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NOTE ON A THEOEEM OF PH. GILBEET'S EEGAEDING 
THE DIFFEEENTIATION OF A SPECIAL JACOBIAN. 
By Thomas Muik, LL.D. 
(Eeceived November 5, 1913.) 
1. The theorem in question is the subject of a memoir presented to the 
Belgian Academy in 1869." As stated by its author, it may be formulated 
as follows : i/fj, fa, f^ he functions of u^, Ug, u„, and if the latter be 
implicit f unctions of ttoo sets of independent variables 
Uj, tta, a,j 
and be such that 
tohere each f is an explicit function of the u's, then 
A I -,/ .) I ^ 1 1 ...,/,.) ) _ f„ 0.) 
'dXi \ c)(ai, ttg, a^) ' ^^ttj i c)(ai, a^, •••^a^) ) ^(ai> a^i' ^i) 
2. For the sake of clearness, and, indeed, of accuracy, it would have 
been better not to have used in the concluding equation the same sub- 
scripts fj, and i as are used in the hypothetical equation, unless an indica- 
tion were given on both occasions as to the range which each of them 
was understood to cover. Doubtless what Gilbert had in his mind could 
have been expressed by annexing to the hypothetical equation the 
explanation 
/x=l, 2, n ^ 
i = 1, 2, n ] 
* Sur une propriete des determinants fonctionnels, et son application au developpe- 
ment des fonctions implicites. Nouv. M&m. de I' Acad, roy. de Belgique, xxxviii, 
pp. 1-12. 
