88 
Dr. Waring on 
1.2. Let tf'nX'and X — b, which is fnppofed to correfpond 
nearly to the Slate of our atmofphere, then will v = — 
J 4aX * X Xx =-2xW iat*x J* ( fa ~ — 
2 e~~ zatX ~ b x (^j'e zai * Jrb '}tbx-\- A), £ being the number, whofe 
hyperbolic log. is i, and and A quantities to be affumed 
according to the conditions of the problem. 
1. q. Let X = X 7 , and it becomes Xx = — and i — 
V 
X(l+aV) # 
2. Let X be an homogeneous fundlion of one dimenfion of.v, 
that is, ~ax, and V a limilar fun&ion of n dimensions of v 9 
that is = bv n , and X 7 a fimilar function of r dimensions of a.' 1 and 
v, and n + rm i ; then by fubSlituting %x and its fluxion for v 
and its fluxion, can be found the fluent of the fluxional equation 
(X + tf VX 7 ) x~ — vv, and confequently the velocity and time 
by the quadrature of curves in terms of the fpace ; and in like 
manner of many other cafes. 
3. Fig. 4. Let a body moving in a given curve be a&ed on 
at any point P by a force f tending to a given point S, and 
refilled by a medium proportional to V a function of its velo- 
city multiplied into its denfity X 7 a fun&ion of the distance 
SP = D ; to find its velocity, time, and di (lance from the given 
point S in terms of each other. Let F =/x the force in 
the direction of the tangent PY, and confequently (F + 
VX 7 ) A.= - vv, and v = f C xf, where A is the increment of 
the arc, and C the chord of curvature in the direction SP ; but 
fince the curve is given, the chord of curvature may be de- 
duced from the distance, &c. and the increment A of the arc 
from a function of the distance multiplied into the increment 
4 of 
2 X 2aX * x / 
