242 J NEW and UNIVERSAL SOLUTION 



rejedled, 



X = tan 2-ii^ = J + ^ X w + T . -^^ X w\ 



Ailume now w=:Cxx + Dx>^'; C and D being indetermi- 

 nate quantities, not depending on x ; then, by fubftitutlon, 



X ^.J/ + ^ X C X + I -^ D + ; • -^ X C = | X X^ 

 But i =±±^ ; and I . I :.lii±Z:) : therefore, 



a 2 2 z' 4 



.^ -j; + ? X (I + J'^) X >+^ + -ly\'^ (!+/') X X'. 



We muft now determine C and D, fo that this value of x, and 

 its value already found, may be identical : Thus we have, 

 C y = 



2 I +J = 



therefore, 



- + — X J = B X ^2 ~ X -y' 5 



2 ' 4 -^ I + y 



C=: 2 Axfin^-x J/, 



D = j"2 B fin' I -^ 2 A^ fin4 ll xy\ 

 and, taking the values of A and B, (Art. 17.) 



D = [if fin= ^ - 788 finV?^-'^ fin« ^- ^ fin' ^Xy^, 

 US 2 315 2 ' 45 2 225 23 



and finally, if we fubftitute for the powers of fin -, their values 



In the cofines of the multiples of the arch, we fliall find, 



Cr= J— X [7 — 6cofz— cof2zJ X/^ 



.I> = -^ \5^° 78 Cof 2 — 341 Cof 2Z— 84 Cof 3s — 7 Cof 42 I J/5 



Having 



