On the Variation of a Differential Coefficient, Sc. 415 
Arr. XLIV.—On the Variation of a Differential Coefficient of 
a Function of any number of Variables; by A. D. Sranvey, 
Professor of Mathematics in Yale College. 
Waren a quantity is a function of a single variable, the varia- 
tions of its differential coefficients are, as is well known, easily 
determined. But the variations of differential coefficients of a 
function of éwo variables have not been ascertained without diffi- 
culty. Lagrange, the inventor of the Calculus of Variations, 
never obtained them by a method duly general. Poisson was 
the first to do this ; which however has since been done in a some- 
what simpler manner by Mr. Ostrogradsky. But for the general 
case in which we have any number of independent variables, or 
even for the third case in point of complexity, viz. that in which 
three variables are concerned, the variations in the differential co- 
efficients of a function have not been determined by either of these 
ciple in algebraic language, 6dV=ddV. (‘That this statement is 
inadmissible, we shall, at another time, attempt to show.) There 
is however-an article on the Method of Variations by Mr. Pagani, 
published in the fifteenth volume of Crelle’s Mathematical me 2 
We use them in a manner considerably different from his. Ef 
the other method is essentially different, and is much more sim- 
Ple and concise. Still as the problem concerned is of funda- 
mental importance in the Calculus of Variations, it may not be 
oe to state the former as well as the latter method of so- 
ution, 
