J $ 
Centripetal Forces . 
In thefe and the fubfequent cafes the lines Py, Py', P y", 
&c. are to be taken negatively or affirmatively, as thev are 
fituated on the fame or different fides of P; and in the fame 
manner the lines P/, P P &c. are to be taken negatively 
or affirmatively as they are fituated on the fame or different 
fides of the tangent yVy' y &c. 
3 . Let the centers M, 3VT, M", M'", See. of forces be 
points not fituated in the plane of the given curve HPI, Sec. 
and the forces f"\ j , &c. tending to each of the centers 
M //7 , M"", &c. (except three M, M', and M ') be given ; to find 
the forces/, and J" tending to thofe three points M, 
and M // . 
Draw MS, M'S', M^S", See. perpendicular to the plane 
HPI, &c. from the above-mentioned points, and afl'ume the 
MS _ r, M'S' r// M"S" 
: / X — — =r=— — / 
J \ l'C'2 1 c m2 J 
equation f X — 
=*=/'" X 
Ms -f- SP 
M"'S'" 
v'M'"S'"* + S"'P 
• r P/ . r / P/' /*, 
tions — = / x — — / x — - =+=/ 
PO ^ PM ^ PM' 
VM'S ' 2 + ST a v' M"S ' 2 + S' F 2 
+ &c. rr £ 9 and tlie two preceding equa- 
P r __ P __ , — r Pv 
■// 
PM' 
&c.and — =/x - v -<- 
A- ^ PM~ 
/*' x x -£f_r±;&c. ; from the may be found all the 
^ PM' ^ PM" y 
quantities / v// , v / //// , &c. ; and confequently from the above 
mentioned equations may be deduced the forces f f' y and \f'\ 
4 . Let the body move in different planes, that is, in a curve 
of double curvature at the fame points; draw PR a tangent to 
the curve at the point P, and PQ an arc of the curve of double 
curvature; draw alfo two planes PRV and PRT, cutting one 
another in the line PR ; from the point QJet fall QV and QT 
perpendicular to thofe planes refpe&ivelv, and from the points 
V and T d raw Vv and T7 refpectively perpendicular to the 
line PR ; let v be the velocity of a body moving in the given 
Vol. LXXVIII. L curve 
