242 A NEW and UNIVERSAL SOLUTION 

 rejected, 



x — tan 



z -f- to y 



= y + - x w + - . 4- x w 2 . 



2 s 2 s* 



Attune now w = C x x -f D x X 2 ; C and D being indetermi- 

 nate quantities, not depending on A • then, by fubftitution, 



But i ^±±11 • and - • % i ~iLiL±Zl) \ a * 

 * 2 ? and I £ T - 4 : therefore, 



* = J' + j X (i + y 2 )x A + !5 + <± 2 ^ 7 (i + - ^ ^ 



We mud now determine C and D, £o that this value of x, and 

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



c a y 2 



D . C * r> y* 



r- +— xj = Bx— 4- . 



2 4 i -\-y 



+~ xy =3 B x ■ _ -^ ^ x J 3 

 therefore, 



C = 2 Axfin J !xj/, 



->= 



2 



X/ 5 



2Bfin 2 *_ r . 2 A* fin 4: 



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



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

 in the cofines of the multiples of the arch, we mail find, 

 C — j^X 1 7 — 6cofs— cof2%| Xy 



^ = ^l 51O ^ 78coCz ~^ lC0{2z -Uco{ 3 z^ 7 co{4z]^ 



Having 



