than any given difference, then it is clear that the ratio will differ from the ratio by a difference which will be less than b a any given difference; consequently, is a limit to the pro b posed ratio; and it is clear that the limit can be immediately obtained by putting = 0. and will be diminished indefinitely, and differ from 0 by x differences less than any given differences, when x is unlimitedly great, since a and b are supposed to be finite; conse quently, when a is unlimitedly great, it is clear that 1 1 = 1 is the limit of the proposed ratio. And it is clear that this ratio can be immediately obtained from the proposed ratio by omitting a and b, on account of their minuteness in comparison to x, when x is unlimitedly great, which reduces the = 1. ratio to It is clear, from what has been done, that the given ratio has two limits; one of which results from making x = 0, or an infinitesimal, and the other from making a infinite, or unlimitedly great. It is hence easy to perceive how we are to proceed in any case to find the limits of any variable ratio. To illustrate what has been done, take the following by rejecting 4 and 2 in comparison to 9x and 3x, the second 7 3 Putting 0 for ø, the first limit is ; and putting = ∞, the x=∞, it is clear that 3 + 5 must be rejected in comparison must be rejected in comparison to to 72, and that 9-11 Erasing the factor x from the dividend and divisor, we get 5c+dx b Hence, putting x = 0 and x = ∞, α we get and = an infinitesimal ( being in5c dx finite), for the required values. 6. To find the limits of 3 + x 4 and that correspond 4 5 2x (46.) To find the true value of a ratio or fractional expression, whose dividend and divisor are reduced to 0, by assigning certain particular values to some of their terms; so that the expression becomes of the form Thus, if we take comes a2 — a2 0 0 ; then, by putting a for x, it be or, since a a2 = 0, a-a0, and that -, a2 a(a — a) a × 0=0, it becomes divisor is erased, as it evidently ought to be, since it makes no part of the value. Hence, if we put a for x in a + x , we when a is put Again, by putting x = a + y, we get a2 = a2 + 2ay + y2, and thence is reduced to 2ay + y2 =2+ y conse a2 - x2 a(a — x) quently (since a = x reduces y to 0), by putting y = 0, we get 2 for the true value of the expression; which is the same as found before. And it is clear that we may proceed in a similar manner in all analogous cases. Hence, if by either of the preceding methods we reduce the dividend and divisor of the given expression to their lowest terms, then, by proceeding with the reduced expression according to the conditions of the question, we shall get the required value. Dividing the dividend and divisor by ax, the reduced 1 1 = α α 0 = infinity, since 1 contains the infinitessimal 0 an unlimited number of times. 4. To find the value of (112 2acx + ac2 b2bcx+be2' when e is put for x. Because a 2cx+ is a factor of the dividend and divisor, the expression equals, and does not depend on any particular value of x. See page 247, Vol. I., of the Calcul. Differential, etc., of Lacroix. (x2 — a2); 5. To find the value of when a is put for x. Putting ≈ = a + y, the expression becomes (2ay + y2)3 3 x x2 =(2a)*, (2a + y); consequently, putting y = 0, the answer = which is manifestly correct, since (x2 — a2) 1⁄23 3 a2) = (47.) OF THE CHARACTERS 0 AND ∞, AND THEIR USES. 1. A number or quantity that is contained an unlimitedly great number of times in the number or quantity that is taken for the unit in any calculation, is called infinitely small, or an infinitesimal, which is (generally) denoted by 0, called naught, zero, or cipher; noticing that the naught may be taken with either of the signs or, according to the nature of the case. Again, a number or quantity that contains the unit of number or quantity an unlimitedly great number of times is said to be infinite, and is frequently represented by ∞, called the sign of infinity; noticing that the sign of infinity may be taken with either of the signs -, accordingly as the number or quantity represented, is regarded as positive or negative. or It may be added, that any number or quantity that is contained in the unit (of number or quantity), or which contains it a limited number of times, is said to be finite. 2. It is clear (from Ax. VII., and from the nature of the case) that an infinitesimal which is connected with a finite number or quantity by or must be rejected, and that a + finite or infinite number or quantity which is connected with an infinitely greater number or quantity by + or – must also be rejected. Thus, if we take the expression 3 + 2x + 4x2, then, if x is an infinitesimal, it must be reduced to 3, and if x is infinite (since a2 will clearly be infinitely greater than x) it must be reduced to 4x3. 3. Supposing to be an infinitesimal of the first order, then a2, which is (clearly) infinitely less than x, x3 infinitely less than 2, and so on, together with all the infinitesimals that have finite ratios to these, will clearly be infinitesimals of the second, third, etc., orders; also, if a is an infinite of |