74 Overflowing Well at Chiswick. 
contains a sufficient quantity of carbonate of soda to render it very 
manifestly alcaline ; this is also the case with the waters of the 
other deep wells in and about London. 
J. L. 
Art. IX, On the Taylorian Theorem. 
[To the Editor.] 
Trinity College, Dublin, 
October 11, 1823. 
Sir, 
I lately communicated to you a demonstration of the Taylorian 
Theorem given to me at lecture in this University by Mr. Edward 
Wilmot, a gentleman-commoner, and under-graduate of this col- 
lege. The same gentleman has since given me an extension of 
this to functions of several variables, which I now enclose. The 
simplicity of the proof, and the elementary nature of its principles, 
must render it very valuable to the student. You will observe 
that it is independent of the Theorem of Maclaurin, and gives it as 
a corollary. It is also free from the functional reasoning so ob- 
jectionable in other proofs of this theorem. 
I am, Sir, Sc. Sc., 
Dionysius LARDNER. 
Let u = F (2,2',2",....+), , 2,0", §c., being independent 
variables, 
Let =F (r+ Aaa’ + Ax, a" + AB", ....0-) 
And let this be supposed to be expanded according to the di 
mensions of x + A a, Sc. 
w= A(x + Ax) + A’(2’ + Ax’) + A’ (a" + Aa")....- 
+ A, (a+ Ax)? + Ay (u’+ Ac’) + A," ike ey alee 
+ B’ («+ Ax) (a’+ Ax’) + B (w+ Az) ("+ Ax’) + B 
Ce ACE AE) i avetis's 
+ A, (z+ day + Ay (2'+ Avy + A,” (2° + Ax’) ytite. 
