110 REPORT — 1870. 



whence 





Section 10. — The expression for 0i_ i (z) may be written as follows : — 



2 ~ 

 ^1—222" cos— +2^" 



^ W 1 (1 — 2"")'^ 



l_2r/« + lcOS— +2^«4-2 



"...(--)= "^ ^^:~^ 



2TrZ 

 o3 1 + 2o'2« COS h f/ '" 



O,,(.) = cos-, ^^^^--^^ . 



27rr 

 „ 1 + 22271 + 1 cos + q^» + 2 



^o,o{~)-^ (l + 22„+l)2 



As we are going to enter on investigations in which the values of w and 

 fc»' are transformed, wc shall write d^^y(z,w\ w) instead of 0|^_y(z). 



Then if ^=-, we shaU find :— 



■A. 



0^, V (z, (o\b)j)~6^^J — , w', w I, where jur =o 

 01, 1 (^, to\u),)=^e^, ,{^-^, 0,', w)j 

 Professor Ectti then shows (A. D. M. 3. 148) that if 



where p is a prime, 



ra + sy^O, 9'/3+s5^0 mod. p, 

 where r and s are less than p, we shall have 



when 



fx'=Sfx+ry + y(i "1 , 9* 



v'^m + ^/S + a/Sj"'^^- ' 



when ^(r) is a function which has no finite roots. 



* To understand these congruences, see A. D. M. 3. 140. The congruences ml—ny=fr, 

 na—m^=t's (A. D. M. 149) are easily obtained from the preceding by multiplying them 

 by n and m, and subtracting. 



