AND THEIE CONNECTION WITH THE THEOKY OF INTEGRAL EQUATIONS. 431 



clearly points of 2 both on its right and on its left. The argument we have just 

 employed will then establish that, by reason of the former, 6 (j) is zero, and that, 

 by reason of the latter, <(i) is zero. 



1 6. Conversely, let us suppose that K (s, t) is defined in terms of continuous 

 functions 6 (s), <fr (.s - ) which have the properties mentioned in the preceding paragraph ; 

 and let us consider the function 



1, S 2 , 



(12) 



where $1, s 2 , ... , s n are variables each confined to the interval (a, V), We may remark 

 that, as this function is symmetric, it will take all possible values iu the domain 

 Si S A - 2 ^ iV 3 < ... < ts n . Thus, since we are only concerned with the sign of the function, 

 we may always siippose the variables to satisfy these inequalities. Firstly, let us 

 suppose that one of the variables has a value not belonging to the domain of the 

 discriminator of K (H, t). If such a value belongs to (a, a), the point s 1 must evidently 

 lie in this interval ; hence, since 



K(*I,*,) = (*i) <!> (O (r = l,2,....,n), 



and 6 ((<i) vanishes by our hypothesis, it is evident that all the elements of the first 

 row of (12) are zero. In a similar manner it may be proved that, when one of the 

 variables has a value belonging to the interval (b, /8), all the elements of the last row 

 vanish. Again, if one of the variables, say ,, has a value belonging to the open 

 interval (a, /3), but not to 2, we shall have 



K) = < (") = o 



by our hypothesis. It is thus easily seen that the elements of the m th row of (12) all 

 vanish. Summing up our results so far, we conclude that the function (12) can only 

 take values different from zero when the variables s l} ^, ... , # are each confined to 

 the set 2. 



17. Let us next consider the case when the variables are restricted in this manner. 

 The function (12), when expressed in terms of the functions 6 and <f>, is 



e (*,) * (*,), e (*) 



8 (s>) + (*,), 8 (,)<!>(*,), 



8 ( 



*(,)*(*.). *(*)*(), 



< S n ), 



