256 Mr. L. F. Richardson on a Freehand Graphic way 
In the following pages, except where specially indicated, 
dY 
we will consider only the case -r — =0. 
As this result does - not appear to be given in the text- 
books, it may be well to confirm it by a slightly different line 
of reasoning, as follows. The tangent of the angle between 
the tangent to any screw-threads and a plane normal to the 
axis of symmetry is ~— - . 
Therefore the first space rate of "any function of position 
Y along the tangent to a screw-thread is 
If ^— = a function of r only, alone- every screw-thread 
then we have a function of r only +27r^—r =— Z^— 
throughout the whole region. Therefore this last -equation 
will still remain true after differentiation by <j> or by z, thus 
2tt d 2 Y d 2 Y 
d 2 Y I d 2 Y 
B<£dz 2ttB^ " 
B 2 Y 
Equating" the two values of ^ -. thus obtained we have 
^V_ l' 2 p 2 Y 
_ d</> 2 " 4tt^ 2 J 
which on substitution in the expression for V 2 Y gives 
0^ r or \ 4:7r 2 r"Joz 
But it is now to be observed that if the distribution of Y 
on any plane passing through the axis of symmetry is known, 
then Y is determined everywhere. And on such a fixed 
plane the contours of z are identical with those of co. So 
that we may replace z by to in the last equation, and the 
previous result is confirmed. 
We have shown that if we make V 2 Y=/(V, r, co) over 
any surface intersecting all the screw-threads, the same will 
