and its Application to the Molecular Motions of Gases. 103 



scribed by the surface-element; and the probability that the 

 point is situated exactly in this space is the same as the proba- 

 bility that the point in its motion strikes the surface-element ds. 



For all cases in which the supposed motion of the surface- 

 element is outwards from the enclosed space, so that the small 

 space described by that element lies outside of the given space, 

 the probability that the point will be in this small space is equal 

 to zero. On the contrary, where the supposed motion of the 

 surface-element is inwards, so that the small space described by 

 it forms a part of the given space, the probability that the point 

 will be found in exactly this part of the space is represented by 

 a fraction whose numerator is this part, and its denominator the 

 entire space. 



If 6 is the angle which the direction of the motion of the ele- 

 ment makes with the normal erected inside upon the element, 

 the magnitude of the small space is represented by the expression 



cos 6ds dl, 

 which becomes positive or negative according as the small space 

 is within or without the given space. Hence, if the entire given 

 space be denoted by W, then with respect to the probability 

 which is to be determined we can say : — For directions of motion 

 with which the preceding expression becomes negative the proba- 

 bility is equal to zero; and for those with which the expression 

 becomes positive the probability is 

 _ cos ds dl 

 ~ W 



Now, in order to calculate the mean probability for all possible 

 directions of motion, we must take account of the law of proba- 

 bility with respect to the angles. The likelihood that the angle 

 which the direction of motion makes with the normal lies be- 

 tween a given value 6 and the infinitesimally different value 

 6 + d6 is represented by the ratio of the area of a zone having 

 the polar angle 6 and the breadth d6 to the total surface of the 

 sphere, therefore by the fraction 



27rsm0d0 _ sinfljfl 

 4tt ~ 2 



We have to multiply the foregoing fraction by this, and then to 

 integrate for all values of 6 for which cos 6 is positive, conse- 



quently from to -. The probability that the point, if it tra- 



verses in any direction the distance dl, strikes in its motion the 

 element of surface ds, will consequently be represented by 



f = ^ co& ed s di^ a 0d6 dsdic ~\ in$cos0rW= 

 J 9 =„ w a 2Wj, 



dsdl 

 4W' 



