OF MECHANICS AND GEOMETRY. 277 



I observe first, that Euclid's last corollary seems to be the easiest proposition to grasp, and will be 

 admitted without formal demonstration as soon as its meaning is apprehended. For if we consider the 

 changes of direction which a straight line undergoes by successively coinciding with the sides of the polygon 

 it is clear that when it has been made to coincide with all in succession, it will at last come into its original 

 position, but a line which has revolved and come into its original position must have described four right 

 angles; whence the proposition is manifest. From this of course the first corollary and the proposition 

 itself immediately follow. 



It appears therefore that Euclid i. 32. is only a form of the self-evident proposition, that a straight line 

 being made to deviate from its original direction cannot assume it again until it has deviated through four 

 right angles. 



Now the condition of forces being such as will produce equilibrium, is simply that the lines respecting 

 them shall form a polygon. And this proposition is I believe only an expression of the fact, that two fores 

 cannot counteract eacli other unless they act in the same straight line, or, to express myself more in con- 

 formity with the geometrical proposition, that if a force has been made to change its direction it cannot 

 produce the same effect as before unless its deviation has been through four right angles ; but this thought 

 I have not yet fully developed. 



Note (C), page 275. 



It may be well to remark here that the symbol re^^-' being the complete expression of magnitude and 

 direction is also the complete expression of linear and angular distance, if r be measured from a fixed point, 

 and 6 from a fixed straight line: consequently re^^^^ may be taken as the complete position-index of a 

 point, or of a physical particle, in a given plane. 



If we consider a particle whose position is changing with the time (t), then the symbol -r-.re*^'-^ will 



express the complete variation of the position-index, and will therefore give the magnitude and direction of 

 its velocity. 



(f 



In like manner the symbol -7-5 . re*"'-' will give the complete variation of the velocity, and will therefore 



be the symbol for the accelerating force in both magnitude and direction. 



Now suppose the particle to be in motion under the action of any forces, the complete expressions for 



which are P.e^^^~-^, P'.^'^^, Sec: then if M be the mass of the particle we shall have for its equation 



of motion 



M-^.rp«^-""'=J'e*>^ + P'e*'^^' + 



at 



This equation I have given merely in illustration of the principles of the preceding memoir, but in the 

 Cambridge Mathematical Journal, (Vol. iv. p. 177.) I have shewn that the symbol re^^^^ may be applied 

 to mechanical investigations with considerable practical convenience. 



H. GOODWIN. 



