...(2.) If the series A + в.г + ел? + т>з? + &c. be equal to the series A' + в'х + c'j? + v'x\ whatever be the value of x, then the coefficients of the like powers of x will be respectively equal the one to the other ; that is, A "-• A', R = B', С ~ с', &С. For since...

...the series A + вх + car2 + Da;3 + &c. be equal to the • series A' + в'х + c'a;2 + n'ar3 + &c., whatever be the value of x, then the coefficients of the like powers of x will be respectively equal the one to the other ; that is, A = А', в = в', с = с', &с. For since...

...since A + A' must be regarded as a single constant. If this equation be cleared of denominators, and the coefficients of the like powers of x in the two members placed equal to each other, we shall evidently form three independent equations, with only two unknown...

...scale is b'x f. clearing of fractions, 1' + Qx + Rx> = 1 + 2 z -4 - 4 — 1 r' Ac ; whence, by equating the co-efficients of the like powers of x in the two members, we have P=l, Q = 0, R = - 9 ; hence, the fraction is 1 - 9z* 494 CYCLOI'KMA OF MATHEMATICAL SCIKNCE....

...is lx° ; and x° must be supplied in the second member, with the coefficient 0. Hence, by equating the coefficients of the like powers of x, in the two members, we should have 1 = 0; 3A=Q; 3B-A=Q; &c. Now, the result 1=0, is impossible ; and hence the given fraction...

...necessary to transpose the terms to one member ; for if neither member is zero we have simply to equate the coefficients of the like powers of x in the two members, according to the third property of identical equations. The method of Indeterminate Coefficients is...

...and (2J.+5.8) in the second, may be considered as coefficients of x°. Now according to (368, III), the coefficients of the like powers of x in the two members must be equal ; and we have, therefore, A+B = 8, (3) 24+5.5=31. (4) From these equations we readily...

...form A+Bcc+Cx 2 + , etc. ; clear the equation of fractions, or raise it to its proper power, and place the coefficients of the like powers of x in the two members equal each to each. Then find from these equations the values of A, B, 0, etc., and substitute these...

...the form A+Bx + Cxz+, etc.=A'+B'x+C'z2 + , etc., must be satisfied for each and every value given to x, then the coefficients of the like powers of x in the two members are equal each to each. For, since this equation must be satisfied for every value of -x, it must be satisfied when x=0. But...

..." ' " • Clearing of fractions, and reducing, x = Bx* + (A + B)x + A + Cx* + G ..... (2) Equating the coefficients of the like powers of x in the two members, and solving the resulting equations, we have, A=\, 5=1, and 0— J Substituting in (1), and multiplying...