330 



REPORT 1865. 



Table A. 



. c+clCl 7 7 .. 



w= : ad — 6c=l. 



a + bLl 



It would be easy to write these equations so as to express <p(Q,) in terms 

 of (f>(h>), thus completing the solution of the Problem of Transformation of the 

 first order ; but it is more convenient to retain them in their actual form. 



Similar formula? exist expressing ip(o>), ( f-^l, in terms of <p(Q.) and \J/(£1)*. 



The propositions implied in the equations of the Table may also be enun- 

 ciated conversely. Thus to case I. corresponds the theorem " If u> and £1 are 



imaginaries in which the coefficient of i is positive, and if <f>2"(u>) = <p 2 ''(£l), 

 where v=l, 2, or 3, four integral numbers a, b, c,d can be found satisfying 



the relations w= ; ad—bc=l ; a=efe=l, mod 2 ; 6=0, mod 2 ; c=0, 



a -(-oil 



mod 2*-'." 



" If <p(<o)=<p(£l), four integral numbers a, b, c, d can be found satisfying 



the relations w = = — ; ad — bc=l; o=0, mod 2; and either a=d= + l, 



a + bO, — 



mod 8, c=0, mod 16, or a=rf=+3, mod 8, css;8, mod 16." 

 * M. Hermite has also shown that the function 



whichisacube root of 0(w)x»//(w), possesses a similar property; viz. if w= f , a^— 5c=l, 



x (oj) can be expressed in terms of x(Q), ,4(0), and ^(Q). (Sur la theorie des equations Mo- 

 dulaires, p. 15.) 



