ELIMINATION. 



KI.I/.A 



AN AKCIHTF.cTCKi:. 



nation is always theoretically possible. But whether the rwulU pro- 

 duoed will bi convergent, and otherwise convenient fur use, uiuat 

 depend upou the circumstance* u{ each CMC. When, however, 

 elimination is spoken of, elimination iu hniu- terms ii gem-rally 

 understood. 



Let us now take simple algebraical equations, and suppoee a peir of 

 then, containing x and y, a* x# 1 -j-V-2 = 0, . x>y- = 0. Let 

 theeebeP'O, Q=0, and proceed as in finding the niot complicated 

 common divisor of r and <j. And first, let it appear that there is a 

 common dirt** , and lot r - B A. o. - K n ; so that A and B have not 

 any TITVTI algebraical divisor. Thene equationa may then be both 

 satisfied by making B = 0, which being only one equation between 

 x and y, admit* of an infinite iminW of solutions. But they may also 

 be satisfied by A = 0, B=0, the method of treating which falls under 

 the next case. 



Secondly, let there be no algebraical common diviitor. Then, if we 

 take y a* the princi|ial letter, there will at last be a remainder which id 

 a function of x only : let this be X. Now it is easily proved that when 

 r and y vanish, x must vanish also : that is, no values of y can make 

 both r and <J vanish, except in combination with values of x which 

 make x vanish. All the required values of x, then, satisfy x = 0. But 

 the converse true, and are all the roots of x = capable of com- 

 bination with values of y, so of to satisfy both equations ! This 

 question will require some consideration. 



In the preceding process, when y is made thu principal letter, we 

 hare generally this alternative, either to introduce fractional functions 

 of jc into the result, or else to employ multipliers beforehand to avoid 

 it : these multipliers will be functions of X. Let us allow of fractional 

 functions of .r, and suppose thu process to bo as follows : 



' ) 1 ( MI 



-', ) ' ( M, 



s, ) N, ( a, 



X 



We have then 



Q =M, P 



Take a value of .< which makes x vanish, and with this value of jc, 

 tind a value of y which makes P vanish. We have then <J= N, = M 3 x,= 

 M,J,s,, or >',(! + UjMj) = 0. If then 1 4 M ? w s should not happen to 

 vanish, we must have N, = 0, or Q = 0; that is, a root of x = 0, com- 

 bined with a value of y which makes p=0, also makes (j = 0. But if 

 it should happen that 1 + M.M, = is satisfied by the values of x and y , 

 we have not this assurance that (j = 0. Again, if M , M 2 , or M 3 , be 

 made infinite by the value of .r, the whole process fails. All we can 

 say then is, that though x = must contain among its roots all the 

 values of x required, it may possibly contain other values. 



Next, suppose we introduce a multiplier to avoid fractions. Let it 

 be convenient to begin with <J z and r, instead of Q and 1*. Then when 

 vi z and r both vanish, x vanishes ; but we can now only say that the 

 rooU of x = may, with proper values of y, satisfy one or the other of 

 the systems g = 0, p=0, or z = 0, P=0. But if we take values of x 

 which make K, = 0, x 0, it is easily seen that we satisfy both i>--0 

 and y = 0, if no one of the quotients become infinite. 



We shall take the following example, to show how comparatively 

 complicated the results of a very simple instance may be. Let the 

 equations be 



Multiply Q by r, to prepare for division without fractions, and 

 divide by r : the remainder is x*(x s -l-l) y (6x 2). Multiply p by 

 j?(x l +l) 1 for a similar reason, and divide by the remainder. The 

 second remainder, x, is 



-' ' + 6x + 2x + 6x 36*' + 24r- 4 



One of the roots of this is x=l, with which it will he found that 

 u = 2 satisfies the equations p = 0, Q = 0. And in this instance it does 

 happen that all the seven roots of x are capable, each with its proper 

 values of y, of satisfying the given pair of equations. 



If we take the principal letter to be x, the problem of ordinary 

 algebraical elimination is reduced to eliminating jc from two such 

 equations as ajT + bj.-' ... =0, an.l lat^-t-^jf- '+ ... =0, where 

 a, p, 4c., are functions of the other letters. The mode which wu have 

 always adopted as moat convenient in practice is as follows : Supp.m- 

 toe equations to be 



+ ').r' + r.,- fc = 



yx 3 + qx* + rjc + 



subtract, which elves au conation of the 



Multiply i 

 form 



Multiply by an.1 <. cnbtrvt, nn.l .livid- by .r, which give* I 

 equation of the i 



B n 

 lte|*t the MUM nort of proeew, in.ikinx the first term* destroy eacJ 



other, and then the latt ; we are thus led to two equations of the first 

 degree, from which z may be at once eliminated. The same n. 

 will 1 1. 1 when the equations an ' degrees, but in that case it 



is more convenient to bring down the higher to the lower, first, in the 

 following manner: Suppose the equations to be ox' + 4x*+fx* *ex* 

 ... = 0, and px* + ^.r 1 + . . . = 0. Multiply by p and by ox 1 , and 

 subtract, which gives a new equation of the form AX* + BX* + . . 

