The roots of equations of the fifth and higher degrees are, properly speaking, transcendental : there is no mode of expression except by infinite series. And, generally speaking, and with the exception of a few cases in which modes of expression have been invented and studied, INVERSE functions are transcendental. And a result of such inversions, even though, from our ignorance of its real properties, it may be expressible by ordinary means, is transcendental so long as that igno rance Lasts. And it is useful to observe that forms of the most different kind may bo connected together by such a relation as this, that both are cases contained under the same transcendental.
To exhibit the arrival of one of these transeendentals of inversion, as they might be called, let us take the equation ¢x. rp's=0(rax), where means the differential coefficient of se.r. A largo class of solution. may be ebtained as follows :—The equation y logy= c has an infinite number of roots, two at most being real, and all the rest of the form a + s/-1. Let a, 1), r, &c., be any of these roots, and let 4se bo a function of x formed as follows :— ipx = ace + Bbs + . . . .
where one, two, or any number of roots may be taken at pleasure : and a, &c. are any quantities independent of .r. Let be the
inverse function of *.s., so that is x; then * I) is a solution of the original equation, or *.r= (sfr—lx—l) gives ass. elo'.c= so (car). Now *six is, when more than one root is used, inexpressible except by infinite series: that is, not merely inexpressible in common algebraical terms, but even with the assistance of logarithms and trigonometrical functions. Nevertheless, as particular cases of this solution, both ax and are found.
As science advances, quantities which are now called transcendental will lose the name, and be received among the ordinary modes of expression of analysis. One of the first of these will be the well-known function of n, which is generally designated by Tn, and is sometimes called the gamma-function, sometimes the factorial function. Its expression lei •—• dx taken from x=0 to and when n is an integer it is simply 2x lx3x x n. But when n is a fraction it can only be calculated by series. Nevertheless, as tables are now formed of its values, and as many properties and consequences of it are known, it stands in as favourable a position for use as ordinary at the end of the 17th century.