L 461 J 
XII. Ellipsoidal Harmonic Analysis. 
Bjj G. H, Daewin, F.R.S., Plumian Professor a7id Fellow of Irmity College, in the 
University of Cambridge. 
Received March 23,—Read May 2, 1901. 
Introduction 
Table of Contents. 
Rage 
162 
Part I. —Formation' of the Functions. 
§ 1. The principles of ellipsoidal harmonic analysis.462 
§ 2. Xotation ; limits of ^ so as to represent all ellipsoids. 463 
§ 3. The ditierential ecpiations.466 
§ 4. The forms of the functions. 469 
§ 5. Preparation for determination of the functions.472 
§ 6. Determination of the coefficients in the fmictions.477 
§ 7. Rigorous determination of the functions of the second degree.488 
§ 8. Approximate form of the functions.489 
^ 9. Factors of transformation between the two forms of P-function and of C- or S-function . . 492 
§10. The functions of the second kind.497 
Part 11. Aitlicathjn uf Ellipsoidal Harmonic Analysis. 
§11. The potential of an harmonic deformation of an ellijisoid.oOb 
§12. The potential of a homogeneous solid ellipsoid.508 
§ 13. Preparation for the integration of the squ ire of a surface harmonic over the ellipsoid . . . 514 
§ 14. Integration in the general case.516 
§ 15. Integration in the case of s = 2.523 
§16. Portion of the integration in the case of s = 1.526 
§17. Portion of the integration in the case of s = 0.529 
§ 18. Pi'cjiaration for the integrations when s = 1 and 0.530 
§19. Evaluation of the integrals jcos-"dAdf^ .533 
§ 20. Reduction of preceding integrals; disappearance of logarithmic terms.538 
§21. Integrals of the scj^uares of harmonics when s = 1 and s = 0 ..." .541 
§ 22. Table of integrals of squares of liarmonics.547 
Part III. 
Summary ■.549 
Appendix. Taljle of functions.551 
(298) 
7.12.1901 
