101 



compose in general two series of rectangular developable pencils, and 

 are tangents to two caustic surfaces. Let us therefore denote by 

 ^,, y^^ z^, three rectangular coordinates so chosen, that the axis of z^ 

 coincides with some given ray, and that the planes of x, z^ and y^ z^ 

 are the tangent planes of the two developable pencils to which that 

 ray belongs ; and let a /3 y denote, for any proposed ray of the sys- 

 tem, the cosines of the angles which the ray makes with the axes of 

 x,y,z^. The equations (^) (B) (C) (D) (E) will apply to the 

 coordinates thus chosen, by simply changing x y z to x,y^z,; and 

 by changing y to its value 



y = ^ 1 — «2 _ ^2 _ 1 _ * + ^^ _ y(4) _ yC6) __ &c. , 



in which 



■ ..+.^-l- 3-5....( 2i + l) («' + ^n' + ^ 

 ' ~ 2.4.6. ,..(2t • + 2) 2i + 4 ' 



the function W will in general admit of being thus developed, 



W^''\ A, B, being constants, and JV^^\ W^*\ W^'\ being rational 

 homogeneous functions of the two small variables a, /S, of the dimen- 

 sions 3, 4, i, respectively. The constants A, B, are here the dis- 

 tances upon the ray, from the points in which it touches the two 

 caustic surfaces, to the origin of the coordinates x^ y^ z^ ; and the 

 terms proportional to a, /3, a/3, disappear from the development of W, 

 by the choice which we have made of these coordinates, and by the 

 principles of the former Supplement. In this manner the function 

 U becomes 



U= [t,WyO) — fAz, + -^1 (z, + A)»' + (2, + B) /3^} + |xt/'(3) + ^.I/W + &c. , (O) 

 VOL. XVI. Q 



