Discussion of: 

E. Carvallo, 
Theorie du mouvement du monocycle et de la bicyclette. 

The comments are made on the reprint version as it is available from 
The University of Michigan, 193 pp. The memoir was published in the 'Journal 
de l'Ecole Polytechnique', second series, volume 5 (1900), pp. 119-188, and 
volume 6 (1901), pp. 1-118. To convert page numbers, add 118 to the present 
pages 1-70 and subtract 70 from the present pages 71-188. 

Summary of the memoir. 

At the end, pp. 186-188, Carvallo gives a summary of the results. 
The first part (pp. 1-70) is about the hoop and the unicycle, and the 
second part (pp. 71-188) is mainly about the the bicycle. 

The first chapter discusses the general equations of the hoop and the 
stability of the stationary rectilinear and circular motion. The treatment 
is fairly complete, but most of the results were probably known at the time 
of writing. 

The second chapter is about the unicycle. It is assumed that the rider always 
stays in a vertical position in the plane of the wheel, so stability in the 
fore-aft direction is left out of consideration. The means by which this 
can be achieved are not discussed. For the stationary motions, this 
configuration can always be achieved. The product of inertia of the rider 
is neglected, but its effect will be investigated in the second part. 
As for the hoop, the stationary rectilinear and circular motions and their 
stability are investigated. Some ill-advised simplifications in the 
equations of motion are made, which do not change the qualitative results, 
but make a comparison with the hoop as a limiting case impossible. 
It is found that for a sufficiently high forward speed, the motion can be 
stable. A 'paradox' of a cycle is discussed: if a pedal in the lowest 
position is pulled in the backward direction, the cycle moves in the forward 
direction. A digression about the analogy between gyroscopic systems and 
electromagnetic systems is made. The first part of the memoir is concluded 
with a section that draws attention to a change in notion in the second part. 

The first chapter of the second part deals with the geometry of the bicycle. 
The simplification is made that the two wheels are equal. The non-linear 
equations to calculate the pitch angle from the lean and steer angle are 
derived. The equilibrium positions for zero forward speed and negligible 
mass of the front wheel assembly are derived: the lean angle is zero and 
the steering angle is either zero or 180 degrees or in two intermediate 
positions with fairly large steer angles (between 60 and 70 degrees). 
Beyond this angle, the trail becomes effectively negative. 
All of these equilibrium positions are unstable. 

The second chapter considers the kinematic relations; that is, the 
non-holonomic conditions of pure rolling, which yield relations between the 
velocities and between virtual displacements. A second paradox is mentioned: 
if the rotation of the wheels is suppressed by brakes and the steer is kept 
in a fixed position, one would expect to have suppressed the three degrees 
of freedom and the bicycle would not fall down. Of course, the constraints 
are dependent for this case. 

The third chapter discusses a modification of Lagrange's equations for 
systems with non-holonomic constraints. This section is remarkable, because 
it predates the classical papers by Boltzmann and Hamel. It points out the 
very ground why the classical equations of Lagrange are not applicable for 
this case: taking virtual variations and taking derivatives with respect to 
time do not commute and the order is important. Then the general method for 
investigating the stability by means of the eigenvalues of the linearized 
equations is discussed and applied to the hoop and the unicycle. It is found 
that the product of inertia of the rider, depending on its sign, can lead 
to asymptotic stability or instability of the unicycle. 

The final fourth chapter deals with the dynamics of the bicycle. It consists 
of five sections, which are discussed in order. The basic simplifying 
assumptions made are that the wheels are equal and the front fork with 
handlebar has no mass. The first section deals with stationary motion with 
the steer in a fixed position. Some simplified expression are also discussed. 
The second section considers stationary motion with free steer. On page 143, 
in equation (1)', a factor sin(omega) is  missing (compare with equation (1) 
on the same page). This error leads to the false conclusion that no 
stationary circular motions are possible with a free steer (page 144). 
The third section discusses the stability of the stationary motions with 
fixed steer. It naturally leads to the conclusion that all of these 
motions are unstable for bicycles of the usual construction. The fourth 
section discusses the stability of the rectilinear motion with free steer. 
This leads to the linearized equations of motion. Note that the entries in 
the first column on page 171 should have their signs reversed. When the 
terms are collected, op page 173, this is corrected. The fifth section 
discusses the stability. The capsize speed and the weave speed are 
identified and a stable speed range between them is found. The influence 
of varying bicycle parameters on the stability is discussed. Furthermore, 
the influence of friction on the stability is discussed in a qualitative 
way. It seems that Carvallo was not aware that friction may actually 
destabilize the bicycle in some cases. 


List of errors. There are apparently more than listed here, and only those 
that are considered important and did not escape my attention are listed. 

