LESSONS IN ALGEBRA. 



with its adherent hydrogen much M a metal plato in an ordinary 

 cell will behave toward* its follow : in other word*, it in 



load. The partial 



.urn nt, wliirh in INI, ntato tlui ooll in able to afford, cauBc-H 

 : oxide to assume a motollio < <l spongy t 



11 is now charged, but in the opposite direction, with tho 

 result that the other plate is poroxidised. ThU change of 

 direction is repeated over and orer again until each plate in 

 ,.)!! into, the one having at hut a thick coating of peroxide, 

 ; - I the ..'li.-r l.i-ing equally changed to the spongy 001 



ion, before it is complete, occupies some months 



of constant care. Thus, tho first day, tho alternate charging 



must be conducted at intervals of about half-an-hour, and the 



..11 nut -t IK) discharged between each step in the process. The 



next day those intervals may be increased to an honr, a short 



time afterwords to two hoars ; so that after a few weeks 



.1 hours may bo made to elapse between each operation. 



When the forming of the oell approaches completion, tho 



charge is sent through it every time in one direction only, 



otherwise tho whole of both plates would be transformed to 



!<>, and they would of course fail in tho object of their 



. -fire. In order to obviate this tedious process of 



forming, M. Fauro hit upon the plan of giving the plates a 



! : Hminary coating of red load. 



Although thin secondary cell has now been known to 

 electricians for more than twenty years, the general public had 

 no idea of tho existence of any such means of storing up for 

 t .tun- use the energy derived from an ordinary electric battery. 

 But a few years ago, when Fanre introduced his improve- 

 ments into the secondary battery whereby its cost was lessened 

 and its efficiency greatly improved, the world was startled by a 

 strange announcement. Sir William Thomson published a state- 

 ment to the effect that ho had brought what he called " a box 

 of electricity " all the way from Paris to Glasgow. On arrival 

 at the latter city he was enabled to use the energy which had 

 boon stored up in this box (containing a few Fanre secondary 

 colls) some days previously in the French capital. Since this 

 announcement, the secondary battery has become well known, 

 and many improved forms have been introduced. To these and 

 their various applications we shall direct attention later on. 



LESSONS IN ALGEBRA. XXXIII. 



ADFECTED QUADRATIC EQUATIONS. 



EQUATIONS are divided into classes, which ore distinguished 

 from each other by tho power of the letter that expresses the 

 unknown quantity. Those which contain only the first power of 

 tho unknown quantity are called simple equations, or equations 

 of tho first degree. Those in which tho highest power of the 

 unknown quantity is a square, are called quadratic, or 

 equations of the second degree; those in which the highest 

 power is a cube, are called cubic, or equations of the third 

 degree, etc. 



Thus x = a + 6 is an equation of the first degree ; x- = c, and 

 z 2 + ax = d, are quadratic equations, or equations of tho second 

 degree ; a; 3 = h, and z 3 + cue 2 + bx = d, are cubic equations, or 

 equations of the third, degree. 



Equations are also divided into pure and adfected equations. 

 A 'pure equation contains only one power of the unknown 

 quantity. This may be tho first, second, third, or any other 

 power. An adfected equation contains different powers of the 

 unknown quantity. Thus, 



C x 2 = d - b, is a pure quadratic equation. 



^ x 2 + bx = d, an adfected quadratic equation. 



I x 3 = o - c, a pure cubic equation. 



( x 3 -f- a 3 ? + bx = h, an adfected cubic equation. 



In a pure equation, all the terms which contain tho unknown 

 quantity may bo united in one, and the equation, however com- 

 plicated in other respects, may be reduced by the rules which 

 have already been given. But in an adfected eqnation, as tho 

 unknown quantity is raised to different powers, the terms con- 

 taining these powers cannot bo united. 



An adfected quadratic equation is one which contains the un- 

 knoivn quantity in one term, and the square of that quantity in 

 another term. 



The unknown quantity may be originally in tecerai terms ot 

 the aquation. But all these term* can be reduced to two, one 

 containing the unknown quantity, and the other its sqnan. 



It has already been shown that a jrur quadratic U solved by 

 cj the root of both fide* of th* ^/tuition. An adfected 

 quadratic may be solved in the same way, if the member which 

 contains the unknown quantity is an etatt u/uart. 



Thus the equation 3 + 2a* + o > = 6+ * may b* reduced by 

 M. Fr the first member is the *ptar* of a binomial 

 quantity, and its root U + a. Therefore, 



4. a = */i -f h, and by transposing a, 



a = */6 + h - a. 



Bat it is not often the ease that the member of an adfeeted 

 quadratic containing tho unknown quantity is an exact iquart 

 till on additional term is supplied, for the purpose of making the 

 required reduction. 



In the equation ** + 2ax = b, the side containing the un- 

 known quantity is not a complete square. The two terms of 

 which it is composed are indeed snob as might belong to the 

 square of a binomial quantity. Bat one term U \canting. We 

 have then to inquire, in what way this may be supplied. From 

 having two terms of the square of a binomial given, bow shall 

 we find the third ? 



Of tho three terms, two ore complete powers, and the other 

 is twice tho product of tho roots of these powers, or, which if 

 the name thing, the product of one of the roots into twice the 

 other. 



In the expression x 3 + 2ax, the term 2<u consist* of the 

 factors 2a and x. The latter is the unknown quantity. The 

 other factor 2a may be considered the co-efficient of the un- 

 known quantity ; a co-efficient being another namo for a factor. 

 As x is the root of the first term x 1 , tho other factor 2a is twice 

 the root of tho third term, which is wanted to complete the 

 square. Therefore half of 2a is the root of the deficient term, 

 and a* is the term itself. 



The square completed is tr 2 + 2ax -f a 2 , where it will be 

 seen that the hist term a? is the square of half of 2a, and 2a is 

 the co-efficient of x, the root of the first term. 



In the same manner it may bo proved that the last term of 

 the square of any binomial quantity is equal to the square of 

 half the co-efficient of the root of the first term. 



From this principle is derived the following 



METHOD FOB COMPLETING THE SQUARE. 



Take the square of half the co-efficient of the first power of the 

 unknown quantity, and add it to loth tides of the equation. 



It will be observed that there is nothing peculiar in the sola- 

 tion of adfected quadratics, except the completing of the tqvare. 

 Quadratic equations ore formed in the some manner as timple 

 equation's ; and, after tho square is completed, they are reduced 

 in the same manner as pure equations. 



EXAMPLE. 

 Reduce the equation z* -f- Gax 6. 



Completing the square, x* + 6ox + 9o 3 = 9a* + 6. 



Extracting the root of both sides, x + 3a = */9a* + t. 

 And * = - 3o 4; V9a + 6. Ant. 



Here tho co-efficient of x, in the given equation, is 6a. 



Tho square of half this is 9a s , which being added to both 

 sides completes the square. The equation is then reduced by 

 extracting the root of each member. 



EXERCISE 62. 



1. Reduce the equation * 86* = h. 



2. Reduce the equation *' + or = b + h. 



3. Reduce the equation ** * = k d. . 



4. Reduce the equation x* + 3* = d + 0. 



5. Reduce the equation z* abx = ab cd. 



6. Reduce tho eqnation * + i? O. 



b 



7. Reduce the equation ** ? = To. 



8. Reduce the equation ** + a* t = b. 



In these and similar instances, the root of the third term of 

 tho completed square is easily found, because this root is the 

 same half co-efficient from which the term has just been 



