122 Miss A. Everett on a Projective Theorem 



hence 



yu/L/— pL _ ///(cosi/r'— a' . sini/r') — /J, (cos yfr~ a. sin yjr) 



/jb'cos\jr'— /jucos^ /Jb' cosi// — fju cosyjr 



u! Sill -dr'itx — a) , , , , 



= 1 + *-j -7-77 ,- a = 1 + a tan ^=l + a' tan ilr. 



■ ft cosy' [a — a.) T T 



Now if fi' > fju, tan ty may become infinite, but tan i/r' 

 cannot numerically exceed the tangent of the critical angle, 

 a finite limit. The equality atan^r' = « ; tan ty, which may 

 at first sight appear paradoxical when i|r = 90°, is explained 

 by ol then becoming of the second order, as follows from 

 fju' cos sjr' . dyjr' = /jl cos i/r . dyfr, so that a! tan ^ is not of 

 order x co , but 2 X go , i. e. of first order of smallness. 



When tan-^r' is infinitely small, atan^' is of the second 

 order ; but for finite values of tani//, atan^' is of the same 

 order as a, that is the first order by hypothesis. If fi < p, 

 the same reasoning will apply by transferring the accents. 



Thus Lippich's theorem applied to the sagittal projection 

 requires neglect of small quantities of the first order, unless 

 incidence is nearly normal ; whereas in the tangential pro- 

 jection only small quantities of the second order have to be 

 neglected. 



In the following numerical examples u = dyjr is taken as 1", 

 or '00000484814 in circular measure, so that millionths are 

 regarded as first order quantities. Also /jl' = 1'5, //,= !. 



fc 





*'. 



a'—d\p'. 



a tan i// = 'a' tain//, 



O ' 



20 





13 10 4851 



00000312 



•00000114 



20 



1 



13 10 4916 







30 





19 28 16-39 



•00000297 



00000172 



30 



1 



19 28 17-01 







80 





41 2 11-08 



•00000074 



•00000421 



80 



1 



41 2 11-23 







88 





41 46 44-776 



00000015 



•00000432 



88 



1 



41 46 44-807 







89 59 59 



41 48 37-135 



•00000000 



•00000434 



90 













Taking a as 1' or -000291, and ii/=45°, would give 

 a tani// = «' tan t|/= "0001555. 



