INTRODUCTION. In course of an attempt to apply direct vector methods to certain problems of Electricity and Hydrodynamics, it was felt that, at least as a matter of consistency, the foundations of Vector Analysis ought to be placed on a basis independent of any reference to cartesian coordinates and the main theorems of that Analysis established directly from first principles. embodied in the present paper and an attempt is made here to develop the Differential and Integral Calculus of Vectors from a point of view which is believed to be new. In order to realise the special features of my presentation of the subject, it will be convenient to recall briefly the usual method of treatment. In any vector problem we are given certain relations among a number of vectors and we have to deduce some other relations which these same vectors satisfy.