260 Mr. L. F. Richardson on a Freehand Graphic way 
where A is an absolute constant, we must have 
1 
/- , 4ttV 
H a 
~ A H, 
So that /3 is a stream function analogous to the forms in use 
when the guiding lines are parallel straights or circles with 
their centres on, and their planes normal to, a common axis. 
In types previously studied, when the graph was drawn 
on a surface normal to the guiding lines, H a and H were 
proportional to the length and breadth of a chequer and 
could be measured directly. But here we must first compare 
the linear dimensions of a freehand chequer with those of 
the standard oblique section of the tube bounded by two 
stream-surfaces and two equipotentials, and then refer to the 
normal section of the same tube in the right-hand margin of 
the chart. 
The standard chequers were obtained in the following 
manner : — 
H , . . / Air 2 r 2 
" being equal to constant X W l_j _ some other re- 
lation is necessary to determine H a and H separately. The 
relation H a xH =1 was chosen for this purpose, as this 
gives a neat appearance to the standard chart. It was also 
found convenient to make the constant such that -~£ = 1 
r 
when -= =0'5. The values of H a and H^ were calculated 
and are given in the accompanying table. The sides of the 
rectangles in the right-hand margin of the standard chart 
were drawn proportional to 2H a and 211^ . 
To obtain the slant section, the tangent of the angle 
between the tangent a guiding-line and the plane normal to 
the axis of the screw, was first calculated. It is equal to 
~ — , and is given in the table under that head. The rect- 
angles were then projected with ruler and compasses in a 
manner which is perhaps sufficiently indicated by fig. 5, 
r 
which shows the construction when y-=O05 and the angle 
between a radius from the axis of symmetry and the tangent 
plane to the surface « = constant meeting at the point con- 
sidered is 45°. 
