A Simplified Method of Finding the Equation of Equal

(SEE No. 123.)

H.B. Goodwin.—With reference to the interesting paper by Mr. G.W. Littlehales entitled "A Simplified Method of Finding the Equation of Equal Altitudes," which appeared in the September number of the PROCEEDINGS may I be allowed to point out that even previously to the publication of the extended Azimuth Tables, the advantages of the single term expression for the "Equation," in preference to the very cumbrous double barrelled form generally employed, had not altogether escaped observation. Some twenty years ago in a paper by the present writer contributed to the Nautical Magazine the same form of expression now advocated by Mr. Littlehales was suggested, and the simplification which resulted, more particularly with regard to the ease with which the algebraic sign of the expression is determined, was pointed out. The extension of the Azimuth Tables to include all latitudes up to 70°, which is indeed an epoch- making work, has still further simplified the problem, since, as was mentioned in the article above alluded to, the Azimuth Tables could formerly be made available to calculate the position angle in tropical latitudes only. Under the new conditions it is probable that the super- session of the old formula by the new one is only a matter of time.

The following extract from the third edition of Harbord's Glossary of Navigation published by Messrs. Griffin, of Portsmouth, may be of interest:

"A simpler form of the "Equation" is

E^{8} = 1/30(sec d cot PXZ) dp”

where PXZ is the angle at the body contained by the circles of altitude and declination, sometimes called the "angle of position." In this case we have for sign the simple rule: "When the body's distance from the elevated pole is increasing add, when decreasing subtract. "If PXZ is greater than 90° the rule is of course reversed, on account of the cotangent changing sign, but this can only occur when (1) latitude is less than declination and of same name and (2) when the observation is taken between the point of maximum azimuth and the meridian. For an observation taken at the maximum azimuth in such places the equation vanishes, a fact which is worth remembering, since at this instant, the Sun is moving in altitude most rapidly, and is therefore best suited for observation.

In a short treatise on "Types de Calculs" contributed by Captain Guyon of the French Navy to the" Annales Hydrographiques" for 1895, published by the French Hydrographic Department, this form is adopted for the equation.

In a valuable work on Navigation by Mrs. Janet Taylor, an expression in one term is given as follows:

E= cot h cosec ? cosec p sin (p- ?) dp

Where tan ? = cot l cos h, with the same rule for signs as that given above.

The extract given above shows the position ten years ago, which has not changed materially at the present date.