216 THE KILLING VECTOR AND THE GENERALISED KILLING EQUATION IN FINSLER SPACE by R. B. Misra and R. S. Mishra (Allahabad, India) SUMMARY Recently we have studied the infinitesimal deformation in Finsler space [8] 0). Here we have obtained the necessary and sufficient condition for the infinitesimal change to become a motion in Finsler space. The condition is termed as the generalised Killing equation and the vector vi(x, x) occurring in this motion has been called the Killing vector. Various alternative forms of this equation have also been obtained. These forms are the generalisations of the correspon- ding equations obtained by Hokari [5], Knebelman [6] and So6s [10]. 1. INTRODUCTION Let us consider an n-dimensional Finsler space F, in which a function F(x, x) (i----- 1, 2, ..., n) of 2n independent variables x i and x i and positively homogeneous of degree one with respect to the variables x i is given. The infinitesimal arc-length of a curve in this space is given by (1.1) ds = F(x, dx). 0) The numbers in the square brackets [] refer to the references given at the end of the paper.