382 



Lord Kelvin 



[April 27, 



the staple. [Boards of a considerable variety of shape with cords thus 

 wound on them were shown as illustrations of the lecture.] 



§ 36. A very easy way of drawing accurately the path of a particle 

 moving in a plane and reflected from a bounding wall of any shape, 

 provided only that it is not concave externally in any part, is fur- 

 nished by a somewhat interesting kinematical method illustrated by 

 the accompanying diagram (Fig. 4). It is easily realised by using 

 two equal and similar pieces of board, cut to any desired figure, one of 

 them being turned upside down relatively to the other, so that when 

 the two are placed together with corresponding points in contact, 

 each is the image of the other relative to the plane of contact re- 

 garded as a mirror. Sufficiently close corresponding points should 

 be accurately marked on the boundaries of the two figures, and this 

 allows great accuracy to be obtained in the drawing of the free path 



Fig. 4. 



after each reflection. The diagram shows consecutive free paths 

 74 • 6—32 • 9 given, and 32 ■ 9—54 • 7, found by producing 74 ■ 6—32 • 9 

 through the point of contact. The process involves the exact measure- 

 ment of the length (I) — say to three significant figures — and its in- 

 clination (0) to a chosen line of reference X X'. The summations 2 I 

 cos 2 6 and 2 I sin 2 6 give, as explained below, the difference of time 

 integrals of kinetic energies of component motions parallel and per- 

 pendicular respectively to X X', and parallel and perpendicular 

 respectively to K K', inclined at 45° to X X'. From these differ- 

 ences we find (by a procedure equivalent to that of finding the 

 principal axes of an ellipse) two lines at right angles to one another, 

 such that the time-integrals of the components of velocity parallel 

 to them are respectively greater than and less than those of the com- 

 ponents parallel to any other line. [This process was illustrated by 

 models in the lecture.] 



