47-’ rUOFESSOli G. H. DAKAVIX OX ELLIPSOIDAL HAPMOXIC AXALYSIS. 
Type B, or L)(JS ; since we now have cos (t — 1) cf) sin </), 
-- - y/ «in G/>. 
1 
Tv])e (J. or ()E ('; C' = ^ y/ cos (/ — 1) d>. 
Type ABO, or OES ; each term is of tyj)e <5^ cos (t — l)<T')sin cf) cos wliich gives 
[sin (t + ]) f/> — sin (f — 3) (f] cp. Hence 
S' = ^ yf sin (O— 1 ) 9 - 
Tt is clear that ^ve may equally ^vell regard the lower limit as unity. 
Type CA, or EOC-; each'term is of type cos (f — 2) 6 cos 6. Hence 
C' = *Sy/ cos(^ - 
Type ( T>, or EO 8 ; each term is of type (I) cos (f. — 2) f/)sin (/). Hence 
S' = E y,. sin (^ — l) fj). 
AVhen i and s agree as to evenness or oddness we have the forms independent of cp^ 
when they differ in this respect the fiictor <E> occurs. 
Therefore (in alternative fbrinl for PlECs EES, OOCf f)OS 
tf' fees , , ^ - 
^ ■ I sin 
and for oEC’, OES, EOC, EOS 
sin ~ + 
( 0 - 
c 
■ S' 
= (Ij> 
+ E AV>'< - 2 ,- j (-^ -2n)cp + I (s + 2»)<f> 
sin ' |_sin ■ • [sill ■ ' ^ 
In these series n jn'oceeds hy intervals of one at a time, beginning with unity. In 
both forms the iqiper limit of tlie first E is As or T (s — I) according as .v is even or odd. 
In the first form the upper limit of tlie second E is — .s), and in the second form 
it is T (' — -s — 1). 
A"e shall ultimately put and p',. wliicli may he regarded as arbitrary constants, 
eipial to unity. 
^.1. Preparaiion for Je(c)‘}!iination offhe Ftincfionr. 
In order to determine the coefficients q, (/, p. p' and cr, Ave liave to substitute tliese 
assumed forms in the differential equations. 
A here the functions involve O and tp as factors, tlie forms already o-iven for the 
7 ^ O 
