42^ Exponential and Logarithmic Theorems. 



(when put under the form (6),) than the formulae deduced (as 

 above) from (9). Before proceeding further, it will be conven- 

 ient to premise the following principle, viz. if we have the ex- 

 pression \-{-kz-{-^z'^ -{-Qz^ -{-^c. where A, B, C, &c. are quan- 

 tities that are independent of ^r, we may evidently (without sup- 

 posing the solution of equations,) assume \-\-Kz-\-]iz'^ -^Gz^ -\-&ic. 

 — il-\-az).[\-\-hz).[l-\-cz) x&c, {a'), supposing the number of 

 factors in the second member of the equation to be equal to the 

 index of the highest power o( z, in the first member of the equa- 

 tion ; for by actually multiplying the factors together, the second 

 member of the equation has the same form as the first member, 

 and since the equation is to be identical, we must have a+i-fc 

 4-&c.=A, ah-\-ac-\-ad-]-&LC.-\-hc-\-hd-\-&LQ,.~B^ abc-\-ahd-\-&LQ,. 

 -\-hcd-\-&cc-. =C, and so on, {b'). Since [a') is true for all values 

 of z, we may change z into z-\-y, and we shall have \-\-Az-\- 

 Bz^^Cz^ + &,c. + [A + 2B^ + 3C;22 + &c]y + &c.=(l + a;^). 

 (1 + 6;^) . ( I + cz). &c. + [(1+ hz) . ( l + c^).&c.]ai/ + [{\ + az). 

 {\-\-cz).&LC.]by-]-[{l-\-az).{\-{-hz).&.c.]cy^&Lc.^ or reducing by 

 (a'), and equating the co-efficients of y, (since the equation is to 

 be identical,) also dividing the two members of the result- 

 ing equation by the corresponding members of (a'), we get 



A-{-2B;z + 3C;g^+ &c. a b c 



1^ Az + Bz^ -^Cz^ -Y^.^ 1+ az'^ V+ bz^ I -\- cz^^^-' ^^'>' 



Since a-{-b-\-c-\-hc.—k, if we put a^+i^+c^ -f &c. = Q,, 

 a3^j3^c3_|.j§^c.=R, a^4-6*+c''-|-&c. = S, and so on, and then 

 convert the fractions in the second member of (c) into series ar- 

 ranged according to the ascending powers of z, and then multi- 

 ply by l + A2;4-B2r^+&c., we shall get the identical equation 

 K+2Ez-\-ZCz^+iViz^ +&C. = [l + A^-|-B^^-fC;r=' + &c.] X 

 \k.—Q,z-\-'Rz'^ — ^z'^-\-&LC^^, .'. performing the multiplication 

 and equating the co-efficients of like powers of z, we get (after 

 slight reductions) A=A = a -I- 6 + &c., Q, = A2-2B, R=ACl — 

 AB-^30=-A3-3AB+-3C, S = AR - Ba + AC-4D = A* - 

 4A2B-1-2B2+4AC — 4D, and so on, (e), whose law of continua- 

 tion is evident; these results agree with those given by Euler at 

 p. 129 of his Analysis of Infinites. 



We now observe that if in (10) we put N=e= the hyperbolic 



x'^ x^ a;* 



base, since L.e = l, we shall have e' = l-f-:r+r2 + T"2^+ i 2 3 4 



-f- &c., (/), and if in this we put successively x=v\^ — \, 



