# The New International Encyclopædia/Sturm, Jacques Charles François

**STURM**, stụrm, Jacques Charles François (1803-55). A French mathematician, born at Geneva. He was educated at the Academy of Geneva, and in 1827, with his friend Colladou, took the *Grand prix de* *mathématiques* for the best memoir on the compression of liquids. The famous theorem that bears his name was discovered in 1829. A statement of the results secured by this theorem requires the definition of Sturm's functions: If *f*(*x*) = 0 be freed from equal roots, and *f*(*x*) be divided by *f'*(*x*) (the derivative of *f*(*x*)), and the last divisor by the last remainder, changing the sign of each remainder before dividing by it, until a remainder independent of *x* is obtained, or else a remainder which cannot change its sign, then *f*(*x*), *f'*(*x*), and the successive remainders, constitute Sturm's functions. The theorem asserts that if, as *x* increases, *f*(*x*) passes through the value zero, Sturm's functions lose one change of sign; if any other of Sturm's functions vanishes, there is neither loss nor gain in the number of changes of sign; the number of roots of *f*(*x*) = 0 between *a* and *b* is equal to the difference in the number of changes of sign in Sturm's functions, when *x* = *a* and when *x* = *b*. In 1838 Sturm began teaching in the Ecole Polytechnique, and two years later was elected to the chair made vacant by the death of Poisson.