MATHEMATICAL SECTION. 105 
Jacobi’s imaginary transformation consists in assuming 
sin 9 = 7 tan ¢ 
or C08 PS coe @ 
Ege SNES 1 
of Seg ley = coe o 4 A -7 (18) 
 d es 
tre Wy: ST in 
then u =| Ag Sam 
0 ° 
or u= vy =i $B | (14) 
therefore, by (13), | 
; : U 1 ‘ 
sin U_y = 2 tan (+) -8 = tan (ut)_g 
1 1 
COS U-y = TF a LS Tae Pn OA SLs 
a (+) te cos (ut)_B 
A uy= ae A (+) aie: Da" (ui)_g (15) 
Using these relations in (7), we obtain the following formule 
for elliptic functions, with complex arguments and complementary 
moduli: 
sin (w+ w)-y=sinu-y Avg +icosu_y A u_ysin v-f cos v_gB 
+ 1—A%*u-_y sin 7v_g 
cos (wu = vi)-y = Cos U_y Cosv_B F isin u_y AU_ysinv_B Av_B 
--1—Au-_y sin *v_g 
A (ut v)-y= Au-_y cosv_pAv_B + y’isin u_y cos U_y sin v_Z 
+1—A*u-_y sin *v_B (16) 
We have sin (_|¢)-g=1 
cos (_|a)-p = 0 
4 (_\s)-=7 17) 
