In this way, i would come to be associated with the direction which the rotation produces, instead of with that of the axis which produces the rotation. 706. The student should very carefully compare the equations In the first a and b are mere algebraical quantities; and as such are incapable of containing within them either length or direction. But if these quantities be applied to geometry, we must assign an arbitrary (or unit) length and an arbitrary (or standard) direction. Taking (OA) as our unit and standard, we may write a. (OA) = a, and b (OA) = ß. Here a and ẞ are not algebraical quantities at all, but vectors. 707. The method of interpreting i explained in the last two articles was historically prior to the method upon which this chapter has been based. Wallis, Buée, Warren, Argand are the chief mathematicians whose names have been associated with this interpretation. Hamilton's grand invention of quaternions may be said to be the immediate result of associating the direction of i with the axis of rotation, instead of with the direction produced by the rotation. The following article may be read by the student who wishes to connect the study of Quaternions with the interpretation of i as explained in this chapter. 708. Let X, Y, Z be three axes perpendicular to one another. Suppose them to be of indeterminate length, but having a determinately positive sense. Let them be so placed that the rotation from X to Y is left-handed for a person looking at the plane of XY from the positive side of Z. Then it will be found that the rotation from Y to Z is also left-handed to a person looking at the plane YZ from the positive side of X; and the rotation from Z to X is left-handed to a person looking at the plane ZX from the positive side of Y. We take such left-handed rotation to be positive. Thus, using the notation explained, we have the following Substituting for Y from (1) in (2), we have = if these operations are performed upon X. This equation should be compared with (6). By the same process we shall obtain the following six equations of operation: (1) i2ix=iy; (3) işiz=ix; (6) ixiz=-iy; (5) iix=-iz; (4) izi-ix This second set of equations is immediately obtainable from the first by substituting i for X; i, for Y; i for Z. In (1), (2), (3) the circular order x, y, z, x is maintained; in (4), (5), (6) this circular order is reversed. It should be noticed that to prove the second set the operand to be chosen must be that which corresponds to the operator which stands first. The equivalence in form between the first and second set of equations led Hamilton to use the same symbol to denote two things; viz.: (1) An undetermined (or unit) vector in any direction. (2) The operation of turning a line about this vector as axis through a right-angle. It is essential to observe that the multiplication of the operators i, iy, iz is not commutative. MISCELLANEOUS EXAMPLES. 1. Show from the definitions of the trigonometrical ratios which apply to angles of any magnitude, that a knowledge of the tangent of half an angle determines all the trigonometrical ratios of the angle without any ambiguity. Also write down the value of each ratio of an angle in terms of the tangent of the half-angle in order to illustrate the above. 2. If A', B', C' be the external angles of a triangle ABC, bc vers A' + ca vers B' + ab vers C' = (a + b + c)2. 3. In any triangle ABC, a2 cos 2 (B − C) = b2 cos 2B + c2 cos 2C + 2bc cos (B − C). cos A cos X + cos B cos Y + cos C cos Z = cos2 A + cos2 B + cos2 C = cos2X+cos2 Y + cos2 Z=1, cos A sec X = cos B sec Y = cos C sec Z = 1. then 6. Eliminate У = sin 30 sin 0 = α. - 9. If a sin a = b sin 0; and a sin ß=b sin &; and a- ẞ= 0 -; and a and b are unequal; then and x sin + y cos 0 = 2a sin 20, (x + y)3 + (x − y)3 = 2a3. 11. Eliminate from 12. b cos2 0 + a sin2 0 = b cos2 (0 + $) + a sin2 (0 + $) = 0. ax sec - by cosec = ax sec 0 – by cosec 0 = a2 – b2 ; = 13. Prove that the equations (x + x-1) sin a = yz−1 + zy−1 + cos2 α, are not independent, but are equivalent to x + y + z = x−1 + y−1 + z ̄1· 14. Solve for a and B = sin a. sin a cosh ẞ = 3 and cos a sinh ẞ = 1. 15. If a and ẞ are the two values of sin ✪ which satisfy the equation a cos (+4) + b cos (p − y) + c = 0, and if 0, 4, are all unequal, then a2 − 0) + c = 0, – b2 + 2bc = 0. 18. and if then then If a, ẞ, y be unequal and each less than 2π, 19. Given x cos a + y sin a + z + cos 2a = 0, x cos 20. If a, ß, y be unequal system of equations = (p − a) sin (ø – ẞ) sin (p − y). and each less than π, sin (2a-ẞ-y) sin (2ẞ-y-a) __ sin (2y- a - ẞ) is equivalent to ( 21. = cos (2ẞ+y+a) cos (2y+a+B)' cos 2 (B+ y) + cos 2 (y + a) + cos 2 (a + ß) = 0. If A + B + C lie between 0 and 2π, - + cos A cos B sin B sin C cos C sin C sin then the 1 + cos B +(2 1 + cos C sin 4 sin B (1+cosA-cos B-cosC) (1+cos B-cos C-cos A) (1+cosC-cosA-cosB) 2 sin A sin2B sin2 C 22. The area of ABC = s2. 2a cos 10 cos 1 cos 14, N are the feet of the perpendiculars from A, B, C on the opposite sides, and P1, P2, P3, -p, the perpen |