294 UNIVERSITY OF COLORADO STUDIES 



+ky8z' ' ^^^^ p"^-^ +a ijz' '-y "z) p-^+kxy' 'p-' 

 _-^z + )8'z' Qy + /3'y' px [! + </' p-^T + /3' x' p-^ 



+a/3'y'zp-^+ ^^^^g~^^ ay'z'p-^ (17) 



R^'Qy'pa[y'^"-y"^']p'";LR^'Qy'p[^'+^'^]p'°"^. (ig) 



A comparison of the members of these equations gives six 

 congruences between the original and the transformed constants 

 and the nine unknown quantities. 



I. ^xz'+a/gx ^ -+k/8z' ~^ — ^+a(yz'— y'z)+kxy' 

 ^ k'x (mod p) 



II. y8xz' = (mod p) 



III. y8'z'= O(modp) 



IV. /3'y'= /3xz" (modp) 



V. «^xz"+a/9x —^ ^+ky8z' '-A__J^a(yz' '— y' 'z) + 



kxy"=«'x+^'x'+ay8'y'z+^— ^^^ -^ay'z' (mod p) 



VL a(y'z"— y"z')^x'+a'x(modp). 



In case two of these groups are simply isomorphic, either one 

 may be transformed into the other; the necessary and sufficient con- 

 dition for this is that these congruences should be consistent and 

 shall admit of solution for x, y, z; x', y' , z' ; and x", y", z" ; where 

 X not=0 and y'z' ' — y' 'z' not=0 (mod p). 



In discussing these congruences the simplest cases are considered 

 first, and we associate with them all cases into which they may be 

 transformed. 



The groups are divided into two sets, A and B. Each one of 

 these sets is subdivided into 8 cases. 



