of Exponential Equations, 327 



— , which conies under the form of the first order of the former 

 class ; and, therefore, by comparing the two, we have z = 

 Mc+L ^• 

 - -, ■ , (where c= — = approximate value of z, and <;onse- 



quently h oi x). By substitution z is therefore equal to 



M-^ +1,— M + l1 



h_ « , -_ i. _ ^ _ Z'(M-Lt) 



M+1. — — +L- - 



b ha 



X. 



Example. Let.r^=l*17. Leti'=l*2; then we have 



l-2(-1884044 — 182S'216) -0072994 nnonn j .u 



V= — -;—- rT-p^T:; ^ = -„,,...;; ='00899, and then 



^ I — lS-^4044 -SI 1 5956 ' 



x= 1*20899, which only errs in the last figure. 



It is evident that this class may be easily converted into the 

 second ; and tlius tiie following solution of the second order may 

 be, shortly though indirectly, obtained. 



(x ) ^"^ ^ 1 (x ) ^^ 



Let X —a, then ( —) = — . Let — = z, and c = 



- = an approximate value of z : then by comparing this whit 

 the second order of the first class, 



M-i+^^: + L"l-L"l 

 JLf b 



M+LC+- M+L- + - 

 quentlv X— — = rr ^^ — . 



b L,b a b 



When the order is infinite, this class admits of an easy solution. 



■ &c. 



Thus x" Lo, then ( \)^^^ = 1 , and ( i )^ = i , therefore 



Before finishing this subject, I may mention that several of 

 this species of equations are subject to the laws of maxima and 

 minima. To illustrate this I shall take an example. 



Thus if .r* = o, then a or .r" admits of being a minimum. By 



putting the ratio of the differentials =0, then f!!£lii±i!^) ==0, 

 or HLx=-l.=0»l = HL^-:J^, and x= ^^ ='36788; 



X 4 an 



