Dr. Rittenboufe, to Mr. Patterfon, relative to a method: 
of jinding the fum of the feveral pawers of the Sines, Gc. 
DEAR SIR, 
Read May 9 Had difcovered a very elegant theorem: for de- 
*8, 279% A termining the times of vibration of a pendulum: 
in given arches of a circle; but it included a problem the: 
folution of which I do not remember to have met with, 
though I cannot fuppoit that it has efcaped the notice of 
mathematicians. It is, to find the fums of the feveral 
powers of the fines, cither to a radius of unity or any 
other. : 
I was induced to attempt the means of doing this folely 
by its ufefulnefs, but in profecuting the enquiry I found 
much of that pleafing regularity, the difcovery of which 
the veometrician often thinks a fufficient reward for his 
labours. 
The fums of the odd powers of the fines bear a very fim- 
ple relation to each other, and fo do the fums of the even 
powers. But all the fums of the odd powers are incom-= 
menfurable to all thofe of the even powers. 
If we take the radius equal to unity the fum of all the: 
fines, or their firft powers, will be=1, and the fum of all: 
their fquares= + multiplied by the archof go*. The fum 
of all their cubes is=*, and the fum of their fourth pow- 
ers=: multiplied by the arch of go’. The fum of the: 
fifth powers is= ,*,, and the fumof the 6th powers= J, x 
by the arch of go’. 
I have not been able ftria&tly to demonftrate any more 
than the two firft cafes. The others were inveftigated by 
the method of infinite feries fo far as to leave no doubt of 
U2 the 
