TRACT 38.

PROBLEMS.

369

We may farther add a remark here concerning the methodof finding the fluents of the three fluxional forms that occurin the solution of this problem, viz, the three forms mz =( 2 f + z)^/(x — z)x, and mz = (12 + z)V{x — z)*, andmz = \/ {x — z)x, the fluents of which are found by assum-ing the fluent mz in an infinite series ascending in terms ofx with indeterminate coefficients a, b, c, &,c, which coeffi-cients are afterwards determined in the usual way, by equat-ing the corresponding terms of two similar and equal series,the one series denoting one side of the fluxional equation,and the other series the other side. By similar series, ismeant such as have equal or like exponents; though itis not necessary that the exponents of all the terms shouldbe like or pairs, but only some of them, as those that are notin pairs will be cancelled or expelled by making their coeffi-cients iO or nothing. Now the general way to make thetwo series similar, is to assume the fluent z equal to a seriesin terms of x, either ascending or descending, as hereZ = x T x r + s + x r + ri &c for ascending,or z — x r + x r ~‘ + x r ~ z ‘ &c for a descendingseries, having the exponents r, r ± s, r ± 2 s, &c, in arith-metical progression, the first term r, and common differences ; without the general coefficients a, b, c, &c, till the valuesof the exponents be determined. In terms of this assumedseries for z, find the values of the two sides of the givenfluxional equation, by substituting in it the said series insteadof z; then put the exponent of the first term of the one sideequal that of the other, which will give the value of the firstexponent r; in like manner put the exponents of the two2d terms equal, which will give the value of the commondifference s; and hence the whole series of exponents r,r ± s, r ± 2 s, 8tc, becomes known.

Thus, for the last of the three fluxional equations abovementioned, viz, mi = \/(x—z)x, or only z — */(x — z)x ;having assumed as above z = x r x r + s 8tc, and taking the

fluxion, then Z = x r - I x + z ,r + I— + &c, omitting the co-efficients ; and the other side of the equation \Z(x — z)x —

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