ioo Mr. Woodhouse on the Truth of Conclusions 
pressions ; for the same series results, whether the terms of the 
developement for e x v ~ \ e— x ^ \ ey v — 1 be connected to- 
gether after the manner pointed out by the signs -f- — , and 
then combined with A, or whether A be first separately com- 
bined with each term of e x — *, e~ xV — l , e y v — 1 i and then 
the resulting terms added together : again, e x ^ — 1 x J , 
and + — \ are equivalent expressions; for the same 
series results, whether the terms for the developements of 
e xV — 1 and — 1 be connected together after the manner 
of quantities in multiplication, or whether e( x + y ) v — 1 be im- 
mediately developed, by putting (a: +y) >/ — i for a:, in the 
series i 4 - x -] — - — \ — f & c. 
1 1 1.2 1 1 . 2.3 
for e xV ~ l is the symbol for 1 -{- x V — 1 — ~ 4 " & c - 
y V— 1 - s for 1 -f- y s/ — 1 — ^ ^ 4 “ & c * 
e x ^ ~ 1 x ey ^ ~ 1 (the symbol x indicating that the several 
terms are to be connected together according to the rules of 
multiplication} equals 1 + (•£-(- y) — 1 — ( ) — &c. 
which series is abridgedly expressed by the symbol x/ ~ 1 - 
e xV ~ 1 x e yV ~ l , and <?(■* + ?) ^ ~ % are symbols alike sig- 
nificant; or, since it must now be evident in what sense the 
equality of imaginary expressions is to be understood, 
e x^~ x e yS~ — e (*+y) 
After this explanation of the nature of the operations directed 
by means of certain signs x,-i-> &c. to be performed with the 
symbols e xV ~ l , e yV &c. the following propositions may 
be clearly and strictly proved; 
