## Resultant Sentence Examples

- If the primary wave be represented by = e-ikx the component rotations in the secondary wave are '1'3= P (- AN y) N r2 ' cwi= r x D y N 'y)' lw2=P (- AD + 6,N z2 - x2 ' D r N r2 where ik3T e-ikr _ P - 4 r The expression for the
**resultant**rotation in the general case would be rather complicated, and is not needed for our purpose. - Moreover, if OP = r, and AO=x, then r 2 =x 2 + p2, and pdp=rdr. The
**resultant**at 0 of all the secondary vibrations which issue from the stratum dx is by (3), with sin ¢ equal to unity, ndx f ? - Through the
**resultant**scarcity of labor, much land fell out of cultivation. - The radiations interfere in an optical sense of the word, and in some directions reinforce each other and in other directions neutralize each other, so making the
**resultant**radiation greater in some directions than others. - The main branches of the
**resultant**" tree " may be rendered as follows: [[Coraciomorphae Odontolcae..Colymbo-+Pelargoalectoromorphae..Ratitae Morphae Morphae ' 'Neornithes]] The Odontolcae seem to be an early specialized offshoot of the Colymbo-Pelargomorphous brigade, while the Ratitae represent a number of side branches of early Alectoromorphae. - It is also the custom to balance a proportion of the reciprocating masses by balance weights placed between the spokes of the wheels, and the actual balance weight seen in a driving-wheel is the
**resultant**of the separate weights required for the balancing of the revolving parts and the reciprocating parts. - Another of Roberval's discoveries was a very general method of drawing tangents, by considering a curve as described by a moving point whose motion is the
**resultant**of several simpler motions. - He treated the
**resultant**electric force at any point as analogous to the flux of heat from sources distributed in the same manner as the supposed electric particles. - The general character of the country,
**resultant**on these conditions, varies according to elevation and latitude. - Temperance and the
**resultant**health and vigour). - When the solutions of two substances are mixed, similar considerations to those given above enable us to calculate the
**resultant**changes in dissociation. - Thus, the thermal equivalent of the unit of
**resultant**electrochemical change in Daniell's cell is 5.66 - 3.00 =2.66 calories. - R is a function of the coefficients which is called the "
**resultant**" or " eliminant " of the k equations, and the process by which it is obtained is termed " elimination." - If al, a2, ...a, n be the roots of f=o, (1, R2, -Ai the roots of 0=o, the condition that some root of 0 =o may qq cause f to vanish is clearly R s, 5 =f (01)f (N2) ï¿½ ï¿½;f (Nn) = 0; so that Rf,q5 is the
**resultant**of f and and expressed as a function of the roots, it is of degree m in each root 13, and of degree n in each root a, and also a symmetric function alike of the roots a and of the roots 1 3; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 4.. - This expression of R shows that, as will afterwards appear, the
**resultant**is a simultaneous invariant of the two forms. - +al pxp = 0, a21x1 +a22x2 + ï¿½ ï¿½ ï¿½ +a2pxp = 0, aplxl+ap2x2+...+appxp = 0, be the system the condition is, in determinant form, (alla22...app) = 0; in fact the determinant is the
**resultant**of the equations. - Forming the
**resultant**of these equations we evidently obtain the**resultant**of f and 4,. - Thus to obtain the
**resultant**of aox 3 +a i x 2 +a 2 x+a 3, 4, =box2+bix+b2 we assume the identity (Box+Bi)(aox 3 +aix 2 +a2x+a3) = (Aox 2 +Aix+ A 2) (box2+bix+b2), and derive the linear equations Boa ° - Ac b o = 0, Boa t +B i ao - A 0 b 1 - A 1 bo =0, Boa t +B 1 a 1 - A0b2 - A1b1-A2b° = 0, Boa3+Bla2 - A l b 2 -A 2 b 1 =0, B 1 a 3 - A 2 b 2 =0, = = (y l, y2,...ynl `x1, x2,...xnl for brevity. - Taking the same example as before the process leads to the system of equations acx 4 +alx 3 +a2x 2 +a3x =0, aox 3 +a1x 2 +a2x+a 3 = 0, box +bix -1-b2x =0, box' +b i x 2 -{-h 2 x = 0, box + b i x + b:: = 0, whence by elimination the
**resultant**a 0 a 1 a 2 a 3 0 0 a 0 a 1 a 2 a3 bo b 1 b 2 0 0 0 bo b 1 b 2000 bo b 1 b2 which reads by columns as the former determinant reads by rows, and is therefore identical with the former. - Bezout's method gives the
**resultant**in the form of a determinant of order m or n, according as m is n. - He first divides by the factor x -x', reducing it to the degree m - I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers xo, x, - of x, and regarding these as the unknowns of a system of linear equations the
**resultant**is reached in the form of a determinant of order m. - Put (aox 3 -}-a l x 2 +a 2 x +a 3) (box' +b1x'+b2) - (aox'3+aix'2+a2x'+a3) (box' + bix + b2) = 0; after division by x-x the three equations are formed aobcx 2 = aobix+aob2 =0, aobix 2 + (aob2+a1b1-a2bo) x +alb2 -a3bo = 0, aob2x 2 +(a02-a3bo)x+a2b2-a3b1 =0 and thence the
**resultant**aobo ao aob2 aob 1 aob2+a1b1-a2bo alb2-a3b0 aob 2 a1b2 - a 3 bo a2b2 - a3b1 which is a symmetrical determinant. - There is no difficulty in expressing the
**resultant**by the method of symmetric functions. - = 0, we find that, eliminating x, the
**resultant**is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations. - Hence, finally, the
**resultant**is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second equations respectively. - They do not recognize it as a power inherent in heroes and rulers, but as the
**resultant**of a multiplicity of variously directed forces. - In their exposition, an historic character is first the product of his time, and his power only the
**resultant**of various forces, and then his power is itself a force producing events. - To find component forces equal to the composite or
**resultant**force, the sum of the components must equal the**resultant**. - This condition is never observed by the universal historians, and so to explain the
**resultant**forces they are obliged to admit, in addition to the insufficient components, another unexplained force affecting the**resultant**action. - The historian evidently decomposes Alexander's power into the components: Talleyrand, Chateaubriand, and the rest--but the sum of the components, that is, the interactions of Chateaubriand, Talleyrand, Madame de Stael, and the others, evidently does not equal the
**resultant**, namely the phenomenon of millions of Frenchmen submitting to the Bourbons. - And therefore to explain how from these relations of theirs the submission of millions of people resulted--that is, how component forces equal to one A gave a
**resultant**equal to a thousand times A--the historian is again obliged to fall back on power--the force he had denied--and to recognize it as the**resultant**of the forces, that is, he has to admit an unexplained force acting on the**resultant**.