Here follows a rambling commentary on Whipple's 1899 paper. 

I cite first Truesdell's quip about errors: 

`Newton relinquished the diplomatic immunity granted to nonmathematical 
philosophers, chemists, psychologists. etc., and entered into the area 
where an error is an error even if it is Newton's error; in fact, 
all the more so because it is Newton's error.' 

The paper contains more printing errors than real errors. 

Whipple was 23 years old in 1899, and we are still poring over his article. 
I was 27 when my first paper was published. And though it is still worth 
reading, it was not as good as Whipple's. 


p.312 and ff: Spelling of tyre: Fowler, and Webster, do not agree. 

p.313: the stick balancing problem. It turns out that it is an easy task if 
the length of the stick is larger than 1 m, corresponding to unstable 
eigenvalues of 2. For sticks shorter than 25 cm (a pencil), balancing is 
impossible. So my definitions of 'mildly unstable' and 'strongly unstable'. 

The traite by Bourlet was published in 1896. I have not seen a copy, but the 
Bibliotheque Nationale in Paris has a copy. 

'intructions' should be 'instructions'. 

I have not seen McGaw's paper, although it must be easy to find it. 

I like this :- punctuation. Note the space before the colon. 

p.314: the distance p' is not indicated in fig.2, but I assume that it is 
the distance Q'O'. My assumption is wrong: it is minus this distance. 

Figure 3 is puzzling. The hundred words in the text say more than the 
picture. Apparently, it is a top view, with V pointing towards us. B_2 is the 
forward direction, or x-direction. The H-system is a little awkward. 
H_1M_2B_3 is more convenient, with H_1 along the steering head hinge, M_2 
in the direction of OQ and B_3 in the direction of the rear wheel axle. The 
H-system is found from this system by a rotation with an angle \phi about H_1. 
\theta is another inconvenient angle, for us. For Whipple, the choice of 
coordinates is quite natural. 

p.315: the four goniometric formulas are correct, as far as I can see. I 
derived them with the help of rotation matrices. 
Eq.I is not correct. The left-hand side should be multiplied by \cos \psi. 
(and \cos \psi' for the primed terms) 
If you take the time derivative, you should get \cos \theta times the first 
plus \sin \theta times the second of Eq.II. After some pages of calculations, 
this appears to be the case for the corrected formula, and not for the 
original one. If you linearize, as on p. 320, the linearized result is the 
same in both cases. 

I came up with the idea: can you eliminate the dependent terms in the 
constraint equation in closed form? Yes, it should be possible. It should 
lead to a polynomial equation of degree four, for which the solution can be 
obtained exactly by a complicated formula. I have not tried to do this, 
however, but someone might have a go. 

Eq.II: some printing errors: a square bracket is missing in the second line; 
       a minus sign should be added in the sixth line. 

p.316: Eq.III: \theta_2 should be \theta_3 in the last term on the fourth 
line. 
Eq.IV: the second a in the third line should not be there. 
last line: force Q_1Q_2Q_3, not F_1F_2F_3. 
It is poor practice to use g for the centre of gravity as well as for the 
acceleration of gravity: do not be confused.  And do not confused by B_3: it 
is an axis and a moment of inertia. 

p.318: line 4: H_1 is meant, I presume. 
4th and 5th line of displayed equation: the expressions between parentheses 
must be interchanged. 
before Eq.VII: VI.(3) should read V.(3). 

Eq.VII: second line: remove subscript 1 from \gamma. 
last line: there should be a \delta before W: compare with Eq.VIII. 

last line: there should be an a between the ( and \cos. 

p.319: Eq.VIII: remove subscript 1 from \gamma. 

forcive: is it in the Dictionary? Yes, it is. But I think that this is a typo. 

Eq,IX: there is some confusion in the order of the indices in the inertia 
tensor E; this tensor is symmetric, of course, so the order does not matter. 

p.320: 3rd line from below: III should read II. 

p.321: 4th line: In the figure, R is called S. 

Eq.VIII: the equation number is spoilt: dots should be before V. 

Eq.XI: subscript 1 is missing in first P, and/or a prime is missing. 

