sin (x+y)=sin x cos y + cos x sin y.

cos (x+y)= cos x cos y sin x sin y.

sin (y)=sin y; cos (-y) = cos y..

sin (y-90°) sin (90° - y) = cos y.

=

(34)

(35)

or

In eq. (38), let x=mz, and let y=z, then (38) becomes

cos (m+1)z-cos (m − 1)z = 2 sin mz sin z;.

= -

cos (m −1)z—cos (m+1)≈ = 2 sin mz sin z.

This last formula will be found useful hereafter.

[6.]

Recurring to Fig. 511, let the points of the compass, beginning with North, be denoted by 0, 51, 52,.. , then any two courses, diametrically opposite, will have sines and cosines of the same numerical value but of opposite algebraic sign, and hence their sum is zero. For example,

sin sin a.. sin + sin 21=0.

- COS 14 = COS (30;. COS (30+COS 11 = 0.

That the sines and cosines of courses differing by 180° are equal, may be seen by Fig. 510; for x=x'; AB,(sin) = — A'B',(sin); and CB, (cos) = − CB′,(cos).

Let indicate summation of quantities: then if 32 denote the sum of like quantities for each of the 32 points of the compass, by analogy with eqs. (44) and (45), we

Σ32 sin cos 2=0; 32 COS 【 sin 25=0. . (49)

32 COSCOS 2=0; 32 sin 25 cos 2=0. (50)

sin2 。=sin2 16=0; sin2 (=sin2 (=1. . (51)

sin2 = cos2; sin2 = cos2 (2; sin2 = cos2 1• (53)

32 sin2 = 16; 232 COS2 = 16; 32 Sin2 25 = 16;

(=

[7.]

Formulas (46) to (54) will be found of use later.

The transformation of coördinates will be explained, as it enters into the formulas of the heeling error. Let CX and CY, Fig. 512, be rectangular axes to which the point P is referred; turn them through the angle a, and draw PD and BF parallel to Y, and EB to X. Then the angle EPB = a; x=CD; y=PD; x' =CB; y' =PB.

x=CD=CF-DF=CF-EB.

CF=x.cos a; EB =y'.sin a, whence.

x=x.cos a-y'.sin a. '.

y=PD=ED+PE=BF÷PE.

BF=x.sin a; PE=y.cos a, whence . (59)

Multiply (57) by cos a, and (60) by sin a; add the results, and by means of (27) we find

x' = x.cos a+y.sin a.

(61)

Then multiply (57) by sin a, and (60) by cos a; subtract the results, and by means of (27) we have

y'

=

x.sin a+y.cos a.

(62)

Equations (57), (60), (61), and (62) give the old coördinates in terms of the new, and conversely.

To pass from spherical to right-line coördinates, consider Fig. 513: P is a point on the surface of a sphere, and E its projection on the horizontal plane through the equator; the arc FP (= angle a) and the angle HCL=HFL=ß

are the spherical coördinates, and CD=x; ED=y; EP= CM-z, the rectilinear: then from the figure,

x=CE.cos ECD, and CE=MP=r.sin a, whence (63)

y=CE.sin ECD, whence y=r. sin a.sin ß. . (65)

z=CP.cos MCP=r.cos a.

(66)

Equations (64), (65), and (66) give the rectilinear coördinates in terms of the spherical.

297. Magnetic Elements of the Earth and Ship, separate and combined.

[1.]

The magnetic elements of the Earth have been so fully treated in Chapter VII and Arts. 196 and 253, that only a diagram for reference, is needed here—Fig. 514. 0=DIP: This has been designated by D, but in view of this

letter being used soon for one of the magnetic coefficients, the Dip will hereafter be represented by 0. T=TOTAL INTENSITY: It has been, and will continue to

be, designated by T; this letter also stands for the

Period of harmonic motion and for the time in which a magnetic needle makes one oscillation; it will continue to represent all three quantities, as the context in each case will show which is meant.

DYNE: This is defined in Art. 180.

21: This means the EXACT LENGTH of a magnetic needlenot that between the estimated location of poles.

M represents the MAGNETIC MOMENT of the needle. m is the POLE-STRENGTH of a needle.

[2.] The composition and resolution of forces enter fun

damentally into the investigation of the Deviations, and the principles of the process will be now briefly explained. In Fig. 515, let CX, CY, CZ be axes at right angles to each other, and let CM represent a force f: within the solid angle formed by planes through the axes, ƒ may take any one of an infinity of directions making different angles with the axes; the length of the projections of f on the axes will accordingly vary, and upon these projections a rectangular form, as in Fig. 515, may be constructed.

The projection of CM directly upon the axes may be made by multiplying it by the cosine of the angle it makes