Peano on Wronskians: A Translation - On the Wronskians

The addition of such a confusing footnote (Passage 4) certainly obscured the main point of Peano’s first article. He responded with a letter, part of which was published under the name Sur les Wronskiens in the same issue of Mathesis as the original note. The letter consists of three sections: in the first part, Peano argued that Mansion’s suggested language does not lead to a true statement; next, he presented another counterexample to the contested proposition which avoids the issue that at every point of the domain, at least one of the functions will evaluate to zero; finally, he referred to the above “proof” of Mansion’s proposition, and stated the most general theorem that can actually be proved using Mansion’s reasoning.

6. On the Wronskians (Excerpt from a letter by M.G. Peano). The note added to the end of my brief article on the Wronskians (Mathesis, p. 75-76), if it contributes to the clarity, it adds nothing to the rigor of the contested proposition; and I must return to the subject. I. The proposition I contest is the following: If the determinant formed with \( n\) functions \( x_1,\dots, x_n \) of the same variable \(t,\) and their derivatives of orders \( 1,\dots, (n-1) \) is zero for all values of \(t,\) there is between these functions a homogeneous linear relationship with constant coefficients; in other words, one can determine \(n\) constants, \( C_1,\dots, C_n ,\) all of which cannot equal zero at the same time, such that \( C_1 x_1+\cdots + C_n x_n =0 \) for all values of \(t.\)

He then particularly attacked the footnote (Passage 4) appended by Mansion to his previous article. He showed that adding the words “one of the functions is identically zero” (he quoted Mansion’s footnote) does not change the validity of the proposition at all. Lest readers would be confused by the way he phrased the proposition, Peano restated the (false) theorem in multiple ways, including a geometric argument similar to what he used to show that two functions were linearly independent.

If one of the functions, for example \( x_1,\) is identically zero, these functions are connected by the linear relation \( x_1+0x_2 +\cdots +0x_n =0.\) Thus the conclusion of the proposition in question is not at all modified if one adds the words: or one of the functions is identically zero. This proposition may again be expressed in this way: If the Wronskian of the functions \( x_1,\dots, x_n \) is identically zero, the determinant formed with the values of the functions, when one gives the variable \( n\) ordinary values, is also zero. A particular case of the general theorem is as follows: If the determinant formed with the derivatives of orders \( 1, 2, 3 \) of the functions \( x, y, z \) of the same variable \( t\) is identically zero, the curve described by the coordinate point \( x, y, z \) is planar.

Peano then reiterated his previous example (Passage 1), and explained that since neither of the two functions is identically zero, Mansion’s addition does not apply to his two functions, and hence his example still shows that the proposition in question is not true.

However, Peano did note that while neither of his functions is “identically zero,” for every value of \( t,\) at least one of them is.  Realizing that this might be confusing to the reader, he gave a second example, which is merely half the sum and difference of his original.  These no longer have the odd property that one of them is zero at any \( t\) value, yet they still satisfy that the Wronskian is always zero. Recall that \( \, {\rm mod}\, t\) denotes the absolute value of \( t.\)

II. To demonstrate that these propositions are not always true, I have indicated in my previous note, an example in which I consider only two functions \( x\) and \( y\) of the variable \( t;\) the determinant \( x{y^\prime} – {x^\prime} y\) is identically zero; but between them there exists no linear relation. Neither one nor the other of the two functions is identically zero, because \( x>0\) if \( t>0,\) and \( y>0\) if \( t<0.\) But these two functions present this unusual characteristic that, for all values of \( t,\) one or the other is zero. To make this anomaly disappear, we propose \[ X={t^2},\quad Y=t{\, {\rm mod}\, t}.\] These two functions of \( t\) satisfy the condition \( X{Y^\prime}-{X^\prime}Y=0 ;\) they only cancel for \( t=0,\) and between them there is no homogeneous linear relation. The sum and the difference of these functions are the functions of my first example; the coordinate point \( (X,Y) \) now describes the two half-bisectors of the axes.

Passage 5

In the third part of this note, Peano explained that the “proofs” of all the false propositions make the assumption that the Wronskians of the functions taken \( n-1\) at a time are either all zero or never zero, referring explicitly to Mansion’s work as an example of one of these incorrect proofs.  He again suggests that readers start looking for additional hypotheses that make the proposition true.

III. The given demonstrations of the proposition, including yours, only allow us to prove that it is true if one of the minor determinants of the last line is always equal to zero, or if it is never equal to zero. For determinants of the second order, one can demonstrate the proposition, except where, for any value of the independent variable, the two given functions and their derivatives each cancel at the same time. It would be interesting to identify all the cases in which the proposition holds true.

Peano finally noted that he had never seen this fact accurately conveyed, and gave several additional citations. We think he was being very generous, because when one reads these three passages, one sees that the issue is not whether the proposition is “clearly expressed,” but rather that it is clearly incorrect!

I believed it necessary to publish this brief note on the Wronskians, because I have never seen the proposition clearly expressed. (See Hermite, Cours d’Analyse, p. 133; Jordan, Cours d’Analyse, III, p. 150; Laurent, Traité d’Analyse, I, p. 183.)

We turn our attention to these three textbooks now.