1826.] Solutions of the Function ^' x. 169 



2. On the latter of the two hypotheses, /' + ' a:- being equal 

 both toff'' X and/-'/x, we obtain by actual substitution, the 

 following pairs of equal values : 



fl=+^ = a^b + a c, = a b^ + a^ c (1) 



Z>^ ^, = bb, + a d^= b b^ + a^ d (2) 



c.^^ = c c. + a. d = c c. + a d^ (3) 



£^, = b,d+ cd, = bd^ + c,d (4) 



From (2) or (3), ad. = a, d. Assume, therefore, 



a^ ■= a s. .• . d._ = d s^. (5) 



Consequently a, ^., = as,^, and d^j^i — <Zs,+, 



Substitute these four values in either (1) or (4), and we have 



s,^ , = b s, + c, = 6, -r c s, (6) 



Collecting the results of (5) and (6), we now have 



a, = as,, b, = s,^, — c s, \,^. 



d. = ds,, c, = s,^., - bs j^ ■> 



and consequently, 



p ^ _ ""= + <^^+i -cs.)x g 



"^ " (s. + i — bs') + ds- X V / 



or, in a form better adapted for simplifying the results, put q, = 

 ^^, which gives 



J '^ - (q. - b) +dx ^^^ 



3. The solution, we see, hinges entirely on the quantities s ; 

 whose values may be found as follows. Comparing together 

 (2), (3), (7), we find 



K + ^ -bs,_^.,-ibc—ad)s, 

 c, + , = c s.^i — {b c — a d) s^ 



and if these be substituted ins.^^ = 6s^+, + c,+ ,, or i.^i + 

 c s, ^ „ which is in the next gradation to (6), the result is, 



s,^, = (b + c) s,+ , — {b c - a d) s. (10) 



so that the quantities s in question form a recurring series, 

 whose scale of relation is + (b + c), — {b c — a d). 



Now we already perceive that the terms of this scale, which 

 for conciseness we shall denote by A and — B, are not altoge- 

 ther arbitrary. The purpose o^ di general solution will be defeated, 

 for example, if either of them vanishes. For, first, let i c — ad 

 = 0. Thens,.^, = (^ + c) s^ + ,, which produces ^, = i + c, a 

 constant quantity, and, therefore/' x = fx, a constant quantity. 

 In fact, the datum itself gives 



f X = -, comtant ,f ,,,,,,,,,, , « • • (1 1) 



