called the Principle of Least Pressure. 287 



Differentiating the above equations, P being considered a func- 

 tion of x, y, z; multiplying the differentials of equations (1) by 

 the indeterminate quantities A 15 A 2 , A 3 , respectively, 



(2) by B w B 2 , B 3 , 



(3) by j* n jct 2 , ^ 3 , p# &c. &c. &c, 



(4) by * 1$ X 2 , X 3J A 4J & c * & c - & c -> 



and adding the resulting equations (all whose right-hand 

 members disappear) to the differential of equation (5), which is 



^ f dP ^ dF * <*P*1 



we shall obtain an equation in which the coordinates of the 

 several points of resistance x v y l9 z v «r 2 , y 2 , z 2 , &c. &c, and 

 a v $v 7i> a 2> fe 7v & c " & c *5 ma y De considered as independent 

 variables; so that by the condition V = a minimum, the co- 

 efficients of 8 x, hy v 8 z v 8 x<# ly^ 8 z 2 , &c. &c. 8 u l9 8 « 2 , &c. 8 jS 1$ 

 8 /3 2 , &c. 8 y M 8 y 2 , &c. &c, respectively equal nothing. We thus 

 obtain, in respect to each point of resistance, six equations, re- 

 duced to four, by the elimination of the indeterminate quantities 

 A and ju,, which are different for the different points of resistance. 

 Of these four equations, two, involving partial differential co- 

 efficients of P, from which one of the variables x, y, z may be 

 eliminated by means of the equation u = 0, are equivalent to 

 one complete differential equation between the other two va- 

 riables, so that on the whole there are three equations deter- 

 mining the quantities P, a and /3 in terms of the coordinates 

 x,y,z, and the indeterminate quantities A 15 A 2 , A 3 , B 19 B 2 , B 3 

 (which are the same for all the different points of resistance), 

 whilst these are in their turn determined, by substituting the 

 resulting values of P 19 a v V P 2 , a 2 , /3 2 , &c. in the equations 

 (l)and(2). 



Thus the values of P, a, and /3, that is, the amount and di- 

 rection of the pressure upon any fixed point of the system, 

 are completely determined. 



The various analytical operations indicated above, are in 

 their nature too elaborate to be here brought under the eye 

 of the reader. Those conversant with the methods of analysis 

 will, however, readily supply the deficiency. 



It may be observed that in the case in which the points of 

 resistance are all in the same plane*, the pressure upon any 



* This particular case of the general proposition has formed the suhject 

 of a paper by the illustrious Euler, entitled, " De Pressione Pondeiis in 

 Planum cui incumbit" Mem. Ac. Pet. Novi Commentarii, vol. xviii. His 

 discussion of the question is most ingenious and elaborate. It is impossi- 

 ble, however, to admit the hypothesis on which he has grounded it. 



