"13th. We find in the works of many geometers results and processes of calculation analogous to those... we... employed. These are particular cases of a general method, which... it became necessary to establish in order to ascertain... the mathematical laws of the distribution of heat. This theory required an analysis... one principal element of which is the... expression of separate functions [f(x)], or of parts of functions... f(x) which has values existing when... x is included between given limits, and whose value is always nothing, if the variable is not included between those limits. This function measures the ordinate of a line which includes a finite arc of arbitrary form and coincides with the axis of abscissae in all the rest of its course. This motion is not opposed to the general principles of analysis; we might even find... first traces... in the writings of Daniel Bernouilli...Cauchy...Lagrange and Euler. It had always been regarded as manifestly impossible to express in a series of sines of multiple arcs, or at least in a trigonometric , a function which has no existing values unless the values of the variable are included between certain limits, all the other values of the function being nul. But this point of analysis is fully cleared up, and it remains incontestable that separate functions, or parts of functions, are exactly expressed by trigonometric convergent series, or by definite integrals. We have insisted on this... since we are not concerned... with an abstract and isolated problem, but with a primary consideration intimately connected with the most useful and extensive considerations. Nothing has appeared to us more suitable than geometrical constructions to demonstrate the truth of these new results, and to render intelligible the forms which analysis employs far their expression."