TABLE II. For determining the Probabilities of Life at NORTHAMPTON, as deduced by Dr. Price from the mortality of that town in the years 1741...1780. The preceding Tables require but little explanation. The former commences by stating that out of 1000 persons who were born at the same time and attain the age of 1 year, 145 die before they attain the age of 2 years. Consequently at 2 years of age there are left 855 out of 1000. Of these 57 die between 2 and 3 years of age; and so on. Thus, of 1000 persons who attain the age of one year, the Table indicates that 346 live to be 50 years of age; &c. The latter Table commences a year earlier by taking 1149 persons born together, that is, at the age 0; and then proceeds in the same manner as the former. Thus, we have given by this Table, that after 50 years, out of 1149 persons born together, 284 are then alive, and that of these 9 die before attaining the age of 51; and so on. It has been objected to both the preceding Tables, although the latter is very generally used by the Assurance offices, that they make no distinction between male and female life, and yet that a very material distinction can be proved to exist. TABLE III. Shewing the Expectation of Life, as deduced by Professor De Morgan from the Statistical Returns of the whole of BELGIUM made by M. Quetelet and Smits. The extent of the error which arises from not distinguishing between the sexes may be seen in Table III. constructed by Professor De Morgan from the statistical returns of the whole of Belgium for three successive years, as given by M. Quetelet and Smits, in the Recherches sur la Reproduction, &c. Brussels, 1832. This Table is calculated to shew the "expectation of life," that is, the average number of years remaining to any individual, at intervals of five years, from the age of 0 to 100. It distinguishes not only between male and female, but between town life and rural life; and the middle column gives the general average for the whole kingdom, male and female, town and country. DISCUSSION AND INTERPRETATION OF ANOMALOUS RESULTS. 463. Negative Results. It often happens, that the result of our operations for the solution of a proposed question or problem appears in a negative form, although strictly speaking, there can be no such thing. existent as an essentially negative quantity. But it will always be found, when such a result occurs, that there is something in the nature of the question which will either dispose of, or supply a meaning to, the negative result. Thus, to take a simple example; suppose a man wishes to ascertain the amount of his property-he puts down what he has, together with what is due to him, as positive, and all his debts with a negative sign. If then he finds that by taking the sum of both positive and negative quantities, the result is negative, its meaning will be sufficiently obvious, viz. that his property is so much less than nothing, that is, he is so much in debt. See Scholium, p. 44; and Art. 214. Also, see Art. 282. Ex. 4. In this and like cases it is true that two solutions may be found for the equation, that is, two values of n; but when either of those values is fractional or negative, it is clearly inadmissible as a solution of the question proposed. It may be observed also here generally, that when in solving a problem, expressed algebraically, we find it necessary, as in the above Example, to extract the square root of a quantity, the double sign ±, (that is, + or −), need not to be prefixed to the root, at least for the object before us, if we have sufficient data beforehand for determining which sign the problem requires. Is it to be wondered at, that we produce an anomalous and unintelligible result, if we wilfully make a quantity negative which we know to be positive, or vice versa ? Oftentimes, however, the negative solution, whether it results from carelessness or necessity, will satisfy another problem cognate with the proposed one; which may be determined by substituting the negative quantity for the positive in that step of the process which most clearly expresses the conditions of the question; and then interpreting the resulting equation with the assistance of the given problem. This has been done in the cases above referred to. 464. Interminable Quotients. Strictly speaking a Quotient can only exist when after the division by which it is determined there is no Remainder; but the term is applied to those cases also where a remainder is left which cannot be got rid of. Thus we say generally, that the quotient of 1÷1-x is 1+x+x2 + x3 + . ..; whereas the true quotient is ...... Thus, whatever be the value of x, N.B. By taking the Remainder into account no unintelligible result can arise from substituting any particular value for x. COR. 1. If x <1, then the Remainder may be neglected, if a sufficient number of terms of the series are taken. (Art. 290, Cor. 2.) 1 If x=1, then =1+1+1+1+&c. in. inf. = an infinitely great number. 1 COR. 2. If a be negative, we have · = 1−x+x2 − x3 +&c. in inf.; in 1 1+x which if we put 1 for x, we get =1-1+1-1+&c.=0, or 1, according as an even or an odd number of terms is taken; both of which results are obviously impossible. Now, taking the Remainder into account, we have without which fractional Remainder no arithmetical equality subsists between the series and 1 And it may be observed generally, that no equality subsists, for purposes of calculation, betwixt any infinite series without the "Remainder", and the primitive quantity from which it was derived, unless the series is convergent, so that we can make the Remainder after r terms, by increasing r, as small as we please. 465. To explain generally the results which assume the forms a×0, ao, 0o, a 0 (1). Since axb signifies a taken b times, it is clear that, if 0 is to follow the same rule as other multipliers, a×0 signifies a taken 0 times, and is therefore equal to 0. b (2). Since expresses the number of times a is contained in b, a will signify the number of times a is contained in 0, that is, 0 times, or 0 a =0. (3). By the general Rule of Indices, axa"=a"+", and a"÷a"=a"-". Now in the first of these let m=0, then ao.a"=a", if the same rule holds when one of the indices is 0; therefore a°, as far as regards the rule for multiplication of powers, is equivalent to 1, or ao=1. a Also, since ¡=a"-", 1=ao, if the rule for division of powers holds when the powers are equal; therefore it also accords with the general rule for division that ao=1. Hence ao, bo, co, &c. are separately equal to 1, if 0 be admitted as an index subject to the same rule as other indices. (4). Since it has been already explained, that any quantity raised to a power represented by 0 may be safely expressed by 1, it follows that (a-6)=1, whatever a and b may be. If then a=b, we have 0°=1. a (5). When an algebraical quantity is made to assume the form, it is said to be infinitely great, and its value is expressed by the symbol. All that is meant is, that if the denominator be made less than any assignable or appreciable quantity the fraction becomes greater than any assignable quantity. This is easily shewn by taking any fraction, as which=10a for if the denominator be successively diminished one-tenth, we obtain the series of quantities 100a, 1000a, 10000a, &c., proving that as the denominator of the fraction is diminished, the value of the fraction is increased, and without limit. a 0.1 (6). Suppose that an expression involving a assumes the form when some particular value (a) is substituted for x, then it is clear that the expression is capable of being reduced to the form p(x-a)" where Р and q have no factor x-a in them; and by dividing numerator and denominator by their highest common factor, the value of the fraction may be found when x=a. (See Art. 387.) Thus it appears that a quantity which assumes the form may have a determinate value. And, conversely, since p is equal to p. that is, any quantity may be |