in the form ax~ + by- + oz- -f dti^ 21 



(8-3) n = 3 (mud 4). 

 If \ =1=1, take A = 1. Then 



M - 4»,A - 4^ (8ya + 7) - hi 

 is of one of the forms 



Si. + 3, 4(4i/+l). 

 If \ = 1, take A = 3. Then 



M - 4mA = 4 (8/* + 7) - 12/i 



is of the form 4(4^- + 2). In either of these eases M — 4?iA is of 

 the form j/-^ + -tf + z-. 



This completes the proof that there is only a finite number of 

 exceptions. In order to determine what they are in this case, we 

 have to consider the values of M, between 4?i and 12w, for which 

 A = 1 and 



M - 4h A = 4 (8;i + 7 - n) = (mod 16). 



But the numbers which are multiples of 16 and which cannot be 

 expressed in the form x- + y^ -\- z- are the numbers 



4''(8z/+7), (/c = 2, 8, 4, ..., y=0, 1, 2, ...)• 



The exceptional values of M required are therefore those of 

 the numbers 



47« + 4*^ (8i. + 7) (8-31) 



which lie between 4fi and \2n and are of the form 



4(8/iA + 7) (8-32). 



But in order that (831) may be of the form (8"32), k must be 

 2 if n is of the form 8/^' + 3, and k may have any of the values 

 3, 4, 5, ... if n is of the form 8A; + 7. It follows that the only 

 numbers greater than 9« which cannot be expressed in the form 

 (7"1), in this case, are the numbers of the form 



9w + 4« (16i^ + 14), {v = 0, 1, 2, . . .), 



lying between 9w and 25?i, where k=2 if n is of the form 8/v + 3, 

 and /c > 2 if ?i is of the form 8^- + 7. 



This completes the proof of the results stated in section 7. 



