296 
ON THE ADDITION OF THE ELLIPTIC FUNCTIONS. 
[834 
The points 1, 2, 5, 6 are coplanar, hence the line 56 meets the line 12, say in 
the point A; and the points 3, 4, 5, 6 are coplanar, hence the line 56 meets the 
line 34, say in the point B. We can, through the curve and any arbitrary point 
in space, draw a quadric surface 
(a + Aa') x 2 + (b + Ab') y 2 + (c + Ac') z 2 + (d + Ad') w 2 = 0. 
Hence we have such a quadric surface through the point A ; and this surface, passing 
through 5 and 6, will contain the line 56, and therefore also the point B; hence, 
passing through 3 and 4, it will contain the line 34; viz. we have the lines 12, 34 
as generating lines, obviously of the same kind, on the last-mentioned quadric surface. 
I say that if, on such a surface, that is, on any surface 
Ax 2 + By 2 + Cz 2 + Dw 2 = 0, 
we have 
(a, b, c, f, g, h), (a', b', c', /', g', h'\ 
the coordinates of two generating lines of the same kind, then 
a _ a b _ b' c _ c' 
f-f’ rtf' 
This is at once seen to be the case; for, taking 0 an arbitrary parameter, we 
have for the equations of a generating line 
[x f{A) + iy f{B)} + 0 [z y'(C') + iw f(D)} = 0, 
0 [x \J(A) — iy *J{B)) — [z v'(C') — iw V(J9)} = 0, 
and the coordinates (a, b, c, f, g, h) of this line are 
i J(AD) (1 - 0-), <J(BD) (- 1 - 0 2 ), i \/{CD) 20, 
i*J{BC){-l + e% >J(CA)( 1 + n i \/{AB) (- 20) ; 
that is, the quotients j, j are each of them independent of 0; and they have 
consequently their values unaltered when for the original line we substitute any other 
generating line of the same kind. Or, to prove the statement in a different manner, 
the equation of the quadric surface through the line (a, b, c, f, g, h) is 
aghx 2 + bhfy 2 + cfgz 2 + abcw 2 — 0 ; 
hence, if this contains the line (a', b', c\ f, g', h'), we must have 
agli : blif : cfg : abc = ag'h! : b'hf : c'f'g' : ab'c, 
equations which give either 
af + af = 0, bg' + b'g = 0, cti + c'h — 0, 
af — af = 0, bg' — b'g = 0, ch! — c'h = 0. 
In the former case, the two lines are generating lines of different kinds; in the 
latter, they are generating lines of the same kind.