PROFESSOR Ct. it. DARWIX OX ELLIPSOIDAL HARMOXIC AXALYSIS. 
481 
When s = 0, the types are EEC and OEC. The “ middle ” of the series is now 
also an end, and the condition is 
- - 8.2^/3^,- . . . + - 2crP -f- (h 0] - l}P-=] 
+ Aio[F - 2crP-^ + {{, 2]lf, 1}P] + /3hy,[F - 2 (tP^ + {/, 4][/, 3]P^] + ... = 0. 
Writing P^ for [i, 0} [i, — 1} P~“ and equating the coefficients of P and P' to 
zero, we have 
1}{l 2] 
\%] ’ 
Therefore /3cr 
<h _ _ _ J- _ 
?o 4 + /3cr - 3} {fi 4} 
i/3Ch,3}{i-,41 
4.+ /So- - 
4.2" + /Sc 
4.3" + /Scr — ... 
ending with - -E E lu ^ -E ^ 
Next when . 9=1 the types are OOS, EOS ; the ‘‘middle” is again an end, and 
the condition is 
-8.1.2r73P3 -8.2.- . . . + [P3 - ScrP^ + [/, 1} {q 0] P-i] 
+ /8/?3[P^-‘2crP3+[q3}[q2]Pi] + /3-^V5[P'”2c7F+ [L5}[q4]P3]+ ... = 0 . 
Writing ^ [i + 1 ) P^ for [i, 1 ] [i, 0 |- P“^ and equating to zero the coefficients 
of P^ and P^, we have 
+ 1) = i+fi. 3}{i, 2} (E , 
E._1 
9i 
Therefore 
^cr — ^/3i {{ + 1) == 
4.1.2 + /So- - i/ 32 K 5 }{h 4}(-?5 
9-i 
FTo E If 2} _ 
4.1 . 2 + /So- - 4.2 . 3 + /S. 
FEo E {/, G[ 
4.3.4 + /Scr — ... 
— 1/3-[i, i] {i,i — 1] 
- - Elh ' - E 
endine: with — 4 /^ ^ ^ OOS, and witli — - — F ' -—^ for 
i~ + /Scr 
(i — 1)" + /3a 
i/3.) W e liave next to consider the other form of P-function for types EES, 
OOC, EOC, namely. 
where 
P'^ = n [r/'.p-^ + t/3^q\ _ .,.,P^ - + t/3"q', , ,„P^ + , 
i_±J' 4 
Q — I ' _ 1 ^ 
3 Q 
YOL. CXCVII.-A. 
