9S CELESTIAL MECHANICS. I. i. 1. 



if jt' be in the direction of z, x" must be perpendicular to 

 it; and supposing?/ to be similarly composed of y^ and y'\ 

 \( being in the direction of r, and y" perpendicular to it; 

 x" must be equal and contrary to y" ; and i^ and y^ toge- 



ther must be equal to z : but y'-=y y, and 'if'zz.-y: so that 



z z 



X V 

 x^ + y^zz-x + '-y^z, ^nd x^ + y^ := z^ ; consequently z is 



equal to the diagonal of the rectangle, the sides of which 

 are x and y. 



It must however be shown that z coincides with this 

 diagonal in position as well as in magnitude. For 

 this purpose we must consider one of the forces y as in- 

 creasing from nothing to its actual magnitude, and we 

 must trace the effects of its combination with x through 

 the intermediate steps. Now if an elementary force 8y 

 be combined with a finite force x^ the variation of the 

 angular direction of the result, which may be called ^d, will 

 be inversely as x and directly as some constant multiple or 

 submultiple of ^y, since the evanescent increments of two 

 quantities, related to each other, are initially in a constant 

 ratio, (248), so that the cl>ord of the angle Sfi may be called 



k^y, and the angle itself — '—: the elementary chord k^y 



obviously depending on x and on the variation of the angle 



x^d 

 Sd, in such a manner, that 3y may be expressed by —-p 



and S9 by — ^ It is indeed sufficiently obvious that the 



chord can in this case be no other than ^y itself, since a 

 force in the direction of the radius could scarcely influence 

 another in the direction of the circumference, but Laplace 

 dots not think it right to take this for granted without 



