Now, if we suppose x, y, z, to be the co-ordinates of the point in which the perpendicular meets the plane, equations (1), (2), and (3), will hold good together, and we shall have M MB MA I + A' + B» Y − y =1+A+B = I + A+B But the distance (7) of the two points whose co-ordinates are x, y, z; L', I', z', is To find the distance (▲) from a point in space to a straight line. Let the co-ordinates of the point be (x, y, z) and let the equations of the straight line be x = az + α .............. y = bz + B (1) (2) } The equation to a plane passing through the point x, y, z, and perpendicular to the given straight line, will be (z — x') + a (x − x) + 1 (y — y) = 0 ...... (3) Now, if we suppose x, y, z; to be the co-ordinates of the point in which the plane meets the given straight line, the equations (1), (2), (3), will hold good together, and if we find values of (x — x'), (y — y'),(z — z'), from these equations and substitute the values thus obtained in the general expression for the distance of two given points in space, viz. In order to effect this, let us put the equations (1) and (2) under the form (x − x') = a (z -z) + α — x + az' Substitute these values of (x — x′) and (y — y') in equation (3), which will then become (2x)(1 + a2 + b2) + a (α — x' + ax') + b ( ~ y + bx') = 0 whence we find = 1 + a2 + b2 - - z' if N = a(z— a) + b (y — ß) + %. squaring these quantities and adding, we find N2 (1+a2+b2) (1 + a2 + b2 )2 + (x —∞)2 + (y'—ß)2 + z2 — 2N ̧‚2'+a(x'—a)+b(y'—ß) 1+a2+b2 N2 1 + a2 +.b3 • To determine the angle between two given planes. Let the equations to the planes be z = Ax+ By + C ... (1); % = A'x+ B'y + C ............ (2) If we let fall from the origin two straight lines perpendicular on these planes, the angle contained by the straight lines will be the same as the angle contained by the planes, let the equation to these straight lines be But, in order that the straight lines may be perpendicular to the given planes, we must have A+ a = 0, B + b = 0, A' + d′ = 0, B + b = 0 Substituting therefore, the values of a, b, d, b', derived from these equations, we find that the expression of the cosine of the angle between the two planes, 1 + AA' + BB' cos. 4 = √(1+A2 + B2) (1 + Aa + B2) In order to find the angle which any plane makes with the co-ordinate planes, we have only to suppose that one of the above planes assumes in succession the position of the different co-ordinate planes, thus let us suppose, that (2) is the plane of xz, then its equation becomes y= 0, so that, A' = 0, C = 0 and therefore, if we denote by the symbols (xz), (yz), (xy), the angles which the given plane makes with the planes xz, yz, xy, we have cos. Q = cos. (xz) cos. (x ́z') + cos. (xy) cos. (x'y') + cos. (yz) cos. (y'z) To find the angle (4) contained by a plane and straight line in space. The angle sought is that which the straight line makes with its projection on the plane. If from any point in the given straight line we let fall a perpendi cular upon the plane, the angle contained between these two straight lines will be the complement of the required angle. Let the equation to the given plane be z = Ax + By + C ........ (1) (3) The equations of the line let fall perpendicular on the plane will be of the form x = dz + a' y = bz + ß (4) (5) But in order that this may be perpendicular to the given plane, we must have Now, the cosine of the angle contained by the two straight lines, is 1 + da + bb cos. @= √(1 + a2 + b2) (1 + a22 + b2) It appears from what has been said above, that, in the present case = 90°-6, and.. cos. = sin. 6. Substituting therefore for a, V, these values in terms of A and B, we find DIFFERENTIAL CALCULUS. CHAPTER L DEFINITIONS. IN considering the relations which exist between different quantities, those which during the whole of any investigation are supposed to retain the same value are called constant quantities, those to which different values may be assigned are called variable quantities. Constant quantities are usually represented by the first letters of the alphabet, a, b, c, &c. variable quantities by the letters u, x, y, z, &c. When two or more variable quantities are connected in such a manner, that the value of one of them is determined by the value assigned to the other, the former is said to be a function of the other variables. where the value of y depends upon the value assigned to x, y is said to be a function of x. In like manner if we have y= Az2 + Bx2 + CÃ3 + D where the value of y depends upon the values assigned to ≈ and z, y is said to be a function of x and z. The words "function of r," are usually expressed by the symbols, ƒ (x), © (x), 4 (x), or similar abbreviations, and the above equations expressed in general terms would be written y = f(x) If y = f (x), and a change takes place in the value of ƒ (x) such that x becomes x + h, x being quite indeterminate, and h any quantity whatever, either positive or negative, a corresponding change must take place in the value of y, which may then be represented by y'. If the quantity ƒ (x + h) be now developed in a series of the form f(x) + Ah + Bh2 + Ch3 + in which the first term is the original function ƒ (2), and the other terms ascend regularly by positive and integral powers of h, and A, B, C, &c., are independent of h ;* then the co-efficient of the simple power of h in this series is • We shall, in the mean time, take for granted that ƒ (x + h) can always be developed in a series of the above form, (showing, however, as we advance, that this is actually the case for all the parti. eular functions which fall under our notice) and defer the general demonstration of this principle until we proceed in Chapter V. to the discussion of Taylor's theorem. called the first differential co-efficient of y or f(x). This is the fundamental definition of the differential calculus. This we at once perceive is a series of the required form, the first term ar is the original function y, and the other terms ascend by integral and positive powers of h; hence, according to our definition, 2ax the co-efficient of the simple power of h in this series is the first differential co-efficient of y or f(x). Here again we perceive that the series is of the required form, and, there. fore, 3x2 the co-efficient of the simple power of h is the first differential co-effi cient of 23. Again, let y = ax3 + bx2 + cx + d Let z become (x + h) and y become y ·._y' = a (x + h)3 + b (x + h)2 + c (x + h) + £ expanding = ax3 + 3ax2 h + 3axh2 + h3 + bx2 + 2bxh + bh2 + cx+ch + d arranging according to powers of h =(ax+bx2+cx+d) + (3ax2 + 2bx + c) h + (3 ax + b) k2+hi a series of the required form, for the first term is ax3 + bx2 + cx + d, the original function, and the succeeding terms ascend regularly by powers of h. Hence, 3ax+2bx + c the co-efficient of the simple power of h in the developement of y' is the first differential co-efficient of y or a3 + b22 + cz +d. If* y = f(x) the first differential co-efficient of y is denoted by the symbol d dy thus in the dx = 3x2 * In this treatise the principles of Lagrange have been almost exclusively adopted, but although that writer has with great propriety denominated this branch of Analysis "The Calculus of Funetions," yet it has been thought expedient to retain in the present work the nomenclature and notation of the Differential Calculus, since it is employed almost universally in the scientific publications both of this country and of the Continent, |
