in the for in ax- + by" + cz" 4- da- 13 



Of these 55 forms, the 12 forms 



1, 1, 1, 2 1, 1, 2, 4 1, 2, 4, 8 



1, 1, 2, 2 1, 2, 2, 4 1, 1, 3, 8 



1, 2, 2, 2 1, 2, 4, 4 1, 2, 3, 6 



1, 1, 1, 4 1, 1, 2, 8 1, 2, 5, 10 



have been ah-eady considered by Liouville and Pepin*. 



8. I shall now prove that all integers can be expressed in each 

 of the 55 forms. In order to prove this we shall consider the seven 

 cases (2*41) — (2*47) of the previous section separately. We shall 

 require the following results concerning ternary quadratic arith- 

 metical forms. 



The necessary and sufficient condition that a number cannot be 

 expressed in the form 



x^-Vy--\-z' (3-1) 



is that it should be of the form 



4^(8ya + 7), (\ = 0, 1,2..., /x = 0, 1,2, ...) (3-11). 



Similarly the necessary and sufficient conditions that a numbei- 

 cannot be expressed in the forms 



x'+ tf+2z- : (3-2), 



.T^+ f + '^z' (3-3), 



.r 2 + 2 2/- + 2^'^ ( 3 • 4 ) , 



a;-^ + 23/- + 3^2 (3-5), 



x- -^ "lif + ^z- (3-6), 



^2+ 2y^+ bz- ...(3-7), 



are that it should be of the forms 



4^(16/i + 14) (3-21), 



9M 9/*+ <)) (3-31), 



4^( 8/.+ 7) (3-41), 



4^(16ya + 10) (3-51), 



4^(16/^+14) (3-61), 



25^(25/x+10) or 25^(25/* + 15)t (3-71). 



" There are a large number of short notes by Liouville in vols, v-viii of the 

 second series of his journal. See also Pepin, ibid., ser. 4, vol. vi, pp. 1-67. The 

 object of the work of Liouville and Pepin is rather different from mine, viz. to 

 determine, in a number of special cases, explicit formulae for the number of 

 representations, in terms of other arithmetical functions. 



t Results (3-11)— (3-71) may tempt us to suppose that there are similar simple 

 results for the form ax- + hy- + cz-, whatever are the values of a, b, c. It appears, 

 however, that in most cases there are no such simple results. For instance. 



