merwin: equations with one unknown constant 467 



in which the geographical pole has been chosen as the point of 

 tangency of the plane of projection. In this, however, since the 

 parallels of latitude are represented by a series of concentric 

 circumferences whose radii are equal to r x cotangent latitude, 

 where r is any convenient linear magnitude and represents the 

 radius of the earth, the equator can not be represented because 

 the cotangent of 0° of latitude is an infinity; and hence the 

 intersection of the straight line representing the great-circle 

 arc passing through the geographical position of the observer 

 and the geographical position of the observed celestial body 

 must be sought on some parallel of latitude near the equator, 

 such as the parallel of 20° of latitude which is made the boundary 

 circle in Godfray's chart. The required azimuth is then found in 

 the column of azimuths tabulated for that declination in the 

 solar azmiuth tables which is equal to the latitude of parallel 

 upon which the intersection is found, which, in the case of inter- 

 sections determined on the boundary circle of Godfray's chart, 

 would be the column of azimuths for 20° of declination. 



The condensed tables as here presented may be arranged to 

 stand complete at one opening of a book of fair size; and 

 hence, in addition to being free of the usual interpolations for 

 declination, are rid of the disadvantage of turning from page to 

 page in interpolating for latitude. 



MATHEMATICS. — Equations containing only one unknown 

 constant to represent the parabola, the rectangular hyperbola, 

 and certain exponential curves. H. E. Merwin, Geophysical 

 Laboratory. 



By completing the following expression — — = — - — 



y2-y 2/2-2/1 

 for three points in rectangular coordinates, equations can be 

 formed for the parabola, the rectangular hyperbola, and certain 

 exponential curves, each equation containing only one unknown 

 constant. In each case after solving for the unknown constant, 

 C, any point on the curve can be located by substituting its 

 known coordinate, Xs or j/3, for x or y of the equation. The 

 curves, represented by the equation, do not pass through the 

 origin. 



