CHANGE 2/11 



The generation and properties of such a series must now be 

 considered. 



Suppose the second transformation of S.2/3 (call it Alpha) has 

 been used to turn an English message into code. Suppose the coded 

 message to be again so encoded by Alpha — what effect will this have ? 

 The effect can be traced letter by letter. Thus at the first coding A 

 became B, which, at the second coding, becomes C; so over the 

 double procedure A has become C, or in the usual notation A-^ C. 

 Similarly B-^ D; and so on to Y-^A and Z^B. Thus the 

 double application o^ Alpha causes changes that are exactly the same 

 as those produced by a single application of the transformation 



,A B ... Y Z 

 ^ C D ... A B 



Thus, from each closed transformation we can obtain another 

 closed transformation whose effect, if applied once, is identical with 

 the first one's effect if applied twice. The second is said to be the 

 "square" of the first, and to be one of its "powers" (S.2/14). If the 

 first one was represented by T, the second will be represented by T^; 

 which is to be regarded for the moment as simply a clear and 

 convenient label for the new transformation. 



Ex.\:\fA:\'^ ^ ^whatis/42? 

 ^ c c a 



Ex. 2: Write down some identity transformation; what is its square? 



Ex. 3 : (See Ex. 2/4/3.) What is A^l 



Ex. 4: What transformation is obtained when the transformation n' = n + \ 

 is appUed twice to the positive integers? Write the answer in abbreviated 

 form, as «' = ... . (Hint: try writing the transformation out in full as 

 in S.2/4.) 



Ex. 5: What transformation is obtained when the transformation «' = In 

 is applied twice to the positive integers ? 



Ex. 6 : If A^ is the transformation 



what is A'2? Give the result in matrix form. (Hint: try re-writing K in 

 some other form and then convert back.) 



Ex. 7: Try to apply the transformation H^^ twice: 



^ g h k 

 2 17 



