in Mechanics and Physical Astronomy,

and A' cos. 7 r z, to the respective forces P sin. pz -j- QJin.gz &c. and P' co i.pz -j- Q' cos. qz &c. whereby the motion ofthe body is disturbed. What this alteration, or correctionought to be, is easily discovered from the general equation onp. 151 ; from whence, by substituting tt, A, and A'instead ofp, P, ahd P', respectively, the new terms affected by 7 r, en-tering into the said equation, will appear to beA .. 2 , A'

1 A

-V.

+

x

x cos.

7 T Z

7T—3

4-3*

r 2 A

i-t

w+3

4-i* 4-

“ 2A ^

3

i-(3)3

AB

e*f

AB

>/

X cos. 7T - [3. Z

X cos. 7 r-]-/ 3 . z. Scc.

and consequently the increase in the value of i-

Cos. 7TZ

-W an

sing

therefrom — e into

A_

A

x

-h

A

X

_f-L

«r—3 ^

3*2A

i -33

9T

X

AB cos.w-X ——

■3.*

-V

+

+

3 A'

2 A

4- ^

1 2

-33

AB

x w x

cos. 7T-(-3. Z

4- &c.

W +3 1 2A I -2 it ^ e\p n 1 —^ 43 s

where, in many cases, the first term alone, will be sufficient.

In making the different corrections above pointed out, itwill be found necessary to have a particular regard to suchmultiples of the arch z, as are, either, very small, or nearlyequal to unity (of which two kinds, those whose exponentsare^>—2/3, and p —/3, will be found the most considerable.)For, though the coefficients of such terms should appear, atfirst, to be small, they ought not, therefore, to be immediatelyrejected; because the divisors which they afterwards receive(the former in obtaining the value of 1 —w, and the latter, infinding the anomaly from thence) are such as may render theeffect, or quotient, too considerable to be intirely disregarded.

And it may be easily conceived, without the help of cal-culation, that a term, or force of the former kind, expressedby the sine or cosine of a very small multiple of the longitudez, must necessarily have a much greater effect, than another(having the lame coefficient) which is proportional to the sineor co-sine of a large multiple of the same angle : because, when

the

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