round in a curve. The instant the stone leaves the sling it proceeds (for a short time) in the direction which it had at that instant. A pellet of mud leaving a carriage wheel gives another familiar example. 93 APPENDIX I. Assuming the exponential theorem, where a*=1+ + + 1 12 |3 A=(a-1)-(a-1)2+(a-1)3-etc. In (1), put x=1, then Again, in (2), put A=1, then the series becomes ..(2). .(3), and this is called e, and is the base of the Naperian system of logarithms. Again, in (1), put A=1, and then e represents the and Nap. log a (a-1)-(a−1)2+(a−1)3- etc.; = .(4) or, reducing this to logs with base 10, or log a log e{(a-1) − (a− 1)2 + } (a−1)3 - etc.}; = log a = M {(a− 1) − 1 ( a − 1)2 + }(a − 1)3 — etc.}; in this put a=1+n, and therefore a then log{1+n}=M{n − 1 = n, n2, n3 3 log (1 + 1) = M(1), approximately, and M is found to be 4342940. APPENDIX II. MACLAURIN'S THEOREM. Assuming the ordinary working of Indeterminate Co-efficients. Let there be any function of x, and suppose that this function may be expanded in ascending powers of x and constants which do not contain x, but which have to be determined; and let these constants be A, B, C, etc., then the function A+Bx+Сx2+ Dx3 + etc.,......... (1) .. dc1=B+2Cx+3Dx2+ etc.,. dc2=2C+3×2Dx+etc.,. dc3=3x2D+etc,........ .(2) (3) .(4) etc. etc. Now let x, being the variable, continuously diminish and ultimately become 0, then Substituting these values in (1) we have function = (function)+(dc1)x+(dc2)x2 1 + (dc3)x3 + etc.; 2 x 3 which is Maclaurin's theorem. EXAMPLES WORKED OUT. I. If the side of a square increases uniformly at the rate of 5 feet per second, at what rate is the area increasing when the side becomes 10 feet. If then and or the side of square=x, differential co-efficient of x2=2x. Now when the side becomes 5 feet differential co-efficient of x2=2×5 feet, = 10 feet, Therefore, when the side becomes 10 feet— =100 square feet. 2x+5 2. What is the value of the fraction when x 4x+6' becomes infinite? Divide both numerator and denominator by x, and the fraction becomes 5 2018 and 61 Now, when x becomes infinite, each of the fractions becomes nothing. 3. Find that angle which increases twice as fast as 4. Divide a straight line into two parts, so that the rectangle contained by the parts may be the greatest possible. Let a=the line, x=one of the parts, a-x=other part, |