1900.] on the Dynamical Theoty of Heal and Light. 383 



§ 37. Virtually the same process as this, applied to the case of a 

 scalene triangle ABC (in which B C = 20 centimetres and the angles 

 A = 97°, B = 29° -5, C = 53° -5), was worked out in the Royal Insti- 

 tution during the fortnight after the lecture, by Mr. Anderson, with 

 very interesting results. The length of each free path (Z), and its 

 inclination to B C (0), reckoned acute or obtuse according to the 

 indications in the diagram, Fig. 5, were measured to the nearest 

 millimetre and the nearest integral degree. The first free path was 

 drawn at random, and the continuation, after 599 reflections (in all 

 600 paths), were drawn in a manner illustrated by Fig. 5, which 

 shows, for example, a path P Q on one triangle continued to Q R on 

 the other. The two when folded together round the line A B show 

 a path P Q, continued on Q R after reflection. For each path I cos 

 2 6 and / sin 2 $ were calculated and entered in tables with the proper 



Fig. 5. 



algebraic signs. Thus, for the whole 600 paths, the [following 

 summations were found : 



Zl = 3298; 2Zcos2 = +128-8; 2 I sin 2 $ = -201-9. 



Remark, now, if the mass of the moving particle is 2, and the velocity 

 one centimetre per second, 2 I cos 2 9 is the excess of the time-inte- 

 gral of kinetic energy of component motion parallel to B C above that 

 of component motion perpendicular to B C, and 2 I sin 2 is the 

 excess of the time-integral of kinetic energy of component motion 

 perpendicular to K K' above that of component motion parallel to K K'; 

 K K' being inclined at 45° to B C in the direction shown in the diagram. 

 Hence the positive value of 2 I cos 2 indicates a preponderance of 

 kinetic energy due to component motion parallel to B C above that 

 of component motion perpendicular to B C ; and the negative sign 

 of 2 I sin 2 6 shows preponderance of kinetic energy of component 



