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surfaces, while the ray passing through that point is perpendicular to its plane, and the bisector 

 of its vertical angle is perpendicular to the caustic surface; and if we put Q('\ SC', to denote, 

 respectively, the number of the near reflected rays that pass inside this little triangle, and the 

 space over which those rays are diffused, on the perpendicular plane at the mirror ; we shall 

 have the approximate equations, 



Q(0 =/fQ(OKri.dr.dv = -^~- .// 



iV^C -/si 



sm. V 



{,•{2 /. ,- ri.dr. dv 





sm. u 



in which the integrals are to be taken from r =r to r = — : — , and from v = f . w — *, to 



sm. V 



V =: J. TO- + «. Performing these integrations, we find 



S.iViC S.iViCh 



s^f> being the area of the little isoceles triangle ; expressions analogous to those which 

 we found before, for the case of the circular sector, and leading to similar results. 



Returning to the case of the sector, we have yet to determine the boundary of the space (St"^') 

 on the perpendicular plane at the mirror. For this purpose, we are to eliminate ( r) and (v) by 

 means of the following expressions, (T'"), 



a = u.r = (Bj,. sin. v — C.^,. cos. v). i-^. C-'. r, 



b = :±zw. ri = z±z (2- V'2^. C-iVr. sin. v , 



from the polar equations of the boundaries of the sector, namely 



1st. u r: r" — ^zz V, 

 lid. V = d" + 4' = V, 

 Illd. r=r, 



of which the two first represent the bounding radii, and the third the circular arc. Putting for 

 abridgment, 



e,. v/2. C-i = v't, B{t./-J.C-i = e.P->, Cj.zrSj,. tan. t/, 

 conditions which give t = 2f. C-^ P = Sj.^j.B-' ; and supposing, for simplicity, that ^t is 

 real, and that f ' < -5- , that is, supposing C and tan v' positive, a condition which may always 



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