104 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
sin (wu + v)_y = sin w_y cos v_y A v_y + sin Vy cos U_y A U_y 
-+1—/’ sin *u_y sin v_y 
cus (wu + V)—y = C08 U_y COS V_y + sin U_y A u_y sin v_y A v_y 
—-1—/’ sin *u_y sin v_y 
A (ut v)-y= Au_yAv_y + 7 sin u_y cos u_y sin v_y cos v_y 
+1—/’ sin *u_y sin v_y (7) 
We have gin (y= 1 
cos(+ _])=0 
a(+_)=8 (8) 
therefore, replacing v by _ |y, we have 
: as COS U—y 
sin (ust |jyj=+ A uay 
ey sin U_y 
eos (ste iy) = ea wes ay 
B 
ST 7 talaga (9) 
Replacing in these u by u + _|y, we have 
sin (w+ 2_ jy) = —sin u_y 
cos (wu - 2_ |y) = — cos u_y 
A(w+2 _b)= Auy (10) 
It follows, replacing in these u by w+ 2_|y, that 4_|]y is the 
complete period of the elliptic sine and cosine and 2 _ }y that of the 
delta. 
Placing u = v, we have the duplication formule: 
sin (2u)_y = 2 sin w_y cos u_y A u_y + 1—?’ sin tu_y 
cos (2u)—y = cos *u_y — sin *u_y A*u_y + 1 — 7’ sin *u_y 
A (2u)-y = A®u_y—7’ sin®u_y cos*u_y + 1—/’ sin *u—y (11) 
Replacing in these u by + wu and solving, we have the dimidia- 
tion formule: 
sin? (+) ~y = 1— cos u_y +1+Au_y 
cos” ($)+= Au_y + cos U—y +1-+Au_,y 
U 
“W(>)y=P+4u_-y+ 7 cosu_y+1+t+au_y (12) 
3 )-7 y 
