106 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
therefore, replacing in (16) v by _|, we have 
sin(u+ _|gt)-p= Tat 
cos (u + _]e1)-p= + Kile ts! 
7 sin u_y 
A(u _|s1)-p= + 1 cot u_y (18) 
Placing in these u + _|g 7 for u, we have 
sin (wt2_|et)-y= sin wy 
cos (w+ 2_ |g i)-y = — cos u_y 
A(ut2 _|pi)-y=— Auwy (19) 
It follows, replacing in these u byu+2_|g%, that 4_|g7is 
the imaginary period of the elliptic cosine and delta and 2 _|gz 
that of the sine. We have then, if m and » are integers, 
sin (u+4m_|ly+ 24 _|6%)-y =sin u_y 
cos (u+4m_}ly+ 44 _]|p8 t)-y = cos u_y 
A(ut+2m_ly+44_|gt)-y= Auy (20) 
The general problem of transformation may be en thus: 
Assuming 
1 1 77 
Sot —c sing —— sin 2¢" Ue ee et (21) 
then it is required to discover the relations between the given quan- 
tities 9, a, y arid ¢’, a’, 7’. 
Before treating of the special subject of this paper (the quadric 
transformation), a short exposition of some important points of the 
general problem of transformation, slightly modified from Abel 
(see Enneper’s Elliptische Functionen, page 239-246), will be 
given. 
We have, by (21), 
cin g = sin (% of ) y= seine) =sf sin (4 gy) b 2 
where f denotes the unknown relation between sin ¢ and sin ¢’. 
