[ 5^6 ] 
P. S. Since the writing of the above, I have been 
inform’d, that, in two or three houfes, finging 
birds, which were at the time roofting in their 
cages, were thrown off their perches by the fud- 
dennefs of the {hock. 
LXXVI. A?i Invefligation of fo?ne Theorems 
which fuggefl Jome remarkable ‘Properties 
of the Circle , and are of Ufe in rejohing 
FraSlions , whoje Denominators are certain 
Multinomials , into more fimple QJies. By 
Mr. John Landen. 
Read May 2, HAT the principal theorems, bc- 
I 754- low inveftigated, will be of con- 
fiderable ufe in the dodtrine of fluxions, by render- 
ing, in many cafes, the bufinefs of computing fluents 
more eafy, will, on perufal, be obvious to every one 
acquainted wi$h that branch of fcience. Therefore, 
as the facilitating computations in that dodtrine (which 
affords us wonderful affiflance in many phyfical en- 
quiries) may be a means of extending our knowlege 
in philofophy ; it is prefum’d, that this paper will 
not be thought unworthy the notice of the Royal 
Society. 
I. 
Suppofing — —=.= — ===, where x and dc- 
n " Vx 1 — 1 Vy*—i 
note the fluxions of the variable quantities x and y 
re- 
