47 



MATHEMATICS. ALGEBRA. 



[FRoroBTioir. 



of higher equations, when the propoaed ones are both 

 quadratics. There are only two particular classes of 

 such irnultaneoui equations to which general method* of 

 solution, capable of being explained here, apply. These 

 classes are ealled respectively kamog**mi tftatiofu, and 

 tymmdruai epMfcrn*. We ahall Tory bnofly consider 



t ; :.::'. 



JTosMOTMOM* Quadratics.*, pair of quadratics is laid 

 to be a pair of homogeneous equations when each unknown 



term in both U of *o dimeiuwn*; that is, when no term 

 occurs without either the tquan of one of tho unknown 

 quantities, or the product of both ; the presence of tw> 

 unknown factors whether equal or unequal in a term, 

 being the circumstance that rondois that term of (wo 

 dimensions. Tho following U a pair of homogeneous 

 quadratics, each unknown term being of two dimensions : 



The following is the general way of solving 

 such equations : Put ;y for x, and tho equations become 



77 12 

 , i ; from the second, y' = . 



From the nt,y' = 



.... [A] 



By means of this contrivance ; viz , the putting z 

 times one of the unknown quantities for the other, we 

 easily obtain a quadratic equation involving z only : 

 solving this, we have 



- 



Substituting each of those in equation [A], we have 



or 16 . . jtJ-1^2, or +4 



4J.or^Uutis,,. = |, 



, 



You see that each of tho unknown quantities has four 

 values ; for J 2 is cither + or : such is usually tho 

 ease when the proposed equations are each of two 



77 * s 

 dimensions. From the first equation above y= - > 



and if this be put for y in the second, the result, cleared 

 of fractions, would evidently be on equation of the fourth 

 degree ; and it is proved, in the Theory of Equations, 

 that an equation always has as many roots as there are 

 units in the number which marks its degree : this is the 

 reason that tho values of x and y above are four in 

 number. 



Symmetrical Quadratics. An equation is said to bo 

 symmetrical, in reference to the unknown quantities 

 which enter it, when they so enter that they may bo 

 interchanged without producing any alteration in tho 

 equation: thusx+y=o; x*+3xy+y*=b; 



_|-2u 3 -c, <tc., are all symmetrical equations; because 

 for every x yu m:iy put y, and for every y, x, without 

 altering either of the equations. Tho following is a pair 

 of symmetrical quadratics; namely, 



gi-fy* z y 18l Tho general method of deal- 

 sv-f-x-t-ycaiajing with such equations is as 

 follows : 1'ut u + cfor x, and u v for y, nud the 

 equations become 



.... [A] 



, ( ._r)+2=19 i '-*M- 

 BT wldins 2' + H = 28 .'. 4 + 1 = / (224 + 1) = + 15 .-. 



_ or _ 4 



2. 



+ 4 = 9. -.'= U .-.e = ^ 11 

 .-.* = + = L+l, or _ 4S 11; that is, * = 4, 3. 



, 

 or, 4^ 11 



= p = or t+U; that U y = 3. 4, 



Tho two clauses of equations hero exemplified will 

 alwiiys yield to tho foregoing methods : but particular 

 cases of these, as also of other kinds of simultaneous 

 equations for the solution of which no general rules can 

 bo given, may often bo solved more easily by the exercise 

 of a little tact and ingenuity : for instance, you would 

 not think of applying the first method to such a pair of 

 equations as x-+xy = 9, y+a;i/ = 16; because tho 

 slightest inspection of them would suggest to you the 

 better way of adding them together, which would t;ivo 

 you z 2 +2zi/+y 2 =23, whence z+y = +5; and since 

 the first is (jc+tj)z=9, you would have at once -j-5j;=9 ; 



9 



+ -^ /. y = + 5 z= + 5 +_- - 



Or you may proceed thus: tho two 



and thence z 



4-25-4-9 16 



=4-"g". 



5 



equations give x + y= , and z 



= /. 9y=16x.*.y 



*7 



x .: by substitution in the given equations, you 



would have 



.10. 



These equations give no different values for z, since ^ 

 tho second ia nothing but - times tho first : henco the 



only values of are x = 4^ A .-. y = x = + ". 



CHAPTER IV. 

 ALGEBRA. RATIO AND PROPORTION. 



Tlir. abstract number which arises from dividing one 

 quantity by another of tho same kind, is called tho ratio 

 of tho former to tho hitter. Itatio, therefore, is only 

 another name for the qttoticiU of one quantity by another 

 of the samo kind : tho first quantity (tho dividend) is 

 called the antecedent of the ratio, and tho second (the 

 divisor) the eonteyatnt: these are the tcrnu of tho ratio. 



Thus, of tho two quantities o, 6, tho ratio ia ? ; or, which 



is the same thing, a -- 6 ; but tho little mark between 

 the two doU, in the Hynibol for division, is usually 

 omitted in exprcsning a ratio ; so that the i;iti<i of a to b 

 would bo denoted by a : 6. Thus, tho ratio of 8 to 4, 



that is 8 : 4 is J or 2 ; this ratio is of course the same as 

 tho ratio 4 : 2, or tho ratio 2 : 1. 



When there are four quantities such that tho ratio of 

 tho first to tho second is equal to tho ratio of the third to 

 tho fourth, tho four quantities are said to bo in propor- 

 tion : thus, if tho four quantities, o, 6, c, d, bo such that 



- - , or, which is tho samo thing, a :b = e : d, then 

 6 d 



a, 6, c, d are in proportion ; and wo express this by 

 i tliat a is to 6 as c is to d ; or by writitw a:b :: c :d. 

 < If these four terms, the first and last (a and </) are called 

 tho extreme*; and tho intermediate terms (b and c) 

 the meaia. 



