PEOFESSOE G. H. DAEWIX OX ELLIPSOIDAL IIAEMOXIC AXALYSiS. 
54-i 
tSubstitiitiiig for / (whicli the reader must not confuse with the functional / in use 
liere) its value (80), the term of order zero is 2 y,,._ 2 /(/’+!)• By (02) this is equal 
/+i: 
t<.) 
2/ + 1 i-l\' 
'fhe tei-m of the tirst order in /3 is 
/32r,.-,/('-+i)( )+/3yoy\-’)(-i+Ti- 
which niav he reduced to tlie form 
2/3 i + i: 
^Sy2,_,.y‘(/-+l)+h/3%-',-2/(^’), and is e(|ual to + 
1 1 J. ^ — 1 ; 1 
Tlie tei-m of the second order in /3 is 
4r + 5 3t2r + i)/ 
This mav he reduced to the form 
1 
4,- ( ■■:'0^)+A'''+hj 
+^Vo./(-) [~H+f(f d-o^,y+t)]- 
of which the tirst term is 
4'herefoi'e 
iSy.,1 )+A (./+13) Sy,,-. O'AA 
1 ' I 
i i+i: 
2/+1 i-i:" 
— (1 + /^ + i/3') 2 ij^-\ / _ 1 '+[ 1 + i/S (./ +1 -0] (»') — To- 
Now 
and 
y2r-2/( >•) = ( — )' 
+ j 
(At- I I ~h • 
(’ — 1 ! /■ + ! ! /•! (’—/■! ' 
(_)>-+i (/AlH’ G’+l)(i + 2)i((' —1) 
ir+t:r:/-r' 112 
= 1-/’(/+h -0 -h !)• 
+ • • ■ 
It is knoAvn that 
I^{a, h, C-, 1 ) 
Then since 11(—() contains an intinite factor 
1J(J >11(0) 
(93) 
A'(/+L -/,2. 1) 
therefore 
U{-i)n(i+]} 
i /> V i “1“ ■ 
: 0 . 
, i-i : 
* I have again tu tliank Mr. Hobson for this formula, whieh is due to GaI'ss. 
(03). 
