12 



Mr Raniaiiujan, On tlie expreasiuii of a namber 



If c? > 14, 14 is an exception ; and so 



2^cZ^14 (2-421). 



(2-43) x" + y' + 3^2 -1- dii\ 

 If c? > 6, 6 is an exception ; and so 



'^^d^Q (2-431). 



(2-44) X' + ly"" + ^z^ + du-. 

 li d >*1, 7 is an exception ; and so 



2^d^1 (2-441). 



(2-45) x'' + 2?/- + '2>z' + diC'. 

 If (Z > 10, 10 is an exception ; and so 



3^c^^l0 (2-451). 



(2-46) X- -r '±y'' + ^z- + diJb^. 



If c? > 14; 14 is an exception ; and so 



4^c^^l4 (2-461). 



(2-47) x'-^'ly-^-hz^^diC-. 



If c?> 10, 10 is an exception ; and so 



o^cZ^lO : (2-471). 



We have thus eliminated all possible sets of values of a, b, c, d, 

 except the following 55 : 



8 



8 



9 



9 



9 



9 



10 



10 



10 



10 



11 



11 



12 



12 



13 



13 



14 



14 



