19J INVESTIGATION OF THE TRANSFORMATION OF CERTAIN ELLIPTIC FUNCTIONS. 121 
/x, fjb' denoting any integers whatever. Also to given values of ¡x, ¡x there corresponds 
only a single system of values of to, to', r. To prove this we must show that the 
equations 
ma + m'b +rf = fx, 
via' + m'b' + rf = f, 
can always be satisfied, and satisfied in a single manner only. Observing the value of v, 
vm + r (bf — bf') = /xb' — ¡xb ; 
then if v and b'f — bf' have no common factor, there is a single value of r less than v, 
which gives an integer value for to. This being the case, m'b and m'b' are both 
integers, and therefore, since b, b' have no common factor (for such a factor would 
divide v and bf — bf), to' is also an integer. If, however, v and b'f — bf have a 
common factor c, so that v— ab' — a'b = ccf>, b'f — bf — ccf>'; then (af — af) b' = c (</>/' — <£/), 
or since no factor of c divides af — af c divides b', and consequently b. The equation 
for v may therefore be divided by c. Hence, putting - = v / , we may find a value of 
c 
r, say r n less than v f , which makes to an integer; and the general value of r less 
than v which makes to an integer, is r = r / +sv / , where s is a positive integer less 
than c. But to being integral, bm', b'm', and consequently cm' are integral; we have also 
cv{m! + (r, + sv) {af — af) = a/x' — a'/x; 
and there may be found a single value of s less than c, giving an integer value for to'. 
Hence in every case there is a single system of values of to, to', r, corresponding to 
any assumed integer values whatever of ¡x. Hence 
u 
U = uTl (1 + 
(5) 
2/xH + (2/x' + 1) H'i, 
(f>,u being a function similar to cf>u, or sin am u, but to a different modulus, viz. such 
that the complete functions are H, H' instead of K, K\ We have therefore 
cbucf) (u + 2ft>) ...(f) {u + 2 (v — 1) o)) 
(/> (2ft)) . . (f> (2 {v — 1) go) K 
Expressing ft> in terms of K, K', we have vH = b'K — a'K'i, — vH'i — bK — aK'i, and 
therefore v(o = {b'f—bf')K—{a'f—af')K'i. Let g, g' be any two integer numbers 
having no common factor, which is also a factor of v, we may always determine a, b, a', b', 
so that vco = gK — g'K'i. This will be the case if g = bf—bf,g' = af—af. One of the 
quantities f f may be assumed equal to 0. Suppose f = 0, then g = b'f g' = af; 
whence ag—bg'=vf Let h be the greatest common measure of g, g', so that g = kg t , 
g' = kg' t ; then, since no factor of k divides v, k must divide f, or f=kf, but 
g, = b'f g', = af, and a', b' are integers, or /must divide g, g'; whence / = 1, or f=k. 
Also ag / — bg\ = v, where g / and g\ are prime to each other, so that integer values 
may always be found for a and b; so that in the equation (1) we have 
gK-g’K'i 
(7), 
v 
g, g' being any integer numbers such that no common factor of g, g' also divides 
C. 
16 
v.