DR WALLACE ON A FUNCTIONAL EQUATION. 553 



found by (16). Greater accuracy may be obtained: this, however, is sufficient 

 to shew the process of calculation. 



As a table of catenarian co-ordinates and arcs may be made from a table of 

 meridional parts ; so, on the other hand, a table of meridional parts might be 

 made experimentally from a catenary. This would indeed be a singular way of 

 finding the course a ship should steer from a given place, to reach a port Avhose 

 latitude and longitude were known. The solution in this way is evidently possible. 



46. From the analogy which has been shewn to subsist between catenarian 

 co-ordinates and the meridional parts of latitudes, and the properties of the former, 

 we have (by the way) this property of the enlarged meridians in nautical charts. 



Theorem. — Let (p, , (p^ , cps , . . . cf)„ he latitudes of parallels on the sphere ; 



and M ((/>,) , M (<^,) , M ((p.,) , . . . M ((p„) their meridional parts ; 

 Let be a parallel whose meridional parts =M (0,) + M ((p,) + M {(p^)... + M (<p„) ; 

 Then, tan (45° + i 0) = tan (45^ + ^ (p,) tan (45° + i 0o) tan (45° + ^(p,) .. . tan (45" + 1 cp„). 



Example. 



Merid. Parts. Log tangent. 



M(0,=12°)= 725.32 45°+ 6^=51° 10.091631 



M.((p, = U')= 848.49 45+ 7=52 10.107190 



M (03= 20°) = 1225.14 45 + 10 = 55 10.154773 



M (0„ = 30°) = 1888.38 45 + 15 = 60 10.238561 



M (0 = 61°18iO =4687.33 45 + 10 = 75°39' 11" 10.592155 



Here the theorem is verified ; for the sum of the meridional parts of 12°, 14°, 20°, 

 30° is the meridional parts of 61° 18^' =0 ; and the continual product of the tan- 

 gents of the halves of these angles, each increased by 45°, is equal to the tangent 

 of 75° 39' 1 1'' = 45° + 1 </) nearly. 



One obvious use of this last formula would be, to construct a table of en- 

 larged meridians, having a common difference of one minute ; the latitudes being 

 placed against their meridional parts. 



Related Properties of a Catenary and a Parabola. 



47. The ancient geometers, in treating of curve lines, endeavoured to shew 

 how they might be exhibited by an organic construction. It may be supposed 

 that, with this view, they defined lines of the second order by sections of a cone, 

 and conchoids by the motion of a point restrained to a certain course by an in- 

 strument. DiocLES defined his Cissoid by shewing how points might be found in 

 it ; but Newton, probably supposing this imperfect, took the trouble to invent 

 an instrument for describing it by continued motion, like the conchoid. The 

 geometers who first treated of the catenary (viz. Gregory and Bernouilli), 

 VOL. XIV. part II. 6 G 



