MATHEMATICS: A. A. BENNETT 
593 
fo^\h(r) \dr, smd fo' \ x(r) + h(r) - ^{r) \ dr ^ \ h(r) \ dr and by a 
'root' is meant a function x{s) such that \ F[x{r)]dr = 0. 
The above equation (A) we can write symboHcally in the form 
F {x -\-h) = F(x) + {x)h + i F'\^)h\ (AO 
We must regard x, h, and ^ as functions whose independent variables 
range over the same interval, viz., from zero to one. Any linear trans- 
formation of X must be accompanied by the same transformation on h- 
and ^. We shall speak of h. and ^ as covariant and of equal weight, 
viz., one, on the first of the two ranges that we shall consider. On the 
other hand the symbol ¥ or h{ri) h{r2) is covariant with h but of weight 
two. Now h and F are not covariant, but divariant. We shall say that 
F is defined on the second range. Clearly F, F' and | F'' are covariant 
and of weight one on this second range. Similarly F^ and h are contra- 
variant, for only if F^ be subjected to the transformation contragredient 
to that on h, will the term F^h be left invariant. Similarly J F'^ must be 
subjected to the square of the contragredient transformation. The 
whole situation may be succinctly expressed by introducing the notion 
of signature, which denotes the range and weight and whether covariant 
or contra variant. We shall say that x, h, and ^ are of signature (1, 0) 
being defined on the first but not on the second range, and of weight one 
on the first, ¥ is of signature (2,0) being of weight two. F is of signa- 
ture (0, 1), F' of signature ( — 1, 1) being contra variant with functions 
of weight one on the first range, and covariant with functions of weight 
one on the second, J F" is of signature ( — 2,1). A constant we may speak 
of as of signature (0, 0). 
The three expressions (1) max^ | z{s) | , (2) V S^z\r)dr, (3) S^^ \ z(f) \ dr 
we may subsume under the notation li z || for the ranges (1, 0) and 
(0, 1). If the integration be in the sense of Lebesgue, it is true that 
if 2(5) =0 identically in s, then || z || = 0 in each of the cases, but in the 
second and third cases it does not follow from II z || = 0 that s (5) = 0 
identically in s. 
We may speak of x, h, ^, F, F' , \ F" as vectors, this being an exten- 
sion of the familiar notion of a real vector in three dimensions. The 
symboKc expression {A') brings in symboHc products of apparently 
different types. But if the signature be kept in mind, no ambiguity 
results, for in each case, the product of two vectors has for its signature, 
the matrical sum of the signatures of the factors. For example in 
i ^" {^)h\ we have (-2, 1) + (1, 0) + (l, 0) = (0, 1) as desired. The norm 
II z II of the vector z may also be readily defined for F' and \ F" , so 
