T II A 
t n a 
813 
equation, so that the new equation may want its 
2d, or 3d, or 4th, &c. term of the given equa- 
tion .v 3 — ax 2 Lx — c — 0, which is trans- 
formed into the equation (A) in the last article. 
Now to make any term of this equation (A) va- 
nish, is only to make the co-efficient of that 
term=:0; which will form an equation that 
will give the value of the assumed quantity d, 
so as to produce the desired effect, viz. to make 
that term vanish. So, to take away the 2d term, 
make 3d — a ~ 0, which makes the assumed 
quantity d — f- a . To take away the 3d term, we 
must put the sum of the co-efficients of that 
term = 0, that is, 3d 1 — 2 ad -f- b = 0, or 3d 1 — 
Had — — /; ; then, by resolving this quadratic 
equation, there is found the assumed quantity 
d — a Ij-ybP — 3b, by the substitution of 
which for d, the 3d term will be taken away out 
of the equation. 
From whence it appears that, to take away 
the 2d term of an equation, we must resolve a 
simple equation ; for the 3d term, a quadratic 
equation ; for the 4th term, a cubic equation, 
and so on. 
4. To multiply or divide the roots of an equa- 
tion by any quantity ; or to transform a given 
equation to another, that shall have its roots 
equal to any multiple or submultiple of those 
of the proposed equation. This is done by sub- 
stituting, for x and its powers, , or py, and 
711 
their powers, viz. for x, to multiply the 
m 
roots by m ; and py for x, to divide the roots 
byA 
Thus, to multiply the roots by m, substituting 
y 
for x in the proposed equation, 
m 
.■*" — ax n — * -{- bx n ~ 2 &c — 0, and it be- 
comes 
1 &c-=0; 
m n m v — 1 «z" — 2 
or multiply all by m n , then is 
y n — amy n ~ 1 -|- bm 2 y n ~ 2 — cmcy n ~ 3 &C = 0, 
an equation that has its roots equal to m times 
the roots of the proposed equation. 
In like manner, substituting py for x, in the 
proposed equation, &c. it becomes 
an equation that has its roots equal to those of 
the proposed equation divided by p. 
From whence it appears, that to multiply the 
roots of an equation by any quantity m, we 
must multiply its terms, beginning at the 2d 
term, respectively by the terms of the geome- 
trical series, m, m l , m', m", &c. And to divide 
the roots of an equation by any quantity p, that 
we must divide its terms, beginning at the 2d, 
by the corresponding terms of this series p, p 2 , 
p\ p\ See. 
5. And sometimes, by these transformations, 
equations are cleared of fractions, or even of 
surds. Thus the equation 
x 3 — ax 2 \/p -j- bx — c\P p — 0, 
by putting y — x\fp, or multiplying the terms, 
from the 2d, by the geometricals \Pp, p, p\/p, 
is transfciffled to 
• y i — apyb -{- Lpy — cp 1 — 0. 
6. An equation, as .v 3 — « 2 ^- bx — c — 0, 
may be transformed into another, whose roots 
shall be the reciprocals of the roots of the given 
equation, by substituting — for v : by which 
y 
. , lab 
it becomes — — -4 c = 0 ; or, mul- 
y y y 
tiplying all by y', the same becomes 
sy' — by' -j- ay — 1 — 0. 
T R A 
TRANSIT, in astronomy, signifies the 
passage of any planet, just by or over a fixed 
star, or the sun; and of the moon in particular, 
covering or moving over any planet. 
Transit Instrument. See Observa- 
tory. 
TRANSITION, in music, the softening a 
disjunct interval by the introduction of inter- 
mediate sounds. In harmony, transition is the 
changing the genus, or mode, in a sens ble but 
regular manner. Thus, when in the diatonic 
genus the bass moves so as to require in the 
parts the introduction of a minor semi- 
tone, it is a chromatic transition ; and if, we 
change the tone by favour of a diminished 
seventh, it is an enharmonic transition. / 
TRANSMISSION. See Optics. 
TRANSMUTATION, in geometry, de- 
notes the reduction or change of one figure or 
body into another of the same area or soli- 
dity, but of a different form ; as a triangle in- 
to a square, a pyramid into a parellclopiped, 
&c. In the higher geometry, transmutation 
is used for the converting a figure into another 
of the same kind and order, whose respective 
parts rise to the same dimensions in an equa- 
tion, admit of the same tangents, &c. If a rec- 
tilinear figure is to be transmuted into another, 
it is sufficient that the intersections of the lines 
which compose it are transferred, and the 
lines drawn through the same in the new 
figure. If the figure to be transmuted is 
curvilinear, the points, tangents, and other 
right lines by means whereof the curve line 
is to be defined, must be transferred. 