Page 26, table at the bottom: in the middle row, the expression for s_0 is 
not correct: see formula at the middle of the same page for the correct 
expression. 

page 30, table in the middle, second line: sign of n_2: it should be 
s < -n_2 < -n_1 

Page 41: some ill-advised simplification are made in the equations. See the 
discussion above. 

Page 42, displayed equations and text around them: it should be sin(i) and 
not tang(i) .

Page 50, first displayed equation: there is a sign error in the denominator; 
the plus before z should be a minus. This error propagates into the 
subsequent equations. 

Page 56, equation (1), top: prime missing for p: it should be Ap' = ... 
                    middle: missing minus sign between B_1 and mu_1 R^2

Page 57, equation (S)_1: it seems that a term with sn is missing. 

Page 57, second equation from bottom: it should be - n'/p , not -p'/p

Page 64, table: the 0,03 should be 0,63 (n at zero degrees)

Page 69: the equation (E) should be with a script letter E 

Page 70, second line: it should be C/(mu_1 R), not (mu_1 R)/C

Page 78, equation (3): I have my doubts about this equation, but I am not a 
specialist in spherical trigonometry. This equation is not used in the rest 
of the paper. 

Page 79, equation (2)', second line: sin(sigma_0) should read cos(sigma_0) 

Page 86, line 8 of No 58: theta should have a subscript 1 

Page 89, equation (4), line 4: second I should read J 

Page 93, equation (delta nu), second line: last R should be within the 
brackets, before the sin(mu) 

Page 94, equation (1), last part: read x cos(alpha) for alpha cos(x) 
(see errata on page 188). 

Page 100, before equation (7): read I|OM for OM 

Page 100, equation (7): sin(gamma) in the second term is missing: gamma=0 
should give eta'=0 (propagates in equation (8), and on page 101)

Page 107, equation labelled (no 3): O' should be added to the first equation. 
This has no further consequences. 

Page 109, last line: sn should be in italics, simply s times n, not an 
elliptic function. See also on page 110, first equation. 

Page 116, bottom, entry (2,1) in the determinant: cos(theta) with L is 
missing. 

Page 117, bottom line: s missing in middle term; in the last term, 
mu should be mu_1 

Page 126, equation (3): s should not be underlined; there should be a minus 
sign before L 

Page 126, equation (4): again, signs before L are wrong; the last sin(theta) 
should be cos(theta) and the last minus should be plus. 

Page 126, equation (5): a term -B eta' is missing. 

Page 126, equation (6): again, sign before L is wrong. 

Page 129: the errors on page 126 have consequences for the middle line 
for the 'Cadre monte' . 

Page 135, line 3 from bottom: 'eos' should be 'cos' 

Page 136, line 5: in the last quotient, theta and gamma should be 
interchanged. See also errata on page 188; in this equation (2), the right 
quotient is shown. 

Page 139, line 5 from bottom: last N' should read M' 

Page 140: propagation of the errors from page 129. 

Page 141: value of mu_1, 57 kg, is presumably a printing error. 

Page 143, equation (1)': sin(omega) missing; see discussion above. 

Page 150. line 10: capital C should be lower case c 

Page 152, first displayed equation: second minus should be plus; this 
propagates into the next equation. 

Page 159, equation (6), last equation: (1-alpha^2) should read (1-0.5*alpha^2) 
This propagates into page 160. 

Page 152, equation (8), second line: last gamma should be gamma' 

Page 163, line 3: the first cos(eta) should not be there (corrected in 
subsequent equations). 

Page 165, equation (1): js'M' should be jS'M' (capital S) 

Page 168, first displayed equation: in last term, j should read j' 

Page 169, first line of the table: d eta/d theta should read d eta/d gamma 

Page 171, table: signs in the first column; in the heading of the fifth 
part, theta should read gamma (see corresponding heading on page 170). 


JPM, Nottingham, 29 August 2007


To come back to the paper by Carvallo, I have no clear idea about the exact 
time it was printed, as it contains no date. Apparently, it was in or after 
1898, as this year is mentioned in the footnote. There is a hand-written year 
1898, apparently added by a librarian. To me it appears that the 
memoir is essentially the same as the one submitted for the Prix Fourneyron. 
A way to find out more is to use the number at the end (hardly legible), 
and to compare this number with publications from the same publisher around 
the same time. As to the archives of the publisher, the material for the 
Proceedings of the 1946 ICTAM congress in Paris seems to have been lost, 
so these Proceedings have not, up till now, been published. 


As to the modified equations of Lagrange, Carvallo did not give a full 
reduction and his equations still differ considerably from those of 
Boltzmann and Hamel. 

JPM, Nottingham, 12 September 2007