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 point of difficulty. Laplace's proof is perhaps the most satisfactory.

Laplace's work on probability is very difficult reading, particularly the part on the method of least squares. The analytical processes are by no means clearly established or free from error. "No one was more sure of giving the result of analytical processes correctly, and no one ever took so little care to point out the various small considerations on which correctness depends" (De Morgan).

Of Laplace's papers on the attraction of ellipsoids, the most important is the one published in 1785, and to a great extent reprinted in the third volume of the Mécanique Céleste. It gives an exhaustive treatment of the general problem of attraction of any ellipsoid upon a particle situated outside or upon its surface. Spherical harmonics, or the so-called "Laplace's coefficients," constitute a powerful analytic engine in the theory of attraction, in electricity, and magnetism. The theory of spherical harmonics for two dimensions had been previously given by Legendre. Laplace failed to make due acknowledgment of this, and there existed, in consequence, between the two great men, "a feeling more than coldness." The potential function, $\scriptstyle{V}$, is much used by Laplace, and is shown by him to satisfy the partial differential equation $\scriptstyle{\frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2}+\frac{\partial^2V}{\partial z^2}=0}$. This is known as Laplace's equation, and was first given by him in the more complicated form which it assumes in polar co-ordinates. The notion of potential was, however, not introduced into analysis by Laplace. The honour of that achievement belongs to Lagrange.[49]

Among the minor discoveries of Laplace are his method of solving equations of the second, third, and fourth degrees, his memoir on singular solutions of differential equations, his