582 PHILOSOPHICAL TRANSACTIONS. [aNNO 1779- 



rightly the method of solving such problems. Euler, De La Grange, Frisius, 

 Silvabelle, Walmesley, Simpson, Emerson, have each considered the subject, 

 and perhaps the importance of the inquiry would justify a minute examination 

 into the cause of the agreement or disagreement ot their sevenil methods; but I 

 am deterred from entering into such a discussion 1 ly the length of time which it 

 would require ; especially as I think, those who have read those authors will easily 

 conceive the substance of what I should have to observe, and to those who have 

 not read them I should hardly be able to say any thing intelligible. 



§ 12. The above solution, if it had no other advantages, is, I apprehend, 

 much more concise than any that has hitherto been given. Abstracted from 

 what is said by way of illustration, articles 4th to gth contain all the calculation 

 requisite, and as I have studiously avoided the ambiguous use of the terms 

 force, vis, efficacia, momentum, &c. as well as every doubtful representation of 

 times, spaces, and velocities, which are often substituted by authors in equations, 

 I believe the whole process will appear easy, and the evidence on which the con- 

 clusion rests be exactly ascertained. 



§ 13. The principles described in articles 2 and 4 depend on the 3d law of 

 motion, and the property of the lever, and are demonstrated in the following 

 manner. Every thing remaining the same as in art. 2, (fig. 6) let av and br, 

 perpendicular to the right line or axis ae, represent the forces and directions 

 with which those bodies are respectively urged, when at liberty to move freely 

 in those directions ; and let At;, Br, cc, represent the accelerative forces of the 

 respective bodies, as altered by their mutual actions on each other : then, be- 

 cause c X cc is the moving force gained by c, and A X t'v -|- b X tr the moving 

 force lost by a and b, regard being had to the lengths of the different levers ae, 

 BE, we shall have a x f v X ae -|- b X rR X be equal to c X cc X ce, that is, 

 A X AE X (av — At) 4- B X BE X (br — Br) equal to c X cc X ce, and by 

 transposition a x ae X av -j- b X be X br equal to c X ce X cc + a x ae X 

 Ar -f B X be X Br. Let s, s', represent, as in art. 2, the spaces which would 

 be described by the bodies a and b at liberty in any very small portion of time, 

 and let x be the space which a actually describes in that time, when connected 



with B and c by the lever ae. The quantities , , will then be the 



spaces described by b and c respectively ; and lastly, because the spaces described 

 in given times are as the accelerating forces, the above equation gives x equal to 



AXAEX«+BXBEX«' 



A X ae' 4- b X be' + c X ce' 



The same method extends itself easily to more difficult cases, and by its as- 

 sistance several very important theorems are briefly demonstrated. 



^ 14. The reasoning used in art. 6 will appear very evident to any one mode- 

 rately versed in the elements of mechanics and the doctrine of moving forces; 



