517 



of 12 3, and vanishing for other combinations. So we get for the 

 numerator 



4 j/^ dx^ dxJ dxk\^ . . .... (10) 



cfj;» (fxJ öx^ I 

 this being four times the volume of the parallelepiped formed by 

 1, dx and dx. 



This sufficiently explains section 5. We may remark that formula 

 (8) for the displacements of rotation implies a convention as to the 

 direction in which the axis of rotation has to be drawn. The axis 

 of rotation must be orientated in a manner to ensure that the 

 direction of $ is correlated to the directions of 1 and u, i.e. a paral- 

 lelepiped constructed from 1, u and C, in the order thus specitied, 

 nuist have a positive volume- 



Z« l'> I'- I 

 \/g «" u^ u'- I = positive. 

 $^ i'' $^- I 

 This amounts to the same relation which exists between the 

 directions of the positive axes of X, Y, Z. 



One sees from (10) that the measure of curvature will be positive 

 if the direction of the axis of the rotation of curvature bears the 

 above-mentioned correlation to the directions of dx and dx. 



Similarly, in four dimensions, if the axis of a rotation in a special 

 case be a parallelogram on the vectors 1 and m, then the I'otation 

 is given by 



gai 9bi 9ri 

 9aj 9hj 9cj 

 9('k 9f>k 9ck 



where abed {■=) 1234, and the direction of ? is correlated to 

 the directions of 1, m and u, i.e. 



j l^ lf> /<■ 

 m" w* m'- 



C« ;f> ;r 



V9 



I' mJ u^ 



)/9 



Id 



ud 



positive 



My thanks are due to prof. J. A. Schoutkn for his kindness in 

 allowing me to read his treatise on Direct Analysis, which is to 

 be published soon in the I'l-ansactions of the Kon. Akademie. 



U* 



