The nature of the correction which is the subject of the present paper, and of not infrequent application in experimental inquiry, will be best understood from an example, as it is a little difficult to state with full generality. Suppose that our object is to determine the distribution of heat in the spectrum of the sun or any other source of light. A line thermopile would be placed in the path of the light, and the deflection of the galvanometer noted for a series of positions. But the observations obtained in this way are not sharp—that is, they do not correspond to definite values of the wave-length or refractive index. In the first place, the spectrum cannot be absolutely pure; at each point there is a certain admixture of neighbouring rays. Further, even if the spectrum were pure, it would still be impossible to operate with a mathematical line of it; so that the result, instead of belonging to a simple definite value of the independent variable, is really a kind of average corresponding to values grouped together in a small cluster.

For the sake of simplicity, let us suppose that the spectrum is originally pure, and that the true curve giving the relations between the two quantities is PQR. Also let MN be the range over which the independent variable changes in each observation—in our case the width of the thermopile.