38 U. S. COAST AND GEODETIC SURVEY. 



On the substitution of these values we obtain as defini- 

 tions of the coordinates of the projection 



a sin X cos (p 



1 +sin a sui ip + cos a cos X cos (p 



_ a(cos a sin <^ — sin a cos X cos (p) 

 ^ 1 + sin a sin <p + cos a cos X cos (p 



From these equations, by solving for sin (p and cos <^, we 

 find 



X sin a cos X + V sin X 



Sm (p= : ^^ \ : : T 



a cos a sin \ — x cos h — y sm a sm X 



a: cos a 

 cos (p = • — ^ :; = ' — T • 



a cos a sm \ — x cos \ — y sm a sm X 

 By squaring and adding there results 



{x sin a cos \-\-y sin X)^ + x^ cos^ a 

 = (a cos a sin X — aj cos X — -y sin a sin X)^. 



By performing the operations and collecting, we obtain 

 finally 



x^ + y^-\- 2ax sec a cot X + 2ay tan a = a^, 



which may also be written 



(x + a sec a cot X)^ +{y + a tan aY =* a^ sec^ a cosec^ X. 



This is the equation of the meridians, and they are thus 

 seen to be circles. The meridian of longitude X has the 

 radius 



Pjjj = (j sec a cosec X, with its center at the point, 



x= —a sec a cot X, 



y= —a tan a. 



The centers, therefore, all lie on the line 



y= —a tan a. 



