INFINITESIMAL CALCULUS. PRELIMINARY THEOREMS. As the following Algebraical Theorems will be frequently applied in the course of the treatise, it is convenient to prove them once for all, and, for the sake of reference, to place them at the beginning of the work. I. If there be any number of equal fractions each of them is equal to A1 + A2 + A3 + ...... + an +bn ...... and, if m1, m2, M3, m, be any multipliers, to and to ...... {b22 + b22 + b2+....... + b„2 } ✯ Let each of the fractions = r, that is, let The geometrical form of the first of these theorems is the twelfth proposition of the fifth book of Euclid. When three unknown quantities are involved in three (so called) linear equations of the forms a1 x + b1y + c1 z = d1 A 2x + b 2 y +C 2 z = d2 a3x+b3y+C3Z = d3 (1) (2) (3) the following is the most convenient method of determining cach of the unknown quantities in terms of the constants. Multiply (1) by A1, (2) by λ2, (3) by λs, and add; then (α1λ1+α2λ2+а3λ3) x + (b1λ1 +b2λ2 +b3λ3) Y +(C1^1+C2^2 + C3λ3) z = d1λ1 +d1⁄2λ2+d3λ3. As three undetermined quantities, viz. A1 Ag As, have been introduced, we may make three suppositions respecting them: leaving one to be determined hereafter, let two be, that the coefficients of y and z be equal to zero; so that But thus the ratio only of the multipliers has been determined, and therefore any numbers bearing to one another the above ratio would satisfy the requisite conditions; to take however the most simple numbers, let the third condition be introduced, and be, that each of the above fractions be equal to unity; and therefore X= d1 (b2C3—C2b3) + d2 (b3C1 − C3b1) + d3 (b1 C2 − C1 b2) а1 (b2c3−c2b3) + а2 (b3C1−C3b1) + aз (b1 C2—C1b2) ' By similar processes, if the coefficients of and a had been equated to zero, the result would have been y = dı (C2α3—α2C3) + d2 (C3α1−α3C1) + d3 (C1α2—α1C2) b1 (C2 α3 — α2C3) + b2 (C3α1 −а3C1) + b3 (C1α2− α1 C2)' and if the coefficients of x and y had been equated to zero, 2= d1 (a2bз—b2αз) +d2 (a3b1−b3α1) + d3 (a1b2—b1ɑ2) C1 (α2b3—b2α3) + C2 (α3b1−b3α1)+cз (α1b2—b1α2) * This method of elimination is generally known by the name of Lagrange's Rule of Cross-Multiplication, the origin of which term is sufficiently obvious from the form of the multipliers: two examples are subjoined for the sake of practice, but the student is recommended to exercise himself in many others. the values of the multipliers corresponding to the several variables being arranged in the annexed vertical rows: whence the values of the several multipliers being arranged as before: The following is a particular case to which the preceding process of cross-multiplication is applicable. Let there be three equations of the forms which involve only two unknown quantities, say and; bc cause each equation may be divided through by z; hence it is evident that a relation exists between the coefficients: this = relation is to be determined; and may be found in the following manner. Since d1 dad30, if we multiply the first, second, and third severally by the multipliers determined as above, we shall have, after addition, = α1 (b2C3-C2 bз) + а2 (b3 C1—C1 bз) + aз (b1 C2—C1b2) = 0, which is the relation among the coefficients of the three preceding equations, when they coexist. This theorem is one of the simplest in the method of Determinants the general principles of which are explained in Mr. Spottiswoode's "Elementary Theorems relating to Determinants," Crelle's Mathematical Journal, Vol. LI, Berlin, 1856; and in the Appendix to "Salmon's Higher Plane Curves," Dublin, 1852. ...... III. ... If a1, a2, aз, an are quantities of the same sign, and A1, A2, A3, ...... A, are any other homogeneous quantities capable of addition and subtraction, then a1 a1+α2 α2 + A3 A3 + + an an is equal to (a1+ A2 + A3 + ... + an) multiplied into some quantity greater than the least, and less than the greatest, of the quantities a1, a2, A3, ... The proof of this proposition depends on the fact, if both terms of an inequality are multiplied or divided by a positive number, the sign of inequality remains the same; that is, the quantity which was greater before the multiplication is the greater after it; but if the terms are multiplied by a negative number, the sign of the inequality is reversed: that is, a > is changed into a <, and a < into a >. This is easily shewn by an example; as, for instance, 5 is greater than 2; let each side be multiplied by +4, then 20 > 8; again, let each side be multiplied by a negative number, as -2: the >is changed into a<, -10<-4, because -10 is less than -4. Let L be the least and G the greatest of the quantities a, a, aз, an; then, with the exception of the two cases of the greatest and the least of the quantities, whereby however the final result is not vitiated, we have the following inequalities, ...... First, let the quantities α1, ɑ2, aз, an be positive, so that the signs of the above inequalities will not be changed when they are multiplied as follows: a1a is > La1, < Gal az az is > La2, < Ga2 anan is > Lan, < Gan; and therefore by addition ... a1a1+ a2 a2+ +anan is > L (a1 + a2+ ...... signifying by mean value, a quantity greater than the least, and less than the greatest. Secondly, if a1, a2, ...... a, be negative, the signs of the inequalities would have been changed in the first multiplication, and would have been again changed in the final result, because a1+ag+......+a, would be a negative quantity, and thus the same result would follow: and therefore the proposition is proved. a1 а2 ძვ If IV. an is a series of fractions, the numebn rators of which are of either sign, and the denominators all of the same sign, then a1+2+...+an is equal to some quantity b1 + b2 + ... + bn greater than the least and less than the greatest of the given fractions. First, let all the denominators be positive, and let L be the least and & the greatest of the fractions; then |