New Formula for solving the Problem of Interpolation. 459 
are warmed by means of vibrations, and we receive an electric 
shock by the sudden vibrations excited in the elastic fluid 
essentially belonging to our own bodies. 
LXXVIII. On a New Formula for solving the Problem of 
Interpolation in a Manner applicable to Physical Investiga- 
tions. By M. Caucuy.* 
[NX the application of analysis to geometry, physics, and 
astronomy, the questions which present themselves for 
solution are of two kinds. First, it is required to find the 
general laws of the figures or the phenomena, that is, the 
general form of the equations which exist among the different 
variables: for instance, between the coordinates of curves and 
their surfaces; between the velocities, the times, and the spaces 
described by bodies in motion, &c.: secondly, to determine 
the numerical values of the arbitrary constant quantities which 
enter into the expression of these laws, that is, the values of 
the unknown coefficients contained in the equations. Among 
the variables we usually distinguish, as is well known, those 
which may vary independently of one another, and are there- 
fore called independent variables, from those which are de- 
rived from them by the resolution of the several equations, 
and which are named functions of the independent variables. 
Let us consider a particular function of the latter kind, and 
suppose that it is derived from the independent variables by 
means of an equation or formula which contains a certain 
number of coefficients. An equal number of observations or 
experiments, each of which will afford a particular value 
of the function answering to a particular system of values 
of the independent variables, will be sufficient to enable us to 
determine the numerical values of all these coefficients ; and, 
these values being determined, we may easily obtain such, new 
values of the function as will correspond with new systems of 
values of the independent variables, and thus solve that which 
is called the problem of interpolation. If, for example, the 
ordinate of a curve be expressed as a function of the abscissa 
by means of an equation containing three coefficients, it will 
be sufficient to know three points of the curve, that is to say, 
three particular values of the ordinate corresponding with 
three particular values of the abscissa, in order to determine 
the three coefficients. ‘When these are determined the curve 
may be easily traced by points, if we calculate the coordinates 
of so many points in the arcs of the curve lying between the 
given points, as we wish to ascertain. 
* Translated from a lithograph circulated by tke Author. 
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