is anv uneven number, the roots are represented by ( - V 



332 report — 1865. 



If in the equation /(n, v)=0 we put w=i|/(w), the roots are represented 



If we put w=e 4 d>(w), where s is any integral number, the roots are re- 

 - ) e "" *0 ( ^—^-i — ■ )• If » = e S T > there arey'roots represented 



by/ - l €*""*> V denoting any divisor of u. If we put u = e 8 ^4, where s 

 Vy/ w 



8" V 1 / ■■■ 



The equations whose roots are respectively the squares, fourth powers, and 

 eighth powers of the roots of f(ii,v) = 0, contain only the squares, fourth 

 powers, and eighth powers of u ; we shall represent these modular equations 

 by/ 2 (« a , ^)=0,/ 4 OV)=0, or /,(.-, X)=0, and/ g (« 8 ,t; 8 )=0,or/ g (^,\ 2 )=0. 

 The last equation (by what has preceded) remains unchanged if we write (1) 



k for X, and vice versd, (2) 1— k 2 for k 2 , 1— \ 2 for X 2 , (3) - for />•, _ for X. 



k X 



If n admits of a square divisor P,f(n, u, v) is divisible by/( — , u, ( ^V ); 

 for if yy'ss— , r'=|-)^| y( ° , J is a root of/|^, «, vj =0, and 



(I) * (*$^HD • is ' '** °»*- «■ »>=°- 



It is sometimes convenient to suppose that the modular equation has been 

 freed by division from the factors corresponding to the quotients of n divided 

 by its square divisors ; its degree will then be 



*(n)-2*f!L > U2*/'JL.)_ 



\p-J \plpV 



if p lS p 2 , . . represent the primes whose squares divide n, or nil ( 1 + - J, if 



p represent any prime dividing h. The roots of this reduced modular equation 

 are expressed by the same formula as before ; only that y, y', and h are now 

 subject to the condition that they must not have any common divisor. 



"With regard to transformations of an even order, we shall only have occasion 

 to consider the case in which n is a power of 2. If n=2, we have the modu- 

 lar equations, 



v*= 2u \ , j/'*=^— * (38) 



of which, if u=(j>(o)), the roots are given by the equations i/'=W^ ), 



* It is easily seen that v=f((o) is one of the roots of / fY-,) ^ 7 ""*", 16 * )' v ~\ =0 = 

 this establishes the first of the equations (37). The other properties given in the text are 

 deducible from the equations «=0(«), v=(^-^<p(J^ 1 — -Y by applying to w different 

 transformations of the first order, and employing the formula; of the Table A. 



