Centripetal Forces. 
as above .of SP — s/ x +y\ whence the fluxional equation 
yxdzxj/ 
i/ 2 i * 2 
(p (x -\-y ). 
2 . Fig. 8. Let a body be afted on bv nnv number of forces 
See.) in the fame plane tending to the given 
points S, S , S 7 , S" , &c. ; to find ail equation exprefling the 
relation between SP = D and ST = D 7 , and their fluxions, 
where P is a point fltuated in the curve which the body 
deferibes. 
Suppofe YP a tangent to the curve at the point P, and PZ 
perpendicular to it ; and refolve all the forces tending to S, S'* 
S 7/ , &c. refpe&ively into two others; one in the direction PY, 
and the other in the direction PZ ; fubflitute for SP, ST, 
S'T, S //7 P, &c. refpeftively D, D 7 , D 77 , D 7// , &c. ; and fuppofe 
SY, S 7 Y 7 , S /7 Y 77 , S /77 Y /7/ , &c. perpendicular to the line PY; 
then will the triangles PQT and SPY, PQ'T 7 and STY 7 be 
fimilar, where PQ denotes a very fmall arc, and QT and 
QT 7 are perpendicular to the lines SP and S 7 P ; hence PQ=^ 
PT x SP _ D x D 
PY PY 
PT' x S'P D'xD' 
• 
PY' * — ' PY 7 ’ 
and confequently PY : PY 7 
:: DxD : D 7 x D 7 ; and if the quantities D, D 7 , D and D 7 
are given, the ratio of PY : PY 7 will be given ; which being 
given, together with the line SS 7 zrtf, the lines PY and PY 7 , 
SY and S 7 Y 7 , can be found ; for, drawing SL parallel to PY, 
and meeting S 7 Y 7 in L, let PY 7 = w x PY, then YY 7 =* 
(w±i) PY = SL, SY = v /(SP 2 -PY ; )=: v /(D 2 -PY i ), S 7 Y 7 
= (S 7 P 2 - PY 72 ) = (D 72 - x PY 2 ), LS 7 = S'Y'^SY » =t 
v /(D /2 -^PY 2 )rt: v /(D 2 ~PY 2 ); and SS , * = SL a + LS'* an 
equation in which all quantities (except PY) are given, and 
confequently PY is determined by an equation, which will be 
a quadratic ; but PY being found, from thence PY 7 , SY and 
S 7 Y'- 
