882 
took up any more CO than oxygen, in spite of the 
great difference between the effective concentrations 
of the two gases. 
Sir William Bayliss also asks for experimental tests 
of the assumption that oxyhemoglobin is a stronger 
acid than hemoglobin itself. The limitations of the 
hydrogen electrode make the measurement of the 
hydrogen-ion concentration of hemoglobin solutions, 
in the presence of oxygen, a difficult problem. I have 
been able to show, however, that if gas is boiled off 
in a vacuum from dialysed hemoglobin solution, the 
electrical conductivity of the solution is considerably 
increased by shaking with oxygen or CO. (Pre- 
cautions have, of course, been taken to exclude the 
possibility of the increased conductivity being due 
to impurities in the gas used.) 
While this is naturally not a proof that combina- 
tion with oxygen increases the acid dissociation 
constant of hemoglobin, it is nevertheless the result 
to be expected if,this be the case, and is a fact to be 
explained by any theory, chemical or physical. 
Prof. Hill and I have pointed out that the divergent 
results of investigators of the heat of combination 
of oxygen and hemoglobin may be due partly 
to bacterial action, and (in experiments on blood) 
partly to failure to allow for the heat changes involved 
when oxyhemoglobin turns out CO, from carbonates. 
By eliminating these sources of error we have been 
able to get quite consistent results in experiments on 
defibrinated blood. 
Without making any assumptions other than the 
recognised laws of chemical combination and chemical 
equilibria, it is possible to explain the behaviour of 
hemoglobin by regarding its reactions with CO and 
oxygen as purely chemical. Sir William Bayliss has 
said that he doubts whether it is justifiable to apply 
these laws to a system in which the number of the 
phases may be uncertain. Surely the best way to 
decide this is by results, and, judging by results, the 
chemical theory has amply justified its position as a 
fruitful working hypothesis. 
Can the adsorption theory explain the phenomena 
so completely, with so few untested assumptions ? 
Since the paper by Wo. Ostwald in 1908, no attempt 
has been made, so far as I am aware, to put forward 
a complete theory of the reactions of hemoglobin 
as adsorption phenomena. Much experimental work 
has been done since then, and until such a theory is 
put forward it is difficult to weigh up satisfactorily 
the merits of the two views. 
At present the adsorption theory is in danger of 
going by default. W. E. L. Brown. 
Physiology Department, 
University of Manchester, June 4. 

A Puzzle Paper Band. 
AN easy solution of the paper-band puzzle de- 
scribed by Prof. C. V. Boys in Nature of June 9, 
Pp. 774, is obtained as follows: Hold the hand with 
thumb up and palm towards you; place the paper 
band over the index finger, letting the ends hang 
down. Observe which way the original four half- 
twists were applied. Treat the nearest of these to 
the index finger on. the palm side of the hand as 
if it were that of an ordinary single half-twist band ; 
which complete, by looping up one-half of the band 
over the finger (the other twists being pushed out 
of the way into the remaining half). Then apply 
the surfaces one upon another at the finger; and 
turn the other half of the band inside out so as to 
get rid of two of the twists. It will be found to 
fit exactly upon the first half, as required. 
ANNIE D. BETTs. 
Hill House, Camberley, Surrey, June 11. 
NO, 2809, VOL. 111] 
NATURE 

[June 30, 1923 . 
- 

Paradromic Rings. 
Pror. C. V. Boys, in his letter “‘ A Puzzle Paper 
Band” in Nature of June 9, p. 774, gives scant 
credit to the geometers. Forty years ago they 
described the endless band of paper with a half-turn 
twist in it, and found that if “ut down the middle 
line it gave a single endless band with four half 
twists. But they were so obsessed, he says, with : 
the consequence of cutting down the middle line 
that they missed the result he now describes. This : 
consists in taking a band with four half-twists and 
converting it by manipulation into a half-twist band_ : 
of double thickness. 
But the difference between the known result and 
the proposed novelty seems not more than trivial : 
for if the medially cut band has its adjacent half- 
widths simply slid sideways, one over the other, 
along the entire length of the band, the double- 
thickness band of half-width is at once produced. 
Or, reversely, if the pulleys of Prof. Boys are made 
twice as wide, and the outer band is teased sideways 
at its entry on to each revolving pulley, the two 
halves of the band will presently come edge to edge 
throughout and are then seen to be nothing but the 
half-twist band medially cut. 
As regards this lateral shifting, it is obvious that 
any endless band, however much twisted and knotted, 
may, when cut down the middle, be continuously 
“ shuffled,’’ in the way in which a “ pack ”’ consisting 
of only two cards may be shuffled. Each neighbour 
slides over the other in perpetual oscillatory contact, 
alternately face to face and edge to edge. Two 
different superpositions and two different edge-to- 
edge positions occur alternately and cyclically. In 
particular, the band with four half-twists may be 
arranged as a two-ply half-twist not in one way 
only but in either of two ways. For either of the 
two different faces of the former may be completely 
exposed or completely concealed in the latter. 
The sheer puzzle of the manipulation Prof. Boys 
plans to make even harder by varying the sense of 
the twists, as right-handed or left-handed. I should 
propose (somewhat on behalf of the geometers) to 
escape this difficulty by letting the paper discriminate 
for itself. The instructions would be these. Strip 
the band along, two-handedly, until the twists are 
concentrated on a short section. They come to 
form roughly a circular cylinder, showing two turns 
of a ribbon screw. Take two adjacent widths, 
touching helically edge-to-edge at any point, and 
fold them together as if closing an open book. Then 
feed the short circuit at the expense of the long loop 
until they come equal, and fit together by stripping. 
These operations may quite easily be done blindfold. 
Prof. Boys says that the double band shows only 
two of the half-turns, and that it is amusing to find 
where the other two have gone. But this is viewx 
jeu: for Tait explained it in his first paper on 
“Knots” in 1877; and he was only following Listing, 
who had these things clear in his “ Topologie ’’ of 
1847. Ifthe paradox is still alive it may be reinforced, 
for those who do not know that torsion and curvature 
are convertible; for the double-twist may be hung 
over one finger as a festoon of three equal loops, 
with the six pendant planes all (approximately) 
parallel to the finger, and then not merely half but 
the whole of the twist appears to have gone. — 
In a parenthetic confession Prof. Boys admits that 
he made his discovery while lying awake one night ; 
but this may almost be interpreted as an indirect 
testimonial to the day labourers. 
G. T. BENNETT. 
Emmanuel College, Cambridge, June 12. 

