taken upon the supposition that y was variable and a constant, hence are called Partial Differential Co-efficients. Ex. 2. Let the proposed equation be x+2ax'yay3 dy that was variable and y constant, and then the differential co-efficient was ཙི|| du du du dx dy ON FINDING THE SUCCESSIVE DIFFERENTIAL CO-EFFICIENTS OF FUNCTIONS OF ONE VARIABLE. The first differential co-efficient of any function is itself a new function of the variable, and consequently its differential co-efficient may be found according to principles already explained. This differential co-efficient of a differential co-efficient is called the second differential co-efficient of the original function, and if the first differential co-efficient be expressed by the symbol dr the second differential co-efficient is represented by dx dry dy In like manner this second differential co-efficient is itself a new function of the variable and its differential co-efficient may be found, this is called the third differential co-efficient of the original function, and is represented by the symbol d3y ძებ Proceeding in the same manner, the differential co-efficient of is called d'y day the fourth differential co-efficient of the original function, and is written so also we shall have d'y dx and so on to any extent. Thus if Ex. 1. y = ax + bx3 + cx2 + dx3 + ex2 + gå + m The first differential co efficient is dy = 6ax+5bx2 + 4cx3 + 3dx2 + 2x + g dx (1) In order to find the second differential co-efficient of y, we must take the first differential co-efficient of this new function (1), which will be 5.6. ax + 4. 5 bx3 + 3 . 4 cx2 + 2 . 3 dx + 1 day dx2 = 5. 6. ax1 + 4. 5 bx3 + 3. 4 cx2 + 2.3. dx + 1.2. e ... (2) Taking the first differential co-efficient of this new function (2), we shall have dgყ da = 4. 5. 6. ɑx3 + 3. 4. 5 bx2 + 2. 3. 4. cx + 1 . 2 . 3 d In like manner d'y dx1 = 3.4.5.6. ax2 +.2.3.4 . 5 b≈ + 1 . 2. 3. 4. c CHAPTER V. ON INVERSE FUNCTIONS. In the preceding trigonometrical expressions, the sines, cosines, &c., have been considered as functions of the arcs; but we shall now treat of the inverse functions, and consider the arcs as functions of the sine, cosine, &c., and investigate their differential coefficients. A peculiar notation has been adopted to distinguish inverse functions. The arc whose sine is x, is represented by the symbol.... sin —1x; the arc whose cosine is x the arc whose tangent is a. the number whose log is x Ex. 1. Let y = sin—1x. Here the direct function is x = sin y; and, therefore, cos-x; tan -1x; x2 In the preceding expressions the radius of the arc is unity; but they may be readily adapted to radius r, by considering that x and due dy dx are numbers; therefore the numerator and denominator of each differential coefficient must Hence, to radius r the formulas now investigated be of the same dimensions. We may now investigate the differential coefficients of a few of the more complicated inverse functions, as in the following examples : |