A stepped stocking should step most at the ankle

A tube of one size presses the calf harder than the ankle worked out the girth a graduated stocking needs at four stations of a leg — 157 millimetres of knitted girth at the ankle, 200 at the lower calf, 247 at the calf, 248 below the knee — to press 20 millimetres of mercury at the ankle falling to 12 below the knee. It ended on the obvious difficulty. A machine does not knit a girth that rises smoothly. It changes its loop length or its laid-in tension in steps, and between two steps a tube is one size over a stretch of leg that is not.

By the essay’s own argument each step is then a short tube of one size, pressing its thicker end harder than its thinner, and the pressure up the leg is a saw-tooth with a ring at the top of every step. How large the rings are, and where, was left undone.

A step is a tube of one size #

Within one step the stocking has one relaxed girth, $C_0$, and the leg under it widens from one girth to another. An elastic band pulls with a tension proportional to its stretch, $C/C_0 - 1$, and presses with that tension over the leg’s radius. So along the step the pressure is

$$P = 2\pi k \left(\frac{1}{C_0} - \frac{1}{C}\right)$$

with k the band’s stiffness. It rises as the leg widens, because the rise in stretch outweighs the rise in radius — the whole content of the essay on a tube of one size, read over a few centimetres instead of a whole leg.

The change across a step is then exact and short:

$$\Delta P = 2\pi k\,\Delta\!\left(\frac{1}{C}\right) \approx 2\pi k\,\frac{\Delta C}{C^2}.$$

A step’s ring is the leg’s change of girth across it, over the girth squared. It does not depend on the target pressure, the step’s knitted girth or anything the machine is set to; it depends on the band’s stiffness and the leg.

The ring is largest just above the ankle #

Two things make a ring large: the leg widening fast across the step, and the leg being thin. Just above the ankle both are true at once. The illustrative leg widens from 220 millimetres to 290 over the first twelve centimetres — the fastest widening on the leg — and 220 millimetres is its smallest girth.

So with eight steps of equal height, four centimetres each, the ankle step carries a ring of 4.8 millimetres of mercury — a quarter of the 20 the stocking is meant to press there. The rings then fall steadily up the leg: 4.0, 3.3, 2.8, 2.4, 2.1 through the lower calf, where the leg is still widening but is already fatter. Across the top of the calf, where the leg stops widening and begins to narrow, they drop below one.

And where the leg narrows, below the knee, the sign of the ring reverses. A step on a narrowing stretch presses harder at its bottom than its top, so the saw-tooth’s teeth point the other way there. That is visible in the figure as the teeth flattening and turning over past the calf.

Spacing the steps by the leg’s own shape #

If the ring is Δ(1/C) times a constant, then steps that each carry the same change in 1/C carry the same ring. Spacing them that way is the natural repair.

Every ring becomes 2.66 millimetres of mercury. The ankle’s has fallen from 4.8 to 2.7, and the steps across the calf, which were wasting their resolution on a leg barely changing, have taken it up. The last step spans the calf’s widest point, where the leg stops widening and starts narrowing, and its ring partly cancels to 1.8.

The steps that achieve it are anything but equal.

The first step is 2.1 centimetres tall and the last 9.7. The graded stocking spends its changes where the leg changes fastest relative to its size — which is at the ankle — and saves them across the calf.

Why the ring has no target in it #

The ring’s formula has a striking omission. It contains the band’s stiffness and the leg’s change of 1/C, and it does not contain the stocking’s knitted girth or its target pressure at all.

The reason is that the knitted girth sets the level of the whole step and the leg sets its slope. Knit a step a little tighter and every point on it presses harder by the same amount; the difference between its top and its bottom is untouched, because that difference is the leg widening under a fixed tube. So a stocking’s rings are fixed the moment its band and its step heights are chosen, and a class II stocking and a class I stocking of the same band, stepped at the same heights, carry the same rings on the same leg.

