23-1

• v — z“

4m?

THEORV AND PRACTICr; TRACT 37.

; these being substituted in the above va-

lue of x, it becomes x =

n •

sz + r-a

— zv m _ —

2 gmd*

n* w4 m* md*

zz + pz _

2 Z + pZ

z * + —y .,”/' 3md- 1

‘Zxmii*

z 1 + q*

putting P = £, and f = ~- p\ or f + f =

Then the general fluents, taken by the 8 th and 11 th formsvol. 2 pa. 307 of the Course, give x — —— x IT log. (z z +

r ’ ° 2gmd* lx o \ 1

<?) + 7- x arc t0 rad - ? aild tan. z] ~ x H log. ( v *

~ £- v + £ 5 ) + 7 x arc to rad. j. and tang. v-p]. But,

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

0 = S* x [* Io S- -

-v + -^-l-b-^x arcradius q

md* q* 1

tan. v — p\. Hence by subtraction, and taking v = 0 forthe end of the motion, the correct fluent becomes

-—- x [Hog. ( v z

Qtrma* L- & v

1 w \ ,i w . p

- v 4- —) — 4-Io°\ - \~ —

1 ~ md~' * 0 md* ' q*

(arc tan. v — p — arc tan. — p to rad 9 )].

But as part of this fluent, denoted by T 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

fluent is x

4 £inii*

X hyp. log.

for the nt-

md*

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, for the numerical value of the general co-efficient ■ , and the term - ; because the mass of the

4gmd- md* 7

ball to the diameter d, is •S236^ 3 , if its specific gravity be s,its weight will be •L>22>Qsd 3 =iv ; therefore ~^~ = '5236sd, and

— == 69259m?, this divided by 4 g or 64, it gives

md* ' - j -a - -- o 4gmd*

1082 sd for the value of the general coefficient, to any diarne-