MATHEMATICS: F. L. HITCHCOCK 
177 
\AF\^\F[<p{x)-^>P{x)]-Fy{x)]\<e 
whenever is of class C^"\ or continuous and of class C^"^ except pos- 
sibly at a finite number of x-values, and 
\^l^{x)\<b,W{x)\<b, ,\^p'^\x)\<b (a^x^b). (10) 
Further F has a derivative at any value x = ^ which is approached with 
order 2n; that is, if does not change sign and vanishes except on an 
interval of length less than h including x = ^, and if furthermore 
\^P(X)\<8,\^P'(X)\<8, ,\^P^'"^{X)\<8, (H) 
then the limit 
lim — 
5 = 0 f 
h = 0 
exists. Further the absolute value of the quotient AF/8h will be bounded 
for all choices oi 8 > 0, h > 0, xP (x) satisfying the relations (11) and 
such that the values (x, y,y\.. y^*"^) on the sue y = (p -{• ^p, a ^ x ^ &, 
are all in the neighborhood R. 
1 Volterra, Legons sur les equations intSgrales, ch. 1, art. 5: or his Legons sur les fonc- 
Hons des lignes, ch. 1, art. 2. 
2 Volterra, arts, VII and 2,3, respectively, of the chapters referred to above. 
3 See Jordan, Cours d' Analyse, vol. 1, p. 247. 
* See Fischer, A generalization of Volterra's derivative of a fxinction of a curve, Anter. 
J. Math., 35, 385 (1913). 
A CLASSIFICATION OF QUADRATIC VECTOR FUNCTIONS 
By Frank L. Hitchcock 
DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
Presented to the Academy, January 9, 1915 
There is probably no chapter of mathematics more worthy of 
attention, or more neglected at present, than the theory of vector 
functions. In the case of the linear vector function, it is true, a good 
deal has been found out in one way or another, and this by some of 
the very greatest of mathematicians. First investigated in detail by 
Hamilton^ and again appearing as Cayley's matrix of the third order,^ 
the linear vector function is essentially the same as the Grassmann open 
product^ and the Gibbs dyadic.^ In Germany the nonion or three- 
square matrix bears the name Tensor,^ a word used by others in a differ- 
ent sense. On the other hand, we may make a clean sweep of all these 
