equation 
is as follows : Let the biquadratic be rcl + ax* + bx- + ex 
+ c( = Find a root of the cubic y : < by- + (*' ld)<i 
d(a2 tb) c'- = Q. Then the roots of the biquadratic 
are the same as those of the two quadratics 
ax + y) 
4- i/ B Z~4& + 4t/ [z(a2_4& + 4y) + aj/ 2c] = 0. 
Canonical equation, an equation brought into a stan- 
dard form; especially, the Lagranglan and Hamiltonian 
equations of dynamics. Characteristic equation, an 
al"ehraic equation which leads to the solution of a linear 
differential or difference equation with constant coeffi- 
cients. Chemical equation. See chemical.- Circulat- 
ing equation, a difference equation in which the coeffi- 
cients take successive forms of a cycle ot forms for succes- 
sive values of the variable. Thus, if we have the equation 
Ul+1 + Pix = 0, where P = 1 when x is divisible by S, P = x 
when x - I is divisible by 8, and P= 2* when x + 1 is divis- 
ible by 3 the equation given is a circulating equation. 
Clairaut's equation, the equation y-scOi/taa+T{ayl 
dz). Complete equation. See incomplete equation. 
Compound equation. Same as adjected equation. Con- 
nected equations, a system of equations such that one 
of them can be deduced from the rest. Constitutive 
equation the equation which expresses the conditions of 
a problem. - Construction of equations. See construc- 
tion. Conversion of equations. See conversion. Cu- 
bic equation, an equation of the third degree. The alge- 
braic solution of the general cubic equation was discov- 
ered by Scipione dal Ferro (died 1525 ?). His method, com- 
monly known as that of Cardan, and perfected by Hudde, 
is as follows : Let the cubic equation be x* + Bora'- + 600; 
+ 2c = 0. Calculate three subsidiary quantities, p, q, R, 
by means of the equations p = '2b a2, q = a3 3rtb + c, 
R2 = p3 + q2. Then, denoting by p any cube root of 
unity, and by the radical a real quantity, 
= py 
q R a, 
which gives three values for the three values of p. If all 
the roots are real, this method is inconvenient; and we 
have the "irreducible case of Cardan's solution," when 
we may calculate two subsidiary quantities, r and 9, by 
the equations r - 92112, tan239 = R2/j2, and the 
three roots will be x\ = 2r cos a, x% = 2r cos 
( + 120) a, z 3 = '2r cos (9 120) a. Darboux's 
equation, the equation Ada; + Bdj/ + C (yAx xdy) = 0, 
where A, B, C are rational functions of x and y. De- 
pression of an equation. See depression. Derived 
equation the equation which expresses the vanishing 
of the differential coefficient of a given equation. Thus, 
if x + z3 = a;2 + 1 is the given equation, the derived 
equation is 5xi -|- ax? = Zx. Determinate equation, 
an equation containing only one unknown quantity, or 
only as many as there are equations in the system. Dif- 
ference equation, an equation expressing a relation be- 
tween the value of a function (or the values of several 
functions) for all values of the variable or variables and 
the values when the several variables are increased by 
1,2,3, etc. Thus, /(x, y) = /(x+ 1, y) +/(x, y 3) is a 
difference equation. The order of a difference equation 
is equal to the difference between the highest and low- 
est values of the variable it involves. Thus, the equa- 
tion just given is of the first order with respect to x and 
of the third order with respect to y. The degree of a dif- 
ference equation is the degree of the equation in the un- 
known functions as variables. Thus,/(z + 2) (fix + 1)]2 
+ /x = is a difference equation of the second degree. 
