394 



NA rURE 



[^>r.i I r..^iilI^i< I 5, I 923 



The equation of movement of a pendulum is 



Jtf + <?'+D = o. 



9 being the angle of inclination and / the length of the 

 pendulum, D a term introduced by the disturbances. 



|=n« 



Putting 

 and introducing the complex variable 



n 



which may be represented by a vector, the projection 

 of which on the real axis is the angle of inclination 9, 

 the equation assumes the form 



q =tnq + t- 



D 



and after integration 



q={qo + ^*q)e"" (i) 



where 



A*q 





D«-'"*<i/. 



If D=o, the constant vector qn is rotating with a 

 constant velocity n. 



If D + o, q varies by the quantity S*q in the time /. 

 The change which the term D causes in the amplitude, 

 i.e. the length q^ + A^q, and in the period of the 



oscillation, i.e. the time 

 in which ^o + '^*q describes 

 ^^^____^ — -^Xf. the angle w, may be readily 

 •^ ^ inferred from Fig. i . 



Equation (i) enables us 

 to investigate the influ- 

 FiG. I. ence of the different causes 



of disturbance. 



(i) Horizontal Movements. — If the acceleration of 

 the horizontal movement is v", we have 'D=y''ll ; using 

 two pendulums with equal values for n and y" and 

 swinging in the same plane, the value of A'^ is the 

 same for both ; hence the difference of the oscillation 

 vectors is constant. This constant vector may thus 

 be considered as the oscillation vector of an undis- 

 turbed pendulum having the same period of oscillation. 

 The angle of inclination of this hypothetical pendulum 

 is equal to the difference between the angles of inclina- 

 tion of the two real pendulums. 



Each pair of pendulums of the apparatus may thus 

 be substituted by a hypothetical pendulum free from 

 the disturbances caused by horizontal movements. 



(2) Vertical Movements. — The influence of the ver- 

 tical movements is less than that of the horizontal. 

 On the other hand, it is impossible to eliminate it 

 entirely. Since the vertical acceleration is indis- 

 solubly connected with the acceleration of gravity, it 

 is obvious that elimination of the former would imply 

 elimination of the latter. 



From the following reasoning it appears, however, 

 that we are able to eliminate the influence which 

 depends on the phase of the pendulum, so that the 

 result is only affected by the mean vertical accelera- 

 tion. Expressing the vertical acceleration by y", 

 then we have 'D={x''ll)6. If we divide the equation 

 of movement by q : 



in J 

 — tn + — X -, 



q g q 



1^.. 



and represent the phase of the pendulum by <p, 

 &=acos<t> and q = ae*^, 

 NO. 281 I, VOL. 112] 



where a is the amplitude ; thus 

 



^**. 



tn + — x^- 

 2g 



the equation may be written : 



a' in. in , .. 



^ = tn+ x'+ x'e-'**. 

 q ^g 2g 



Each hypothetical pendulum corresp<)ii . 

 a pair of pendulums of the apparatus ki'<'^ ■<■ 

 equation ; the two may be ciistinguishcU one ..... 

 the other by the suffixes i and 2. The followin 

 relation is easily derived : 



(gi7^)-(g .7i?.)g»<*»-»') 



Passing to real quantities and putting the ratio 

 of the amplitudes ajax=p, we get 



« + ^^=^'-ifcot(^..^,). 



For the right-hand member of this equation th 

 observations yield a mean value ; the first term ; 

 the mean velocity of the phase. 



For the computation of n it is necessary to kno-. 

 the mean value of x" during the time between the 

 observations ; obviously we may take for this value 



h ' ' \ 



4 \^ end ~ ■* beginning J • 



If the beginning and the end of the observatior. 

 coincide with the moments when the vertical velocit 

 of the support may be supposed to be o, the same : 

 true for the mean value of x" . These moments canm 

 be accurately ascertained, but we may take tK 

 moments when the vertical movement changes it 

 direction. The resulting error can be reduce 

 ad libitum by extending the duration of the observa 

 tions. 



In this way the horizontal as well as the vertical 

 movements of the support may be eliminated. The 

 influence of the inclination of the support can also be 

 taken into account. In order to obtain the required 

 accuracy, however, it should not be allowed to exceed 

 1° in either direction. 



J. J. A. MULLER, 



Member of the Dutch Geod. Comm. 

 Zeist, August 18. 



Long-range Particles from Radium-active Deposit. 



While studying the H-particles found by Sir Ernest 

 Rutherford to be the first disintegration product of 

 aluminium and some other atoms, under a-bombard- 

 ment, we have developed a new method for obtaining 

 strong and practically constant sources of such radia- 

 tion. The method consists in enclosing dry radium 

 emanation mixed with pure oxygen within thin- 

 walled capillaries of hard (potassium) glass, lined 

 with some 12 m thickness of aluminium foil pressing 

 well against the glass. As a small number of long- 

 range particles were given off from the glass itself, we 

 have also made use of capillaries drawn out from 

 tubes of pure silica. 



Some of the elements not previously investigated 

 for H-particles have been examined in this manner 

 by the scintillation method, the results proving that 

 scandium, vanadium, cobalt, arsenic, and indium — 

 the three first as oxides, the last two as metallic 

 mirror and as chloride respectively — do not give 

 off long-range particles (s-30 cm. of air) to a greater 

 number than 3 or 4 times N . 10-*, where N is 

 the number of a-particles from radium C discharged 



