250 



Prof. G. H. Darwin. 



ellipsoids are comprised in either of these types by making /3 vary 

 from zero to infinity. But it is shown that, by a proper choice of type, 

 all possible ellipsoids are comprised in a range of f3 from zero to one- 

 third. When /3 is zero we have the spheroids for which harmonic 

 analysis already exists, and when ft is equal to one-third the ellipsoid 

 is such that the mean axis is the square root of mean square of the 

 extreme axes. We may then regard /3 as essentially not greater than 

 one-third, and may conveniently make developments in powers of f3. 



In spheroidal analj^sis, for space internal to an ellipsoid v , two of 

 the three functions are the same P-functions that occurs in spherical 

 analysis ; one P being a function of v, the other of The third 

 function is a cosine or sine of a multiple of the longitude <£. For 

 external space the P-function of v is replaced by a Q-fmiction, being 

 a solution of the differential equation of the second kind. 



The like is true in ellipsoidal analysis, and we have P- and Q-f unc- 

 tions of v for internal and external space, a P-function of /x, and a 

 cosine- or sine-function of <j>. Por the moment we will only consider 

 the P-functions, and will consider the Q-f unctions later. 



There are eight cases which are determined by the evenness or 

 oddness of the degree i and of the order s of the harmonic, and by 

 the alternative of whether they correspond with a cosine- or sine- 

 function of <f>. These eight types are indicated by the initials E, O r 

 C, or S ; for example, EOS means the type in which i is even, s is odd r 

 and that there is association with a sine-function. 



It appears that the new P-functions have two forms. The first form, 

 written ^, is found to be expressible in a finite series in terms of 

 p?±2fc w h en the P's are ordinary functions of spherical analysis. The 

 terms in this series are arranged in powers of ,6, so that the coefficient 

 of P/* 27 "' has p k as part of its coefficient. The second form, written 



P/, is such that aA^V P 'W° r a/ ]+5 W \ is 



expressible by a series of the same form as that for . Amongst 

 the eight types four involve ^-functions and four P-functions ; and if 

 for given s a ^/-function is associated with a cosine-function, the 

 corresponding P t is associated with a sine-function, and vice versa. 



Lastly, a ^-function of v is always associated with a ^-function of 

 /x ; and the like is true of the P's. 



Again, the cosine- and sine-functions have two forms. In the first 

 form 5 and i are either both odd or both even, and the function written 

 C* or is expressed by a series of terms consisting of a coefficient 

 multiplied by /3 k cos or sin (s ± 2k)<f>. In the second form, s and i 

 differ as to evenness and oddness, and the function written CV or S' 

 is expressed by a similar series multiplied by (1-/5 cos 2<£)i. 



The combination of the two forms of P-function with the four forms 

 of cosine- and sine-function gives the eight types of harmonic. 



