Defence of the Newtonian Law of Molecular Action. 197 
Clearly therefore the form of the function in this case, and 
where there are no particles of matter, is different, and Mr. 
Earnshaw withdraws his objection that they must be the 
same. I have only to add in answer to the suggestion, “ Perhaps 
the Professor will point out what step of my investigation im- 
lies the existence of the absent particles:” none whatever. 
The symbol = may take in everything. And this puts the 
matter in the most simple light as regards Mr. Earnshaw’s 
inference. If > in a medium is discontinuous, and z2 vacuo 
continuous, then have we the clearest reason why the expres- 
sion does depend on h in the one, and does not in the other. 
The next paragraph has reference to another subject,—nume- 
rical verification. Of course no one considers an error of cal- 
culation as strengthening a theory. And I have already ex- 
plained why the processes employed do so (p. 267). 
3. The paragraph at the foot of p. 440, is a reiteration of 
Mr. Earnshaw’s assertion that he has proved v = Oor 7 = 0, 
&e. As this is of very great importance, the consequences 
being broadly hinted at by Mr. Earnshaw, I deem it requisite 
to state that I find three proofs of it. The first (Phil. Mag. 
for Nov., p. 341) depends on the assumption that v, uv’, and vu! 
are equal, which they are not. The second (Memoir, Art. 8), 
2 
; at Cao sitlene 
on the assumption that G2 dP? which it is not, as I have 
shown above. The third (Phil. Mag. for Jan. 1843, p. 24), 
on the assumption that an exponential function is inconsist- 
ent with the reductions effected by means of a circular one. 
To this I replied in my last*. The objection that v = 0 is so 
important that it ought not to be lightly passed over. If it 
is admitted, then a considerable portion of the writings of 
Cauchy and myself must be incorrect. No one I am sure will 
attach any weight to the arguments which I have mentioned, 
but lest any one should conceive the posszbility of proving the 
function to be zero, I write it down, 
1 38a7\ ., 792 
TOs | | ea 2 
n 23 (4 5 ) sin a 
taken throughout the whole medium. This expression can 
be summed so as to depend on a single definite integral with 
respect to 1, Viz. 
sin k sink 3 7 
= 3(-* fin, Oe fev pass 
k x4 br hex 
Qn 
where k = —. 
A 
* I may add that it assumes the existence of transverse and normal vi- 
brations at the same time. 
