234

THEORY AND PRACTICE

TRACT 37.

~ v = z' 1 — these being substituted in the above va-lue of x, it becomes x —

ZZ +

2 gmd ' 2

X

2 m

5 9,9 w4m 3 m d a

_ — to

2 gmd ' 2

zz + pz

** +

md 2 r

Zgmd 2

X

z UlOs

i l +

• W • 20 <n

putting f = s;, and y* = — - /, or p‘ + ?* =

Then the general fluents, taken by the 8th and 11th formsvol. 2 pa. 307 of the Course, give x = x [|- log. (x 1 +

f) + *Jr * arc to rad - ? and ta "; *] = x Ct lo g- (®*

~ Hr v + laP + ^ x arc to rad - and tar, g- But >

at the beginning of the motion, when the first velocity is vfor instance, and the space x is = 0 , this fluent becomes

° = i^ x Ci lo K- ( v * - 7 T V + £-)+fx arc radius ytan. v — p], Hence by subtraction, and taking v — 0 forthe end of the motion, the correct fluent becomesx = ^ x Wog. (V 1 - -£v + £) - ilog.^

(arc tan. v — p — arc tan. — p to rad </)].

But as part of this fluent, denoted by -£ x the dif. of thetwo arcs to tans, v — p and — p, is always very small in com-parison with the other preceding terms, it may be omitted,without material error in any practical instance; and then the

; + X

fluent is .r = —~ x hyp. log.

4rrm.il* J 1 ©

. V +

md 2

V)

md 2

i

, for the ut-

mu*

most height to which the ball will ascend, when its motionceases, and is stopped, partly by its own gravity, but chieflyby the resistance of the air.

38. But now r , for the numerical value of the general co-efficient 7777 ,, and the term —p‘, because the mass of theball to the diameter d, is '5236 d i ) if its specific gravity be s ,its weight will be •5236sd 3 =a>; therefore -~=-5236sd, and

~ = 69259sd, this divided by 4g or 64, it gives —1082^ for the value of the general coefficient, to any diamc-