834] 
ON THE ADDITION OF THE ELLIPTIC FUNCTIONS. 
297 
Now, considering (a, b, c, f, g, h) as the coordinates of the line 12 and (a', b\ g', h') 
as those of the line 34, the equations just obtained are 
y\%1 y^l y^i y^P3 Z x X 2 — Z 2 0C x Z 3 X 4 Z 4 X 3 X x y 2 ^2^1 3/4 #42/3 
X X W 2 — X 2 W x X 3 W 4 — X 4 W S ’ yi'W,, — y 2 W x y-.iW 4 — y 4 W 3 ’ Z X W 2 — Z 2 W X Z 3 W,j — Z 4 W 3 ’ 
Of course the equations hold good if, instead of the two lines, we have one and 
the same line; the equations 
&x 2 4- b y"- 4- cz 2 + d w 2 = 0, six 2 4- b'y 2 4- c'z 2 + cl'w- = 0, 
considering therein x 2 , y~, z\ w 2 as coordinates, may be regarded as the equations of 
a line; and thus the points (x x , y x , z x 2 , w x ), &c., will be four points on a line. 
And we have thus 
yiW ~ y»V = y 3 V - 2/4V &c 
x?w£ — x 2 2 w x 2 x 3 2 w 4 — x 4 w 2 ’ 
equations which are, by means of the foregoing set, converted into 
2/1^2 + y*h y^Z 4 + y 4 Z 3 ZjXj + ZJG 1 ^ Z 3 X 4 + x x y 2 + x 2 y x _ x 3 y 4 + X 4 y 3 
X x W 2 + X 2 W x x 3 w 4 + x 4 w 3 ’ y x w 2 + y 2 w x y 3 w 4 + y 4 w 3 ’ z x w 2 + Z 2 W X Z 3 W 4 + Z 4 W 3 ' 
If for x, y, z, w we write s, c, d, 1, then the equations are 
c x d 2 c.,d x z 3 d 4 c 4 d 3 
s x s 2 s 3 s 4 
d x s 2 d-2S x d 3 s 4 d 4 s 3 c 2 5 2 Ci $304 S4C3 
Ci — c 2 c 3 — c 4 ’ d x — d 2 d 3 — d 4 ’ 
c x d 2 + c 2 d x _ c 3 d 4 + c 4 d 3 
s x + s 2 s 3 + s 4 
d x s 2 4* d 2 s x d 3 s 4 A d 4 s 3 
Cx 4- Co C3 4" c 4 
s x c 2 4~ s 2 c x s 3 c 4 4“ S4C3 
d x + d 2 d > 4" d 4 
where s x , c x , d x are the sn, cn and dn of u x , &c. ; and where the relation between 
the arguments is u x 4- u 2 — u 3 + u 4 . 
In particular, if u 4 = 0, we have s 4 , c 4 , d 4 = 0, 1, 1; and then writing u for u s , 
and consequently s, c, d for s 3 , c 3 , d 3 , the relation between the arguments is u = u x + u 2 ; 
and we have 
c x d 2 — c 2 d x 
c — d 
d x s 2 — d->Si s 
s x c 2 S 2 C } 
s x s 2 
s 
Ci ~~~ C2 1 c 
d x — d 2 
c x d 2 4~ c 2 d x 
c + d 
^1^2 “t" ^2^] 
s x + s 2 
s 
Ci 4- c 2 1 4- c ’ 
d x 4“ d 2 
The last two pairs give 
1 4- c _ (d x s 2 - d 2 s x ) (Ci 4- c 2 ) 
1 - c (d x s 2 4- dzS 1) (ci - c 2 ) ’ 
that is, 
s x c x d 2 S 2 c 2 d x 
s x c 2 d 2 s 2 c x d x 
d + 1 _ (s x c 2 — 5 2 Ci) (d x 4- d 2 ) 
d — 1 (s x c 2 4- s 2 c x ) (d x — d 2 ) ’ 
^ s x d x c 2 ~ s 2 d 2 c x 
s x c 2 d 2 s 2 c x d x 
C. XII. 
38