Cambridge Philosophical Society. 453 



evident that these equations must apply to every point of the 

 caustic, provided that x have the value corresponding to the point 

 of reflection on the ordinary law ; but it may be shown also that, in 



dUY) 

 general, J" ^ is finite, and admits of being expressed in terms of 



the radius of curvature of the caustic and other lines. Having found 



d (V) rf2 c V) rf3 CV) 



therefore, that in the caustic — ^ = 0, , ^ = 0, , ^ = C, 



dx dx'^ dx^ 



the proper value being given to j?, the author infers that, for a point 



at the distance $p from the caustic, \ ■ will = A.^o, , ■■ 



dx ^ dx"^ 



= B ^p, = C + D . ^p. The values of A and C are easily 



found. Consequently the value of V, for a point at the distance Bp 

 from the caustic, when measured through a point of the reflecting 

 surface, whose ordinate is x-\-z, is of the form 



\<+AJp. '- +BJp . ^^(C-^n.ip) — . 



Rejecting the unimportant parts of the coefficients, and altering z so 

 as to take away the second power, this becomes V + A ? o . «' 



Q 



+ -— z^^ : the expression for the disturbance of ether is 

 6 



which, observing that / , sin [K^p.z^^ "F'^'^J ^^tween — 



infinity and + infinity = 0, may in all cases be shown to be 



proportional to sin ivt —Y] x / cos — {w^—m,w\ the 



integral being taken from w = to w = infinity, and m being ex- 

 pressed in terms of A, C, Bp, and \. Putting S for the definite in- 

 tegral from w = to w = infinity, the intensity of light therefore 

 is proportional to 



[nr 12 



Sm; cos —■ {w^ — m.w) I . 



The author then considers especially the case of the rainbow (in- 

 cluded in the general case of the deviation of rays having a maximum 

 or minimum), and shows that it depends on the same expression. 



An account was then given of the way in which the value of the 

 definite integral had been found for forty-one diiferent values of m (be- 

 ginning with m = — 4*0, — 3*8, &c. and ending with m = -}- 4-0). 

 As far as z^ = 2'0, it was found by summation ; after that, by series. 

 The series possessed the property of diverging indefinitely after 

 some assignable term, yet of having a sum always finite. The pro- 

 cess was one of considerable labour. 



