7, the extent 
of the overlap N 2 N x may be increased, and the limiting station for 
one quadrant may touch that for another quadrant of the curve, 
and thus we may determine the character and height needed to 
ensure that some specified odd number of normals, neither more nor 
less, may be drawn from any point assumed in the axis. 
In the solution of such a problem we have to consider the two 
roots of equation 3 below u = ~ 7r, and recurring in each succes- 
n 
sive quadrant of the curve. When r exceeds 7 r, and is less than 
3 
— 7r , there are three such roots, as in fig. 5, and the discussion of 
2 
the number of normals becomes exceedingly involved. The con- 
sideration of the wheel systems deduced from such curves belongs 
to purely speculative geometry. 
3. Laboratory Note. By Professor Tait. 
On the Thermo-electric Positions of Sodium and Potassium. 
Farther experiments with the apparatus described in the “Proceed- 
ings” of 2d March 1874, and with a similar one containing potas- 
sium, have led to the following values of the tangent of the inclina- 
tion of the corresponding lines in the thermo-electric diagram. I 
have added its value in palladium for comparison — 
Na - *00212 
K - *00066 
Pd - *00182 
To reduce these to the corresponding numerical values of the 
specific heat of electricity, the factor required is 4xl0“ 8 of a 
Grove’s cell. 
The line of Na in the diagram intersects that of Pb at about 
- 20° C, and the line of K intersects that of Arg about the same 
temperature. By the help of these data they may easily be inserted 
in my diagram in vol. xxvii. part i. of the Transactions R. S. E. 
