Undulatoi\y Theory of Light. 425. 



.(*(r) + v};(r)Azf) Aj//j 



— m sin 9 cos 9 E . (^ (r) + v}/ (r) A ^,9) A t, A j/, . 

 The third sum in this equation must be zero in consequence 

 of the supposed arrangement of the molecules round the axis 

 of X, , and therefore, if we denote by c^ and c^^ the products 



of the first and second sums multiplied by — , we have 



Vi, =. \/ (c/ cos- S -i- c„* sin' 6) . (4.) 



(7.) Let C D, D E, (fig. 2,) be elementary portions of a 

 wave-surface diverging from the centre of agitation O; let 

 A D, B E, be planes coincid- 

 ing with CD, D E ; and let P"ig. 2, 

 O A, O B, be perpendiculars 

 to these planes. Then the 

 velocity with which the wave 

 is, at C D, transmitted in 

 the direction perpendicular to 

 C D, must be equal to the ve- 

 locity of a plane wave moving 

 in the direction ofOA; and 

 the velocitj' with which the 

 wave is, at D E, transmitted 



in the direction perpendicular to D E, must be the same as 

 that of a plane wave moving in the direction of O B. Conse- 

 quently, if we conceive an indefinite number of plane waves, 

 which, at the commencement of the time t, all pass through the 

 centre of agitation O, the wave surface will be that touched 

 by all these plane waves at any instant. 



(8.) Now let a number of planes like A D and BE, all per- 

 pendicular to the plane of the angle Xj O j/^ , be so drawn 

 that their perpendicular distances from O, the origin of the 

 coordinates, may be proportional to the values of v^ given by 

 the equation (4.) ; where 9 is the angle which the perpendi- 

 cular .r drawn from O to any plane wave makes with the 

 axis O .t", . Then the curve in the plane of x^ Oj/^ touched 

 by all these planes will, by the property of the equation (4.), 

 bean ellipse, the axes of which are proportional to c^ andc„ . 

 (9.) The phaenomena of chemistry show that molecular 

 attractions and repulsions vary rapidly at particular distances 

 of the molecules from each other. Suppose then the forces 

 w/(r), of the paper at p. 7, to vary rapidly at particular values 



of r. The diHcrential coefficients -, — niav, in coiisctiticnce, 



dr ' ' ' 



