ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
367 
the suffixes being added to the a’s and b’s under the radical sign so as to have 1, 2, 3, 
4 5 for the complete system under any one root-sign. Then 
and therefore 
Again 
and therefore 
al t al rtS =— 1 /rrr [b t V a,.a^bb—a^\/b r b,b^aa] 
(a,. — a s )al t al r< s + ( a s — a t )al,a l s> t + (a t —a r )al s al r< t = 0 
1,01, al r> s = - —Tf [a,. \/ a,.bbbb — h,.\/ baiaaa] 
(a J) —a q )l r al r al, tS + {o q —a r )l v al p al T ^-f (a r —aj,)l (/ al (j al ^ s =0 
( 5 ). 
(«) 
in which p, q, r, s, t may be any of the numbers 1, 2, 3, 4, 5. 
26. Writing —~— — cl s (s= 1, 2), equation (39) of the last example gives 
i (yo 
cd 1 z (u)—al 1 2 (v)=a 2 al 2 (u-\-v){al l {y)al 2 (u)al h2 (v)—al 1 (u)al 2 (v)al h2 (u)} 
\-v) {a l —a^{al l {L^al z (ii)al hZ ii—al l {iL)al 2 jqv)al h?> {v)} 
+ a z ^ —a. ^ a h( v ) a hi u ) a h, z{ u ) ~ a h ( u ) a h( v ) a h, z( v )} 
(d 2 (ii)—al 2 2 {v) = a l ol 1 (ii J rv){al 1 {u)al 2 (v)al h2 (v)—al 1 {v)al 2 (u)al h2 (u)} 
-\-al 2 (ii+v) (a 2 —a s ){al 2 (v)al s (u)cd 2 ^(u)—al 2 (u)al 3 (v)al 23 (v)} 
+ a i 
« 3 — r 
a, — a. 
1 3 
- {a/ 1 (u)nio(v)cd 1] o(?i) — a/ 1 (r)a/ i (u)a/ li o(r)} | 
two equations which determine al^a-^-v), al 2 (u-\-v). 
Assuming these known we have 
al*{u+v)= 1 - ali(u+v)+-^- al 2 2 (u+v) , 
a x — a 3 
a 2 — cq 
al£(u-\-v)— 1— \ al 2 2 (u-\-v) }>, 
r> ™ Ct 2 ^4 
a 
ct Y — a 4 
al 5 °(u + v) = 1 
al^u+v)^ 
The equation (36) applied to this case is when m = 1 
