T R A 



T R A 



equation wanting the third term, by fuppofing y 



3 9 



P + 



3? 



If the propofed equation is of n dimenfions, the value of 

 t, by which the third term may be taken away, is had 



2* 



by refolving the quadratic equation »'' ' -^ 



^' + 



X e + 



^b 



- ■=. o, fuppofing — p and + y to be the co- 

 il X n — I 



eiBcients of the fecond and third terms of the propofed 

 equation. The fourth term of any equation may be taken 

 away by folving a cubic equation, which is the co-efficient 

 of the fourth term in the equation when transformed. 

 The fifth term may be taken away by folving a biquadratic, 

 &c. 



There are other tranfmutations of equations that on fome 

 occafions are ufeful. An equation, as .i;' — ^ x' + q x — 

 r = o, may be transformed into another that fhall have its 

 roots equal to the roots of this equation multiplied by a 



given quantity, as_/", by fuppofing y =^fx, and x = =^t 



and fubftituting this value for x in the propofed equation. 



there will arife 



•^ - ■^ + ^ - »• = O, and multiply. 



ing all by/', y'^ — f p'^ -|- f^ q y — /' r = O, where the 

 co-efficient of the fecond term of the propofed equation, 

 multiplied intoy, makes the co-efficient of the fecond term 

 of the transformed equation ; and the following co-efficients 

 are produced by the following co-efficients of the propofed 

 equation (as q, r, &c.) multiplied into the powers oi f 

 {/"j f^t Sec.) Therefore, to transform any equation into 

 another whofe roots (hall be equal to the roots of the pro- 

 pofed equation multiplied by a given quantity^", you need 

 only multiply the terms of the propofed equation, be- 

 ginning at the fecond term, by f, f~, f^. Sec. and put- 

 ting y inftead of x, there will arife an equation having its 

 roots equal to the roots of the propofed equation multiplied 

 byy"as required. Let it be required to transform an equa- 

 tion, the higheft term of which has a co-efficient different 

 from unity, into one that ffiall have the co-efficient of the 

 higheft term unit. If the equation propofed is a x' — p x' 

 ■i- q X — r = o, then transform it into one whofe roots are 

 equal to the roots of the propofed equation, multiplied by a : 



and there will arife 



a y^ 



i. t. fuppofe^ =. a X, ot X ■=■ 



U U- 



r = o ; fo that y^ — p y' + q a y — r a 



py'" , 9y 



= o : whence we deduce the following rule ; change the 

 unknown quantity x into another y, prefix no co-efficient to 

 the higheft term, pafs the fecond, multiply the following 

 terms, beginning with the third, by a, a', a', &c. the 

 powers of the co-efficient of the higheft term of the pro- 

 pofed equation, refpeftively. Thus, the equation 3 x^ — 

 13 k' -f- 14 X -|- 16 = 0, is transformed into the equation 

 73 — i3jr'* 4- 14 X 3x X fi6x9=o, orji^ — 13 y 

 + 42 X + 144 =: o. Then finding the roots of this equa- 

 tion. It will eafily be difcovered what are the roots of the 

 propofed equation ; fince 3x=j>, orx = 4jr: and there- 

 fore, fince one of the values of j; is — 2, it follows that one 

 of the values of x is — |. By this rule an equation is eafily 

 cleared of fraftions. 



Suppofe the equation propofed is x' ^ x' + - x — 



m m 



r 



— = O. Multiply all the terms by the produft of the de- 

 nominator, you find m n e x x — n e p x x + m eb X x 



— »! n r = o. Then transforming the equation into one 

 that fhall have unit for the co-efficient of the iiigheft term, 

 you find y — n e p x y' + m' e n q x y — rrfl n^ e' r — o. 



Or, negledling the denominator of the laft term — , you 



e 

 need only multiply all the equation by m n, which will give 



mnxx^~npXx^-\-mqx 



72^ n^ i 



= o, and then, y'' 

 After the va- 



— n p X y'^ + m' n q x y = o. 



lues of y are found, it will be eafy to difcover the values of 



X ; fince, in the firft cafe, x = — ^—- ; in the fecond, x = 



m n e 



. E. g. The equation x' * — 4 x ^— 



m n 27 



= o is firft 



reduced to this form 3 x' * — 4 .x — 



146 



= o, and then 



transformed mto y' *— 12 y — 146 = 0. 



Sometimes, by thefe transformations, furds are taken 



r "^ a 



away. E. g. The equation x' 



= o, by putting y =: \'a x x, or x = ''' , is transform- 



p V a X x' -f J X ■ 

 V a 





ed into this equation — p ^ a x 



a V a 

 — r \/ « = o ; which, by multiplying all the terms by 



a ^ a, becomes y^— pay^-^qay— rd^=^o, an 

 equation free of furds. 



An equation, as x' — /> x' + 9 x — r = o, may be 

 transformed into one whofe roots ffiallbe the quantities reci- 



procal of X, by fuppofing ji = — , and y = ■ — ,or by one 



fuppofition X =: — , become z' — qz^ + p r z — r' = o. 

 By this transformation, the greateft root in the one is trans- 

 formed into the leaft root in the other : for fince x = — , 



y 



and y =■ — , it is plain that when the value of x is greateft, 



the value of y is leaft, and converfely. See on this fubjeft 

 Maclaurin's Algeb. part ii. chap. iii. iv. Saunderfon's 

 Algeb. vol. ii. p. 687, &c. See Reduction of Equations. 



TRANSFUGA, m Jntiquity , a deferter. Among the 

 Romans, deferters were commonly punifhed by cutting off 

 their hands, it being thought that living in fuch a miferable 

 truncated condition would ftrike more terror than death it- 

 felf. We find, however, that deferters were likewife cruci- 

 fied, burnt alive, thrown from the Tarpeian rock, or ex- 

 pofcd to wild beafts at public fhows. 



TRANSFUSION, Transfusio, compounded of the 

 prepofition trant, beyond, farther, and fundo, I pour, the 

 aft of pouring a liquor out of one veflel into another. 



In the preparations of chemiftry and pharmacy there are 

 frequent transfufions of liquors, fyrups, &c. 



Tkans- 



