OF PRESSURE AND EQUlLIBUiUM. 99 



proof. We have therefore initially ^Q— — ^. In any other 



situation of the result z, we must suppose the clement hj 

 to be resolved into two portions, one in the direction of 2:, 

 which only affects its magnitude, the other perpendicular 

 to it, which determines the increment of the angle IQ from 

 z, in the same manner as ^y determined it in the first in- 

 stance from X, Now the por- 

 tion of the force ^y perpen- , .-''"'^^ 



X 



dicular to 2: is - Sy: conse- 



quently Sfc ^ ; or, since x 



zz 



Jchi 

 is here considered as invariable, or =1, ZQ— — ^. But 



zz 



the fluxion of the angle ^, of which the tangent is y, is 



dy 

 ^ , as is readily understood from considering the rela- 

 tive situation of the increments ; consequently, since z^ has 



been shown to be equal to x^+ii". J.z=. :---<- — =: 3^ ans: 

 ^ -^ z" ^+yy 



tang y, and 6= k ang tang y-\-c. But since Q—0 when 



yziO, c vanishes, and fizz ^ ang tangy; and when y is 



infinite, fizzQQo, since 2 coincides with y, consequently k 



must be =il, and 6— ang tang y. So that 2; must coincide 



with the diagonal of the rectangle in position as well as in 



magnitude. 



Scholium 1. Laplace has supposed both the forces 

 X and y to vary together: but this is evidently an unneces- 

 sary complication. 



Scholium 2. The principle of the proportional 

 variation of evanescent increments must not be applied 

 without some caution, for in the present investigation, if it 



H S 



