Rev. B. Bronwin on the Uectification of tJie Ellipse. 379 

 from this last by the first of the second set of equations we ob- 

 tain »=*— 4.- -rpA+-r-A = 0. DifFerencinff this 



and eUminating — — , wegetfinally ^'*— 4. — 7 + *T -^ ~ ^* 



1 

 Thus we can find both A and A. 



Mr. Knight in the 3rd volume of Leybourn's Repository', 



has shown us how to find A from A by means of an integral, 

 and seems to intimate that it had not been done before. Per- 

 haps therefore the conclusions here arrived at are found for 

 the first time ; and if we could integrate the above differential 

 equation of the 2nd order in finite terms, we should have a 

 finite expression for the quadrantal arc of an ellipse. I fear, 

 however, this cannot be done. Still the equation is not use- 

 less; we may derive from it a multitude of different series ex- 

 pressive of the elliptic arc, and such as perhaps otherwise 

 could not have been deduced. 



Putting 1/ for A and 2 x for p to simplify, the preceding 



equation becomes a?° — 1 . —^ 4- — - j/ = 0. Assume 7/ =a 



1. 1 _3 2 _7 _ , d^i/ 



ax^ -\- a X ^ -\- ax ^+ &c., and substituting for y and -^-^ 



. IT ' > 12 



m the precednig ; we get a = — v c rr ^ = — tt cf> « = 



_ 3-5 3-5 3 _ 3 -5 - 7 ' 9 „ 



(3-5+i)(7-9 + i) " — — 4^-8^ a, a — — j^^.^i.-y^, a «c., 



where by making e nothing, or x infinite, a is found = -/ 2. 

 Hence if we put t = — we find the length of the elliptic qua- 

 drant = ^ . ^.3, = ^ ^ 1-1 ^ into I- l-f - 



.-T-oT i* — - 1 0, ,0, ^^ &c. ; which is die same as it would have 



been found if we had expanded ^/ 1 — ^^ sin ^ ,r by the bino- 

 mial theorem, and developed the powers of sin x in cosines 

 of the multiple arcs; and is the same, under a form a little 

 dififercnt, with the second series, page 196, vol. i. of Ley- 

 bourn's Math. Repository, and there ascribed to Legendreand 

 Euler. 



Let us now consider y a function ofu = x— v^^^ — 1 = 



•— c , . . „ dy du dv 



1 , > Where c = semi-coniuffate. Ihcii -,- = -7— • -; — 



3 C 2 , £ji 



dx' 



