-166 D'ALEMBERT. 



these were all private paths to one solution, whilst that is a 

 high road to all. The solution of every problem is obtained 

 from an equation involving some principle to which the mo- 

 tions of the system are subject the advantage of D'Alem- 

 bert's step lay in this, that it was the same principle which 

 he applied to each particular case. 



Note to p. 396, line 19, by the author mentioned p. 396, note. 



Since these last forces mutually destroy each other, and 

 that the forces actually impressed were compounded of them 

 and of those (usually called effective) which act in the direc- 

 tion the bodies really move in, so that the force originally 

 applied (usually called the impressed force) is the result of 

 these two forces, it follows that the effective forces would, if 

 they acted in the contrary direction, exactly balance the 

 impressed forces. Problems of dynamics are thus reduced to 

 a general equation of equilibrium and become statical. 



II. 



That Euler, in the Memoir published in 1734, solved an 

 equation of Partial Differences is quite incontestable, though 

 he laid down no general method ; which, indeed, D'Alembert 

 himself never did, nor any geometrician before the publica- 

 tion of Euler 's third vol. of the ' Institutions of the Integral 

 and Differential Calculus.' The problem, as given in the 

 * Mem. Acad. Petersb.' vol. vii., was this ; We have the 

 equation dz = P dx-\- Qda, z being a function of x and a; 

 and the problem is to find the most general value of P arid 

 Q, which will satisfy the equation. QrzrF^-fPE, P 

 being a function of a, and E- a function of a and x, Euler 

 seeks for the factor which will make d x -f- R d a integrable. 

 Call this factor S, and make S d x -f S R d a = d T, and 



make fFda = log, B. 



He finds for the values required 



P = BS/' : T, Q = 1?L? + BES/:T 



