PARTICULARLY THOSE OE TWO VARIABLES. 
801 
and from these the following expressions for the ratios of the c’s are easily obtained : 
r 4 
G l 
o Q 
*1 *8 
K 1 K 3 
1 
05 
II 
IV 
c o 4 
“ / Q 
a: 
/j 4 
C 0 
4*^ 
c ^ 
C 2 
%v 3 3 k 3 * 
c 4 
W 
4 
ns 
ii 
c o 
c o 
' «3 2 K 3 3 
c ^ 
*iV s *K 3 3 
c 4 
z& _ 
/c 1 3 « , 1 2 K 1 3 
0 4 
C 15 
k, 2 V 
li 
4 3 K S 3 
r 4“ 
c o 
/CgVg 2 Kg 3 
/. 4 — 
C 0 
k £ k '£ 
If now these be substituted in the equations of which (41) are a type, a set of 
algebraical identities is obtained; in fact, putting /c l 3 =<x, K. 2 2 =b, k 3 3 =c, a , b, c being 
perfectly independent 
b(a— c)=a('l— c)(b—c)-\-ac(ci— c)+c(t— ct)(a—b) 
(b—c)(a—b)=ac(l — b)-\-(l — a)(l—c)b—b(l—b) 
and from these 
a(l— c)(b— c)(l--6)+c(l— a)(l — b)(a— &)+&(!—c)(l — a)(c— 
= {a—b) (b—c) (c—a) 
and many others similar to these, all of which admit of immediate verification. 
14. Two other equations, which will afterwards be useful, are 
== ^10’^lAAo l ®8^lAAl"l"^2^3 1 ^6^7 *^0^1 ^4^5 
=2(VsVj-WA) by ( xxix ).(49) 
f)—■»is(*+f)'9o(*— £)} 
= $ O S 5 0O&O ~i~ ^8^1S^8^1S 
=2(.J o 3 5 0 o 0 5 —* r >o-J 7 0 2 .#j) by (xxviii).(50). 
Connexion with the hyperelliptic integrals. 
15. Taking the fifteen ratios obtained by dividing all the functions but one by 
that one, it follows from the relations already established as (31), (32), (33), . . . that 
any thirteen of them can be expressed in terms of the remaining two, or that all these 
5 K % 
