of a Number of Unit Vibrations. 507 



so that 



Mean .v 2 = a 2 /12n. 



Accordingly u = 6?i/a 2 , as in (19). 



A similar argument might be employed for the law of 

 facility of various resultants (r) of n unit vibrations with 

 phases entirely arbitrary, starting with Ae~ nr ~rdr, and. 

 assuming that the mean value of r 2 is n. 



My principal aim in attacking the above problem was an 

 introduction to the question of random vibrations when the 

 phases of the unit components are distributed along a 

 circular arc not constituting an entire circle. When the 

 circle is complete the solution has already been given *, and 

 the same solution obviously applies when the circular arc- 

 covers any number of complete revolutions. All phases of 

 the resultant are then equally probable, and the only 

 question relates to the probability of various amplitudes, or 

 intensities. But if the arc over which the representative 

 points are distributed is not a multiple of 2tt, all values of 

 the resultant phase are not equally probable and the question 

 is in many respects more complicated. 



There is an obvious relation between the question of the 

 resultant of random vibrations and that of the position of 

 the centre of gravity of the representative points of the 

 components. For if 6 denote the phase of a unit component, 

 the intensity of the resultant is given by 



B, 2 =(2cos0) 2 + (2sin0) 2 . 



If we suppose unit masses placed at angles 6 round the 

 circular arc of radius unity, the rectangular coordinates of 

 the centre of gravity are 



j: = (X cos 6) /n, ~— (2, sin 6) In ; 



and r, the distance of the centre of gravity from the centre 

 of the circle, is related to R according to r=R/w. And in 

 like manner the phase of the resultant corresponds with the 

 angular position of the centre of gravity. 



The analogy suggests that a mechanical arrangement 

 might be employed to effect vector addition. A disk, sup- 

 ported after the manner of a compass-card, would carry the 

 loads, and the resulting deflexion from the horizontal would 

 be determined by mirror reading. Perhaps there would be 

 a difficulty in securing adequate delicacy. 



* Phil. Mag. vol. x. p. 73 (1880) ; Scientific Papers, vol. i. p. 491. 

 See also Phil. Mag. vol. xxxvii, p. 344 (1919). 



