Ionization of Gases by Alpha Particles from Polonium. 571 

 and thus 



/l»0 = ^-2{J o (r)+|j o (3r) + ^J o (5r)+... }, (16) 



th^ right-hand member being equal to r from r = to r — ir. 



The corresponding vibration is of: course expressed by (16) 

 if we multiply each function J (w) by the time-factor 

 cos nt. 



If this periodic vibration is to be maintained without 

 external force, the initial condition must be such that it is 

 represented by (16) for all values of r, and not merely for 

 those less than it. By (11) from to w, F(f) = ^7r£, from 

 which again by (4) the value of F for higher values of f 

 follows. Thus from it to 2-7T, F(f) = ^ir(2ir — f) ; from lir 

 to 37r, F(f) = j7r(f — 27r) ; and so on. From these./ is to be 

 found by means of (6). For example, from it to 27r, 



/"(*•) =r| sin0</0 + {2>n—r sm9) d6 



Jo t ;sin 9=7r/r 



= r-2\/{r~-7r-)-\-27rcos- l (7r/r) (17) 



where cos -1 (irjr) is to be taken in the first quadrant. 



It is hardly necessary to add that a theorem similar to 

 that proved above holds for aerial vibrations which are 

 symmetrical in all directions about a centre. Thus within 

 the sphere of radius it it is possible to have a motion which 

 shall be strictly periodic and is such that the condensation 

 is initially arbitrary at all points along the radius. 



LXIII. On the Ionization of Different Gas<s by the Alpha 

 Particles from Polonium and the Relative Amounts of 

 Eneray required to produce an Ion. By T. S. Taylor*. 



Introduction. 



IN previous papers f the writer has shown that the air- 

 equivalents}: of metal foils decrease with the speed of 

 the alpha particles entering the foils. For sheets of different 

 metals of equal air-equivalents, the rates of decrease are 



* Communicated by the Author. 



t Amer. Journ. Sci. vol. xxvi. pp. 169-179, Sept. 1908 ; ibid. vol. xxviii. 

 pp. 357-372, Oct. 1909 ; Phil. Mag. vol. xviii. p. 604, Oct. 1909. 



% By air-equivalent is meant the amount by which the range of the 

 alpha particle is cut down bv its passage through the foil. 



2 P 2 



