240 Mr. L. F. Richardson on a Freehand Graphic way 
Take two points A and B on the mid line o£ one tube, and 
from A and B draw normals to the direction of flow cutting 
the mid line of the other tube in D and C respectively. 
Halfway between AD and BC draw a line PQS normal to 
the direction of flow so that PQ is the width of one tube and 
QS of the other. Now if the fluid is incompressible and we 
have drawn the tubes so that the. flow in each is the same, 
then the respective velocities are to one another inversely as 
k k 
PQ and QS. Let the velocities be ^- and ^. Next let 
us take the line integral of the velocity round the small 
rectangle ABCD. The sides AD and CB are normal to the 
flow and so contribute nothing. The sides AB and CD 
contribute 
AJ3X PQ QS *LPQ QSJ- 
AB 
Now p^ is the ratio of the length along the flow to the 
breadth across the flow of the small chequer which has A, Q, 
B, P, at the mid points of its four sides. It will be convenient 
to have a special name for this quantity, and I propose to 
call it the "chequer ratio " with the understanding that 
length along the flow is always in the numerator, and that 
the chequer is so small that its size no longer causes an 
appreciable deviation from the accuracy obtained by using 
infinitesimals. Then we have : — 
I Difference between successive 
Line integral of the velocity around ABCD = £x X chequer ratios in a direction 
( perpendicularly across flow. 
