24 Lord Kelvin on the 



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 measurement of the 

 length (/) — say to three significant figures — and its inclina- 

 tion (0) to a chosen line of reference XX'. The summations 

 2 / cos 20 and 2 I sin 20 give, as explained below, the 

 difference of time-integrals of kinetic energies of component 

 motions parallel and perpendicular respectively to XX ; , and 

 parallel and perpendicular respectively to KK', inclined at 

 45° to XX'. From these differences we find (by a pro- 

 cedure 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 

 components parallel to any other line. [This process was 

 illustrated by models in the lecture.] 



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

 of a scalene triangle ABC (in which BC = 20 centimetres 

 and the angles A = 97°, B = 29°'5, C = 53°-5), was worked 

 out in the Koyal Institution during the fortnight after the 

 lecture, by Mr. Anderson, with very interesting results. The 

 length of each free path (/), and its inclination to BC (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, through 599 reflections (in 

 all 600 paths), was drawn in a manner illustrated by fig. 5, 

 which shows, for example, a path PQ on one triangle con- 

 tinued to QR on the other. The two when folded together 

 round the line AB show a path PQ, continued on QR after 