But some mathematicians would make the degree of a 
difference equation strictly analogous to that of a differ- 
ential equation. A linear difference equation with con- 
stant coefficients is solved by means of its characteristic 
equation (which see, above). Differential equation, 
an equation expressing a relation between functions and 
their differential coefficients. An ordinary di/erential 
equation is one which contains only one independent va- 
riable ; a partial differential equation is one which con- 
tains two or more independent variables. The order of 
a differential equation is that of the highest differential 
coefficient it contains. The degree of a differentia! equa- 
tion is that of the power to which the highest differential 
coefficient is raised when the equation is in rational form 
and freed from fractions. A solution of a differential 
equation is an equation containing no differentials nor 
integrals unless of explicit functions and such that the 
given differential equation can be deduced from it. A 
general solution is one which is as indeterminate as pos- 
sible that is, which contains the number of arbitrary 
constants or functions indicated by the order of the equa- 
tion. A particular solution is (a) with modern writers, a 
solution which is a particular case of the general solution ; 
(b) with older writers, any solution not general. A sinrju- 
lar solution is one which is neither general nor implied 
in the general solution. The complete integral of a par- 
tial di/erential equation is a solution containing the full 
number of arbitrary constants or functions. Disjunc- 
tive equation. See disjunctive. Eminential equa- 
tion. See eminential. Equation of achromaticity, 
an equation between the radii of curvature of a com- 
pound lens, determining it to be achromatic ; also, a simi- 
lar equation determining the distance between the lenses 
of an eyepiece. Equation of condition. See condi- 
tion. Equation of continuity. See continuity. Equa- 
tion Of differences, the equation lor the squared dif- 
ferences of the roots of a given algebraic equation. 
Equation of hydrodynamics, an equation often used in 
solving problems in hydrodynamics, expressing a differen- 
tial relation between the pressure, the components of the 
velocity, and the forces. Equation of Laplace's func- 
tions, the partial differential equation 
1982 
Equation of moments, an equation of rigid dynamics 
expressing the forces of rotation. Equation Of motion, 
the differential equation of dynamics connecting the forces 
and accelerations. Equation of payments, an arith- 
metical rule for the purpose of ascertaining at what time 
it is equitable that a person should make payment of a 
whole debt which is due in different parts payable at dif- 
ferent times. Equation Of rest, a special case of the 
equation of motion, showing the conditions of equilibri- 
um. -Equation Of the argument, in old outran., the 
angle atlhe earth between a planet and the center of its 
epicycle but in the cases of the sun and moon, the dif- 
ference between the true and mean places. (Clamus, In 
Sacro Bosco.) Equation of the center, (a) In old as- 
tron. usually, the difference between the true and mean 
place of the center of the epicycle (Short, Kepler, ); 
but in the case of the moon, generally the angle at the 
center of the epicycle between the true and mean apogee 
(Claviux; Ozanam), but sometimes the first inequality 
(Holm, Almagest, V. vii.). (b) In modern astron., the ex- 
cess of the true over the mean anomaly. ((Jams, 1 heoria 
Motus, I. 7.) Equation of the orbit, in old astron.: (a) 
The total correction of the mean place of a planet to give 
its true place, (b) The equation of the argument. (Kepler, 
DeMotibus Martis, I. iv.) Equation Of time, the reduc- 
tion from mean solar time to apparent solar time. Equa- 
tion of translation, the differential equation for the 
translation of a system. Equation to a curve, surface, 
etc an equation defining the shape and position of the 
curve, surface, etc. Equation to corresponding alti- 
tudes in aitron., a correction which must be applied to 
the apparent time of noon (found by means of the time 
elapsed between the instants when the sun had equal al- 
titudes, both before and after noon) in order to ascertain 
the true time. Eulerian equation, (a) The equation 
expressing the addition theorem of elliptic functions, (b) 
Any one of the usual equations of hydrodynamics, where 
the components of the velocity at fixed points of space are 
taken as variables: so called in contradistinction to the 
Lagrangian equations where the coordinates of a definite 
particle are taken as variables ; these equations, though 
also discovered by Euler, having been used by Lagrange. 
