CALCULUS OF PAETIAL DIFFERENCES. 39 



freely along a revolving rod, at the extremity of which, rod a 

 fixed body moves round in a given plane a locus which the 

 calculus founded on the Principle shows to be in certain cases 

 the logarithmic spiral.* 



No one can doubt that -the Principle of D'Alembert was in- 

 volved in many of the solutions of dynamical problems before 

 given. But then each solution rested on its own grounds, and 

 these varied with the different cases ; their demonstrations 

 were not traced to and connected with one fundamental prin- 

 ciple. He alone and first established this connection, and ex- 

 tended the Principle over the whole field of dynamical inquiry. 



The ' Traite ' contains, further (part 1, chap, ii.), a new 

 demonstration of the parallelogram of forces. The reason of 

 the author's preference of this over the common demonstration 

 is not at all satisfactory. His proof consists in supposing the 

 body to move on a plane sliding in two grooves parallel to 

 one side of the parallelogram, and at the same time carried 

 along in the direction of the other side. This is not one 

 whit more strict and rigorous than the ordinary supposition 

 of the body moving along a ruler parallel to one side, while 

 the ruler at the same time moves along a line parallel to the 

 other side. Indeed I should rather prefer this demonstration 

 to D'Alembert's. 



The ' Traite de Dynamique ' appeared in 1743 ; and in the 

 following year its fundamental principle was applied by the 

 author to the important and difficult subject of the equili- 

 brium and motion of fluids, the portion of the ' Principia ' 

 which its illustrious author had left in the least perfect state. 

 Pressed by the difficulty of the inquiry, which is one of the 

 most important in Hydrodynamics, the motion of a fluid through 

 an orifice in a given vessel, and despairing of the data afford- 



ydx z 



* The general equation is d-y - - (- - ; - --.. in which y is 



a 2 A a 2 + D y 2 



the distance of the moving body D from the fixed point, or the length of 

 the rod, at the end of which is the body A, describing an arch of a circle, 

 and x that arch. The velocity of D is likewise found in terms of the 

 same quantity. 



