VOL. XXVI.] PHILOSOPHICAL TRANSACTIONS. 439 



let Aa be the nautical spiral ; then because the angle sap is given, the ratio of 

 SA to sp will be also given : let that ratio be a to ^ ; then will 



bsA J • 6SA 1 SA X SA aSA TT •.. 1 • 1 



SP = — , and SP = — , and ar = : — = -r-. Hence it plainly ap- 



a a SP *• 



pears, that the evolute of the nautical spiral, is the same spiral in another po- 

 sition. 



Because ar = r-^, it will be — ; = r. And hence ae:ain, 



SP SP' X AR sp» X SA ^ 



from the given relation of sa to sp, the law of the centripetal force will be easily 

 found. 



Example. — Let vab, fig. 8, be an ellipsis, whose focus is s, the greater axis 

 VB = Z), the less axis z=zld, and latus rectum = 1r. And let vaa be another 

 curve, so related to this, that the angle vsa may be always proportional to the 

 angle vsa ; and let sa = sa. Then it is required to assign the law of the 

 centripetal force tending to s, by which the body may move in the curve 

 vaa. 



Because the angle vsa is to vsa in a given ratio, the increments of these 

 angles will be in the same ratio, which let be the ratio ofw to w: hence it will be 



« Tj i ^'SA , nrfSA 



oi = -OT. But OT = -TTT 1 — TT? ; hence ot = — —n ; — ir^* 



m ^ (6sA — SA* — d*) ' mV (6sa — sa* — d*) 



And because sa"* + sp'^ : sp* :: ta^ + ot^ : ot^ :: sa^ -f- 



m^{bsA - SA* - (P) • m\bsA — sa^ - d») " m*(6sA — sa* ~ d^) ' m^ (fcsA — sa* — iP ) ** 

 m*(^SA - sa'^ - ^) + n'd^ : n^(P; hence /w^ (Z^sa — sa* - d') + w*rf* : nd:: 



SA : sp = v^'^(6sA - sa'1 d^) + nH-' '^^ ^^^ ^"^ ^xxxAon of this, put X for 

 m\h%K - sa* - t^) + n*^* ; then will sp = "4^, and sp^ = ^^' : hence 

 X = m*^si - 2m*SA.si, and sp = w^sa^— ^ - \nk%xx "^ = 2;»rfsl;r - nrfsA^ 

 by reducing to the same denominator. And substituting the values of x and .r, 

 and reducing to order, it will be sp = wcfsA X ^ ^^ " Tz "*" » hence it will 



1 SP im*bsA — m*d* + n*d* -n ^ ^t r r ^l i • i 



be r = TTi — z . But the former of these being as the 



SP3.SA " " SA* *= 



centripetal force, therefore the latter is so too: or, because n*c?* in the deno- 

 minator is a given quantity, that force will be as ^*^* ^^ '"^* +^' \ ^^^ p^^^ 



*'^ -k^r for d*, and omitting the given quantity -^b, the force is as 

 ro*SA--m* + rw' »n« , n»* — m* .„ , . , ,, . ., .^, t_ ^ • 



^ ^ sl» "^ — r"3 — * which exactly coincide with what is 



iCB od im i. .xM. ai ^a#o ihsn Qvag do 



