OF PRESSURE AND EQUILIBRIUM. 



97 



and the increment of the ordinate will be, for an infinitely 

 small space, proportional to that of the absciss, whether it 

 be doubled or quadrupled, or in any way subdivided. The 

 truth of the proposition is however shown more generally 

 and conclusively by means of the invaluable theorem of 

 Taylor, demonstrated in Lemma E, for the increment Aw' 

 of the ordinate, beginning from u, is to the increment h of 



the absciss in the constant ratio of t- to 1, as lono- as the 



ax ® 



increment k remains so small, that its square and its higher 



powers may be supposed to vanish in comparison with 



itself. 



Scholium. It is however necessary to except the 



case in which the first fluxion of one of the quantities 



compared becomes =0. (See 249, Sch. 2). ] 



249- Theorem 240, of the Composition 

 of Forces, demonstrated in Laplace's man- 

 ner. 



Case 1. The forces x and 

 y, acting at right angles to 

 each other, will produce a 

 joint result z, of which the 

 magnitude is expressed by the 

 diagonal of the rectangle xy. 

 For we may obviously suppose 



X to be composed of two forces, x^ and /', also at right 

 angles to each other, and in the proportion of x to y, since 

 the same law must apply to forces similarly related, what- 

 ever their magnitude may be ; and the result x must be 

 derived from / and x^^ in the same manner as z from x 



X It 



andy; consequently we have /=z-jrand x^zz^x. Now 



JC 



