u Regula Tertia PMlosophandi." 29 



tilinear propagation takes place along an axis, and that the 

 contiguous motion is such as only a fluid is capable of. To 

 express these conditions, it suffices, after taking the straight 

 line of propagation for the axis of z, to assume that 



(d . /(f)) = udx '+ vdy + wdz, 



/ being a function of x and y only, and cf> a function of z and 

 t only. For, on these suppositions, 



,df ,df j,d<b 



so that if the function/ be such that f=l, j- =0, and 



df . 



-jL =0, where x = 0, and y = 0, the axis of z will evidently be 



an axis of the motion. On reasoning from these antecedents 

 no contradiction is met with like that which occurred in the 

 previous method ; and the reasoning is proved to be legiti- 

 mate by actually conducting to a definite form, expressed in 

 series, of the function <£, and also to a series of definite form 

 for /in case the motion be a function of the distance from 

 the axis for any given value of z. The series for <f> is the har- 

 monic series assumed by Helmholtz and other physicists in 

 the mathematical theory of music. As the motion indicated 

 by this series is not supposed to be dependent on a particular 

 mode of disturbance, it evidently should be derived mathema- 

 tically from the initial hydrodynamical definitions. This is 

 what is done by the above-stated course of reasoning. The 

 other factor / differs very little from unity in aerial vibra- 

 tions : but in a medium of great elasticity, such as we have 

 supposed the sether to be, it is of special significance, serving 

 to account for transverse vibrations in the undulatory theory 

 of light, and the distinction between common and polarized 

 light. As experience shows that polarization is a quality of 

 light depending on the constitution of the medium by which 

 the luminous vibrations are transmitted, being producible 

 under a great variety of extraneous conditions, it is an im- 

 portant confirmation of the present reasoning that it is capable 

 of deducing transverse vibrations from the original definition 

 of the aether. 



By the same course of reasoning, the velocity of propaga- 

 tion is found to be /ca, k being a determinable numerical con- 

 stant. Having concluded that my attempt to calculate the 

 value of k, contained in the Phil. Mag. for February 1853, 

 was erroneous, I made another in the Phil. Mag. for May 

 1865, p. 329, which I have introduced into the ' Principles ' 

 4c. prop. xiv. pp. 214-224. The theoretical value of the 