Exponential equation. See exponential. Fluential 
equation the equation of the fluents: corresponding to 
the solution of a differential equation. Fluxional equa- 
tion, the equation of the fluxions. Functional equa- 
tion, an equation in which the unknown is not a quan- 
tity, but a functional operator. Such, for example, is the 
equation !"-' = I, which means that the operation 1' is such 
that the result of performing it twice is to restore the ori- 
ginal operand. General equation, an equation in which 
no account is taken of initial conditions, or of special or 
exceptional features of a problem. Group of an equa- 
tion, a group of permutations of the roots such that they 
all give the same values for rational functions of the known 
and adjunct quantities, and for no others. Hamiltonian 
equation, one of a certain system of equations for ex- 
pressing problems of dynamics. The equations are ApjAt 
= 6H/5it and du/d( = 511 //>, where u is an element of 
position, p is the differential coefficient of the vis viva rel- 
atively to ', and II is the total energy. Hesse's equa- 
tion, an equation of the ninth degree, expressing the ]K)si- 
tions of the inflections of a plane cubic. Homogeneous 
equation, one of which all the terms are of the same de- 
gree. Identical equation, one which is satisfied by all 
values of the literal quantities. Incomplete equation, 
an equation in which some power of the unknown quan- 
tity lower than the highest does not appear. Thus, x + 
Spx + 2q = is an incomplete equation. Independent 
equations, a system of equations no one of which is ne- 
cessarily satisfied when the others are satisfied. Indeter- 
minate equation or system of equations, an equation 
with two unknown quantities, or a system of equations 
less in number than the unknown quantities. Intrinsic 
equation Of a plane curve, an equation between the 
arc measured from a fixed point upon it and tlie radius 
of curvature. Irreducible differential equation, one 
which admits only of proper solutions. Irreducible 
equation, an equation whose first member, after all the 
terms have been transposed to one side, has no rational 
divisor. Jacobi's equation, the equation 
equational 
tial coefficients. Modular equation, in elliptic func- 
tions, an equation between A and k, where 
Md;/ Ax 
Monge's equation, the equation 
K + 8 + T =-^ = V, 
where R S, T, V are functions of x, y, z, 3z/8z, and 3z/ 
ay Normal equation, in least squares, one of the sys- 
tein of equations equal in number to the unknown quan- 
tities which are formed from the more numerous equa- 
tions'of condition, according to the rule of least squares. 
Numeral or numerical equation, an equation hav- 
ing all itscoefficients individual numbers. Optical equa- 
tion in (inc. nitron., the apparent displacement of a plan- 
et owing to the eccentricity of the orbit ; more precisely, 
the angle at the center of the epicycle between the center 
of the world and that of the orbit. Ordinary equation, 
partial equation. See differential equation. Particu- 
lar equation, an equation which takes account of initial 
positions and velocities or other peculiarities of a special 
problem. Personal equation. () The constant which 
must be added to every time observed by one observer, in 
order to make the mean of such observations agree with 
those of another observer. If, for example, two observers 
note the times of passage of a series of stars over the same 
meridian, it will generally be found that one observer has 
a tendency to note the time later than the other, so that the 
mean difference, say for sets of twenty-live observations, 
presents some approach to constancy. In consequence of 
this, if we have to combine observations of the two ob- 
servers, it will be proper to apply to all the observations of 
one of them a constant, in order to give the times such as 
they would have been observed by tbeother. This constant 
is the personal equation. The absolute personal equation is 
the amount which has to be added to the time as observed 
by any given observer in order to reduce the error of the 
mean of a large number of his observations to zero, or as 
nearly so as possible by any such constant correction. The 
personal equation is said to be eliminated when the ob- 
servations are so treated that it does not affect the re- 
sult. Thus, in determining the difference of longitude of 
two stations by the telegraphic transmission of the times 
of transit of stars over the two meridians, the result will 
be affected by the personal equation between the observ- 
ers at the two stations. But if the observers afterward 
change places and redetermine the difference of longitude, 
the personal equation will enter into this second result 
with the opposite sign to that which it had before. Con- 
sequently, the mean of the two results will give a third 
result which is free from the effect of any constant per- 
sonal equation. Hence, loosely ((/) Any kind of tendency 
to error of a determinate kind and amount peculiar to a 
given observer or reasoner for which it is possible to make 
any approximate allowance. Physical equation, in 
astron., the displacement of a planet from the position 
which an equable circular motion would give it owing 
to the eccentricity of the orbit being only one half that 
of the equant. Primitive equation, any equation from 
which another is derived in any way. Pure equation, 
one in which each unknown occurs to only one degree. 
Quadratic equation, an equation of the second degree. 
Such equations were solved by the ancients. Given A*2 
-f 2Bz + C = 0, the solution is 
B B 
AC 
When B2 is much larger than AC, the two roots are 
2 B 
C ACa 
2B + 8B3- 
(ax + by + cz) (yAz zAy) 
+ (a'x + b';i + c'z) (zAx xAz) 
+ (a"x + b"y + c"z) (xAy i/Ax 
= ft 
uaBi<uisc ciiuaxiu, one of the equations Ax I P =fy /Q 
= &z / R used in the solution of Lagrange's linear equation. 
