110 
A MEMOIR ON THE ABELIAN AND THETA FUNCTIONS. 
[825 
Chapter I. Abel’s Theorem. 
The Differential Pure and Affected Theorems. Art. Nos. 1 to 5. 
1. We have a fixed curve and a variable curve, and the differential pure theorem 
consists in a set of linear relations between the displacements of the intersections of 
the two curves ; in the affected theorem, a linear function of the displacements is 
equated to another differential expression. I state the two theorems, giving afterwards 
the necessary explanations. 
The pure theorem is 
5 (x, y, z) n ~ 3 do) = 0. 
The affected theorem is 
n (ffi y> DvT^do) = _ 8(f), 8(f), m 
012 </>, cf), * 
2. We have a fixed curve f—0, or say the curve /, or simply the fixed curve, 
of the order n, with 8 dps, and therefore of the deficiency | (n — 1) (n— 2) — 8, =p. 
The expression “ the dps ” means always the 8 dps of f 
And we have a variable curve (f> = 0, or say the curve </>, or simply the variable 
curve, of the order m, passing through the dps and besides meeting the fixed curve 
in mn — 28 variable points. 
Moreover, do) is the displacement of the current point 0, coordinates (x, y, z), on 
the fixed curve, viz. the equation f=0 gives 
< &dx + jfdy + ( !fdz = 0, 
dx dy dz 
df df df 
~ x + y + -f z = 0, 
dx 
and we thence have 
dy 
dz 
df : df : df = 
dx ' dy dz 
ydz — zdy : zdx — xdz : xdy — ydx, 
so that we have three equal values each of which is put = do), viz. we write 
y dz — z dy _z dx — x dz xdy — y dx 
df 
dx 
df 
dy 
df 
dz 
do), 
and do) as thus defined is the displacement. 
* For comparison with C. and G. observe that in the equation, p. 47, F=log^^ = \ 0 a x hÈ) 
?)’/'(£)’ 020i 
suppose, their xf/ belongs to the upper limit and corresponds to my <p : the equation gives therefore 
5\b, S\ 
dV = —— + , agreeing with the formula in the text. 
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