ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
for coefficient is 
(u m + v m ) 2 (u r + v r f — ( u r +v t )\u m + v m f — v m V + u/uj — v m W + v r 2 v m 3 
while in (B) it is 
{u m +v m )v m u r s — {u r +v r )v r u m z —%u m v m (u m +v m ) {w r 2 +v r 2 + (w r +v,) 2 } 
+ (u m +v m )u m v r 3 — (u r -\- v r )u r v m z -\-f u r v r {u r + V,) { u, 2 -\-v m 2 + {ll m + V,,) 2 } 
4" U m V m ( Ur + ^v) 3 — U r V r {u m + v m f. 
Adding the latter up in columns, it is 
u r \(u m -\- v m ) 2 —u 2 '] + v r \{u m + v m ) 2 —vj ]+3 u m v m u T v r {u r -\-v r ) for first 
— uJ^Ur+Vr) 2 —u,?] — v m s [( u r -f v,.y — v 2 ~] — 3u m v m u r v r (u m -\- v m ) for second 
— 3 u m v m [u m + v m )(u r + v ,) 3 + 3 u m v m u r v r {u m + v m ) + 3 u r v r (u r +v r ) 
— 3 u r v r u m v m (u r -\-v r ) for third 
= — U m 2 U r S — vjv?+u m H 2 +v m z v 2 + (u m +v m )\u t +v r ) 3 — (u r -\-v r y(u m + v m ) 3 
and therefore, to the order five, (B) and U+V-f-W are equal. 
Thirdly, consider in the order seven the term in U+Y+W which has 
for coefficient; it is 
L ( 2B m A J\ 
P(OV 35 ' 9 / 
(u m +v m y—u„7—vj 
— t MmVm 4“ ^m)\^m 44 ~3u m I 2U m V m (ll m ~\~V m )J 
while in B the term of order seven which is free from all the as and is multiplied by 
' L . 
PK) 13 
L 
P '(Urn) 
P'(O 
I'm. 
pv5 
ka 24-an 
[u m v m (u m + v m ) { uj-j-vj-i- (u m 4-^) 4 }] m 15 —- 
A 2 
g >r$m~\~ ^m) I ^m) 4~ ^V/z) ~\~^m ] 
2B 
— * 'n 
5 
IK 
u m v m (u m +v m )[u m 4 +vj+3ujvj-i- 2u m v m (u„ 2 +v m 2 )] 
and again these terms are equal. 
I have verified the exact agreement of the two expressions for one or two others 
(but not for all, owing to the labour involved) of the terms of the seventh order; and 
this exact agreement leads us to infer the truth of the equation 
3 A 2 
