## Extract

1. The object of this note is to show how ellipsoidal harmonics of the fourth, fifth, sixth, and seventh degrees may be calculated. Some of the fourth and fifth degrees are easily found, depending as they do upon the solution of a quadratic equation. When, however, the type of the harmonic of the fourth degree is Θ_{1} Θ_{2} where, if we employ Greek letters for current co-ordinates, Θ_{1} = *ξ*^{2}/*a*^{2} + *θ*_{1} + *η*^{2}/*b*^{2} + *θ*_{1} + *ζ*^{2}/*c*^{2} + *θ*_{1} – 1, Θ_{2} = *ξ*^{2}/*a*^{2} + *θ*_{2} + *η*^{2}/*b*^{2} + *θ* _{2} + *ζ*^{2}/*e*^{2} + *θ*_{2} – 1, and, in order that Θ_{1} Θ_{2} may satisfy Laplace’s equation, the quantities *θ*_{1} *θ*_{2} are to be found from 1/*a*^{2} + *θ*_{1} + 1/*b*^{2} + *θ*_{1} + 1/*c*^{2} + *θ*_{1} + 4/*θ*_{1} + *θ*_{2} = 0, (1) 1/*a*^{2} + *θ*_{2} + 1/*b*^{2} + *θ*_{2} + 1/*c*^{2} + *θ*_{2} + 4/*θ*_{2} + *θ*_{1} = 0, (2) the elimination of *θ*_{2} leads to a sextic equation in *θ*_{1}. Since, however, the roots occur in pairs as in (1) and (2), if we put *θ*_{1} + *θ*_{2} = 2*u*, *θ*_{1} + *θ*_{2} = 2*v*, the equations for *u* and *v* derived from (1) and (2) will be of lower degrees than the sixth.

## Footnotes

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- Received March 22, 1906.

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