1. This paper is a revision of the paper with the same title which was published a year ago. The paper deals with a space of four dimensions V4 of coordinates xi as i takes the values 1 to 4. The geometry of the space is based upon the asymmetric functions gij and the asymmetric coefficients of connection Prk and upon the Einsteinm equation2 gij,k = gih kj + ghj ri. (1) Here, gi,k denotes the derivative of gij with respect to xk and each of the terrms on the right stands for the sum of terms as h takes the values 1 to 4. Throughout this paper, a quantity followed by a comma and index denotes the derivative of the quantity with respect to x with this index. Also, when there is a repeated upper and lower index, the one term stands for the sum of terms as that index takes the values 1 to 4, as for example h in equation (1). Write equation (1) in the form gij,k + gij,k = (g9 + gh)(rj - rjh ) + (gjh - gjh) (rp + rk), - - - v ~ v w v v i where gij and gij are the symmetric and skew-symmetric parts of gij, and Frj and Fjk: are the symmetric and skew-symmetric parts of Frj. This type of notation is used throughout this paper. When i and j are interchanged in the above equation, the resulting equation is