C 2 45 II 
XVII. JVote respecting the demonstration of the binomial theorem 
inserted in the last volume of the Philosophical Transactions. 
By Thomas Knight, Esq. Communicated by Taylor Combe, 
Esq. Sec. R. S. 
Read April 17, 1817. 
I n looking into Mr. Spence's ingenious “ Essay on Loga- 
rithmic Transcendents/’ a work published in 1809, but which 
I have been so unfortunate as never to have seen till within 
the last fortnight, I was not a little surprised to find that a 
demonstration of the binomial theorem, similar to the one I 
had the honour to present to the Royal Society, had been 
already given by that writer. The same may be said of the 
first proposition of the preceding Paper on the construction of 
Logarithms. 
Having made this acknowledgment, I shall perhaps be 
pardoned for observing, that Mr. Spence is not particularly 
happy in the manner of developing the kind of functions he 
treats of in his preface. I shall endeavour to give the solu- 
tion of a class of equations of which he (Pref. p. vii.) has 
considered a particular case : with this we will begin. 
It is proposed to develope the function which has this pro- 
perty, viz. 
<K* +- r )-HU + #-i +y)+?>(i +*-i +*)+ 
<Ki+jM+s)— ?(i+JM+>i+*). 
Assume <p( t-f x )=A-j-A'.r 4 -A".z* A" n x n -\- , 
and after making the requisite substitutions in the given 
