32b' Mr. Herapath on the Solution of^ n x = x. [Nov. 



in which Q, Q, are the arbitrary constants. Now if we assume 

 q _ _ f^JO. — Y and identify (16) with the well-known theorem 



(cos z ± V — 1 sin z) v = cos v z ± V — 1 sin v z 

 we get 



2 B = (— ?! — V . $ Q (cos v z + v — 1 sin v 2) -f Q, (cos w 



* \2 cos s / l x 



- v r ^~l sin »2) } (17) 



Determining now Q, Q, from the conditions of (17) when 

 v = and v = 1, we shall find after d ue re ductions 

 ^ I _ (ft, - c r ) j - 1 



Q, = l + 



(ft r + c r ) tan z 

 (ft, - c,) ^/^T 



(ft, + c,.) tan i 



And if R, R, be the corresponding arbitrary constants in the 

 solution of (14), we easily perceive that 



R = Q, and R, = Q. 



Consequently 



B = ibr + ClY . \cozvz + -^-=- c>) cos z ; sin " * I (18) 



(2 cos e)" ( U> r + f,. sin 8 5 v 



and C v is the same expression with a negative instead of a plus 

 sign before the second number under the vinculum { }. 



In the same way if P, P, be the arbitrary constants of (12), 

 it appears that P = — P, and 



\2coss / 



Whence 



. /ft, + c,\'-' sin v z ,, n . 



a vr = a,. A„ = - . — a, (19) 



"' ' \2 cos z J sin i v 



Moreover since — ==~jf> we obtain by introducing for d T its 

 value q ^ '' c ' , and substituting for q its value — — — -~- 



n r 02 ^2 cos t y 



m , _ (ft,. + c,y- ' ■ (2 ft, c, cos 2 g - ft,'' - c,") sin v z -,„ 



«, ,— ,, — ^, cos .),- 1 , 2 o,(cos 2 s + 1) sin s " ^ ' 



Having now obtained general values for the coefficients in 

 •4/" x, let us consider a little the limitations of the indeterminate 

 quantity z. In the first place, it is plain that z must vary inde- 



pendently of v ; and in the next, it appears from q = — — 



(ft, + c,f 7 7.1 . 1 >• 3 \ 5 >. 



= - c2 cos a = a r «, — K c r that s must never be -, -y, — , 



See. because then (cos z)- = 0", which, unless b r = — <r,, gives 

 a r d T = — ( oo) 3 , that is, both a r and rf r infinite, the one negative, 

 and the other positive, Thirdly, z must never be a 0, 1 , 2, 3, &c. 



