P. HISTORY OF ' AATHEMATICAL - School of Mathematics

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240 A HISTORY OF MATHEMATICAL NOTATIONS
Ernest Pfannenstiel,' in 1882, wrote D;z, D;.uZ, etc., to represent partial
derivatives in a paper on partial-differential equations.
Duhamel- in ordinary cases used no distinctive device for partial
differentiation, but when many variables and derivatives of any
orders were involved, he designated "par F':':':~~: .o. (x, y) ou par
D::':~~:'" u le resultat de m derivations .partielles effectuees par
rapport a x sur Ill, fonction u=F(x, y), suivies de n derivations partielles
du resultat par rapport ay, lesquelles seront elles-memes suivies
de p derivations par rapport ax; et ainsi de suite." For partial differentials
he wrote similarly, d::':~~:" u. It will be observed that,
strictly speaking, there is nothing in this notation to distinguish
sharply between partial and total operations; cases are conceivable in
which each of the m+n+p .... differentiations might be total.
616. G. S. Carr?' proposed "experimentally" a notation corresponding
to that for total derivation; he would write a::~3 in the contracted
form U:!z311'
617. Ch. Meray.-An obvious extension of the notation used by
Duhamel was employed by Ch. Mcray.' Representing by p, q, .... ,
the orders of the partial derivatives with respect to the independent
variables x, y, .... , respectively, and by the sum p+q+ .... the
total order of the derivatives, he suggests two forms, either
(P. Q•••••J (x
f y ..•.) D(P.Q, 0" ')j( )
:t.y,.... " or Z,II,... x, y, ...••
Applying this notation to partial differentials, Mcray writes,
d(p'Q'·.. ')j=j(p,Q, .. 0 0) (z Y
) dx p dyQ
x, y,.... x, y, ... . " • . • • • ••••
Meray's notation is neither elegant nor typographically desirable.
It has failed of general adoption. In the Encyclopedie des sciences
mathematiques 5 J. Molk prefers an extension of the Legendre-Jacobi
. aku
notation; thus ax p iJyQ ••• .' where k=p+q+ . . . . .
I Societat der Wissenschaften zu Upsala (Sept. 2, 1882).
2 J. M. C. Duhamel, Elements de calcul Infinitesimal, Tome I (3d ed.; Paris,
1874), p. 267.
3 G. S. Carr, Synopsis . . . . of the Pure Mathematic.., (London, 1886), p. 266,
267.
• Ch. Meray, Le~ons 7WlWelles sur l'analyse infiniMsimale, Vol. I (Paris,1894),
p.I23.
~ Tome II, Vol. I, p, 296.