386 MATHEMATICAL APPENDICES 17* 



matical proof; for a gaseous body executes a motion of the 

 centroid and rotations about fixed axes exactly as if it were solid. 

 But if in a gaseous mass there are layers which shift together 

 with unequal velocity or rotate unequally quickly, our formulae 

 which have been used in the proof are no longer valid with 

 absolute strictness, since the energy within any layer need 

 remain constant as little as any other of the observed magnitudes ; 

 for the propositions are only strictly true for the gas as a whole. 

 We may, however, look upon them as approximately valid if the 

 interchange of energy or of velocity between the layers occurs 

 only slowly. With this assumption, which is permissible if the 

 differences in the motion of neighbouring layers are small enough, 

 each of the mechanical theorems remains valid with sufficient 

 exactness even for a single layer within an interval of time that 

 is not too long. At the same time, this interval may be long 

 enough to allow the very rapidly resulting 1 arrangement in the 

 ^^distribution of the velocities according to Maxwell's law to 

 ^ occur. In such cases, therefore, the law must also hold good if the 

 gas is divided into unequally moved layers. In each of these 

 layers, then, Maxwell's law holds good for the distribution of 

 the velocities on this condition, that on each occasion the velocity 

 in which a particle shares by the flow or rotation of its layer is to 

 be subtracted from the value which the particle would have in the 

 state of rest and equilibrium of the gas as a whole. 



18*. Transformation of Coordinates 



In using Maxwell's law it is often of advantage to give it 

 another form by a transformation of coordinates. Naturally 



/it is not always necessary to know how many molecules have a 

 velocity the components of which are u, v, w, that is, a velocity 

 of given magnitude and direction ; far oftener the question arises 

 as to the number of particles which possess a given speed, that 

 is, a motion of given magnitude without reference to direction. 



This question is answered if we pass over to polar coordi- 

 nates from the system of rectilinear coordinates to which the 

 components u, v, w are related, and therefore introduce the abso- 

 lute velocity 



1 Tait, Trans. Eoy. Soc. Edin. xxxiii. 1886, p. 82; Natanson, Wied. 

 Ann. xxxiv. 1888, p. 970. 



