120 
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
[578 
9. It will be noticed that if the coefficients of P + Qx taken in order are 
a, /3, . . p, a, 
then in every case the first and last equations are 
f.hn-1) a 2 = Ho-'-, 
2pa + a- = (a 2 + 2a/3). 
v'- 
Putting in the first of these k = u 4 , il = -• , the equation becomes 
u- n a 2 = v-<r' 
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 — va. 
This speciality, although it renders it proper to employ ultimately u, v in place of 
k, ft, produces really no depression of order (viz. the ftA>form of the modular equation 
is found to be of the same order in ft that the standard or ww-form is in v), and 
is in another point of view a disadvantage, as destroying the uniformity of the several 
equations: in the discussion of order I consequently retain ft, k. Ultimately these are 
to be replaced by u, v; the change in the equation-systems is so easily made that 
it is not necessary here to write them down in the new form in u, v. 
10. The case a = 0 has to be considered in the discussion of order, but we have 
thus only solutions which are to be rejected; in the proper solutions a is not = 0, 
and it may therefore for convenience be taken to be =1. We have then a = u n +v. 
The last equation becomes therefore 
or recollecting that /3 is connected with the multiplier M by the relation ^ = 1 + 2/3, 
that is, 
and substituting for 1 + 2¡3 its value, the equation becomes 
that is, the first and the last coefficients are 1, 
and the second and the penultimate 
v 
coefficients are each expressed in terms of v, M. The cases n = 3, n = 5 are so far 
peculiar, that the only coefficients are a, /3, or a, /3, 7 ; in the next case n = 7, the 
only coefficients are a, ¡3, 7, 8, and we have in this case all the coefficients expressed 
as above.