P. HISTORY OF ' AATHEMATICAL - School of Mathematics

DIFFERENTIAL CALCULUS
231
cannot be here distinctly pointed out, but it would be difficult, I
apprehend, to introduce a similar and corresponding change in the
fluxionary notation: and this is an additional reason, why I have
made a deviation, for which the English reader may feel a propensity
to blame me."
604. A. L. Crelle.-In this restless period in the development of
the calculus, there were many minds, and divided councils. We have
seen that new notations were suggested by Grelle,I who in partial
derivatives has a notation resembling Euler's of 1776; Grelle states
d 'd d
(p. 72): }'When Z= IxYI, then we let -z or -lxyl, or - fxy stand for the
x x x
first partial derivation according to x of the magnitude IxYI, while
~ z or ~ Ixyl or ~ fxy stands for the partial derivation according to y,
y y y ,
of the magnitude Ixyl." Similarly, ~ z or ~fxy or ~ Ixyl indicates
x x x
the second partial derivative with respect to x, while d2 z represents
xy
the partial derivative with respect to z, followed by that with respect
to y. It is to benoted that with Grelle d does not signify "differential,"
but abgeleitete Grosse, which Lagrange called the "derived function"
and Arbogast the "derivation." The partial is distinguished from the
total derivative, by having the independent variable with respect to
which the differentiation takes place written below the d. Thus dz
means the total derivative, ~ z means the partial. On page 82 he
x
introduces also the D for the total derivative.
605. P. Barlow.-A quaint notation was given in England in
1814 by Peter Barlow.t If u is any function of x, y, z, etc., let ~, d, D,
etc., be the characteristics of their differentiation relatively to each of
the variables, then the partial differences are shown here in the
second and third lines,
x, v , z, etc.,
~
dx'
~~u
dx2 ,
du
dy'
d~u
dxdy'
Du
(ii'
D~u
dxdz'
etc.,
etc.
1 A. L. Crelle, Rechnungmit veranderlichcn GriJsscn, Vol. I (Gottingen, 1813).
• P. Barlow, Dictionary oj Mathematics (1814), art. "Partial Differences."