90 Mr. Davies Gilbert on the 



Art. XIII. On the Vibrations of Heavy Bodieu in Cytloidal 

 and in Circular Arches, as compared tvith their Descents 

 through free Space ; including an Estimate of the Varia- 

 ble Circular Excess in Vibrations continualli/ decreasing. 

 By Davies Gilbert, Esq., F.R.S. &c. &c. &c. 



To the Editor of the Quarterly Journal of Science and the Arts. 

 Dear Sir, 



I AM really not able to determine in what degree the follow- 

 ing investigations may be thought worthy of attention. They 

 were made about twenty years ago, and the impression left by 

 them on my mind mainly contributed to my subsequently mov- 

 ing the House of Commons, on the 1.5th of March, 1816, for an 

 Address to His Majesty, praying that directions might be given 

 for determining the length of the Pendulum ; which has led to 

 all the important theoretical and practical discoveries of Capt. 

 Kater, and to the highly interesting observations of Captain 

 Sabine, and of others : on this account, at least, I may be ex- 

 cused for laying them before the public. 



They exhibit a curious integration, by which a very simple 

 result, conformable to that of Euler, is derived from a great 

 apparent complexity. 



The correction for variable circular excess in a free pen- 

 dulum, beginning its vibrations from an arc comparatively 

 large, and ending with one very small, differs from those already 

 given by mathematicians ; but the deductions seem to rest on 

 solid principles. 



The whole possesses one quality throughout, which, in my 

 opinion, has not been sufficiently regarded ; and that is, a strict 

 preservation of the Harmonia MensuraruiM. 



I have constantly used the words Fluxion and Fluent, not- 

 withstanding that I am fully satisfied with the acknowledged 

 superiority of the new method over the old ; and that the deve- 

 lopement of functions is far preferable, as a general principle, 

 to considerations of motion ; but there appear to me no 

 stronger reasons for changing established expressions, or no- 

 tations, on that account ; than might be supposed to exist for 



