traced upon the Surface of the Sphere. 345 



upon a plane. The second is the equation of the curve traced on the sphe- 

 rical surface. The third simply expresses that the equator becomes the axis 

 of x in the development ; and the fourth is the relation between the radius- 

 vector of the curve at that point, and the said functional projection of its 

 complement on the cylinder. 



Equation (3.) is always given ; and most commonly (1.) and (2.) are 

 given to find (4.). This is the case we propose at present. To effect it, 



Put (1.) in (3.), and equate the result to (2.) This gives 



=<; or = 



(5.) 



Applying this to find the law of projection which shall so divide the me- 

 ridian as to represent the loxodrome by a straight line on the developed cy- 

 linder, we have 



x cos a. +y sin a =0; or x = tan a .y ............... (6.) 



Q cot a = log tan ^ (f>; or <9=:tan a log tan 5 (f) ......... (7.) 



The former of these is the equation of the straight line cutting the prime 

 meridian under an angle a, and lying in the (+ <, 6) region of the 

 sphere. The second is the equation of the loxodrome (xxxi. 4.) resolved 

 for' a 



Inserting these in (5.) we have 



y = i /~ 1 rf t (f> = cot a r tan a log tan ^ 

 = r I tan ^ (f) ; or 

 y = rl tan ^ (j). 



y_ 



Hence, e r = tan | 0, and 



6 r = cot ^ 0, or 



/ =r/cot4 (p, (8.) 



which is the usual formula given for this purpose, and which is the solution 

 of equation (4.) above given. 



Though objections have often been made to this chart, for alleged 

 inaccuracy of principles, yet, with two exceptions, they are undeserving of 



