Prof. ChalhVs Explanations of Phenomena of Light, 463 



antecedently to the supposition of any particular arbitrary dis- 

 turbance. The axis of z being supposed coincident with an axis 

 of free motion, V being the velocity transverse to the axis at the 

 point xyz distant from it by r, and w being the velocity parallel 

 to the axis, and a the condensation at the same point, the values 

 of V, w, and cr are given with sufficient approximation for the 

 proposed applications by the following equations : — 



-~-=m sin — - (icat— z + c), /= 1 — — -% (arts. 42 and 48), 

 dz A A 



Tr . df 4<mr 2tt , . x 



V=<6-p-= —--cos—- (Kat — z-b-c), 

 T dr irX a v ' 



w=f-^ = m{l---^}sm — (teat-z + c), 



The distance r from the axis is supposed to be always very small 

 compared to a (see art. 44), for which reason /is very nearly 

 equal to unity in the expressions for w and cr. That supposition 

 has been made because the investigation of the above equations 

 depends on the integrability of udx + vdy-\-wdz, and this condi- 

 tion is more strictly fulfilled in proportion as the distance from 

 the axis is smaller (art. 20). It is, in fact, assumed in this 

 theory that that analytical circumstance is the exponent of the 

 physical limitation of a ray of light. Accordingly I now proceed 

 to account for the known properties of individual rays of light 

 by means of the above equations. 



The circumstance to be first noticed respecting the expressions 

 for V, w, and <r, is that they are all functions of the quantity 

 Kat — z + c, and that consequently the velocities and condensa- 

 tions of the undulations which they represent are propagated 

 through space with the constant velocity ica. Now as the theory 

 supposes that by the agency of these undulations the sensation 

 of light is produced, the uniformity of their propagation corre- 

 sponds to the ascertained fact usually described as the uniform 

 propagation of light. This fact is rather the uniform propaga- 

 tion of conditions of physical force by the action of which on the 

 parts of the eye light becomes sensible. 



It is proper to state here that the velocity of propagation is 

 the constant ica when only terms of the first order are taken 

 into account ; and that if the approximation be carried further, 

 the expression for the rate of propagation involves the quantity 

 m, as is shown by the value of a^- given in art. 31. Unless, 

 therefore, m be the same for different undulations, the rates of 



