P. HISTORY OF ' AATHEMATICAL - School of Mathematics

DIFFERENTIAL CALCULUS
229
represents ((Euvres, Vol. IX, p. 158), by F'(x), F'(y), F'(z) the primes
jonctions of F(x, y, z) taken with respect to x, y, z, respectively, as independent
variables. The danger of this notation is that F'(x) might
be taken to be the primitive F' function of the single variable x.
Lagrange also (ibid., p. 177) denotes by u', Ul, and lU the prime functions
(first derivatives) of u with respect. to z, y, z, respectively.
Furthermore (ibid., p. 273), u", Un, uf stand for our ::~, :~, a:~y'
respectively. In 1841 C. G. J. Jacobi declared that "the notation
of Lagrange fails, if in a function of three or more variables one
attempts to write down differentials of higher than the first order."!
But Stackel 2 remarked in 1896 that the Lagrangian defects can be
readily removed by the use of several indices, as, for example, by
aa+fJ+-rj
writing jath for our a
x a y
fJa
z-r
.
The historical data which we have gathered show that as early
as 1760 Karsten had evolved a precise notation even for higher
partial derivatives. It was capable of improvement by a more compact
placing of the symbols. But this notation remained unnoticed.
That Lagrange was not altogether satisfied with his notation of
1797 appears from his Resolution des equations numeriquee (1798),
page 210, where he . introduces (Z') a' , (Z') V, (Z") a'2 , (Z" a'b' ) to designate .
. I deri . az ez a 2z a 2 Z d h "C
the partia erivatives aa' ab' aa 2 ' aa ab' an t en says: ette notation
est plus nette et plus expressive que celIe que I'ai employee dans la
Theorie des jonctions, en placant les accens differemment, suivant les
differentes variables auxquelles ils se rapportent." Lagrange adds the
further comment: "In substituting the former in place of this, the
algorithm of derived functions conserved all the advantages of the
differential calculus, and will have this additional one of disencumbering
the formulas of that multitude of d's which lengthen and
disfigure them to a considerable extent, and which continually recall
to mind the false notion of the infinitely little."
It will be noticed that Lagrange's notation for partial derivatives,
as given in 1798 in his Resolution des Equations numerique,and again
in his Leeone sur le calcul desjonctions (Paris, 1806), page 347,3 closely
1 C. G. J. Jacobi, "De determinantibus functionalibus," Journal far d. r. & a.
Mathematik, Vol. XXII, p, 321; Jacobi, Ges. Werke, Vol. III, p. 397.
2 P. Stackel in Ostwald's Klassiker der ezakten Wiss., No. 78, p. 65.
3 J. Lagrange, (EuvreB, Vol. X.