SECTIONAL TRANSACTIONS.—A*. 353 
The ultimate physical aim, and the relation between physical results 
obtained by the various methods. Illustrations from a comparison of the 
results of general relativity and classical theory. 
Dr. P. Dienes.—On spaces with quadratic connection (10.30). 
The experimental verifications of relativity theory refer to quantities of 
the second order, whereas the underlying geometrical structures, Riemann 
geometry, metric spaces, are all derived from spaces with linear connection. 
In the Pascal-Vitali theory local (osculating) spaces of higher order have 
been systematically studied, but the local spaces are still dove-tailed by 
a linear connection. In this paper, by the example of quadratic connection, 
I show that such a theory can be completed into a homogeneous construction 
involving new geometrical tensors. Another advantage of our method is 
that it applies equally well to continuous and to discontinuous distributions. 
A vector v*, a= 1, 2,..., m, issued from the point 
OG Fe date) joa + de) 
viewed from P(x1, . . . x”) appears as the vector 
(1) v(Q || P) = vt+ V§(P)v’dxe + 40% 4(P)v’dxdx4 at P, 
where I'?. and I, are two sets of arbitrary functions called linear and 
be bed yi 
quadratic connection parameters, respectively. If we substitute (1) at P 
to v* at QO, we also say that we transport v? to P by ‘ parallel’ transport. 
In a change of ree Tj4 are transformed by the formula 
Ill 
hb Peet 38x04 a xb 8x? a 82x? 
“ee 8x2 Sx° Sxt Sah T b6 Sxb' Sx’ Sx0" bd 3x0 Sxe" 
a 8x §2ye a 9x? 8xe ey 
= 8x Sx" Sx% bed Sb" Sixt’ Sx’ 
For covariant vectors, and then for tensors of any type and rank, 
transport and parallelism can similarly be defined by two fresh sets 
‘T%, and ‘I%,;. The necessary and sufficient condition that transport and 
contraction be interchangeable is that 
(3) ie tas cee gu at tne baday lublon Lyauan case 
_ These definitions lead to the corresponding extensions of the idea of tensor 
derivatives. 
The fundamental fact of the theory is that in a change of variables 
sry 
«@) Steenitg Lee Ty 
is transformed as a tensor of the type indicated by the position of the suffixes. 
Since I},, = V3.2, the alternating (skew-symmetric) part of S%, in c, d 
is the Riemann-Christoffel tensor. Its symmetric part appears in the 
theory of metric spaces defined by the condition 
(5) G,,(Q || P) = G,,(P) for every x* and dx*. 
In fact, the necessary and sufficient condition for (5) is 
(6) S\(ab)(cd) = 0. 
Another geometrical significance of the symmetric part S} (cd) IS given by 
the formula 
(7) v4 (P || Q || P) = 0% + Sh dxcdxt. 
