### §33.

Third Method.—Profiles described by Secondary
Centroids.

We have already mentioned ( 9) the employment of secondary centroids instead of the original primaries, and have found that by their means problems which otherwise contained some difficulties could be easily solved, that in certain cases we could use the secondary centroids interchangeably with the principal ones. Among these secondary cen- troids one class especially is useful in the delineation of element profiles. This class consists of those in which two curves and their tan- gent are used to represent the motion. Such centroids are obtained if through a sufficient number of points in the two primary centroids we draw a series of secants making a constant angle to the tangents at the points through which they are drawn; these secants enve- lope a pair of curves which, touched by a line rolling upon them, form together with it secondary centroids. If for example and Z) be the centres of curvature for the

Fig. 109.

the

C 1 7?

portions of the centroids touching at 0, the ratio __ of

JJ -F

C perpendiculars is equal to that into which the point divides

the line of centres; for any very small motion of the line E F, therefore, the same small angular motion occurs as if the primary centroids were rolling on each other. If now any point P in the straight line describe a curve relatively to A and another upon