4th line from below: there should be an extra dot over \phi, or \phi should be 
\tau. 

p.322: apparently, it is assumed that the products of inertia E_{13} and 
E_{23} are zero, which is the case for symmetric frames. No, the terms 
containing them are not important: the first-order terms may be neglected. 

line 8: there should be a prime on the second W. 
3rd line from below: \gamma' should be \kappa' 
2nd line from below: the minus before W' should not be there. 

p.323: fourth line: there should be an opening parenthesis. 
5th line: the same expression with dashes, but with a minus sign before 
\Lambda. \Lambda and also b are odd, so they change signs with primed terms. 
Whipple calls the primes dashes. 
11th line: the corresponding primed coefficient has +\mu'\Pi . 

Eq.XV: 
2nd line: dot over \phi missing. 
3rd line: two dots over first \phi' and one dot over second \phi' missing. 

So there we are: the linearized equations. Note that \tau could be eliminated 
with the help of Eq.XIII, yielding two second-order differential equations. 
For the tricycle, in section 23, this is done. 
Do these equations really agree with Papadopoulus, Schwab and Meijaard? 
Yes, they do in the following way. 
First use the lean angle and the steering angle as independent coordinates. 
Then we have \phi = \psi/(\sin \theta) and \phi' = \delta + \psi/(\sin \theta) 
Use Eq.XIII to eliminate \tau and \dot{\tau}. 
And hurray! the first equation is identical to SMP. To get the second equation 
of SMP, multiply Eq.XV by (\mu\mu')/(b \cos \theta) = (f \mu)/(\cos \theta), 
and subtract (f \sin \theta)/(\cos \theta) times the first from it. This gives 
the desired result, after some further substitutions. 
(f is here according to SMP). 

The elegance of Whipple's paper is that he presents the equations in a fully 
symmetric form. Can we do something similar with our equations? No, it seems 
that you always un against a singularity for \theta=0. But we can define the 
auxiliary quantities in a symmetric form. This does not make the equations 
simpler; only the comparison with Whipple's results becomes simpler. 

p.324: Eq.XVI is our steering angle = \phi' - \phi. 
our lean angle is \phi \sin \theta. 

p.326: the determinant. Element (2,1): subscript 2 should be power after 
\lambda. 
Element (2,3): no prime at first W; closing parenthesis is superfluous. 

p.327: Element (1,3): \cos b should read \cos \theta. 
two lines down: the terms with \Pi b \cos \theta, and the closing parenthesis 
should not be there. 
next line: in expression between brackets with \mu's, prime should be 
with the other \mu. 

p.329: Eq.XXVI, third line, last term between (): in the first term, \mu 
should be \mu' (or is it with the second?). 

p.330: Whipple makes some simplifications for the calculations of the 
moments of inertia: the mass of the wheels is not considered separately. 

p.331: Whipple, without a Millonaer, gets his calculations 
basically correct. I have one objection: 1116.6/264.7 = 4.2 and not 4.6 

p.332: there should be a dot before 839 somewhere. 
The gear is the effective wheel diameter. 

p.333: The idea of a very heavy rear wheel is interesting. The bicycle 
becomes a rolling penny. You can also make the centre of gravity of the 
rear frame very low, as in a bicycle on a tight-rope with its centre of 
gravity below the rope. 
20 revolutions of the pedals? I found about 5 revolutions, or 3 seconds. This 
seems more in agreement with experience, but a little too unstable for a 
good bicycle. Note that the trail in Whipple's example is small: 5 cm or 
5.5 cm. For the SMP-bicycle it is about 6 seconds, which seems to be right. 

p.334: Eq.XXIX: \omega should be W; lower case v should be upper case V ; 
lower case \pi should be upper case \Pi . 

p.335: Eq.XXX: in first line, minus 7.5 should be + 7.5 . 

It is funny to see that Whipple rounds the same numbers differently in 
different places. 

p.336: I come out at Z_2 = 0.24 (\epsilon=0.8) and Z_1 = -3.1 (\epsilon=0.1). 

p.338: he calls it an eliminant. 

p.340: .86 should be 8.6; the coefficient before \epsilon^2 should be 
about 1.45 instead of 2.4. 
Near the bottom of the page: the conversion from \chi to A\phi is not done 
correctly. The error is small, however. 

p.341: .46 should be .36 in the equation. This is a typo, because 0.36 is used 
further on. 

p.343: Whipple now writes 'can not' instead of 'cannot': does he mean to put 
emphasis on this? 

p.347: There should be a prime at C_3 . 

JPM, Nottingham, 25 Oct. 2005.

Whipple nog een keer. Je weet toch dat zijn niet-lineaire vergelijkingen niet 
goed zijn. 

JPM, Nottingham, 17 Oct. 2005.