128 Lippicli's Projective Theorem in Geometrical Optics. 



The denominators were also derived by means of the 

 relations 



aL + 6M-|-6*N = 0, al + bm + oi = 0, a\ + b'fi + cv=Q, 

 IX ■\-m/i + nif = ) 



where \, \x, v are the direction cosines of the projection, and 

 a, b, c of the normal to the projecting plane. 



The result might also be obtained by changing the co- 

 ordinate planes so as to make the plane of projection one of 

 them, but it seems simpler to proceed directly. 



For the following much neater method the writer is 

 indebted to Mr. T. Smith. 



Make rise of the point in which the line meets the plane. 

 (In the particular case where there is no intersection the 

 projection result is obvious without investigation.) If the 

 point of intersection is 



x' + L P , y + M/o, t' + 'Np, 



then W + my' + nz ! + p cos 0= p. 



The projection lies in the same plane as the line and the 

 normal, and therefore its direction cosines are of the form 



ah+fil, aM + Pm, «N + /3n. 



Since it is perpendicular to the normal to the plane, 



acos0 + £ = O ; 



.*. the direction cosines are 



«(L — 7cos0), a(M — 771COS0), a(N — /ICOS0), 



from which it at once follows that a=l/sin 6. 

 The projection is then 



x — x' H -Mx' + my' + nz' — p) 



- = &c.= &c, 



or, adding to each fraction — 



L — I cos 6 



lx' + mi/ + nz' —p 



ces.0 '"' 



x-x' -{-Ulx' + my' + nz' — p) ... 



— r _/ n — = two similar expressions. 



The writer desires sincerely to thank Mr. T. Smith for 

 reading and criticizing the manuscript before publication, 

 and suggesting (besides the matter just mentioned in the 

 note) the clearer expression for the final analytical condition 

 in the case of the sagittal projection, which led to the dis- 

 covery that the theorem is less accurate for this case than for 

 the tangential projection. 



