360°, the elementary triangles yy contain those already computed. It would be easy to supply this defect by calculating the elementary trapezia comprised between two adjacent spires. The same inconvenience will be found to occur in the common formula fux, if several ordinates correspond to one abscissa. ON THE RECTIFICATION OF 111. If we conceive the point m to be taken indefinitely near to M, Mm will be the fluxion of the arc AM, and considering this as a right line, we shall have flux AM = √(x+y). Consequently AM=ƒ√(x2+y3)+C; a formula which equally applies, whether the ordinates are parallel to one another, or whether they proceed from a fixed point. 112. Example I. In the circle y= √(aa—xx); and y2 = √(aa-xx) x2x2 aα-xx 1 +&c.; therefore the arc QM=S ax =x+ have shewn (71) that. fx √xx + aa = 2 C+} x√(xx+aa)+1⁄21⁄2 aal (x+√xx+aa) Therefore AM=C+ y ✓ (yy + Pi') + pl {y+√(yy+\pp)}. Make y=o, A m PP and we shall find C = — § pl1⁄2 p. Consequently AM = Y √( YY+\PP)+\ pl (Y+√yy+‡PP). P P N' D 1 2.4 2.4.6 cf- -&c 1.3.5 2.4.6.8 quantities that enter into this series, of which the law is easily per ceived. riphery of the ellipse is to that of the circumscribed circle as transverse axis a). This series will be very convergent when the foci are near to one another. For example, if c=. 1 10 -a, the circumference of the ellipse will be to that of the circumscribed circle as 0.997495 292 861 261: 1 The rectification of the 'hyperbola may be found in nearly the same manner. Ex. IV. The equation of the second cubical parabola is y3=az2. Consequently ƒ √(x2+ y2) = ƒÿ√(1+2)=2a (1+2y)3 +C 8 4a 27 4a Making y=o, we have C=—-a, and any arc of this curve com 27 Ex. VI. In the logarithmic curve yx=ay, and ‚√(s2+ÿ2) = 1⁄2 y ✔(yy+aa). Let √(yy+aa)=z, and we shall have yy=zz-a6, a integral is z+12 , or✅ (aa+yy)—al { a+√(aa+yy) } +C; 2 z+a y the expression for any arc of the logarithmic curve, in which C is easily determined. √(+yy). Therefore CN=COM. From which we may conclude, that there exists some analogy between the Archimedean spiral and the parabola. Ex. VIII. In the hyperbolic spi ral, the arc COM=ƒ1⁄4 √ (bb+yy). y Therefore if we describe a logarithmic curve NK, whose subtangent 6 to that of the spiral, we shall have MOC the infinite arc NK, by taking the ordinate NR=CQ= CM. But if we desire an expression for any arc of the spiral, or of the logarithmic curve contained between two ordinates y, y', we shall find it to be A √ (bb+yy)—√ (bb+y'y')+bly (b+√bb+yy |