22 
perpendicular to the normal PI, it becomes mna in the 
direction KI and in the plane PIK. It is not difficult to 
show that the cosine of the angle between these component 
velocities will be correctly represented by the following 
formula 
Cos.KIH = 
COSaCOS)/ 
sinasinv 
( 6 ) 
Therefore, the velocity of the element (a) in a tangent plane, 
that is, in a plane perpendicular to its normal line at P, is the 
resultant velocity of rminv,vsina acting in directions making 
the angle KIH, and, by the ordinary formula for calculating 
the resultant it becomes equivalent to the formula 
V rWsin^j/ + + 2mycosacosj/ (7) 
9. The above result may be obtained in another way, viz. 
find the resultant of rw and v in the plane HPK, then re- 
solve this resultant parallel and perpendicular to the 
normal line. 
The formula (7) admits of the following form, easily ob- 
tained, viz. : 
V tV + - (rwQOBy - vcoBaY (^) 
10. The point P moves with a velocity + v% and 
describes the common helix whose radius is v and pitch 
(2ttv) upon (w). 
11. Taking 6080 feet to represent a knot or nautical mile, 
V = feet per second = K x 1 *688 
where K is the knots per hour. 
If, n = number of revolutions of screw per minute. 
u = angular velocity = ^ 
On the Slip of the Element (a) of the Propeller Blade. 
12. Let it be supposed that the element (a) rotates in 
water perfectly at rest, and also that its motion is the same 
as the- motion of the nut of a screw. This condition is 
evidently fulfilled when the normal velocity of the element 
