428 ME. J. MERCER: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE, 



at each point of the interval (a, b) ; and it can be proved in a similar way that the 

 remainder of the functions 



, a a , .... a r _,, *, 



fv ti n ri 



U\, a 2 , ..., (*r- 1) w r 



,, ..., a n \ / = l 2 n ) . . . . (ll) 



ft I * ' * * 



have the same property. 



Again, because the determinant (9) and the functions (l 1) all vanish, it is easily 

 seen that the function (10) vanishes identically. Accordingly, if any one of the 

 functions (8) vanishes for all values of the variables, so must all those which follow it. 

 it appears, therefore, that, when K (s, t) is of positive type, the determinant of the 

 integral equation (7) is either an infinite power series in X whose coefficients are 

 alternately positive and negative numbers, or else it is a polynomial whose coefficients 

 obey the same law. 



Another property which is worth noticing is that, if L is the upper limit of the 

 function (*, *) in the interval (ri, b), then 



in the whole of the square Q. This follows immediately from the fact that, since 



we have 



12. We have so far confined ourselves to the consideration of functions of positive 

 type, but the reader will easily perceive that the results obtained for these functions 

 may be made applicable to those of negative type by a simple device. In fact, if 

 K(.S V , ) is of negative type in the square Q, and we suppose that 



it is evident that K (s, t) is of positive type in Q. Applying then what we have said 

 about functions of positive type to K' (s, t), we may deduce the analogous properties of 

 K (s, t); for instance, the necessary and sufficient condition that a continuous symmetric 

 function K (s, t) defined in tJie square a^s^b, a^t^b may be, of negative type is 

 that the functions 



should never be negative when the variables s l} s 2 , ..., s n , ... each range over the closed 

 interval (a, b). 



We may remark that this result and that of 10 prove the classes of functions of 



