4G2 
PROFESSOR G, H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
IXTRODUCTIOX. 
Lx4Me’s tuiictioiis or ellipsoidal liarmoiiics have been successfully used in many 
investigations, but the form in which they have been presented has always been 
such as to render numerical calculation so difficult as to he })ractically impossible. 
The object of the present investigation is to remove this imperfection in the method. 
I believe that I have now reduced these functions to such a form that numerical 
I'esults will l^e accessible, although Ijy the nature of the case the arithmetic will 
necessaiily remain tedious. 
Throughout my work on ellipsoidal harmonics I have enjoyed the immense 
advantage of frequent discussions with Mr. E. W. Hobson. He has helped me 
freely from his great store of knowledge, and beginning, as I did, in almost complete 
ignorance of the sul:)ject, I could hardly have brought my attempt to a successful 
issue without his advice. In inany cases the help derived from him has been of 
immense value, even where it is not possible to indicate a specific point as due to him. 
In other cases he has put me in the way of giving succinct proofs of propositions 
which I had only proved by clumsy and tedious methods, or where I merely felt sure 
of the truth of a result without ligorous proof In particular, I should have been 
(|uite unable to carry out the investigation of § 19, unless he had shown me how the 
needed series were to be determined. 
My original object in attacking this problem was the hope of being thereby enabled 
to obtain exact numerical results with inspect to M. Poincare’s pear-shaped figure 
of equilibrium of a mass of liquid in rotation.^'" But I soon found that a partial inves¬ 
tigation with one particular point in view was impracticable, and I was thus led on 
little by little to cover the whole field, in as far as it was necessary to do so for the 
purpose of practical application. This paper has then grown to such considerable 
dimensions that it seemed best that it should stand by itself, and that the discussion 
of the specific problem should be deferred. 
A paper (.)f this kind is hardly I'ead even by tlie mathematician, unless he happens 
to be working at a cognate subject. It appears therefore to be useful to present a 
summary, which shall render it possible for the mathematical reader to understand the 
]iature of the method and results, without having to pick it out from a long and 
conqdex train of analysis. Such a summary is given in Part HI. 
PAPT I. 
EoRMxVI'ION of the Eunctions. 
j 1. TJte Pflnciples of Ellipsoidal Harmonic Ancdysis. 
The basis of this method of analysis is expounded in various works on the subject. 
I begin with a statement of results in m}^ own notation. 
* A paper giving the recpiired result will he presented to the Society in the autumn. — \Juhj 2, 1901.] 
