On dividing both members of (233) by the factor of B within parentheses, we obtain eq. (234), after rejecting quantities of the fourth and higher orders:

By (232) it is seen that as a result of squaring (230), B2 is equal to B2 plus a term B2.D; but that in (234)— which is the quotient of the second member of (233) by the factor of B in the first member-the term B2.D has disappeared, because it produced terms of the fourth and higher orders.

Now, comparing the quantity within parentheses of the last member of (234) with the factor of B in (230), it is seen that, with some changes of sign, they are the sameD2

only, that B' has replaced B2, and has disappeared,

4

because in passing from (233) to (234), it also produced terms of the fourth and higher orders.

If (209) be arranged as (208) was, an equation for C, symmetrical with (230) for B, will result; and operating on this in the same way as on (230), we should obtain an equation for C entirely analogous to (234) for B: therefore, Band C-as well as B and C which are coupled respectively with in (208) and (209)-can be replaced directly by B2, C2, B, and C; because, while their values when found as in (232) are connected with other terms, still these disappear in the operations, on account of producing higher orders than the third. As a result of these facts, we may therefore write at once an equation for C derived from (209), analogous to (234) for B derived from (208); and at the same time we may replace D by D and E by E; the equa

tions for B and C respectively, both similar to (234), then are

Substituting in (235) the necessary values from (224) to (229) and then performing the indicated operations on (235), we obtain (237), after, however, rejecting all terms of the fourth and higher orders, of which several will result from squaring and multiplying as required in (235). The sum of the exponents of a term determines its order: sin B, sin C, sin D are each of the first order and first power, so that the product of all three would constitute a term of the third order; sin A, sin E, and the versines of B, C, D, are each of the second order, and any two, or one of them connected with two of the first order, would constitute a term of the fourth order; it is such terms, and others containing higher orders, that are rejected in passing from (235) to the following equation:

B = sin B [1 + } versin B + 1⁄2 sin D – } sin2 B

— § sin2 C] + 1⁄2 sin C.sin E. (237) Substituting in this the arc for its function in the case

B =sin B [1 + sin D+ versin B- versin C]

And by a similar procedure with eq. (236) we obtain the following:

C = sin C [1-sin D+ versin C - versin B]

+ sin B. sin E. (241)

Eqs. (227), (228), (229), (240), and (241) are those given on "Form 10, Analysis of Deviations," for computing the Exact Coefficients; see page 921.

The value of the remaining exact coefficients F, G, H... N, in terms of A, B, C, D, E, will now be obtained from eqs. (212) to (218), which, as already stated, are the comparison of the coefficients of corresponding terms in (148) and (149); the procedure need not be given in detail-it is entirely analogous to that practised upon eq. (208) to obtain (240). Since B.DB(D+D2), from multiplying (228) and (235), member by member, and rejecting in the result the fourth and subsequent orders, we hence have from (212)

Similarly, from (228) and (236), C. D=C(D − D2), whence, by substitution in (213), this becomes.

G = C[{D + √ 1C2 — }B2 + }D2] + E.B.

(243)

Then from (214) to (218), respectively, we have directly, to third order of small quantities included:

The computation of the coefficients F, G, H... N will be illustrated in the next section.

Section Three: Analysis and Computation of the Deviations.

312. Compass Report Form 7.-This was devised in the Compass Office in 1882, and, until recently, was known as Form [I].

It was designed to be a convenient form for recording all the data relative to the observations and deviations, either while steaming in a circle or swinging at anchor, using the Sun or a terrestrial mark as the object of observation. With but few alterations consequent upon the changed conditions on shipboard, it is the same that has been in use in the Navy for the past twenty years, and therefore requires no special explanation. Besides, the Form itself indicates the mode of procedure: it is reproduced on pages 918-19 with a complete series of observations made at sea on board the U. S. S. CONCORD.

313. Method of analyzing the Deviations-Form 10. Like the Form of the preceding article, Form 10 has long been in use in the Navy, and known as Form [IV].

Comparing Form 10 with eqs. (195) to (201), Art. 310, it will be seen that its columns are but a tabular arrangement of the quantities in those equations: the mathematical basis of the Form being thus pointed out, its practical use will now be described.

A series of observations having been made for Form 7, pages 918-'19, the resulting deviations, col. (9) of Form 7 are transcribed to cols. (2) and (4) of Form 10, pages 920-'21: to illustrate the use of the Form, the deviations of the steering compass of the U. S. S. ATLANTA will be analyzed. This ship is a protected steel cruiser of 3000 tons, with a battery of eight heavy guns: she was swung at the buoys of the Compass Station, Newport, R. I., Sept. 18 and 19, 1886, and the observations were made on the 32 compass-points.

The case of the steering compass is selected because of its unfavorable location as regards directive force and deviations, the latter attaining a maximum of 50°; it will therefore be a severe test of certain matters to be illustrated.

Form 10 consists of three parts, headed Tables I, II, and III; and the chief liability to error in its use is the manipulation of the algebraic signs: therefore it must be remembered that plus multiplied by plus, and minus multiplied by minus, both produce plus; that plus multiplied by minus produces minus; and that the "sum" of quantities means their numerical difference when of opposite sign, but their addition when of the same sign-the result in the former case receiving the sign of the greater quantity. On account of the varied signs of the parts composing the exact coefficients, especial care is necessary in Table III of Form 10. Upon entering the deviations in cols. (2) and (4), Form 10, pages 920 and 921, mark all that are easterly plus, and all that are westerly minus. The multipliers So, S1, S, . . . S。 have their signs determined by the quadrants in which their angles fall-Fig. 511.

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In the case of the ATLANTA, sin B and sin C are minus, although both apparently belong to angles less than 90°, and should therefore be in the first quadrant, where the sine is plus; this seeming inconsistency is not real; for the starboard angle a, calculated from B and C, is 199° 53', therefore in the third quadrant, where the sine is minus, and thus the signs of B and C are in accord with the Trigonometrical fact.

By eq. (23), page 856, versin x=1-cos x: as the cosine is always less than unity, the versin is therefore plus, whatever be the actual sign of x.

Column (5), Form 10, p. 920, is formed by taking half the algebraic sum of the quantities on the same horizontal line in cols. (2) and (4), and giving the result the sign of the greater: col. (11) is obtained from (9) and (10) in the same way.