n= 7. septic transformation, 
Y=u-\-yx 2 , Q=|3-{-&r 2 . 
Or ELLIPTIC JUNCTIONS.. 
403 
# 3 a 2 =I2& 2 , 
^2ay+2«i3+/3 2 )=Q(y 2 +2yH2^), 
y 2 -f- 2/3y -f- 2aci -f- 2j3£ = 0&(2ay -f- 2j3y + 2aS +/3 2 ), 
S 2 + 2y£=0£ 3 (a 2 + 2«/3). 
%=9, enneadic transformation. 
P = a -{- yx 2 -f- sx 4 , Q=f3+lv 2 . 
k 4 oc 2 = Os 2 , 
k\2oc,y + 2«/3 +/3 2 ) = 0(2y £ + 2sS +S 2 ), 
2a £ +y 2 +2aH2y/3+2/3S=0(2as + y 2 +2yH2sj3+2/3^, 
2y£+2yH2£j3+^ 2 =0^(2oiy+2 a H2yi3+i3 2 ), 
£ 2 -f- 2<k = O k 4 (u 2 -j- 2aj3). 
?i=ll, endecadic transformation. 
P— a -j~ y^ 2 d - bx 4 } Q == 3 -f - ^x 2 “l - '^x 4 . 
ta 2 =Of, 
^(2ay+2^+i3 2 )=Q(£ 2 +2<+2^), 
£(2a£ + y 2 + 2*1 + 2y0 + 20S) = Q(2ys + 2y £+ 2^ + 2(3 ■ + S 2 ), 
2y£ + 2<+2yH2^+23^+^=0^(2«£+y 2 +2a^+2yH2£3+2^), 
£ 2 + 2 y£ + 2 £tS + 2 ££ = D,k 3 (2ay + 2a<5 -{- 2 y0 + 0 2 ), 
2<d-^ 2 =m 5 ( a 2 +2^), 
and so on. 
9. It will be noticed that if the coefficients of P -f- Qx taken in order are 
«, £ f, <r, 
then in every case the first and last equations are 
£i<»-l) a 2 =0ff 2 5 
2 f <r + ( r 2 =a^ ( ”- 1) (« 2 + 2 a j3). 
Putting in the first of these k=u 4 , 12 = U, the equation becomes 
u 2n a 2 =i ;Vj 
where each side is a perfect square ; and in extracting the square root we may without 
loss of generality take the roots positive, and write u n a~m. 
This speciality, although it renders it proper to employ ultimately u, v in place of 
k , 12, produces really no depression of order (viz. the O/t-form of the modular equation is 
found to be of the same order in 12 that the standard or uv- form is in v), and is in 
another point of view a disadvantage, as destroying the uniformity of the several equa- 
tions : in the discussion of order I consequently retain 12, k. Ultimately these are to 
