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, 
3 

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 already considered by Liouville and Pepin
^{[1]}.
3.
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 (2Β·41)β(2Β·47) of the previous section separately. We shall require the following results concerning ternary quadratic arithmetical forms.
The necessary and sufficient condition that a number cannot be expressed in the form


(3Β·1) 
is that it should be of the form


(3Β·11). 
Similarly the necessary and sufficient conditions that a number cannot be expressed in the forms


(3Β·2), 


(3Β·3), 


(3Β·4), 


(3Β·5), 


(3Β·6), 


(3Β·7), 
are that it should be of the forms


(3Β·21), 


(3Β·31), 


(3Β·41), 


(3Β·51), 


(3Β·61), 

or ^{[2]} 
(3Β·71). 
 β There are a large number of short notes by Liouville in vols. vβviii of the second series of his journal. See also Pepin, ibid., ser. 4, vol vi, pp. 1β67. 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.
 β Results (3Β·11)β(3Β·71) may tempt us to suppose that there are similar simple results for the form , whatever are the values of , , . It appears, however, that in most cases there are no such simple results. For instance,