1877.] Magnifying -power of the Half '-prism. 17 



A', A", A'" being the dispersion at the first, second, and third surfaces 

 respectively ; also 



= ty' + &//, = ^xp + d x \ and a D=50— ax- 

 Eliminating d<f>' and 



m"<ty+m'chp==m"A' + m'A"; 

 and eliminating and £x'> we have 



m"m"'<ty — m , fy=m''m'''A'-\-m'm'"A''~-m'A''' t 

 with the three further relations, 



df—m'$<f>' =A', 

 $lp—m"ty' =A", 

 cx—m"'dx=A"'. 



3. Purity. — Putting U', n", II'" for the purity of the first, second, and 

 third surfaces respectively, 



.i= a :. n»-C, n-=c, 



mm m 



and we have 



ir«few, n"=f^-^\tanf, n'" = Mtan x ', 

 m"m"'$f—m'Sx=i m'm"m "'(n'+n")— mWn"'. 

 Taking the three cases as before, we have : — 



i. 20=0, /. A=wA'+m"'A"-A"'=m"m"'(n' + n") — m"'n"' 



=m"m"' tan 0' + ^- tan f 1 -m" , ^'tanv , j 



ii. fy'=-Sf = and — ^ x ' = 0, 



/. 20=A' = m'n', ^=A"=m"n", ax = A" ; =m''"n"', 

 and £D=A'-A"'=m'n'— m"'n"'. 



iii. S D=0, .*. (ty=c)x= angle through which prism is turned =6. 

 Then 



n mA' + m'"A" — A'" A m „ 



y= = = — =-= — i u - 



m — 1 m — 1 m — 1 



Taking the three forms of compound prism, we have 

 I. Half-prism magnifying. 



i n/r m"m"' eos\b' cosv' 



1. Magmrying-power m = = -. . — — . 



° J & r ' ^ cos cos x 



2. Dispersion . . . . A y = m"'A"~ A"'=m / /u^— ^^tan^/' — ?^tan x'. 



3. Purity n =/,fn"-5Q= - 



\ m J m t 



VOL. XXVI. C 



