139 



the density of the reftected light at the point (r, v) upon this latter plane, 



r.ar.av 

 which nearly agrees with the formula (R'") of the preceding paragraph, because, as we have seen, 



\ dv dv J «-\/(x C. sin. v) 



and therefore 



€^'».r-i = 



More accurately, the density At") being equal to the sum of the two quotients obtained by 

 dividing the quantity of light corresponding to each of the little parallelogranas (W'"), by the 

 space over which that quantity is perpendicularly diffused at the point (r, v), has for expression 



y^.r.clr.dv yi.r.dr.dv r.dr.dv 



= r-*.( t/W+ t/(l). r -j- ...) (AC) + aC*)!. r -[- ...) 



+ H.( l/«»,4- t/('V r+ ...) (AJ'*) + A(« . r + ...)• (D'"') 



[2) Ksi 



The first term of this development being the same as the approximate expression (C""), and 

 therefore agreeing nearly with the formula (R'") of [67.]) «'e see, by this method, as well as by 

 the less accurate one of the 67th paragraph, that the density upon the plane of aberration varies 

 nearly inversely as the square root of the perpendicular distance from the caustic surface : a con- 

 clusion which might also be deduced from the general theorem [4-3.], that along a given ray 

 the density varies inversely as the product of the distances from its two foci. But the present 

 method has the advantage of enabling us to take into account as many of the remaining terms 

 of the density as may be necessary, by means of the formula (D"") ; it gives also, by integration 

 of the formulEE (B"") and (X'"), the whole number of the near reflected rays which pass within 

 any small assigned space f/r. dv, upon the plane of aberration, and the whole corresponding 

 space on the perpendicular plane at the mirror; since this latter space is expressed by the sum 

 of the following integrals : 



■\ff f^^*'- ■r^- dr.dv + &c. ( E"") 



and the corresponding quantity of light is expressed by this other sum, 

 Q(') = Jf Q{») = ff 0(0). ri. dr.dv 



+ // Q"^« r%. dr. dv i- &c, ( F"") 



the integrals in these developments being taken within the same limits as the given integral 



