. 416. If x denote any quantity then - is called the reciprocal of x. One quantity is said to vary inversely as another when the first varies as the reciprocal of the second. Or if A = -=, where m is constant, A is said to vary inversely as B. 417. One quantity is said to vary as two others jointly when, if the former is changed in any manner, the product of the other two is changed in the same proportion. Or if A = mBC, where m is constant, A is said to vary jointly as B and C. 418. One quantity is said to vary directly as a second and inversely as a third, when it varies jointly as the second and the reciprocal of the third. Or if A = —— , where m is constant, A is said to vary directly as B and inversely as C. 419. If A c= B and B cc C, then A cc C. For let A = mB and B = nC, where m and n are constants; then A = mnC, and, as inn is constant, A cc C. 420. If A cc C amiBccC, tlhen A ± B cc C, and V(AB) °= & For let A = mC and 5 = "O, where m and ft. are constants; then A±B = (m ± «) C; therefore A ±B ccC. Also J{AB) = ,J{mnCt) = C,J{nm); therefore J{AB)<^C. 421. if AccBC, then Bcc^ and Cc=^. 1 A A For let A = mBC, then B = — -=• • therefore B <x — . Simi m C C larly C'cc — . 422. If A cc B and C=cD, then AC « BD. For let A=mB and C = nD, then AC=mnBD; therefore AC'ccBD. 423. 7/acqb, rten A" cc B". For let A = nt5, then A" = m"B"; therefore A" cc 424. 7/"AccB, then APccBP, where P is any quantity variable or invariable. For let A = m5, then AP = ra5P; therefore hp cc BP. 425. 7/" A cc B when C is invariable, and A cc C when B is invariable, then will A cc BC when both B and C are variable. The variation of A depends upon the variations of the two quantities B and C; let the variations of the latter quantities take place separately, and when B is changed to b, let A be A B changed to a'; then, by supposition, —7 = changed to c and in consequence let a' be changed to a; then, by a' C supposition, — = —. Thus, 426. In the same manner if there be any number of quantities B, C, J), &c. each of which varies as another A when the rest are constant; when they are all changed, A varies as their product. EXAMPLES ON VARIATION. 1. Given that y varies as x, and that y = 2 when x = 1, what will be the value of y when x = 21 2. If a varies as b and a = 15 when 6=3, find the equation between a and 6. 3. Given that z varies jointly as x and y, and that z = 1 when a; = 1 and y = 1, find the value of z when a; = 2 and y = 2. 4. If s varies as mx + y, and if z = 3 when * = 1 and y = 2, and a = 5 when a; = 2 and y = 3, find m. 5. If x varies directly as y when z is constant, and inversely as z when ff is constant, then if y and s both vary, x will vary y as —. s 6. If 3, 2, 1, be simultaneous values of x, y, z in the preceding example, determine the value of 7. The wages of 5 men for 6 weeks being £14. 8. If the square of x vary as the cube of y, and x = 2 when y = 3, find the equation between x and y. 9. Given that y varies as the sum of two quantities, one of which varies as x directly, the other as x inversely, and that y = 4 when x = 1 and y = 5 when x = 2, find the equation between x and y. 10. If one quantity vary directly as another, and the former be 2 when the latter is £, what will the latter be when the former is 9? T. A. 15 11. If one quantity vary as the sum of two others when their difference is constant, and also vary as their difference when their sum is constant, shew that when these two quantities vary independently, the first quantity will vary as the difference of their squares. 12. Given that the volume of a sphere varies as the cube of its radius, prove that the volume of a sphere whose radius is 6 inches is equal to the sum of the volumes of three spheres whose radii are 3, 4, 5 inches. 13. Two circular gold plates, each an inch thick, the diameters of which are 6 inches and 8 inches respectively, are melted and formed into a single circular plate one inch thick. Find its diameter, having given that the area of a circle varies as the square of its diameter. 14. There are two globes of gold whose radii are r and r; they are melted and formed into a single globe. Find its radius. 15. If x, y, z be variable quantities such that y + z — x is constant, and that (x + y — z) (x + z — y) varies as yz, prove that x + y + z varies as 16. A point moves with a speed which is different in different miles, but invariable in the same mile, and its speed in any mile varies inversely as the number of miles travelled before it commences this mile. If the second mile be described in 2 hours, find the time occupied in describing the n '* mile. 17. Suppose that y varies as a quantity which is the sum of three quantities, the first of which is constant, the second varies as x, and the third as x*. And suppose that when x = a, y = 0, when x = 2a, y = a, and when x = 3a, y = 4a. Shew that when x = na, y = (n— 1)' a. 18. Assuming that the quantity of work done varies as the cube root of the number of agents when the time is the same, and varies as the square root of the time when the number of agents is the same; find how long 3 men would take to do one-fifth of the work which 24 men can do in 25 hours. (See Art. 426.) XXIX. SCALES OF NOTATION. 427. The student will of course have learned from Arithmetic that in the ordinary method of expressing integer numbers by figures, the number represented by each particular figure is always some multiple of some power of ten. Thus in 347 the 3 represents 3 hundreds, that is, 3 times 10!; the 4 represents 4 tens, that is, 4 times 101; and the 7 which represents 7 units, may be said to represent 7 times 10°. This mode of representing numbers is called the common scale of notation, and 10 is said to be the base or radix of the common scale. 428. We shall now prove that any positive integer greater than unity may be used instead of 10 for the radix, and shall shew how to express a number in any proposed scale. We shall then add some miscellaneous propositions connected with this subject. The figures by means of which a number is expressed are called digits. When we speak in future of any 429. To shew that any positive integer may be expressed in terms of any radix. Let |