cular upon the plane, the angle contained between these two straight lines will be the complement of the required angle. The equations of the line let fall perpendicular on the plane will be of the But in order that this may be perpendicular to the given plane, we must have A + α = 0 B+ b = 0 Now, the cosine of the angle contained by the two straight lines, is It appears from what has been said above, that, in the present case @ = 90o - 6, and.. cos. sin. 4. Substituting therefore for a', V, these values DIFFERENTIAL CALCULUS. CHAPTER I. 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. that When two or more variable quantities are connected in such a manner, 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 x," are usually expressed by the symbols, ƒ (x), Q (x), ↓ (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 f (x + h) be now developed in a series of the form in which the first term is the original function ƒ (x), 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 f (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. cular 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 ara2 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-efficient of x3. Again, let y = ax3 + bx2 + cx + d Let x become (x + h) and y become y y' = a (x + h)3 + b (x + h)2 + c (x + h) + d 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) h2+h' 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, 3x2+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 ax3 + bx2 + cz +d. If* y = f(x) the first differential co-efficient of y is denoted by the symbol above examples dy * 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 Func tions," yet it has been thought expedient to retain in the present work the nomenclature and nota. tion of the Differential Calculus, since it is employed almost universally in the scientific publications both of this country and of the Continent. in like manner if u = ƒ (z) the first differential co-efficient of u or ƒ (z) will be We might obtain the first differential co-efficient of any function presented to us by following a process analogous to that exhibited above, but we shall materially abridge the labour of our operations by establishing certain general rules, which will enable us at once to determine the first differential co-efficient of any variable quantity, without the necessity of having recourse to the substitution of x + h for x, and the subsequent expansion. The investigation of these will form the subject of the two following chapters. dy Note.—Since a constant quantity is not susceptible of change, it is manifest that it can have no differential co-efficient, or if y = a, dr = 0. CHAPTER II. ON FINDING THE FIRST DIFFERENTIAL CO-EFFICIENT OF SIMPLE FUNCTIONS OF ONE VARIABLE. 1. To find the first differential co-efficient of any power of a simple The first differential co-efficient of any power of a simple Algebraic quantity is found by multiplying the quantity by the index of the power, and then diminishing the exponent by unity. dy dx dy dy da dy qx¬(9+1) m n In the system of logarithms whose base is e, p = 1. If the logarithms be taken in the system whose base is Let 4. To find the first differential co-efficient of sin. x. y= sin. x Let a become x + h and y become y y' sin. (x + h) sin. x cos. h + sin. h cos. x Substituting for sin. h and cos. h their developements as found in p. 587. |