592 Professor CHALLIS, ON A MATHEMATICAL THEORY 



= l-eix' + f) + - (^.v* + y*) - &c. 



This value of / agrees with that given by equation (23) only to two terms. Consequently the 

 exact integral (18) may be employed only for small values of r and y. With this limitation, it 

 cives a value of / definitely expressed, and at the same time satisfying the hydrodynamical 

 conditions. These results point to the inference that the phEenomena of light depend exclusively 

 on the motions contiguous to the axis of z ; for it may be presumed that so far as the motions 

 correspond to the phsenomena of light, they admit of being defined by exact expressions. The 



V Y 



ratios — and - as applied to the lumitious ray, will each be very small. 



A l 



13. It may here be remarked, that in my Paper on the Polnrixation of Light, the equation 

 /■=cos\/2er corresponds to common light, and the equations /=cos2\/e,r, f—cos2\/ey, to 

 light polarized in the planes of wx and yx, subject to the limitation of taking r, x, and y, very 

 small. The first equation was obtained by assuming / to be a function of r, because common 

 light is observed to have the same relations to space in all directions perpendicular to the direction 

 of its propagation, and the other two were deduced from the first, by assuming the bifurcation of 

 a ray of common light to take place, so that the sum of the condensations at corresponding points' of 

 the two parts, is equal to the condensation at the corresponding point of the original ray, and the 

 velocities are the parts of the original velocity resolved in directions at right angles to each other. 

 Since in the present Paper the same values of / have been arrived at by a priori considerations, 

 that particular property of common light, and its resolution in that particular manner, may be said 

 to be accounted for on hydrodynamical principles. 



14. The foregoing theoretical conclusions serve to explain some general phenomena of light. 

 In Article 7. it was argued that the motion transverse to the axis of the fiuid Jilament, must be 

 defined by a particular form of / independent of the arbitrary disturbance of the fluid, and in 

 Art. 9, a form of this function was found without assigning particular forms to the arbitrary 

 functions, which in Art. 10. was proved to be consistent with the hydi'odynamical conditions. As 

 this form indicates that the condensation is arranged alike in all directions about an axis of pro- 

 pagation, it follows that light which comes directly to the eye from its origin, of whatever kind 

 the disturbance may be, is common light, the distinctive property of which is, that it is alike 

 affected in all directions perpendicular to the direction of propagation. This inference is confirmed 

 by the fact that Light from the Sun, from Stars, from a lamp, from the electric spark, from 

 lightning, Sec. is common light. The dispersed light by wliich objects are rendered visible, which 

 originates in the disturbances passively caused by the presence of the individual atoms of the 

 medium on which any ray impinges, should according to the theory be common light : and such 

 it is found to be. Moon-light and light from the Planets come under the same description. 



Again, the form which the ray assumes at its origin determines it to have direction, for it is 



clear that the direction of its propagation must from the first be coincident with the axis about 



which the condensation is symmetrical. Hence as direction is determined without reference to the 



mode of disturbance, there may be an unlimited number of directions of propagation, as there may 



be an unlimited number of rays, (see Art. 5), due to the same disturbance. In fact, the state of 



the fluid at the first instant, whatever it may be, can be satisfied by having at disposal in the 



2 7r 

 equations V=aS = m sin — {at - z + c), the quantities m, X, e, and by an unlimited number 



A 



of rays unlimited as to direction, notwithstanding that tlic functions (p and /' are defined for each 

 ray. This agrees with the fact, that liglit coming immediately from its origin, is seen in all 

 directions. 