Lagrange's linear equation, the equation ftz/tx 
+ Q Sz/S.v = R, where P, Q, R are explicit functions of x, 
y, z. Lagrangian equation, (a) An equation of the 
form 
'! ( 
sin ' 
where T is the living force, Y the positional energy, an 
element of position, and t the time, (b) A general equation 
of hydrodynamics, in which, instead of considering the ve- 
locity at each fixed point of space, the motion of each par- 
ticle is followed out. This is called a Lagrangian equa- 
tion because used by Lagrange in his ''Mechanique Ana- 
litique," though invented by Euler. Lame's equation, 
the equation A-y I dz2 [m (m + 1)*2 sn2 x + A] ;/ = 0, where 
m is an integer and k is the modulus of the elliptic func- 
tion sii x. Laplace's equation, the equation 
32u 82 U 32 U 
3x2 + 3y2 + 9z-' 
Also called Laplace's principal equation. See equation of 
Laplace's functions, above. Legendre's equation, the 
equation 
Also called Laplace's secondary equation. Equation of 
light, (a) In older writings, the sum of those equations 
of the moon's motion which depend on its distance from 
the sun. (b) In modern writings, the correction to be 
applied to the position of a planet or to the time of an 
eclipse, et., owing to the finite velocity of light. Equa- 
tion Of living force (vis viva), an equation derived from 
the immediate application of the principle that the liv- 
ing force added to the potential energy is a constant. 
Linear equation, an equation of the first degree. Lit- 
eral equation, one in which all the quantities are ex- 
pressed by letters. Local equation, the equation of a 
locus. Lunar equation, the correction of the Grego- 
rian calendar for the error of the lunar cycle, which adds 
1 to the epact in 1800, 2100, etc. See epact. Mixed equa- 
tion of differences, or equation of mixed differences, 
an equation which contains both differences and differen- 
Quadrato-quadratic equationt, a biquadratic equa- 
tion. Quartic equation, one of the fourth degree. 
Quintic equation, one of the fifth degree. The general 
equations of the fifth and higher degrees cannot be solved 
by means of radicals. Reciprocal equation,an equation 
which is satisfied by the reciprocal of the unknown quan- 
tity. Resolvent equation, an algebraic equation which 
has to be solved in order to solve another equation. Thus, 
the cubic which has to be solved in order to solve a bi- 
quadratic is a resolvent equation. Riccati'S equation, 
the equation Ay /Ax + by2=cx<". Root of an equation, 
a number or known quantity which substituted for thr un- 
known quantity in the equation satisfies the latter identi- 
cally. Secular equation, the equation of the secular 
inequalities. Simple equation.au equation of the form 
Aim -(_ B = o. Simultaneous equations, two or more 
equations which are true at the same time. Solar equa- 
tion the correction of the epact in the Gregorian calen- 
dar for the fact that three out of every four century -years 
are not leap-years. See epact. Solution of an equa- 
tion. See di/erential equation. Symbolic equation. 
(a) A functional equation, or an equation whose members 
are not quantities, (b) An equation of analytical geom- 
etry in which certain curves are represented by single let- 
ters. Thus, if U = 0, V = 0, W = 0, represent the equa- 
tions of three circles, UV = W2 is the symbolic equation 
of a bicircular quartic. The equation of a quantic, 
the equation formed by putting the qnantic equal to zero. 
Cat/lei/, 1854. Theory of equations, that branch of al- 
gebra which seeks those functions of the roots of any given 
equation that are expressible rationally as functions of its 
coefficients and of certain given irrationals called the ad- 
juncts of the equation. Gauloi.t. To eliminate the per- 
sonal equation, to remove from the results of an obser- 
vation or calculation the amount of error to which the 
person making it is found to be liable; hence, in a general 
sense, to make allowance for personal prejudice or bias in 
considering a statement or an expression of opinion. See 
personal equation, zlx>\e. Total differential equation, 
one which has only one independent variable, but two or 
more dependent variables. Transcendental equation, 
one in which the unknowns enter in a mure complicated 
way than in algebraic equations. Transforming equa- 
tion. See equation <:f ?/m<7\ aliovr. --Vector equation, 
an equation between vectors. (See also /ormuto, theorem, 
leriei, Inn:) 
equational (f-kwa'shon-al), (i. [< equation + 
-Z.] In much., equalizing; adjusting: equiva- 
