the Elliptic Equation. 199 



With great geometric simplicity, my method has the disad- 

 vantage of using a transcendental ordinate ; but the only other 

 that I know, has less simplicity — 1 mean Dr. Booth's sphmcal 

 parabola, which primarily belongs to the integral of the third 



class, 1 — Tj ^^^ requires reduction to 



J (l-/tsin2 «/>)(!- Fsin2</))^ 



pass to the simpler integral Y{6, <p). 



As a particular case, I have discussed the integral j — ^^ in 



a way free from some of the complexities attending the trigono- 

 metry of the parabola, which relates more directly to the integral 



r # 



J COS^ <\) ' 



For the purposes of Mercator's chart, circles divide themselves 

 into three sorts. Let a. be the distance of the centre from (say) 

 the north pole, and /3 the radius, a and /3 being both arcs of 

 great circles. Let the meridian of reference be the one passing 

 through the centre. 



(1) Ifa<yS. The circle has one pole within and one pole 

 without it. Of these the great circle is the type. Let 



; = sin ^ be called the modulus ; then it is evident that the 



sin/3 



pole is a centre of similitude to all circles having the same mo- 

 dulus, and, from the principle of Mercator's chart, their projec- 

 tions will all be similar and equal, each being symmetric to a 



COS Q 

 parallel of latitude /x, such that sin^= -. This we shall 



call the mean latitude. 



(2) a =/3, a critical case. In this case the cii-cle passes through 

 a pole. These curves have the same simplicity on the chart that 

 great circles have on the sphere, or right lines on the plane. If 

 s be the arc and the longitude, both measured from the me- 

 ridian of the centre, ds= — ^, whence s=loge ^^'^(4 + %)> *^^ 



common formula for meridional parts. We shall therefore call 

 its projection on Mercator's chart the meridional curve. 



(3) a:?- /3. In this case the circle is necessarily a small one, 



• 1 1 1 Tj? si" /3 a T cos a 

 and does not include a pole, li -: = sm o, and 7,= sin/i. 



Sin a 



COS/S 



it will be projected into an oval divided symmetrically by the 

 j)arallcl of fx, and touched by the meridians of 6 and —6. Evi- 

 dently, provided 6 is constant, the curves will be similar and 

 equal. 



If (/) be the current longitude and \ the latitiuln, the formula 



