VOL. LXI.] fHILOSOPHICAL THANSACTIONS. 19 1 



4. Taking, in the last article, r and s each = ^,^ = |-, a = -- <i = - i == 1, 

 and c = «S we have 



It appears therefore, that y beine:= n' X ^xr; — T-"^~ "^ ^^ — 



VnM^l^i' IS = imn z- 4 ^ X (-— "^^ " =-^;;==-^^=; 



which, by art. 2, is = the fluxion of the tang. dt. 



Consequently, taking the fluents by art. 1, and correcting them properly, wc 



find DP — AD -f PR — AF = L + dt. 



cp, in fig. 7, being = Vm»z« ; cp, in fig. 8, = « \/ ^; 

 CR, perpendicular to the tangent pr, = v'w^'; 

 DP - AD = the fluent of ^^^fe^^; 



V n' + 2/z — a* . , , 



PR - AP = the fluent of -;^-^-=. ; 



and L the limit to which the difference dp — ad, or pr — ap, approaches, on 

 carrying the point d, or p, from the vertex a ad infinitum. 



5. Suppose y equal to z, and that the points d and f then coincide in e, the 

 points d and p being at the same time in e and q respectively. Then cv b^ing 

 perpendicular to the tangent ev, that tangent will be a maximum and equal to 

 eg — ac= */otJ^~^2_„. the tangent Ea, in the hyperbola, will be = 



'^nF+'vTi the abscissa bc = w \/(\ + -7^-=—) ; the ordinate be = n X 



n " 



V ^ ~,~r^i ; and it appears that Lis = 2Ea — 2ae — ev =n+ \^ m^ + n^— 2ae1 



Thus the limit proposed to be ascertained, is investigated, m and ?i being any 

 right lines whatever ! 



6. The whole fluent o f ; == , erenerated while z from O becomes = m, 



being equal to l; and the fluent of the same fluxion (supposing it to begin 

 when z begins) being in general equal to l + ad — dp = pr — af — dt ; it 

 appears, that, k being the value of z corresponding to the fluent l + ad — dp, 

 —^ r will be the value of z corresponding to the fluent l + ap — pr, and 



PR — AP will be the part generated while z from — r"T"~r becomes = ?». It 



follows therefore that the tangent dt, together with the fluent of -^=1===_, 

 generated while z from becomes equal to any quantity ft, is equal to tl^e fl,uent 



"t 



.V *".v. 