That makes the rings a proportionally worse problem for the lighter stocking. A ring of 2.7 millimetres of mercury is thirteen per cent of a 20-millimetre ankle target and seven per cent of a 40-millimetre one. The gentler a graduated stocking is meant to be, the more finely it has to be stepped to keep its rings in proportion — the opposite of the instinct that a light stocking can afford to be made coarsely.

It also says where the band’s choice enters. A stiffer band makes a larger ring for the same step, in proportion to its stiffness, which is the modulus a knit has instead of one turned into a manufacturing tolerance: a stiff band needs more steps than a soft one to leave the same rings.

A step in courses #

A step’s height becomes a count of courses on the machine, and the conversion is the loop’s own dimensions. A hosiery knit at a 3.5-millimetre loop, fully relaxed, has about sixteen courses to the centimetre.

So eight equal steps of four centimetres are about 63 courses each. Eight graded steps are about 33 courses at the ankle rising to 152 across the calf. Twenty-four graded steps — the number that holds every ring under one millimetre of mercury — would put the first change about eleven courses above the welt and the last about fifty below the knee. None of that asks a modern machine for anything it cannot do; it asks the programme to change its settings on a schedule set by the leg’s shape rather than by a fixed count.

The ring falls as one over the step count #

Both placements leave rings that fall in proportion to the number of steps, because each ring is a fixed share of the leg’s total change in 1/C divided among them.

Graded steps are worth nearly half the steps. With equal steps the worst ring is set by the ankle, the worst place on the leg; with graded steps it is the average over the whole leg. At every step count the graded ring is about 55 per cent of the equal one, and to bring every ring under one millimetre of mercury — five per cent of the ankle target — takes about 24 graded steps against more than 32 equal ones.

That is a statement a machine can use. At twenty-four graded steps the first is under seven millimetres tall and the steps across the calf are about four centimetres — and a fixed pitch as fine as the ankle’s would need forty-seven steps. A seamless hosiery machine changes its settings course by course, so a step is a number of courses, and grading the steps means changing the loop length or the elastomer’s tension more often near the ankle and less often across the calf — the same number of changes, placed where the leg needs them.

The leg is illustrative and the rule is not #

Every number above is for one illustrative leg, and a different leg moves all of them. What does not move is the rule the numbers come from, and it is worth stating for any leg a stocking might be made for.

A step’s ring is the band’s stiffness times the change of one over the girth across the step. So the total ringing a leg can produce is fixed by the change of 1/C from ankle to knee, and a leg with a slender ankle and a heavy calf has more of it: going from 200 millimetres to 380 changes 1/C by more than going from 240 to 340. That leg needs more steps for the same rings, and it needs them concentrated even harder at the ankle, because a slender ankle’s 1/C changes fastest of all.

The rule also names what makes a stocking easy to step. A leg that widens slowly and uniformly, or a stocking that starts higher up the leg where the girth is already large, carries little change of 1/C per centimetre and forgives coarse steps. The ankle is where the leg is least forgiving, and a graduated stocking is specified by its ankle pressure precisely because the ankle is where it matters most — so the part of the stocking the specification is about is the part the steps treat worst.

Why a ring is worth caring about #

A ring of a few millimetres of mercury sounds small against a stocking’s twenty. It is not small where it lands.

A band presses where the limb turns: the pressure at a point of a leg’s section is the band’s tension times the curvature there, and on a shin’s ridge that is about ten times the round-limb number. A ring is a jump in the band’s tension, so it is multiplied by exactly the same factor wherever the section is curved. A 2.7-millimetre ring on the round-limb number is a ring of nearly 30 on a ridge of five millimetres radius, and a stepped stocking draws a line of such jumps up the front of the shin at every step.

That is the answer to the question the limb essay ended on — where the worst point on a leg is — and it is a combination rather than either factor alone: the worst point is on the ridge of the shin, at the top of a step, low on the leg where the steps are steepest. Grading the steps lowers the ring there by the same 45 per cent; nothing about the steps changes the ridge’s multiplication.

The same staircase, at a panel’s edge #

This is not the first knitted shape to turn out as discrete steps approximating a continuous one. A fashioned edge has a quantised angle: a panel narrows by whole wales at whole courses, and its edge is a staircase whose treads and risers are loops. A tube can only be shaped by its loop found the continuous alternative bounded and costly.

