78 REPORT — 1856. 



we shall have 



fsecwrfw=ysec0rf0+Vsecx<^X + p^^'/"^^ 5 • • • (1^) 

 and if U(m,w), n(»i.^), n(m.)^), and n(m . v//) are four conjugate para- 

 bolic arcs, 



n(m.w)-n(m.(p)—U(m.x) — Tl(m.\P)= 



2wtan(^-Lx)tan(0-J-v^)tan(x-^4'). • • • • (16) 

 which gives a simple relation between four conjugate parabolic arcs*. 



When there are five parabolic arcs, whose normal angles 0, ^, »//, v, Q, are 

 related as above, namely 



we may proceed to obtain in like manner a formula which will connect five 

 parabolic arcs, whose amplitudes are connected by the given law. 



XVI. To exemplify the foregoing formula. Let us assume the following 

 arithmetical values for the angles w, (j), x^ 4' '• — 



- 10 + 4^^5 1 V^ , ^ 



tan w= , tand)=-, tanv= , tan 4-= ;^> 



3 ' ^2 2 3 



_ 84-5-/^ . v/? 3 I 5 



secw=^iZ___:^ sec (6= ^, secY=_, sec 4/=—. 



3 2 '^ 3 



Hence 



n(»i.tan-'ri^^ilA^ I j=m(20+9\/5) + w— + mlog(6 + 3i/5) 

 Il(m . tan- |) = m^+mlog (1+^) 



\ 2 / 4 ^V 2 / 



/ 4\ 20 



ni m.tan-1- 1= m — + mlog3. 

 V 3/ 9 ^ 



* This latter theorem may be proved as follows : — Since w is conjugate to w and i//, we 

 shall have by (8), 



II(»w .w) — n{m.oj) — U(m .Tp) = 2m tan H tan w tan >// ; 

 and since w is conjugate to (p and x> 



n(OT. w) — n(m .0) — n(w . x) = 2?ntan w tan (/>tan ^. 

 Hence, adding these equations, Tl(m . oi) will disappear, and 



n(»n.<j) — n(m.0) — n(m. x) — n(m.i^)=2m tana>[tana(tani^+ tan^ tan x}- 

 Now tan w = tan (w-i-v^). 



Therefore tan w = tan w sec >//-)- tan >// sec w. 



But tan w = tan sec x+ tan x sec 0. 



Substituting this value in the preceding equation, and multipl}'ing by tan xp, 

 tan oi tan ;// = tan <p sec x sec j/; tan vp-)- tan x sec sec »// tan ^/z 



+ sec <p sec x tan^ i|/+ tan (p tan x tan^ \p, 

 and 



tan tan x = sec^ xp tan tan x — tan^ >// tan tan x^ 

 Consequently 



tan hi tan ^-\- tan (p tan x = (sec 4' tan ^ -|- sec ^ tan i|/) (sec x tan »//+ sec i// tan x) 

 = tan (0-i-i|') tan (x-'-'/'), and w = 0-|-x. 



(IV) 



