550 
ME. B. GOMPEETZ ON THE SCIENCE 
and where, as will generally be the case, Aj, A2, A3, &c., or B,, Bg, B3, &c. are a series of 
terms which are very convergent, a very few terms of the said coefficients -^511 be found 
necessary, which is a very advantageous circumstance for our purpose. 
Another mode of finding the anti-Napierian logarithm of the expression 
Ai^ + + A^x^ + &c. 
is to consider it 
Af^'’+&c.^ 
X 1 + A2.r^-j-^A2^^+&c. X l+A3a’^+-|A3.2;®+&c. x &c. ; 
=c-AiXX<^A2X^Xcr-AsX^x, &c.= ^l+Ai^fi- 2 
this being multiplied out at length would give what might be considered an easier way 
than the last for effecting our purpose, if many terms were thought requisite. But an- 
other mode of finding the anti-Napierian logarithm of Ai^+A2^^+&c., which would be 
the anti-Napierian logarithm of Ao-fAi^-{-A2;a?^+&c. if Ao= 0 , and therefore Bo=l, 
would be by result of the equations of A,^+A2 ^p^-1-&c.= the Napierian logarithm of 
l + Bi^+Ba^^ffi&c., by putting the equation in fluxions, which would give 
A, -{■ 2 Aa^’-j- 3A3^^+ &c. = 
B j + + SBga:^ + &c. 
1 + Bja; + -f &c. 
this will give the following equation for finding Bj, B2, &c.. 
A,+2A2^ ■\■^ A ^ x ^ +&C. 
-f A,B 2 A2Bi.r^ -j - 3 A3Bi.r^ + &c. 
-1-A,B2^^ -l-2A2B2a;^+&c. -=0; 
-f- AjBg^* -j- &c. 
— Bi — 2B2^r — 3B3^^ — AQ^x^ + &c. ^ 
and consequently 
Bi=A, ; B2=A2 +^AiBi; B3=A3+fA2Bi+|^AiB2 5 
I^4=A4+f A3Bi+f A2B2 +|^AiB 3 ; B5=A5+fA4Bi-}-f A3B2+|-A2B3+yA,B4? &c., 
where B,, Bg, B3, &c. are found respectively one after the other by a very simple law ; 
and this method may also, if many terms be thought requisite, be preferable to the first 
method, and the equation reversed would give 
A,=:Bi, A2=B2— ^AjBj, A3=B3— fA2Bi— IA1B2, A4=B4— fA3B,— fAgBa— :^AiBi, &c. 
N.B. The calculations which these sheets required were laborious, and necessitated the very frequent, in 
fact the continual use of a table of Logarithms, and the searching for the Anti-logarithms ; and I feel 
pleasure in stating that I found the last edition of Anti-logarithms, published by Mr. H. E. Eiltpowskt, a 
very serviceable assistance, and I have no doubt that calculators in the same path must be glad to possess 
these Tables. 
