134 ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [23 
By eliminating w, v from these equations and the former system, it is easy to obtain 
(o' = a 0 + b U, 
v = a'O + b' U, 
where 
a' = - (a'B' - g'A'), V = - - (a'B - €'A). 
p p 
a = 
The coefficients a, b, a!, b' are integers, as is obvious from the equation p,(a'B— £’A) 
= p{L’a — lA'), and the others analogous to it; moreover, a, b' are odd and a', b are 
even, and 
ab' - a'b = - 1 - {A'B - A'B) (a§' - a'g) ; 
ab' — a’b = 1. 
that is 
Hence the theorem,—“ The general values <u', v of the complete functions are 
linearly connected with the particular system of values 0, U by the equations, 
(o' = aO + bU, v'=a'0 + b'U, in which a, b' are odd integers and a', b even ones, 
satisfying the condition ab' — a'b = 1.” 
With this relation between 0, U and co', v', it is easy to show that the function 
is precisely the same, whether 0, U or &/, v be taken for the complete functions. 
In fact, stating the proposition relatively to (fix, we have,—“ The inverse function <f)x 
is not altered by the change of w, v into &/, v, where w' = aw + §v, v = a'tw + §'v, 
and a, £, a.', §' satisfy the conditions that a, §' are odd, a', £ even, and a€' — a'§ = 1.” 
This is immediately shown by writing 
mw + nv = m'w' + n'v , 
m = m'a + n'a', 
n = m'S + n'G'. 
or 
It is obvious that to each set of values of m, n there is a unique set of values of 
m, n, and vice versa : also that odd or even values of m, m' or n, n' always corre 
spond to each. It is, in fact, the preceding reasoning applied to the case of p= 1. 
Hence finally the theorem,—“ The only conditions for determining &>', v are the 
equations 
where a, £' are odd and a', § even, and 
a§' — a'§ = p, /x§' — va' = I'p, — va= Ip, 
l and V arbitrary integers: and it is absolutely indifferent what system of values is 
adopted for w', v, the value of fax is precisely the same.”