492 
PROFESSOE G. TI. DAEWIX OX PE.LIPSOIDAL HAEMOXIC AXALYSIS. 
Therefore for OOS, EOS 
7i 
— /o “t“ 1)]> — 3V 5 — 25V0 
For 000, EOC 
*L\ — ~ 4 V vh -} {h 3] [1 + + 1)]> — 9~6 j — 7 (3'Vo • 
For 000, OOS, with upper sign for cosine and lower for sine, 
/'i = itg {h ± + 1)] y 2^-0 — ~ sVih > 
'P~, ’25Go{h ^l{h hj . 
For EOO, EOS, with upper sign for cosine and lower for sine, 
Vi = -iV {h [1 dz -f 1)] , iV = - 3T [h 5] , 
n ' —_i-_ s j o');; T') 
7h — i 5 0 0 (.h ^ J Ih O 
) (30). 
It will save much trouble to note that if we were to admit neo-ative suffixes to the 
p’s, the general formula would give us the term /3'p_i F~\ where 
12 s. 2.1 
Thus this term is ‘ exactly that part of 
the term in (30) which arises from ydp^P^, but which is not included in the general 
formula. 
Similarly the general formula gives for p'_|, 2^-i^ those parts of the terms 
arising from p/, 7>i, p/ which are not included in the general formula. 
It follows that in much of the subsecpient work we need not devote special 
consideration to the case of s = 3. 
§ 9. Factors of I'ransfrrrnation hetiveen the two forms of P function and 
C~ or Sfunction. 
The rigorous exiDressions and P' always differ from one another, but a])proxi- 
rnately they are the same up to a certain power of yd, })rovided that .9 is greater than 
a certain (piantity. 
/r- — i ' 2 I3 f ,,, , 
Since n = ( pn- --— j “ ( ^ _ I) —"P)/ ’ is legitimate to develop Q in powers 
of 1 '(w — 1) up to a certain power, say t, provided that it is to be multiplied bv a 
fanction involving at least {v- — ])'' as a factor; for this condition insures that there 
shall be no infinite terms when 1. At present, 1 limit the development to 
so that 
yS + /3- ip- 
V- -1 {V-l)-' 
n = i 
