208 0)1 the Geometry of the Elliptic Equation. 



Now logtau (x+ 9) is the ordinate of the chart. The only 



constant the equation contains is additive. It is evident, there- 

 fore, that so long as the scale of the chart is unaltered, the 

 curve does not change either size or direction, but simply shifts 

 north or south as its extreme latitude alters, or east and west as 

 we shift its axis. If then we cut a pattern for any given chart, 

 we can apply that pattern as easily as we can the ruler in plane 

 geometry, merely keeping the axis in the meridian. The geo- 

 metrical simplicity of Mercatoi-'s chart thus appears in a new 

 and very singular light. 



The easiest way to draw the curve is to find the ordinate, on 

 the supposition that the curve touches the equator. In this 



case a = Tj and therefore logtan« = 0. The negative value of 



4 



log cos <^ merely means that the ordinate must be measured 

 towards the pole through which the curve passes. 



Supposing the chart to be developed into a plane, the curve 

 in its general form very nearly resembles the catenary. It has 



two asymptotes, the meridians of +^. 



The function log tan ( -r + ^ ) = 1 — , is one of the most 

 ° V4 2J jcoscp 



remarkable that occurs in analysis. It fulfils the functional 



equation \/ — 1 .f{u) =f{u V — I). This property was, I believe, 



first remarked by Baron Maseres, although I suppose that the 



old gentleman, who had a horror of the negative sign, especially 



when standing by itself under the radical, would be very much 



shocked at the above expression of it. His observation was, 



that if u be the function, and the variable, 



^^6^24^5040^72576^ 



u" 



eivF 277u^ 



24 5040 ^ 72576 

 the coefficients being numerically the same for both series ; but 

 all positive in the one, and alternately positive and negative in 

 the other. 



With I'egard to Lagrange's scale of moduli, it should be re- 

 membered that in this case, as in the other extreme of the cir- 

 cular arc (where ^ = 0), the application of the scale alters neither 

 modulus nor amplitude. 



In applying my theorem to this case, it is obvious that there 

 is only one auxiliary circle, and that it and the tangent circle 

 pass through opposite poles. Any two meridional curves, of 