TRANSOM, among builders, denotes the 
piece that is framed across a double-light win- 
dow. 
Transom, among mathematicians, signi- 
fies the vane of a cross-staff, ora wooden num- 
ber fixed across, with a square whereon it 
slides, &c. 
Transom, in a ship, a piece of timber 
which lies athwart the stern, between the two 
fashion-pieces, directly under the gun-room 
port. 
TRANSPORTATION, the act of con- 
veying or carrying a thing from one place to 
another. 
Transportation is a kind of punishment, or 
more properly an alleviation or commutation of 
punishment, for criminals convicted of felony ; 
who for the first offence, unless it is an extra- 
ordinary one, are generally transported to 
the plantations (at present to New South 
Wales), there to bear hard labour for a term 
of years; within which if they return, they 
are executed without further trial than identi- 
fying their persons. 
Transportation of plants. In sending 
plants from one country to another, great 
cautions are necessary. The plants sent from 
a hotter country to a colder, should be always 
put on board in the spring of the year, that the 
heat of the season may be advancing as they 
approach the colder climates ; and, on the 
contrary, those which are sent from a colder 
country to a hotter, should be sent in the be- 
ginning of winter. The best way of packing 
up plants fora voyage, if they are such as will 
not bear keeping out of the earth, is to have 
boxes with handles, filling them with earth, 
and planting the roots as close together as 
may be ; the plants should be set in these 
boxes tliree weeks before they are to be put 
! on board ; and in good weather they should 
| beset upon the deck, and in bad removed or 
| covered with a tarpauhn. If they are going 
j from a hotter country to a colder one, they 
must have very little moisture; if, on the 
contrary, they are going from a colder to a 
warmer, they may be allowed water more large- 
ly, and being shaded from the heat of the sun, 
they will come safe. 
A great many plants, however, will live out of 
the earth a considerable while ; as the sedums, 
euphorbiums, mesembryanthemums, and o- 
ther succulent ones. These need no other care 
than the packing them up with moss in a close 
box ; and there should be a little hay put be- 
tween them, to prevent them from wounding 
or bruising one another, and holes bored in 
the boxes to keep them from heating and pu- 
trefying. In this manner they will come safe 
from a voyage of two or three, or even four 
or five months. Several trees also will come 
safe in the same manner ; taking them up at 
a season when they have done growing, ami 
packing them up with mess. Of this sort 
are oranges, olives, capers, jasmines, and 
pomegranate-trees. These, and many others, 
are annually brought over to us from Italy ; 
and, though they are three or four months in 
the passage, seldom miscarry. The best way 
of sending over seeds, is in their natural husks, 
in a bag, or packed up in a gourd-shell, keep- 
ing them dry, and out of the way of vermin. 
TRANSPOSITION, in algebra, the bring- 
ing any term of an equation over to the other- 
side. 
TRANSUBSTANTIATION, in theo- 
logy, the conversion or change of the 
substance of the bread and wine in the 
eucharist, into the body and blood of Jesus 
Christ, which the Romish church hold is 
wrought by the consecration of the priest. 
This is a main point in the Romish religion,, 
and is rejected by the protestants, the former 
maintaining the transubstantiation to be real, 
the latter only figurative; interpreting the text 
hoc est corpus nietuii, “ this signifies my 
“ body:' 5 but the council cf Trent stood up 
strenuously for the literal sense of the verb est, 
and say expressly, that in transubstantiation 
the body and blood of our Lord Jesus Christ 
are truly, really, and substantially, under the 
species of breatl and wine. The controver- 
sies about this point are almost innumerable. 
TRANSVERSE MUSCLES, in anatomy, 
are certain muscles arising from the trans- 
verse processes of the vertebrae of the loins. 
See Anatomy. 
TRAPA, a genus of the tetrandria mono- 
gynia class of plants, the corollh u hereof con- 
sists of four petals, vertically ovated, and 
larger than the cup : the fruit is a hard os- 
seous capsule, of an oblong oval figure, con- 
taining only one cell, and armed with four 
sharp thick spines, placed oppositely in the 
middle of the sides, and pointed ; these be- 
fore were the leaves of the calyx: the seed is 
a covered single nucleus, of an oval figure.. 
There are two species, aquatics. 
TRAPEZI U M, in geometry, a plane figure 
contained under four unequal right lines, 
1. Any three sides of a trapezium taken toge- 
ther, are greater than the third. 2. The two 
diagonals of any trapezium, divide it into four 
proportional triangles. 3. If two sides ot a tra- 
pezium are parallel, the rectangle under the 
aggregate of the parallel. sides and oue-Jialt' 
.* 
