Mr. Cayley on certain Elliptic functions. 353 



where v extends from to v ~ 1 inclusively, the single com- 

 bination 7W = 0, w' = 0, ;•= being omitted in the numerator. 

 We may write 



w K + w' K' » + r fl = ft H + ft' H' *, 

 fA, i*} denoting any integers whatever. Also to given values of 

 /*, ft' there corresponds only a single system of values of /«, m\ d. 

 To prove this we must show that the equations 



ma + m' b ■{■ rf^= f*, 



ma' + m^ V + r/' = ju,, 



can always be satisfied, and satisfied in a single manner only. 

 Observing the value of v, 



vm + 7' [b'f- bf) =ix,b' -[^'b; 

 and if V and b'J"— bf have no common factor, there is a single 

 value of r less than v, which gives an integer value for m. This 

 being the case, w' b and 7n' b' are both integers, and therefore, 

 since b, b' have no common factor (for such a factor would 

 divide v and b'J"— bf ), m' is also an integer. If, however, v 

 and Uf — bf have acommon factor c, so thatv = o6'— a'6 = c^, 

 b'f-bf = c^; then {af-a'f)U = c. i<^'-<p!/), or since no 

 factor of c divides af —a'/, c divides i', and consequently b. 

 The equation (7.) may therefore be divided by c. Hence, 



y 



putting — = Vy, we may find a value of r, r^ suppose, less than v^, 



which makes m an integer; and the general value of r less 

 than V which makes m an integer, is r = ry4- svp where s is a 

 positive integer less than c. But m being integi-al, b m\ b' m', 

 and consequently c nJ are integral ; we have also 



V /»' 4- (^'/ + svy {af — a'f) =: ay.' — a' [x,; 

 or dividing by Vp 



cjn') +s.{af' -a'f)=:l 



an integer, where c and af — a'/ are prime to each other. 

 Hence there may be found a single value of s less than c, 

 giving an integer value for m'. Hence in every case there is 

 a single system of values of m, m\ r, corresponding to any as- 

 sumed integer values whatever of ju,, j«,'. Hence 



■^" (^ + VH + (2"f*' + l)H',) =^'"-J 



(5.) 



<f)yM being a function similar to ^u, or sin a mu^ but to a dif- 

 ferent modulus, viz. such that the complete functions are H, H' 

 instead of K, K'. We have therefore 



Phil Mag. S. 3. Vol. 25. No. 167. Nov. 1844. 2 A 



