VOL. LXXXVI.] PHILOSOPHICAL TRANSACTIONS. 705 



= 1 4- AX + — - a- + — ^^ 23 4 '^^— 2+ H ^^ z' + &c. SO that this 



_i-t-A^-r^2~l.2.3'l.2..4. ^1.2... 3 ' 



series is the limit of the binomial (l + — )". And hence also a^ is the limit of 



the same binomial (1 + — )'; and a-» is the limit of the binomial (1 — ^Y; and 



the quantities ° ~° , are the liniits of the quantities (l + a ^)" + (l— A^)«. 

 ^ 6. To pass from the expression of x to a and a. 

 Since a« = 1 + as; + ^— ^a' + 77^ «3 + &c. 

 Let s = nAs ; then we also have 



A* ' a' 



and ti" = a"^'- = (a-^-)" = 1 + AAa + y;^ As'' + YT^Ta ^^^ + &c.= 1 +v. 



Hence aAz + 7^ Ax^ + t-^ ^*^ + &c. = (1 + i;)" — 1, 

 and A77A^ (1 + Vh ^~ + TTTrH A*' + &c. = n ( (I + v)n - 1) 



I — I — 2 - — 



= ^ - 772" "^ + TT^ 



Now suppose n log. (l + aAz + 775^2' + ^ .^ _ 3 ^z' + &c.) = log. 

 (1 + i;), or, n log. n''^ = log. (1 + w); and «Az log. a — log. 1 + v; or, making 

 a the base of the system, and thereof log a = 1. 



A log. (1 + ,.) X (1 + Y^ A2 + ^ Az^ + 7^ Az^ + &o. 



2 



,3 



= v- -^^v- + 17^ .-^ t^' - &c 



A2: AiO 

 I 2 



= w — -775- z;''+ 7^ . -3- v^ — &c. 

 This equation always taking place, it takes place in particular between the 

 limits of its members; which 



are a log. ( 1 ■\- v), and t; — i t;'' + i w^ — i t;* + i w' — &c. 



thereof; a log. (l + t^) = v - \v'' -^ \v^ - \v^ -^ \v' — h.c. 



A log. 1— t)=~ V — iiV^ — \V^ — ^V" — \V^ — &C. 



A log- ^ = '^(^ +Tt^' +ij;' + &c.) 



Alog. '/~^= '^ +1^' +-'t;' + &c. 



-2 



Let 1 + t; = « ; the base of the system 



A = (a - 1) - -i- (a - 1)' + T (« - 1)' - 4- (« - 1)' + &c. 

 _ aa- I 1 .^'^ - 1x3 . i /«" - '). + &c. making i±^ = a^ 



Which is the relation by which the modulus is determined in the base. 



§ 7. Mr. L'H. does not stop at the consequences that flow from these known 



VOL. XVII. 4 X 



