346 



THE POPULAR EDUCATOR. 



tion, of certain definite colours or rates of vibration. Such 

 spectra are formed by the light of the sun and stars." 



If the light producing the yellow lines in sodium by the 

 electric arc is allowed to pass through the vapour of metallic 

 sodium, the yellow lines change to black lines. The sodium 

 vapour absorbs the same kind of light as it emits ; and it was 

 by this remarkable discovery that Kirchoff identified many of 

 the dark lines in the solar spectrum with the bright lines ob- 

 tainable from terrestrial substances, and ascertained that in the 

 solar atmosphere there existed sodium, calcium, barium, mag- 

 nesium, iron, chromium, nickel, copper, zinc, strontium, cadmium, 

 cobalt, and hydrogen. If the evidence depended only on the 

 coincidence of one or two dark solar lines with the bright bands 

 from the vapours of the terrestrial metals, it would be worth 

 little or nothing ; but in a complicated series of sets of lines, 

 such as would be produced by the above metals, all the lines 

 coincide, and in speaking of one of these metals, viz., iron, 

 Kirchoff remarks that " the observations of the solar spectrum 

 appear to me to prove the presence of iron vapour in the solar 

 atmosphere, with as great a degree of certainty as we can 

 attain in any question of natural science." Messrs. Huggins and 

 Miller have continued these observations with the planets, the 

 stars, the nebulae, and the comets, and added largely to our 

 knowledge of the constitution of these distant heavenly bodies. 



LESSONS IN ALGEBRA. XXIII. 



THEEE UNKNOWN QUANTITIES. 



IN the preceding examples of two unknown quantities, it will be 

 perceived that the conditions of each problem have furnished 

 two equations independent of each other. It often becomes 

 necessary to introduce three or more unknown quantities into a 

 calculation. In such cases, if the problem admits of a deter- 

 minate answer, there will always arise from the conditions as 

 many equations independent of each other, as there are unknown 

 quantities. 



Equations are said to be independent when they express 

 different conditions. 



They are said to be dependent when they express the same 

 conditions under different forms. The former are not con- 

 vertible into each other ; but the latter may bo changed from 

 one form into the other. Thus b x = y ; and b = y + x, are 

 dependent equations, because one is formed from the other by 

 merely transposing x. Equations are said to be identical when 

 they express the same thing in the same form expressed or 

 implied ; as 4x 6 = 4x 6, or 2 (2x 3) = 4 6. 



EXAMPLE (1). Given x -f- y -f- z = 12, x -f- 2y 2z = 10, and 

 x -f- y z 4 ; to find the values of x, y, and . 



From these three equations, two others may be derived which 

 shall contain only two unknown quantities. One of the three 

 unknown quantities in the original equations may be extermi- 

 nated, in the same manner as when there are at first only two, 

 by the rules already given. Thus, if in the equations given 

 above, we transpose y and z, we shall have, 



From the first, x = 12 y z; 

 From the second, x = 10 - 2y + 2z ; 

 From the third, x = 4 - y -f z. 



From these we may now deduce two new equations,, from 

 which x shall be excluded. 



By making the first and second equal, we have 



12 y z = 10 2y + 2z. 

 By making the second and third equal, wo have 



10 2y + 2z = 4, y + z. 

 Beducing the first of these two, we have 



y = 3z 2. 

 Eeducing the second, we have 



y = a + 6. 



From these two equations one may be derived containing only 

 one unknown quantity. 



By making the one equal to the other, we have 



3z 2 =+6, 



/hei-efore, z 4. Hence, y = 10, and x= 2. 

 To solve a problem containing three unknown quantities, and 

 producing three independent equations. 



EULE. First, from the three equations deduce two, containiny 

 only two unknown quantities. Then, from theso two deduce one, 

 containing only one unknown quantity. Lastly, Jind the values 

 of the other unknown quantities as before. 



For making these reductions, the rules already given are 

 sufficient. 



EXAMPLE (2). Given x + 5y + 6z = 53, x + 3y + 3z = 30, 

 and x + y + z = 12 ; to find the values of x, y, and z. 



Here, from these three equations, in order to derive two con- 

 taining only two unknown quantities, 



Subtracting the second from the first, we have 



2y + 3z = 23 ; (the fourth equation) 

 Subtracting the third from the second, wo have 



2y + 2z = 18. (the fifth equation) 



Next, from these two, in order to derive one, 



Subtracting the fifth from the fourth, we have 



To find x and y we have only to take their values from the 

 third and fifth equations. 



Eeducing the fifth, we have 



i/ = 9 z = 9 5 = 4. 

 Transposing in the third, wo have 



a; = 12 z y=12 5 4 = 3. 



In many of the examples in the preceding lessons, the pro- 

 cesses might have been shortened. But the object was to 

 illustrate general principles, rather than to furnish specimens of 

 expeditious solutions. The learner will do well, as he passes 

 along, to exercise his skill in abridging the calculations here 

 given, or substituting others in their stead. 



Ho must also exercise his own judgment as to the choice 

 of the quantity to be first exterminated. It will generally be 

 best to begin with that which is most free from co-efficients, 

 fractions, radical signs, etc. that is, the quantity least involved. 



EXERCISE 39. 



1. Given x + y+x = l2,x + 2y + 3z = 20, and Jz + %y + z = 6 ; to find 



the values of x, y, and z. 



2. Given x + y = a, x + z = b, and y + z c ; to find the values of x, y, 



and 5. 



3. Three persons, A, B, and C, purchase a horse for 100 dollars, but 



rjeither is able to pay for the whole. The payment would 

 require the whole of A's money, together with half of B's; or 

 the whole of B's with one-third of C's; or tho whole of C's, 

 with one-fourth of A's. How much money had each ? 



4. The sum of the distances which three persons, A, B, and C, have 



travelled, is 62 miles ; A's distance is equal to four times C's 

 added to twice B's ; and twice A's added to three times B's, is 

 equal to 17 times C's. What are the respective distances ? 



5. 3iven %x + frj + i* = 62, %x + y + z = 47, and fa + %y + z = 38 ; 



to find the values of x, y, and z. 



6. Given xy = 600, vz = 300, and ijz = 200 ; to find the values of x, y, 



and z. 



FOUR OK MORE UNKNOWN QUANTITIES. 

 The same method which is employed for the reduction of 

 three equations, may be extended to four or Jive, or any number 

 of equations, containing as many unknown quantities. 



The unknown quantities may be exterminated, one after 

 another, and the number of equations may be reduced by suc- 

 cessive steps from Jive to four, from four to three, from three to 

 two, and so on to one. 



EXAMPLE (1). Given $y+ z + -lw = 8, (1) 



x + y + w= 9, (2) 



E + V + s = 12, (3) 



x + w + z = 10; (4) 



to find the values of iu, x, y, and z. 



Here, clearing the first equation of fractions, we have 



y -f 22 -f- iv = 1G ; (5) 



Subtracting the second from the third, we have 



z w = 3; (6) 



Subtracting the fourth from the third, we have 



y w = 2. (7) 



Next, adding the fifth and the sixth, we nave 



y + 3z = l9; (8) 



Subtracting the seventh from the sixth, we have 



y + z=l. (9) 



