764 



BOSCOVICH'S THEORY. 



Bodcovich's C the two opposite force*, equal also to HA X A and 



theory of LD X D. Wherefore 



Natural 



l'liiioso- The point of A his two forces, AI, AH. 

 P h y- The point of D has two forces, DM, DL. 



' "~ ' The point of B has two forces, A X AH, D X LD. 



And C four, A X IA, D X MD, A X HA, D X LD. 



Now let the line BC express the magnitude of the 

 force compounded of CN and CR parallel to DB, 

 AB. BN and BR will express the magnitude of 

 these forces as well as their directions, and therefore 

 RC, NC, equal and parallel to them, will express the 

 third and fourth forces of the point C. Produce AC 

 and DC till they meet, in T and O, the lines RT, 

 NO drawn parallel to VF, GZ, or KX, and drop the 

 perpendiculars AF, DE, RS, NQ. 



Since IAG, CTR are similar, having their sides 

 parallel, and also CON and MDK, therefore as IG 

 or AH, to CR or BN or Ax AH, (that is to say, 

 as 1 is to A,) so is AG to TR, and A* to Tc - 

 TR is therefore equal to GA, (or AZ,) drawn into 

 A, and CT=IAx A. The former consequently ex- 

 presses the sum of the forces AZ of all the points of 

 A ; the latter the first part of the force of the point 

 C, viz. A X I A. For the same reason, NO will ex- 

 press the 6um of all the forces DX of all the points 

 in D, and OC the second force of the point C, viz. 

 D X DM. Wherefore 



The sum of the parallel forces in An TR. 

 The sum of the parallel forces in D= NO. 

 The two forces in B= BN, BR. 

 The four forces in C= CT, OC, RC, NC. 



Now it is obvious, that the first CT and third RC 

 compose the force RT=the sum of parallel forces in 

 A, and that OC and NC compose NO the sum of 

 the like forces in D. Wherefore it is also evident, 

 that the fulcrum C is urged by the point C alone, 

 with a force which has the same direction as the pa- 

 rallel forces in A and D, and is equal to their sum ; 

 that is, it is urged in the same manner as if all the 

 points in A and D were in the point C alone, and 

 acting with these forces immediately on the fulcrum. 



Besides, from the same parallelism of the sides 

 we have the following triangles similar, viz. CNO 

 and DPC ; CNQ and PDE ; CPR and VCN ; 

 CRS and VNQ ; CVA and TCR ; VAF and 

 CRS. 



These exhibit the following proportions : 



ON : CP : : NC : PD : : NQ : DE. 

 CP : CV : : CR : NV : : RS : NQ. 

 CV : RT : : VA : RC : : AF : RS. 



In which, comparing the first column with the last, 

 we have by perturbate equality ON : RT : : AF : 

 DE. That is, the sum of all the parallel forces in D, 

 to which ON is equal, is to the sum of all the forces 

 in A~TR, as the perpendicular distance AF (from 

 , the latter point to the line which passes through the 



fulcrum parallel to the direction of the forces) to the 

 perpendicular distance DE from the former point to 

 the same line. Wherefore the ratios required are now 

 found ; and we have a demonstration of the funda- 

 mental property of the lever in the commonly suppo- 

 sed desperate case of parallel forces. 



Kg. SI. 



We shall only take notice, in this place, of the Bo*covich' 

 mode in which the theory is applied to the pressure riory of 

 and velocity of fluids. Let the points lying ( Fig. 20.) N '.' ur;,1 

 in any straight line AB, tend in that direction by any _,,_ 

 external force the action of which these points de- ' _ . 

 stroy by their mutual forces, so that they are in equi- Prejim 

 librio. Between the first point A, and the second fl'J'U^. 

 next it, there must be a repulsive force equal to the 

 external force acting on A. The second point will 

 therefore be urged by this repulsive force, as well as 

 by its own tendency. The repulsive force between 

 the second and third must be equal to this, and so 

 on, increasing towards B, which will be urged for- 

 ward by the sum of all the external forces of the 

 points before it. 



But if the points are not in one straight line, 

 but dispersed throughout a parallelopiped, Fig. 21. P^ate. 

 whereof FH is the base perpendicular to, and EFHG i 

 a section parallel to the direction of the external 

 force, it may be shown by the composition of 

 forces, but it is sufficiently evident, that the repul- 

 sive forces with which the base acts on the particles 

 next it, is in this case also equal to the sum of all 

 the external forces, and this either in solids or fluids. 

 But since the parts of a fluid are free to move in any 

 direction, the reason of which we shall see hereafter, 

 each particle will be pressed in every direction with 

 the same force j so that in any plane IL, the forces 

 are every way equal, and any particle N in LM will 

 be urged towards FH as towards EG ; and hence 

 we see at once the reason why the base FH receives 

 the same pressure from the fluid FLMACKIH, 

 as from the whole FEGH ; and the superficies LM 

 receives a pressure upwards from the particles N, 

 equal to what it would receive in the opposite direc- 

 tion from the mass LEAM. In this way, therefore, 

 the hydrostatic paradox, bellows, See. admit of ex- 

 planation. If to increase the repulsive force consi- 

 derably, much change, of distance be requisite, the 

 compression of the mass will be sensible, and like- 

 wise the increase of density among the lower par- 

 ticles ; such is the case with air. I# the repulsive 

 force be powerful at small changes of distance, the 

 mass will appear as if incompressible ; which is the 

 case with water, mercury, &c. 



When a free exit is allowed to the particles of a 

 mass of this kind, by means of an orifice, they will 

 escape with velocities corresponding to the forces by 

 which they are impelled. The first particle will be- 

 gin to move by the repulsive force with which it is 

 pressed by the neighbouring particles ; then the se- 

 cond, being more distant from it than from the third, 

 will also move away with a force corresponding to 

 the difference between the repulsions, and therefore 

 more slowly : the particles, therefore, will separate 

 by the first being accelerated, until at length the re- 

 pulsive force ceases, or an attraction begins, so that 

 there will be some oscillation ; but this only during a 

 \ery short space of time. The velocities will depend 

 on the area of a curve, the axis of which expresses the 

 space passed over from the beginning of the motion. 

 We know, that in the efflux of water the velocitk 9 

 arc as the square roots of the heights, or compr 

 ing forces. This may be expressed by the logistic 

 curve, as well as by many others. Whether the 



