648 " PHILOSOPHICAL TKANSACTIONS. [aNNO J 775. 



pears, that in the ellipsis aed (fig. 1 2) whose semi-transverse axis cd is = m -\- n, 

 semi-conjugate ca = 2\/ mn, and abscissa cb (corresponding to the ordinate be) 



=: ~ — - t; the arc ae is equal to the fluent of -/," ^r—r .« X i. 



m — n ^ (m — w)* — r 



3. In the ellipsis aefd (fig. 13), the semi-transverse axis cd being i= m; the 

 semi -conjugate ca. := n; and the abscissa cb (corresponding to the ordinate be) 

 = a:; if ep, the tangent at e, intercepted by a perpendicular (cp) drawn to it 

 from the centre c, be denoted hy t; gx X v/^-^ — 5 (as is well known) will be 

 = /, g- being as in the preceding article. 



Hence aP = -^—^ \/ ^ ^ — - — ^~ . From which equa- 



2g _ 2g ^ 



tion, by taking the fluxions, we have, xx 



But i: being = */ "* ~ ^1 X i, as observed in the preceding article, it appears 

 that ^ X a^'i" is = z. It is obvious therefore that i is = -J-< -|- -i X 



, ^ (*»'+"•)• Xt -<'/ _ , 1 , ^^ (ct-«)' X i-^i 



Vliim - n)^ - f) X ((m + ny - 1')-] ^' ' * '^ ^^m - n)' - t') x {(m + n)' - t')"] 



(OT + n )— X < . Whence, taking the fluents by the theorems in art. 1 and 



(m — n)' — t' ° ■' 



2, we have z = ae (fig. 1 3) = 4-/ -f °''~'^° (fig. 1 1 ) 4- ^ (fig. 1 2); consequently 

 the hyperbolic arc ad is = dp -|- ae + 2i — 4ae. Thus, beyond my expecta- 

 tion, I find, that the hyperbola may in general be rectified by means of two 

 ellipses. 



Writing e and f for the quadrantal arcs ad, ad (fig. 12 and 1 3) respectively, and 

 I, for the limit of the difference dp — ad, while the point of contact (d) is sup- 

 posed to be carried to an infinite distance from the vertex a of the hyperbola 

 (fig. 11), we find 2f — E = L, the value of ae being = -lf -\- ±m — ^7^ when / 

 is = m — n; that is, when e coincides with d (fig. 12), and p with c (fig. ll), by 

 what I have proved in the before-mentioned paper, art. 10. 



4. From what is done above, the following useful theorems are deduced. 



, _ - + 3 



Theorem 1 . The fluent of 4a*z ^Z'/- is = de. 



' a — z 



Theorem 2. The fluent of ia^z'^ X /^^^ = (^+ l)de-(^ + 2)ef: 



-r: - +z 



a 



1 H- 

 Theorem 3. The fluent of -ttt—^ti^ r: = 2ef- de = 2f - e -f- ad - dp. 



Theorem 4. The fluent of —Mr^r-A — ^. = 2 (de — ef.) n.b. ^ = " " 



V(i' -1- 2/2 - z') ^ / • • 2a 



