Ellipsoidal Harmonic Analysis. 



251 



Corresponding to the two forms of P-function there are two forms 



in a series of ordinary Q-f unctions ; but whereas the series for 33/ and 

 P* are terminable, because P ; s vanishes when s is greater than i, this 

 is not the case with the Q-functions. 



In spherical and spheroidal analysis the differential equation 

 satisfied by P 4 * involves the integer s, whereby the order is specified. 

 So here also the differential equations, satisfied by 33/ or P/ and by 

 C*, C ; s ,or Si*, involve a constant; but it is no longer an integer. 

 It seemed convenient to assume s' 2 - fiv as the form for this constant, 

 where s is the known integer specifying the order of harmonic, and 

 <r remains to be determined from the differential equations. 



When the assumed forms for the P-function and for the cosine- and 

 sine-functions are substituted in the differential equations, it is found 

 that, in order to satisfy the equations, f3cr must be equal to the 

 difference between two finite continued fractions, each of which 

 involves f3cr. We thus have an equation for f3<r, and the required root 

 is that which vanishes when j3 vanishes. 



For the harmonics of degrees 0, 1, 2, 3 and for all orders a- may be 

 found rigorously in algebraic form, but for higher degrees the equation 

 can only be solved approximately, unless /3 should have a definite 

 numerical value. 



When /5a- has been determined either rigorously or approximately, 

 the successive coefficients of the series are determinable in such a way 

 that the ratio of each coefficient to the preceding one is expressed by 

 a continued fraction, which is in fact portion of one of the two frac- 

 tions involved in the equation for f3cr. 



Throughout the rest of the paper the greater part of the work is 

 carried out with approximate forms, and, although it would be easy to 

 attain to greater accuracy, it seemed sufficient in the first instance to 

 limit the development to /3 2 . With this limitation the coefficients of 

 the series assume simple forms, and we thus have definite, if approxi- 

 mate, expressions for all the functions which can occur in ellipsoidal 



In rigorous expressions 33* and P/ are essentially different from 

 one another, but in approximate forms, when s is greater than a 

 certain integer dependent on the degree of approximation, the two 

 are the same thing in different shapes, except as to a constant factor. 



The factor whereby P/ is convertible into 33,% and C ; s or S, s into 

 C ( s or §&* are therefore determined up to squares of /3. With the 

 degree of approximation adopted there is no factor for converting the 

 P's when s = 3, 2, 1. Similarly, clown to s = 3 inclusive, the same 

 factor serves for converting C * into C f s and S,' into §2?. But for 

 s = 2, 1, one form is needed for changing C into C, and another 



VOL. LXVIII. T 



of Q-f unction, such that and 



1 - 13 



are expressible 



analysis. 



