traced upon the Surface of the Sphere. 333 



ready, in the case of a ratio of equality by MM. PARENT * and GODIN f , 

 as it has also been by most authors who have written on such subjects since. 

 The following solution is different, I think, from any that have preceded it ; 

 and though it might, in some points of view, be deemed very elementary, 

 yet as it is short, it might not be improperly introduced here, as another 

 specimen of the method. It may be remarked that GODIN professes to 

 dispense with the differential calculus ; but he does so only inform, as the 

 elementary spherical triangle is in reality the one that is employed in his 

 method " par les Infiniments Petits." 



The circles being represented as above, we have 



cosjc = cos(p sin A 4- cos A s\n(p cos 6 ................... (1.) 



cot(p = tan A cos 6 ......................................... (2.) 



ndQ = dx, ..................................................... (3.) 



The first of these is the expression for the arc of the ecliptic intercepted 

 between the first point of Capricorn and a point in the ecliptic, whose co- 

 ordinates are (f>, 6. The second is the equation of the ecliptic, referred to 

 the equator and winter colure ; and the third is the given relation between 

 the velocities of the sun in right ascension and longitude. 



From (1, 2) we get 



and hence d Y, = -- sn (5.) 



2 cos'(f> 



Differentiating (2), we find 



dO=- _ co .. (6.) 



sin Vsin* A cos* (J> 



Inserting (5, 6) in (3), we get 



sin* (p = cos A (7.) 



which gives the north polar distance of the sun at the time in question, and 

 agrees with the results obtained by other methods. 



* Mem. 1704, p. 315. -j- 1730, p. 27. 



VOL. XII. PART I. U U 



