the Molecular Velocities in Gases. 65 



a convergent series for every value of x. The value of <j)(x) 

 which satisfies all these conditions is the following: — 



4>(x) = ux 2 e-^ 2 , (19) 



e being the base of the Napierian logarithms, a and /3 two 

 positive arbitrary constants. It is expansible into series for 

 all the values of x : and. according to what we have just seen, 

 to verify that it satisfies the equation ^•{x^)--^-'(x) suffices: 

 to suppose a.=l will be sufficient: the common factor a 2 

 vanishes everywhere. The values of/ &c. then become 



g = j Ve-P" 3 dv, g 1 = ( Ve~^ civ 



%. «- 



(which cannot be found in a finite form), then 

 for/' the indefinite partial integral is 



e-Pt+^ie-Vvdv, 



tr 



or between the limits 



f 2{3 e ' /8 V2/3 ' 2/3V 



On substituting these values of/, / v , formulas (18) become 



and on putting 



we get 



^%r)=(^-|).^ + |/ + ||.-^-2 l rX^. 



Then formula (17) becomes 

 \=[ X U e-^ 2 Y\u)du, F / (m)=|* _ v / ( li 2 + r ->_ ir 2)g-^ l . rfl . < 



Taking for the variable 



\/ir + v 2 — x-=y. whence v 2 = x 2 — n 2 +f, vdv—ydy, 

 Phil. Mag. S. 5. Vol. 13. No. 78. Jan. 1662. F 



