388 PHILOSOPHICAL TRANSACTIONS. [aNNO 1/88. 



and assume the equation / X ,, ^^ — -r + /' X ,, ^, r — r-r + /" X 



^ -^ ^/(ms* + sp*) — *^ VCm s^ 4- s'p') —J ^ 



M S , r-f,, 



■7(;^V-T1^ ±/" X ^(^-s"- + s'"p^) + &c. = O, and the two preceding 

 equations ^ =/x — ± /' X ^ ± /'' X ^, + &c. and ^^ =/x ^ + /' 



^ PO '^ PM *^ PM "^ PM — A* *^ PM —^ 



X — , ± /" X ~7 ± &c. ; from the data may be found all the quantities f"\ 

 f""i &c. ; and consequently from the above-mentioned equations may be deduced 

 the forces y,y', andy ^'. 



4. Let the body move in different planes, that is, in a curve of double curva- 

 ture at the same points; draw pr a tangent to the curve at the point p, and pa 

 an arc of the curve of double curvature; draw also two planes prv and prt, 

 cutting one another in the line pr; from the point a let fall qv and qt perpen- 

 dicular to those planes respectively, and from the points v and t draw vi; and Tt 

 respectively perpendicular to the line pr; let v be the velocity of a body moving 

 in the given curve at the pomt p, and assume — = 2c and — — = 2c' respec- 

 tively; from the given centres of forces m, m', m", m'^', &c. draw ms, ms', m'^s''', 

 m"V^^ &c. ; M*, mVj m"^'''', m'V, &c. respectively perpendicular to the two planes 

 Rpv and rpt; and pl and p/ perpendicular to the line pr in the same two planes 

 RPV and rpt; and also sp, s'p, s'^p, s'^'p, &c.; jp, /p, s"y^ &c.: from the points 

 s, s', s'-', s''', &c. s, /, s", &c. draw the lines sh, s'h', s^'h'-', s'^'h'"', &c. sh, sk\ s"h", 

 t"'k'", &c. respectively perpendicular to the lines pl and p/; and sk, s'k^ s'k'''', 

 h'"Yi"\ &c. sk, s'k\ s"k"i s'"k"\ &c. perpendicular to the line rp; and let the forces 

 f'",f"'\ &c. tending to all the points m"'", m'"'", &c. (except three, m, m', andM''') 

 be given; then from the three given equations — = — X / ± ^ X /' ± -5- 



X/^^+&c.and^! = -^X/+^ X/'+ ^ X /'' + &c. and Ili^ = ^ X 



•' ~ C MP*'~'MP-'~MP-'~ A' MP 



/ + Lf: x/ + ^' x/" + ^' x/" + &c. = ii X/+ i:^ X/+ 4^ X 



-' — MP '' ~ MP •'—MP •' MP -'— MP -^ — MP 



/'''' + &c. which contain only three unknown quantities, can be deduced the 

 forces/,/', and/', required, tending to the points m, m', and m''. 



Prop. 5. — Let a body, acted on by forces tending to any given points s, s', s'''', 

 &c. move in a given curve; to find its velocity in any point of the curve. — Find 

 the fluent of the fluxion {f x ^ -^ f X% + f" X % + f" X^-, -\- &c.) ; 



V-' PS ~ -^ PS ~ -' PS ~ -' PS ~ ^ 



= /d ± /i*' ± fa" ± &c. = — t'v when the forces are all contained in the 

 same plane; or the fluent of (/ x ~^' + f X — ; + /' X -^ + &c.) X 1 



r ' ^-^ PM — -^ PM ~ -^ PM ~~ / ^^ M. 



(when contained in different planes) = / x pm ± / X pm' ± /'' X pm'''' ± &c. = 

 f X ii ± f X li ± f" X h" ± &c.; but since/, /, / , &c. are given functions 

 of the quantities d, d', d'^, &c. the fluents of /X d,/' X n'if" X h", &c. can 

 be found ; which, when properly corrected, will be as 4-v^ = 4- the square of the 



