504 riiOFEWSOIi (4. ]l. J)AI44VIX OX ELLIPSOIDAL IIAEMOAIC AXALASIS. 
When .S' — 1 we have 
V .-2 = 0 , f/, + 2 = -,V[l - YV/3^■(^■ + 1 )], 50+i = yk 
7., _,i = 0, 7,5-2 = h, 7,^0 = Py[ 1 + + 1)]> ?'« + 4 = Yes' 
The terms in ^<ls + -2 /Bq's + .j now contribute to tlie terms in /Sn 
For the term in inside the bracket is 
• 8 “i [(^' ~ ^)(^' ~ ^) + (^' + 2) (?. + 3)] = + l) + 4], 
The term In (i~, of which the first portion is cai'ried over from the term in /S, is 
“ d- - 1)(^‘ - 2) + (f + 2)(i + 3)] 
d~ T 7 s’ 7 Ty[(^' ~ f)(' “■ 2)(i ~ 3)(i — 4) + (i + 'l){i + 3) (/- + 4) (^ + 5) 
-p ; 3 (/ _ !)(/ _ 2 )(i + 2)(f + 3)]. 
This is efpial to — yo-g [^~(^' + i )" ~ 53^ (^' H“ U “ 180]. 
As we camiot now use tlie al)ridg’e(l notation with Xd, wliicli is infinite. I write 
j — + ^)- 'j 
V A 1 , ^.w .. ... ..... r • • (-^3). 
Tims 
C’ = - .3Y4[1 + J/3(7 + 4) - - 50/ - 180)] I 
For Ed tlte coefitcient of is tliree times as great as l:)efore, and the coefficient of 
p- is 
tI8 ^ + 1)D^'('' + 1) + 4] + Ty8~2T3[5(*' ~ 1)(^' “ 2) (f ~ 3)(i ~ 4) 
+ 5 (f + 2) {i + 3) {i + 4) {i -f 5) + 27 (f - 1) {i - 2) {i + 2) (/ + 3)]. 
On effecting tlie reduction 1 find 
c_> 
■/ + 1 ! 
■/ - 1 ! 
[1 + 8 ^(./ + 4) + yd's 
+ 3707 + 1044)]. . (40). 
When s ~ 0 we have only to determine. Here 
7 , _4 = 7»-2 = 0, 7 ,,+., = i 7 ,+ 4 = yds. 
The term in /3 is 4[f (f — 1) + (f + l) (i + 2)] = 4 {j + 1). 
That in is 
~ l)(^' ~ 2) (f — 3) + (i + I)('i + 2)(< + 3) (■? + 4) 
+ 8'/(f — 1) (f + 1)(^ + 2)] =: -d4(57' + 14y + 12). 
