ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 105 



F. Since /i^+i is of the form ^yq"+^'R^{q), aud /xr of the form \/q^^'R,(q), 

 we see the irrationality to be the same in each case. Hence, as the equation 

 necessarily admits of a term c^"+^, it must also admit of a term of the form 

 c'/x»'. 



G. Since the modular equation remains unaltered when k and I are inter- 

 changed, it follows that it must also remain unaltered when ^t and + v arc 

 interchanged. The modular equation is of the form 



v"+i+ . . . +anv+i.i"+^ = 



if n is of the form 8rHhl. Here we must manifestly interchange fi and v, 

 as the equation cannot be rei^roduced if fi is placed instead of r, and — v in- 

 stead of fi. On the other hand, the modular equation is of the form 



y^+l-H . . . +OJUJ/ — ^"+1 = 



if (n) is of the form 8r + 3. Here we must place jj. instead of >', — i- instead 

 of /y, as the equation cannot be reproduced if we interchange fx and r. 



H. Hence Sohnke shows that the coefficients of fx'"yP and v^fiP are equal 

 always in magnitude, although differing in sign when n=8r + '3, and ^ is 

 even. Also that the coefficients of it'^t'P and m"+i-'»v"+^-1' are always equal 

 in magnitude, although differing in sign, when n=8r + 3. 



J. Lastly, Sohnke proves that when ^=1, the equation necessarily takes 

 the form (»/ + l)«(>'— 1)=0 when n is of the form 8r + 3, and (v — l)"+i 

 when (n) is of the form 8)' + l. 



Section 4. — The method of ascertaining the form of the modular equation 

 now becomes manifest. 



"We determine the indices of fx and v by E. Then H, J give us relations 

 between the coefficients, which greatly diminish their number considered as 

 independent quantities. Finally, we determine the remaining coefficients by 

 substituting the values of p. and v expanded in terms of q in the equation, 

 and then equating the coefficients of the powers of q thus obtained to zero. 



This method is fully illustrated by Sohnke by an example. He has also 

 added a modification of the process, which will bo found useful in practice. 



Section 5. — The discriminant of the modular equation is of the form 



TC(l-K^)'''(Ao + A^nHAjt'°+ +kpu'p). 



For a proof of this the reader is referred to the concluding section of Pro- 

 fessor Betti's Mouografia on Elliptic Functions, contained in the third and 

 fourth volumes of the ' Annali di Matematica,' which I am now going to 

 bring under the notice of the reader. Professor Betti has founded his 

 theory on the geometrical basis adopted by Riemann and his followers, and 

 which it is not my object to consider in the present Report. I shall there- 

 fore explain at once the connexion between the notation in the Monografia 

 with that we have already employed, and so lead the way to some new 

 aspects of elliptic functions. 



Putting w=2K, w' = 2iK', and therefore q=e " , we have, according to 

 Professor Betti's notation, 



0i..(^)= ^^ -, e,„(.o=0(-j .... (2) 



