90 4. WOMEN MATHEMATICIANSSince Berlin would not award the degree, he wrote to the more liberal University
of Gottingen and requested that the degree be granted in absentia. It was, and one
of the three papers became a classic work in differential equations, published the
following year in the most distinguished German journal, the Journal fur die reine
und angewandte Mathematik.
The next eight years may well be described as Kovalevskaya's wandering in
the intellectual wilderness. She and Vladimir, who had obtained a doctorate in
geology from the University of Jena, returned to Russia; but neither found an
academic position commensurate with their talents. They began to invest in real
estate, in the hope of gaining the independent wealth they would need to pursue
their scientific interests. In 1878 Kovalevskaya gave birth to a daughter, Sof'ya
Vladimirovna Kovalevskaya (1878- 1951). Soon afterward, their investments failed,
and they were forced to declare bankruptcy. Vladimir's life began to unravel at
this point, and Kovalevskaya, knowing that she would have to depend on herself,
reopened her mathematical contacts and began to attend mathematical meetings.
Recognizing the gap in her resume since her dissertation, she asked Weierstrass for
a problem to work on in order to re-establish her credentials. While she was in Paris
in the spring of 1883, Vladimir (back in Russia) committed suicide, leading Sof'ya to
an intense depression that nearly resulted in her own death. When she recovered,
she resumed work on the problem Weierstrass that had given her. Meanwhile,
Weierstrass and his student Gosta Mittag-Lefflcr (1846-1927) collaborated to find
her a teaching position at the newly founded institution in Stockholm.^11 At first
she was Privatdozent, meaning that she was paid a certain amount for each student
she taught. After the first year, she received a regular salary. She was to spend the
last eight years of her life teaching at this institution.
In the mid- 1880s, Kovalevskaya made a second mathematical discovery of pro-
found importance. Mathematical physics is made complicated by the fact that
the differential equations used to describe even simple, idealized cases of physical
laws are extremely difficult to solve. The obstacle consists of two parts. First,
the equations must be reduced to a set of integrals to be evaluated; second, those
integrals must be computed. In many important cases, such as the equations of the
three-body problem, the first is impossible using only algebraic methods. When
it is possible, the second is often impossible using only elementary functions. For
example, the equation of pendulum motion can be reduced to an integral, but that
integral involves the square root of a cubic or quartic polynomial; it is known as
an elliptic integral. The six equations of motion for a rigid body in general cannot
be reduced to integrals at all using only algebraic surfaces. In Kovalevskaya's day
only two special cases were known in which such a reduction was possible, and the
integrals in both cases were elliptic integrals. Only in the case of bodies satisfying
the hypotheses of both of these cases simultaneously were the integrals elementary.
With Weierstrass, however, Kovalevskaya had studied not merely elliptic integrals,
but integrals of completely arbitrary algebraic functions. Such integrals were known
as Abelian integrals after Niels Henrik Abel (1802-1829), the first person to make
significant progress in studying them. She was not daunted by the prospect of
working with such integrals, since she knew that the secret of taming them was to
use the functions known as theta functions, which had been introduced earlier by
Abel and his rival in the creation of elliptic function theory, Carl Gustav Jacobi
(^11) It is now the University of Stockholm.