74 Mr Sang on some prevailing misconceptions 



cerning the original forces, would commingle the means for deter- 

 mining the forces in all positions between the first and last, and 

 would rather be the basis for determining the conditions of the 

 mean state, than of either of the extreme ones. However small 

 may be the displacement, still will there exist an error in the 

 estimate ; yet the more minute the displacement, the more in- 

 considerable will be the error, because there will exist the less 

 difference between the two extreme states of the machine. In 

 order, then, to compute the pressures accurately, we must deter- 

 mine the proportions which subsist between the motions of the 

 parts, supposing that these motions are infinitely small. I know 

 of no other method for effecting these computations than that 

 contained in the Differential Calculus, or in the more abstruse 

 but more satisfactory theory of Lagrange ; and, although the 

 name of fluxions be a bug-bear to thousands, it is absolutely 

 impossible, without its aid, to advance beyond the threshold of 

 mechanical investigation. The advocates for mental indolence 

 may urge that the statical properties of many machines may be 

 examined without the aid of the fluxional calculus. Such is, 

 indeed, the case, and these machines and such methods of cal- 

 culating concerning them may afford lessons to the beginner ; 

 but from such processes few, if any, general conclusions can be 

 drawn, while the instant that motion is contemplated, all these 

 resources fail. 



The principle of virtual velocities, such as I have defined it, 

 holds true of all machines, even though subjected to the retard- 

 ing influence of friction ; and is applicable not merely during a 

 momentary state, but also to the motions of machines. Denot- 

 ing by the word jforce, the result of the combination of a pres- 

 sure with the distance through which it acts ; if the sum of the 

 accelerating forces be just equal to that of the retarding ones, 

 the quantity of motion in the machine will be unaltered, but if 

 the two sums be unequal, then will the speed be changed. Now, 

 in almost all machines the forces and velocities are subjected to 

 periodic variations ; the method then of computing the change 

 in velocity consequent upon a change of force must be clearly 

 understood, ere we can obtain any information as to the general 

 properties of machines in motion. 



The great proposition which connects the statical with the 