 Multiply by /> and by AX* (using the lower of the given equations), and 

 subtract; and continue this process until a second equation of the 

 fourth degree is found : after which proceed with the Brat and hut 

 terms as before. 



The problem may be reduced to that of elimination between purely 

 linear equations, as follows : Suppose it required to eliminate xK 

 ax' + 6x + c=0, andj>x 5 gx <l + rx + i=0. Multiply both these equations 

 by x, and we have then four equations involving the first four powers 

 of .'. Multiply again by x, which gives six equations involving the first 

 five powers of x. Let these five powers be considered ait five distinct 

 quantities, j- ir r.,x.,, 4c., and eliminate these five quantities between 

 the six linear equations, which are 



ax t + 4x, + fx, = 



y/.c a 



+ rx, + w, = 

 , + r.r t > x, = 



Since two equations are introduced at each new step and only one 

 new power, there must be a step at which the number of equ 

 becomes equal to the number of powers, after which, at the next step, the 

 number of equations will be in excess by one, which is what is wanted 

 for elimination. It is also worth notice that if we were to stop at thu 

 i-ti-p at which the number of equations is the same as that of powers 

 and find the powers as independent quantities, we might then elimi- 

 nate between the first and second ]iower, and produce the result in the 

 form A' B=0. 



The following method, troublesome enough iu actual operat 

 the theoretical conquest of the difficulty. Let <t> (x, y) = 0, ii (..-, 

 be two algebraical equations ; and let the values of y from the first be 

 x,, x,, X 3 , &c. The equation which indicates that the second equation 

 is satisfied by some root of the first is^(x, x,) . vH-'', x i) ...... -0; ;md 



the first side, being a Kymmeliie.il function of x,,x,, &c., can be ex- 

 pressed in terms of the coefficients of powers of y in <f> (x, y) = 0, without 

 actual solution. Let this be done, and we have that equation involving 

 x only which is the necessary consequence of the two given equa' 



On this subject see Dr. Peacock's 'Algebra,' vol. ii. cap. 44 ; or Mr. 

 Sylvester's remarkable pajx-rs. Phil. Mag.' Dec. 1839, Feb. 1840, Juno 

 1841; also ' Cambridge Mathematical Journal,' vol. ii. pp. - : '._ 

 vol. iii. p. 183, vol. iv. p. 9. 



KLIXIK OK VITRIOL. [SumiriiicAcm] 



ELIZABETHAN ARCHITECTURE. By this name is commonly 

 distinguished that transition style which prevailed in KngLui 

 about the middle of the Kith to the end of the first quarter of the 17th 

 century, and was accordingly in its meridian during the long n 

 Elizabeth. If it were worth while to disturb a name ah. 

 might with perhaps greater propriety be termed the Sag/ink Jlm-i , 

 it being a style formed out of the Continental Renaissance engrafted 

 upon our own Tudor and Old English domestic styles ; and . 

 applied exclusively to secular and chiefly to domestic buildings, in 

 which respect it was very differently circumstanced from the Cothie. 

 styles, which were almost as exclusively ecclesiastical in chanu ! 

 purpose. The age of Elizabethan architecture was that of palace- 

 building, not church-building; and a style developed itself which was 

 eminently palatial in many of its qualities, certainly not deficient in 

 stateliness (one very important ingredient), nor more delieient in 

 picturesqueneas. 



It is unjust to speak of Elizabethan architecture, as is soin- 

 done, as being marked by the introduction of the Grecian orders, those 

 employed in it having nothing whatever in common with the orders of 

 antiquity but that sort of resemblance which renders us all thu 

 more sensitive of the prodigious difference between the res; 

 styles. Considered as classical or antique, they can be regarded only 

 as grotesque parodies, since the application of them in all the varieties 

 of the Renaissance style is as contrary to the practice of the ancients 

 as their character is dissimilar. The orders are employed merely ta 

 decoration, and then only for the separate stories of an edifice, or for 

 distinct compartments of a front. Hence they are invariably miero- 

 style, and are still further reduced in height by being placed on tall 

 pedestals, and look all the smaller owing to the spacious proportions .,f 

 the windows between them. In fact they are to be regarded only as 

 accessories and decorative filling-up, for they have scarcely even thr 

 apparent character of eolumniation. In point of design, too, they 

 retain little of the orders after which they .-ire n^i ..... I : with the 

 exception of one or two distinctive marks, such as the Doric triglyphs, 

 and thu forms of the respective capitals, they are nil .1 

 character, the Doric being frequently quite as slender and ax mu< h 

 embellished or more so than the Corinthian. Pedestals are mostly 

 panelled and filled in with ornament, the panels themselves l-in^ 

 multiplied in a variety of pattern*. The shafts of columns are fre- 

 i|iiently enriched with one or more bands (either sculptured or plain), 

 and the lower part of the shafte is often covered with arabesque carving. 

 Pilasters are similarly treated sometimes banded, sometimes pan< II. <l, 



