440 MATHEMATICAL APPENDICES 45* 



points of the gas, but the magnitude to vary continuously from 

 layer to layer. I consider, further, that this motion may be 

 looked upon as vanishingly small in comparison with the heat- 

 motions, though not actually small in itself ; for, since the mean 

 molecular speed which will be ascribed to all the molecules is 

 very great, amounting to several hundred metres per second, the 

 forward motion of even a tolerably quick flow, such as occurs 

 with a speed of 10 metres per second, will seem of but little 

 importance in comparison. 



Consider a system of rectangular coordinates x, y, z such 

 that the ^-axis is parallel to the direction of the forward motion, 

 and take the surface of friction, or the plane for which the friction 

 between the gaseous layers on either side of it is to be determined, 

 as perpendicular to the #-axis, and therefore parallel to the 

 2/^-plane, and let this plane pass through any arbitrary point in 

 the medium with coordinates x, y, z. In this plane take an 

 infinitely small rectangle with edges dy and dz, and find the 

 number of particles which pass through it and the amounts of 

 momentum, which I will denote by Q l and Q 2 , carried over it in 

 both directions by these particles. 



For this purpose consider an infinitely small volume-element 

 dx'dy'dz' at another point (x', y f , z') of the gas, and first determine 

 the number of particles which, starting from it in a straight 

 course, meet the surface dy dz and pass through it. If N is 

 the number of molecules contained in unit volume, there are 

 N dx'dy'dz' particles in this volume-element at any moment ; and 

 if T denotes the average interval between two successive collisions 

 of a particle with others, the number of straight paths commenced 

 in unit time by this group of molecules is 



NT- l dx'dy'dz f ; 



this is also the number of particles which issue from the element 

 in unit time in all directions. 



Of these a portion, whose number is 



traverse a path of length r without a collision ; herein J3L = 1, or 

 /3 is the reciprocal of the mean free path L which, on the assump- 

 tion of equal speeds for all molecules, we have to put equal to the 

 value found byClausius 1 , so that /3 = f^A- 3 , s and X denoting 



1 67 of the text, or 33* of Appendix III. 



