8o 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



nut, it can not advance along its axis in a definite direction unless it performs a 

 rotation around this axis in a corresponding definite direction; and vice versa it 

 can not turn around its axis in a definite direction unless it advances along this axis 

 in a corresponding definite direction (see fig. 72) . Thus this screw connects a definite 

 direction of translation with a corresponding definite direction of rotation, and vice 

 versa. We shall agree to give the same sign to directions of translation and of rota- 

 tion which are connected to each other in this way. 



Two vectors in space, A and B, define two rotations which are smaller than two 

 right angles, that from A to B and that from B to A. Both rotations take place 

 around an axis which is normal both to A and to B, and can be represented symboli- 

 cally by arrows pointing along the axis of rotation, in that direction which by the 

 screw-rule is positive in reference to the direction of rotation. 



Positive Translation, < 



Fig. 72. Positive-screw rule. 



Now let us consider a vector F which is normal to the two given vectors A and B, 

 which by its direction represents the rotation from A to B, and which has a tensor 

 equal to the product of the tensors of the given vectors and the sine of the included 



-F=BxA 



Fig. 73. Vector-product. Fig. 74. Positive system of rec- 



tangular coordinates. 



angle. The fact that the vector F has this relation to the two vectors A and B will 



be expressed by the formula 



(d) F = AXB 



and F will be called the vector-product of the vectors A and B. 



