m 



are, at the mirror, disused nearly perpendicularly over two little parallelograms, whose areas are 

 /da db da db ' 



2^) — ( — _ \ ,1 ^ "i 



* ~" \dv' dr dr'dvJ' ' ' f 



^,„. /da db' da/ db' \ , , i 



^'^'^^[-d7--d;;—dVd7}''-''^'S 



(W"0 



a, b, a', b', being the coordinates of the two points, determined by the series (T"). Substitut- 

 ing for these coordinates their values we find that the two areas (W") are the two values of the 

 following expression, 



tdv ^ dv ^ dv ^ \dv^ dv^ J ) 



X [w + S-tm" + 5.r».Hi"" + ... ± 2.H.(io' + 2.y.io"' +...)] 



X [2.(m + 2.r.u" + 3.r\u"'+ ...) ± rK {3.u' + 5.r.u"' + ...)] 

 =i.,..*....^ (^^ + ,.*' +...),»+,.„.^...,-..g + r/-^ + ...)(.+..„.4..,| 



+,,».*.*. ^,.(|+,.fV...)(^+..„.+...,-(^'+^.^+...)(«+=....-+...)^ 

 ±1..*.... {(^+,.*"+ ...)(»+3.,.».+...,-(g + .^+,..),3...+,,.„.+ ,.. } 



± ..*.*. ^ (J- + ..^'+ ... )w+2.,.»..+,..,-(^+..''^'+...)„, + 2.,... + ...) J 

 which is of the form 



SW = i- r^. ( I7W + C/d) . r+ [/(«). r' + ). dr. dv. 



±^. r. (U(%+ m>,r+ [7(%^+ ). dr. dv, (X«) 



the coefficients [/(?), C/(i).... [/m , [/(i)^^ ,,_ being functions of the polar angles (v). The densities 

 of the reflected light, at these little parallelograms (W"), have for developments 



t/ 2 



