123 
functions. [578 
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
; à priori that il is 
atious eliminating il, 
ratio a : ¡3 has four 
il ; viz. il is deter- 
m ; the elimination of 
thence 4.4 —2.2, =12 
> in piano, the curves 
16 points ; but among 
oint a = 0, 7=6 four 
iere remain therefore 
1). 2^ n ~ 3 K Hence il 
12; but the proper 
rejected. To explain 
stion are 
the second a triple 
it 4 of these are the 
thus the point a = 0, 
two equations /3 = 0, 
satisfied if k-a-—y 2 =0, 
y=± ka), and these 
* 2 ) = o. 
:he points in question 
the tangent at the 
); 
this is the line from 
iy for the other two 
Hence among the 12 points are included the point (y = 0, a = 0) twice, and the 
points (/3 = 0, 7 = ± ka) each twice: we have thus a reduction = 6. 
15. Writing in the equations 7 = 0, a = 0, the first and third are satisfied 
identically, and the second becomes /3 2 = fl/3 2 , that is, the equations give fl = 1; writing 
/3 = 0, they become 
k- a? = fly 2 , ay = flay, y- = fl k 2 a. 2 , 
viz. putting herein y- = k 2 a 2 , the equations again give fl = 1; hence the factor of the 
order 6 is (fl — l) 8 , and the equation of the twelfth order for the determination of fl is 
(fl — l) 6 {(fl, l) 6 } =0, 
where (fl, 1) 6 =0 is the fl&-modular equation above written down. 
16. Reverting to the equation 
1 - y _ (1 -x)(P - Qxf 
1+2/ (1 + x)(P + Qx) 2 ’ 
it is to be observed that for a = 0, 7 = 0, that is, P = 0, this becomes simply y = x, 
which is the transformation of the order 1; the corresponding value of the modulus 
A is \ = k, and the equation A, = fl 2 k then gives fl 2 = l, which is replaced by fl — 1 = 0. 
If in the same equation we write /3 = 0, that is, Q = 0, then (without any use of 
the equation y 2 = tea?) we have y = x, the transformation of the order 1; but although 
this is so, the fundamental equation 
(P 2 + 2 PQx 2 + Q 2 x 2 )* = fW (P 2 + 2 PQ + QW), 
which, putting therein Q = 0, becomes (P 2 )* = flk 2 P 2 , that is, (k 2 x 2 a + y) 2 = ilk- (a + yx 2 ) 2 , 
is not satisfied by the single relation fl -1 = 0, but necessitates the further relation 
y 2 = k~a 2 . 
The thing to be observed is that the extraneous factor (fl-1) 8 , equated to zero, 
gives for fl the value fl = 1 corresponding to the transformation y — x of the order 1. 
17. Considering next n = 7, the septic transformation ; we have here between a, (3, 7, S 
a fourfold relation of the form 
( U, V, w, z ) = 0, 
I w, V', W', Z' I 
where, as before, U, U', &c. are quadric functions, and the number of solutions is here 
8.2 2 , =32; to each of these corresponds a single value of fl, or fl is in the first 
instance determined by an equation of the order 32. But the oidei of the modulai 
equation is = 8; or representing this by {(fl, 1) 8 } = 0, the equation must be 
(fi, 1 mn, i) 8 }=o, 
viz. there must be a special factor of the order 24. 
16—2