244

THEORY AND PRACTICE

tract 37-

first velocity v is diminished to any less one v ; and when itis quite extinct, the state of the fluents becomes — x + 2c xh.l. = — x h.l. - V J —> for the greatest height x as-cended.

63. Here, in the quantity h. 1.

the term x is al-

C 7

ways small in respect of the other term c; therefore, by thenature of logarithms, the h. 1. of -- + — is nearly =

2r

+ :2r* —

2c -f- ;

; therefore the above fluents become — x +

c+ \X4rx

or

2c — x W i

-—r—- X — — X h.

2c -f- x 64a

2 c + a?

Now the latter side

of this equation is the same value for x as was found in the5th problem, which therefore put = 5; then the value of xwill be easily found from the formula ——? x = b , by a qua-dratic equation. Or, still easier, and sufficiently near thetruth, by substituting 6 for x in the numerator and the de-nominator of ~ ~ , ■ * , then ~~.x = b. and hence x = ^-—'b,or by proportion, as 2c — b : 2c + b :: b : x ; that is, onlyincrease the value of x, found by prob. 5 f in the ratio of 2c— b to 2c + b.

64. Now, in the first example to that prob. the value of xor b was there found = 2930; and 2c being =: 110000, there-fore 2 c — b = 107070, and 2c + b = 112930; then, as107070: 112930 : : 2930 : 3093 = the value of the heightx in this case, being only 163 feet, or T '-g-th part more thanbefore.

But, for the 5th example to the 5th prob. where .r was —6463; it will be, as 2c — b : 2c + b, or as 103537 : 1164636463 : 7285 the height ascended in this example, beingabout the 8th part more than before. And so on, for anyother examples; the value of 2c being the constant number110000 .

Note. In a similar example to this, at the top of pa. 289vol. 3 of the Course, there are some errors in the numbers,by having used, in the expressions 2c — b and 2c + b, in the2d line, the number 2955 instead of 6420, for the value of b.