PROFESSOR O. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
55'J 
given 6- a p/-fuiictioii is associated with a cosine-function, the corresponding P- is 
associated A^atli a sine-function, and vice versa. 
Lastly, a ^^-finiction 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 fall into two forms. In the^ first form s and i 
are either both even or both odd, and the function, which T write (!]:/ or Sh is 
expressed by a series of terms consisting of a coefiicient multiplied by ^8'^' cos or 
sin [s ± 2/i')^. In the second form s and i difter as to evenness and oddness, and the 
function, written C/ or Sf, is expressed by a similar series multiplied by 
(1 — ^ cos 2(/))'-. 
The combination of the two forms of P-function with the four forms of cosine- and 
sine-function gives the eiglit types of solid harmonic. 
Corresponding to the two forms of P-function there are two forms of Q-h^nction, 
sucli that and Q/ A/ x—expansilde in a series of ordinary Q-functions ; 
but whereas the series for P* terminable, because P* vanislies when s is 
greater than I, this is not the case with the Q-functions. In fact the series f<>r tlie 
(^)-functions begins with Q,- or Qf, and the order of the Q’s increases by two at a time 
up to s vlien we liave the principal or central term ; it then goes on increasing up to 
= i or i — 1, and on to infinity. 
In spherical and spheroidal analysis the difiei'ential equation satisfied l)y P- 
involves the integer s, whereby tlie order is specified. So here also the differential 
equations, satisfied by - or P* and by (£■, or C/, Sk involve a constant, but it 
is no longer an integer. It seemed convenient to assume as the form for 
this constant, where s is tlie knoAvn integer specifying the order of harmonic, and 
(T remains to be determined from the differential equations. 
AVlien the assumed forms for the P-function and for the cosine- and sine-fnnctio]is 
are substituted in tlte difierential equations, it is found (§ G) that, in order to satisfy 
the equations, /So- must be eipial to the difference between two finite-continued 
fi-actions, each of whicli involves /Str. \Ye thus have au equation for y8cr, and tlie 
required root is that wliicli vanislies when /3 vanishes. 
For the harmonics of degrees 0, 1, 2, ;>, and for all orders, cr may lie found 
rigorously in algebraic form, but for higher degrees the equation ca.n only be solved 
approximately, unless ^ should have a definite numerical value. 
When /3cr has been determined, either rigorously or approximately, the successive 
coefficients of the series are detei'ininable in such a vary that the ratio of each 
coefficient to the preceding one is expressed by a continued fraction, v liich is, in facf, 
a portion of one of the two fractions involved in the equation for ^cr. 
’fliroughout tlie rest of the paper the greater part of the work is carried out 
with approximate forms, and, although it would he easy to attain to greater 
accuracy, I have thought it sufficient, in the first instance, to stop at /3''. With this 
