Determinant al Relation of Dynamics, 531 



heat of nitric tetroxide and iodine, already made use of in 

 the first publication of 12th Jan. 1891, in the detailed 

 accounts of my electrical calculations (Miinchener Akademie, 

 Bd. xxiv. p. 1, 13 Jan. 1894, and Wied. Ann. Bd. Hi. p. 385, 

 1894). 



There I have also added the following calculation. Assu- 

 ming that molecular magnetism is produced by the rotation 

 of the valency-charges, we obtain for the specific magnetism 

 at saturation-point values which correspond in the order of 

 magnitude with those found by experiment. 



Helmholtz's electrochemical theory has meanwhile also 

 been confirmed in other respects by the very interesting 

 calculations of Prof. A. P. Chattock (Phil. Mag. [5] xxxii. 

 p. 285, 1891 ; xxxiv. p. 461, 1892 ; xxxv. p. 76, 1893). 



I am, Gentlemen, 



Yours faithfully, 

 University of Bonn, April 1895. P. RichABZ. 



LII. Note on a Simple Graphic Illustration of the Deter- 

 minantal Relation of Dynamics. By GL H. Bryan *. 



IN the whole range of theoretical dynamics there is probably 

 no theorem which is so difficult to fully grasp as the 

 deter minantal relation connecting the multiple differential of 

 the initial coordinates and momenta of a system with that of 

 its final coordinates and momenta. This relation, which 

 may almost be regarded as the keystone to the Kinetic Theory 

 of Gases, is conveniently written in the Jacobian form 



d(Pi'j Pi, • - - ? 9i, ga'> ; ■ =1 

 d(Pi,Z>v-, gi, £»..•) " ' 



where p l9 p%, . . . are the generalized momenta corresponding 

 to the generalized coordinates q iy q^ . . . , and unaccented 

 and accented letters refer respectively to initial values and 

 final values after a fixed interval of time t. 



To my mind the difficulty of grasping this result arises from 

 the want of simple graphical illustrations and verifications 

 from first principles not involving the use of the Calculus. 

 The following illustrative examples of its applications to 

 systems with one degree of freedom have afforded me great 

 assistance in understanding the theorem, and I trust that they 

 may prove useful to others. 



Consider a particle moving in the straight line OX (fig. 1) 



* Communicated by the Physical Society : read April 26, 1895, 



