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The Quarterly Journal of Mechanics and Applied Mathematics 1955 8(4):448-451; doi:10.1093/qjmam/8.4.448
© 1955 by Oxford University Press
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TWO PROPERTIES OF SPHERICAL HARMONICS

HAROLD JEFFREYS

( St. John's College Cambridge )

The integral of the square of the gradient of a solid harmonic over a sphere is evaluated; the corresponding integral for the second derivatives is also evaluated, and the results are applied to an integral that includes the elastic energy in a strained sphere and the rate of dissipation in a viscous sphere.

A natural definition of the irregularity of a function over a sphere leads to the conclusion that the irregularity is stationary for small variations of the function when the function is a surface harmonic and that the irregularity of any function is greater than that of the lowest term in its expansion in surface harmonics.


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