82 Mr Tredgold’s Description of an Improvement 
lowest point, the crank of the small wheel will begin to ascend ; 
and the radius of the cranks being unity, the effective length of 
the stroke of the one pump will be 1 + cos § «, and the other, 
1 + cos a ; consequently, the sum of the strokes is 2 + cos a 
+ cos | a. In the second revolution the effect in length of 
stroke of both pumps is 2 + cos 2 a + cos (1 -j- J) a ; in the 
third we have 2 -{- cos 3 a -f cos (2 + J) ; and in the ninth 
stroke it is 2 -f- cos ?t«| cos ( n — J) a. 
When na ~ 180° its cosine is — 1, and the effect is 1 -f- cos 
180 — 1 a. 
The total quantity of water injected during n strokes will be 
as 2 n + sum of the cos n a sum of cos n — \ a ; and by 
Gregory’s Trigonometry, art. 21, note. 
. n n 4- 1 . n 
sm a. cos — - — a 4- sin — 
% /w 
n -f- g 
-a. cos — a 
sin \ a. 
If we neglect the difference between cos n a and cos, n — \ a, 
the area representing the total effect of the two pumps will be a 
rectangle, of which the one side is equal to the diameter of the 
circle described by the cranks ; and the other, the sum of the 
areas of the pumps, multiplied by the number of strokes neces- 
sary to cause the small wheel to gain half its circumference on 
the other. 
The quantity of water injected at any number n of strokes 
will be very nearly 
. n n -f- 1 
sm - a. cos — - — a 
A / 2 2 \ 
2 n A r ( n + — — = — \ V 
V sm i a. J 
In this formula, A is the sum of the areas of the pumps, r — 
the radius of the cranks, n the number of strokes, and a the arc 
the small wheel gains in one revolution of the larger one. 
To illustrate this subject more clearly, we have annexed the 
diagram Fig. 2., where D' is the crank of the larger, and D that 
of the small wheel ; and we suppose the crank D, in this case, 
to gain half a revolution at the end of 12 strokes. The crank 
D will begin its effective stroke successively at the points 1, 2, 8, 
Sec., and always terminate its stroke at b. The crank D' will, on 
the contrary, always begin its effective stroke at a, and terminate 
