16 Prof. De Morgan- on Continued Fractions. 



'6 



a 9 x a.^x a a bx ex a 



-, &c, or - -— — - — ~, &c, 



*1+ ^2+ h+' *' 1 + 1 + 1 +' 



to the second of which the first may be easily reduced. In 

 reasoning it may be proper to use a v a 2 , a 3 , &c. ; in working, 

 a, b, c, &c. will be found more convenient. If 



a a . , x a . , a , n x 



A „ = rr -iV' &c " A «+i - tt m~! &c " 



we have A n (1 + A M+1 x) = a n . If then/(a TO+1 , *># &c.) 



be called the advanced form oi\f (a , a n , &c), and if A be 



taken to be P + P 1 x + P 2 x* + , &c. ; and if Q , Q 1? &c. be 

 the advanced forms of P , P x , &c, we have, from the equation 

 between A l and A 2 , 



P = a v P, = P Q , P 2 = P Qx + V t Q , &c. 



P W + 1 = P Q n + P ! Q.-1 + - + P n-1 Ql + P « Qo, 



which gives an easy law of formation for a few terms. Thus 

 we have, using a> b, &c. for a v ff 2 » &c, 



P = a, P x = a b, P 2 = a (b c) + a b (b) = a b c + a & 

 F 3 = a{bcd + be*) + ab\bc) + (abc + aW)b, 



= abcd + abc 2 -{-2ab' 2 c + ab 3 , 

 P 4 = abcde + abed 2 + < 2abc i d + abc 3 + 2 a Wed 

 + 3a6 2 c 2 + 3ab 3 c + ab 4 . 



The results of this method would give little encouragement 

 to attempt finding the law of these terms, which is, however, 



very simple, as follows : let such an expression as ab c y d ... 

 in which the order a, b, c, d, §tc. is unbroken, be called conse- 

 cutive ; and let m denote the coefficient of the mth power of 



x in the development of (1 + xf\ then will the coefficient Pjbe 



the sign 2 extending to every way in which /3 -f y -f 8 + ... 

 = I, on condition only that every term shall be consecutive, 

 that is, that no one of the set /3, -y, &c. shall vanish, unless all 

 the subsequent ones vanish also. This law will be evident on 

 a very slight consideration of another mode of development, 

 namely, that of a ■+■ (1 + bx\ followed by the substitution of 

 b ■+■ (1 + ex) for b, followed by that of c ■+■ (1 + dx) for c, 

 and so on. 



The number of terms in P. must be 2 4- , since they are 

 formed by writing over b, c, d, &c. exponents /3, y, &c. in 

 every possible way and order in which i can be /3 -f- y + ... 



