It does not seem possible to state the method of tensors in untechnical language; I am afraid that those philosophers who have not thought it worth while to learn the calculus cannot hope to understand it. Perhaps in time some simple way of explaining it may be found, but none has been found so far.[24] Suppose we have a vector quantity whose components are , , , . (Here 1, 2, 3, 4 play the part of suffixes, not of exponents denoting powers.) It happens in certain cases that, if we transform to any other co-ordinates , , , , which are continuous functions of the old co-ordinates , , , , we shall have, as the components of the vector in the new co-ordinates, , , , , where: [Pg 65] with similar formulæ for , , . When this happens, the vector in question is called contravariant. The simplest example is (). Except in this one case, the "contravariant" property is symbolized by the upper position of the suffix. Again we may have a vector, whose components are , , , , which is transformed according to the law: with similar formulæ for , , . Such a vector is called covariant. The simplest example is the vector whose components are: where is some function which has a fixed value at each point, independently of the co-ordinate system. It is obvious that, if we have two contravariant vectors and whose components are equal in one system of co-ordinates, then their components are equal in any system of co-ordinates; and the same applies to two covariant vectors and . This follows at once from the above rules of transformation. Thus an equality of two contravariant vectors, or of two covariant vectors, when it occurs, is a fact independent of the co-ordinate system. It is, in fact, a tensor equation of the simplest kind. The general definition of a "tensor" is a generalization of those of contravariant and covariant vectors. Instead of a vector with only four components, we may have a quantity with sixteen components: Such a quantity may be denoted by "" where it is understood that and can each take all values from 1 to 4.[Pg 66] Similarly we may have a quantity with sixty-four components, , , etc.; such a quantity may be denoted by "" where and and can each take all values from 1 to 4. Such quantities are called "tensors" if they obey laws of transformation analogous to those of contravariant and covariant vectors. Thus a contravariant tensor with sixteen components, which is written "," is one which satisfies the rule: with similar equations for the other components—e.g.: These equations are comprised in: where , are to take all values from 1 to 4. Similarly a covariant tensor with sixteen components, written "," is one which is transformed according to the rule: and a mixed tensor, written , is one which satisfies the rule: