50 



and finally the dependence of ^ upon 11, that is, the law of the focus 

 by projection will be expressed by the following formula : 



— = — (cos. n)' + — (sin. n)' . (U'") 



P Pi Pi 



When the given ray is one of those principal rays determined in 

 the foregoing numbers, the angle 11 disappears from this formula, 

 and all the foci by projection coincide in the principal focus, the 

 condition (S"0 being at the same time identically satisfied, and fail- 

 ing to determine the planes of extreme projection : but in general 

 these planes can be determined by that condition, and have a remark- 

 able connexion with the tangent planes to the developable pencils, 

 which can be deduced from the equation (!,') of the ninth number, 



S* iy i/i dx 



For, when we suppose h == 0, ^3/ = Sx tan. D, we find from this 

 equation (L') the following quadratic equation to determine the two 

 values of tan. II corresponding to the tangent planes of the two deve- 

 lopable pencils : 



and if the first condition (T'") be satisfied, that is, if the planes of 

 extreme projection be taken for the planes of xz, yz, the product of 

 the two values of tan. 11 determined by this quadratic will be unity ; 

 the tangent planes to the developable pencils are therefqre symmetri- 

 cally situated with respect to the planes of extreme projection, the 

 bisectors of the angles formed by the one pair of planes bisecting also 

 the angles of the other pair. The tangent planes to the developable 



