204 
Proceedings of the Royal Society of Edinburgh. [Sess. 
In framing the purely mathematical part of his law of gravitation, 
Einstein introduces a certain tensoi-, compounded symmetrically from the 
foregoing quantities by the law 
P-) 
P P k 
= ^rAi(/^b li/) + A^3 (/x 3, li/) + Aj4(/x 4, Ir) 
+ A 2 d)«'b *2v) + A 22 (ft 2 , 2f) + A2g(/x3, 2v) + 2v) 
+ Agj(/x,l, 3 j^) + Ag2(/x2, 3v) + Agg(/x3, 3v) + Ag^(/x4, 3i/) 
+ 4i/) + B42 (/x 2, 4v) + A4g(/x3, iv) + A^^{fxi, 4v)]; 
and he introduces ^ these ten quantities as the elements of the tensor 
defining the gravitation field when they vanish. There thus will arise ten 
differential equations of the second order satisfied by the coefficients of the 
quadratic differential form which represents the ‘‘ interval ” in the newly 
defined field of gravitation. 
He deals in particular with the form — it may be called a four-square 
form — 
a dx-^ -t Ij dx^ + c dx^ -}- d dx^^ 
free from product-terms ; and then 
= lr) + f(M2, 2..) + i(A 3.') + ^(M, i’'}, 
abed 
so that there are ten equations 
in all, subsequently found to be connected by relations. For the purpose 
of investigation, he assumes that 
(i) a, h, d are functions of X-^ only, 
(ii) c is a function of X-^ and x^ only, 
including, of course, the special case when a, h, c, d are pure constants. 
(The assumption is valid for the field 
- dx’2 - dlf - d?d + 
or, with spherical polar coordinates, 
- _ y ‘1 
It will be noticed that t, or x^, is not supposed to enter into any of 
the coefficients, particularly not into the coefficient c. With the earlier 
* “ Deshalbs liegt es nalie, fiir das materiefreie Gravitationsfeld das Verscliwiiiden des 
Tensors zii verlangen,” l.c., p. 803. 
