141 



the integral in each expression being taken from vzsv" — 4'> 'o u = «" +^ ; so that we 

 have the relation 



QW = A(''). SW, (H"") 



If the semiangle of the sector be so small that we may neglect its cube and higher powers, 

 the definite integral y U^^^dv, being the difference of the two developments 



^^ dv" 2 - dv'"' 2.3 ^ ' 



^ ^ dv" 2 dv"" 2.3 ^ 



is nearly equal to 2. t/W.^' ; and tlie quantities SW, QW, may be thus expressed, 



3.i.V|cr -/sin. d" S-z.-Z^C.^" 



s" being the area of the little circular sector, and y" being the projection of its bisecting radius 

 upon the normal to the caustic surface ; so that if the sector were to receive a rotation in its own 

 plane round its own centre, that is, in the plane of aberration round the focus of the given ray, 

 the area at the mirror (SW) and the quantity of light (QW) would vary nearly inversely as the 

 square root of the cosine of the angle, which the bisecting radius of the sector made with the 

 normal to the caustic surface. If, on the contrary, without supposing the angles (u") or (i^) to 

 vary, we alter the length of the radius, or transport the centre of the sector to any other point 

 on either of the two caustic surfaces, so as to produce another sector, similar and similarly si- 

 tuated ; it follows from (G"") that the quantities S(''^ and QW will vary as the following ex- 

 pressions, 



j,.j,.ri. i-K C-i, i,. g.. aW. ri. i-K C-i; 



so that if the centre of the sector be fixed, they vary as the sesquiplicate power of its radius ; and if 

 the radius be given, but not the centre, then they vary, the one as the product of the two focal 

 lengths of the mirror, divided by the difference of those two focal lengths and by the square root of 

 the radius of curvature of the caustic curve ; and the other, as this latter quotient, multiplied by the 

 density of the reflected light at the corresponding point of the mirror. These latter theorems, being 

 founded on the formulae (G""), do not require that we should neglect any of the powers of i|/, that is 

 of the semiangle of the circular sector ; they may even be extended, by means of the equations 

 (E"') (P"'),to the case of similar and similarly situated sectors, bounded by lines of any other form. 

 If, for instance, we suppose any small isosceles triangle, having its height =h, and its base =2.h. 

 tan. t, to move in such a manner that its summit is always situated on one of the two caustic 



