212 Mr. Her'apath on the Law of Continuity. [March", 



relations and properties to the distance, some interesting con- 

 clusions may be deduced ; but we shall at present only notice 

 one or two well known theorems of motion, and then proceed to 

 the demonstration of the saltation of states. 



Suppose the perpendicular distance to be always moving 

 parallel to itself, and as the velocity of a moving body, the 

 straight line which it cuts off from a given point being as the 

 time, then will the area described by the distance be as the 

 space or distance described by the moving body. For the incre- 

 ment or fluxion of this area will be equal to the fluxion of the 

 right line drawn into the perpendicular distance, that is, to the 

 fluxion of the time drawn into the velocity. But the product of 

 the velocity into the fluxion of the time is well known to be 

 equal to the fluxion of the space. Therefore the space and this 

 area have always the same rate of increase, and are consequently, 

 equal, or in a given ratio. The same conclusion may be a» 

 easily obtained by the method of exhaustions. 



Again, let the perpendicular R S be drawn any where in the 

 line N O, and let its intersection with the curve be S. Upon 

 R S construct the little rectangle R T, which is the fluxion of 

 the space described by the moving body. Let W be the inter- 

 section of the curve and of the side X T, 

 and let S V be taken equal to twice 

 S T ; and with T X, T V, and T W, 

 T V, construct the rectangles V X, 

 V W. Then because the time flows 

 equably, the equals R X, X Y, repre- 

 sent the fluxions of the time at the 

 moments R X ; and the rectangles S X, 

 W Y, the corresponding fluxions of the spaces. Consequently 

 the rectangle W V (= + WT x R X), is the second fluxion 

 of the space ; T \V being the fluxion of the velocity. And 

 since (v) the fluxion of the velocity is equal to the force (f) 

 multiplied by the fluxion (/) of the time, we shall have + x 



= ft* ; an equation well known in the higher branches of phy- 

 sical astronomy. 



In a similar manner we may proceed to the development of 

 other theorems depending on the principle of continuity in the 

 different branches of pure and mixed mathematics, if we had 

 time and inclination to pursue them, but it is now necessary to 

 resume our original subject. 



The construction in other re- P _J^ 

 spects remaining the same, let 

 the line M P, instead of being 



continued, be broken or inter- N£L_^-^ 



rupted at the extremity b, for 

 instance, of the part a b, and 



suppose c in the other part of the q + x -> ; _ 



line, which, for convenience, we PCBA. N 



