ELISHA MITCHELL SCIENTIFIC SOCIETY. 29 



ordinate to the curve and a solid to be generated by the 

 motion of an area. 



As a point traces a curve DPS (fig. 2), when it reaches 

 the point P, it has the direction of the tangent PT at that 

 instant; for we can make only two suppositions: (i) the 

 direction coincides with that of some chord passing through 

 P, whether the otlier end of the chord precedes or follows 

 P, or (2) it coincides with the tangent; but it cannot have 

 the direction of a chord at the point P without leaving the 

 curve; hence this supposition is false, and as one must be 

 true, it follows that the direction of motion at P must coin- 

 cide with the tangent PT. 



At the point P therefore, by the definition above of a 

 differential, the simultaneous differentials of .r, y and s are 

 what their increments would become, during any time, if 

 at P their rates of change should become constant. This 

 can happen only where the motion takes place uniformly 

 along a straight line and this line must be the tangent at 

 P, as that is the direction of motion at that point. The 

 differentials dx, dy and ds can thus be represented by PQ, 

 QI and PI respectively, or by PR, RT and PT, for a uni- 

 form increase along the tangent would correspond to a 

 uniform rate horizontally and vertically. This agrees with 

 what has hitherto been established. 



The rates of increase, horizontally, vertically and along 

 dx dy ds 



the tangent, are — , — and — respectively. 

 d/ dt dt 



The differential of an area is found as follows: Let area 

 CDPA = 2/, then if A.r = dx ^= AB, du ^=ydx ; for although 

 A?/ = area APSB, the increase of area will not be uniform 

 if the upper end of the ordinate AP moves along the curve, 

 but it wnll be uniform if it moves uniformly along PQR; 

 for then equal rectangles, as APQB, will be sw^ept out by 

 the ordinate AP as it moves to the right, in equal times. 

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