296 UNIVERSITY OF COLORADO STUDIES 



The special forms the congruences take in these cases and from 

 which these conclusions are drawn are given below. 



A I. 



I. k'=0(modp) 



III. /3'z'^O (modp) 



IV. /3'y'=0 (modp) 



V. a'=0(modp) 



VI. x'+a'x^O (mod p) 



III and IV give with condition (15) /9' =0. 



A 2. 



I. kxy' =k'x (mod p) 



V. ky" = « (mod p) 



VI. x'+a'x = (mod p) 



Bl. 



I. k'^0 (mod p.) 



II. /8z' =0 (mod p) z' ^Ofrom condition (15) y' and z' 'not = 



III. )8'z'=0(modp) 



IV. /3'y'=/3xz" (mod p) /9' not = 



V. a'x+/3'x'=0 (mod p) 



VI. x'-|-a'x = (mod p). 

 Elimination between V and VI gives 



a' — a'/S' :^0 (mod p) 

 B2. 

 I. ky' =k' (mod p) 



xfx— l^l 



V. k/9z"-5^-2 — -^+kxy"sE«'x+/3'x' (mod p) 



VI. a'x+x'=0 (mod p) 



B3. 



I. k'=0(modp) 