A stepped stocking is the loop route used in steps, and the lesson is the same as the fashioned edge’s: a staircase approximates a curve best when its steps are spaced by how fast the curve bends, not by a fixed pitch. A fashioning pattern book puts its decreases closer together where a garment’s edge curves sharply; a hosiery programme should put its changes closer together where the leg’s girth changes fastest for its size.

What a cuff’s pressure already implied #

What a cuff presses with computed a single band’s pressure on a wrist and found it tiny beside medical compression. It did not need steps, because a cuff is one girth on one short stretch of limb. The same arithmetic applied along a tapering limb is what produces a ring: a cuff on a wrist that widens by a few millimetres over the cuff’s own depth has a ring too, of the band’s stiffness times the change in 1/C across it. For a cuff that ring is a small fraction of a small pressure, which is why nobody notices it; for a stocking pressing twenty times harder, over a limb widening far faster, it is the part of the design that decides whether the graduation is felt as graduation or as a series of bands.

The model named #

The leg is the illustrative one used for every pressure calculation here — 220, 290, 360 and 340 millimetres round at the ankle, lower calf, calf and below the knee — now given heights of 0, 12, 24 and 32 centimetres, with the girth running straight between stations. They are round figures, stated rather than surveyed. The target is the graduation the tube-of-one-size essay used: 100, 85, 70 and 60 per cent of 20 millimetres of mercury, interpolated in height. The band is its elastic one, sized to press 20 millimetres of mercury at 40 per cent stretch over the ankle. Each step is knitted to the ideal relaxed girth at its own middle height, and the pressure along it is Laplace’s law at the band’s tension for the leg’s girth there.

Required of the arithmetic, not shown: that graded spacing never leaves a larger worst ring than equal spacing, at every step count in the sweep.

What was counted #

Nine step counts from 2 to 32, each in equal and graded placement, with the pressure computed at eight hundred heights along the leg and the ring of each step read off as its range. The ridge’s multiplication is taken from the limb-section essay’s own calf, not recomputed.

What the model cannot show #

A leg is soft. A ring of pressure pushes tissue along the leg away from where it is applied, and the ring a person feels is the ring here spread over some centimetres of limb. How far pressure spreads along a soft leg is not computed, and it is the one thing that would make many fine steps and a few coarse ones feel more alike than their rings say.

The steps are sharp. A machine changing its loop over a few courses makes a short ramp rather than a step, which rounds the teeth without changing their height.

And the band is linear. A knitted band stiffens as it stretches — what a knit gives when it is pulled is soft and then abruptly stiff — so its ring across a step is larger where the stocking is stretched further, which is at the calf rather than the ankle. For a covered-elastane stocking the linear band is the right first model; for a knitted one it understates the calf’s rings.

Who found it, and when #

Graduated compression and the machines that knit it are a well-established field, and step-wise changes of stitch or tension are how those machines work. That pressure is tension over radius is Laplace’s.

What is added here is the ring’s form — the leg’s change of 1/C across a step — and its two consequences: that equal steps waste their resolution across the calf and fall short at the ankle, and that on a shin’s ridge a stepped stocking’s rings are multiplied by the ridge’s own factor. Neither needs anything the cuff and tube calculations had not already computed.

Still open: how a soft leg smooths a ring #

Every ring here is on a rigid leg. A real leg’s tissue flows away from a line of higher pressure and towards a line of lower, so the saw-tooth a stocking applies is smoothed into the pressure the tissue actually carries, over a distance set by how deep the tissue is and how stiff.

That distance decides whether a machine’s step count matters at all. If a leg smooths pressure over more than a step’s height, twenty steps and forty feel alike; if over much less, only the ring counts. Measuring the smoothing distance — with a pressure sensor under a stocking with known steps, on legs of different build — would turn the rings here into the pressure a wearer feels, and it is the one measurement that decides how finely a stocking needs to be stepped.