1)6^ lVpt*AOf^ien m^y^i/»*^olkal Mechanic ^ 



I shall call a^ a unit of translation * ; consequently the nume- 

 rical magnitude of uv is to be found by multiplying the numerical 

 magnitude of ii by that of v, and by the sine of the angle which 

 V makes with w. 



From the result just obtained, it appears that all units of 

 translation in the same plane (or in parallel planes, by former 

 papers) are equivalent to each other; for, if we suppose a?=lj 

 y = l, and ^ = 90^, uv becomes a unit of translation anywhere in 

 the same plane as the unit ayS : since, therefore, these suppo- 

 sitions reduce the equation just obtained to uv=ia0, it follows that 

 all units of translation in the same plane are equivalent to each 

 other. The method employed in statics of representing couples 

 by their axes, suggests a similar sort of representation here; J 

 shall therefore assimie a unit of length di'awn at right angles to 

 the plane of a and ^ to represent the unit of translation otp, 

 which it will properly do, since it completely defines uff as 

 i*egards magnitude and plane of translation ; and this is all that 

 need be defined. 



' Let 7 be the unit of length thus drawn ; then I shall put 7 

 for a(9, or a^ for 7, as the case may require, in any investigation. 



Since /9a =—ayS, it follows that ^u is represented by —7. 

 To determine generally the direction of the unit of length which 

 represents a unit of translation, I shall adopt the following rule, 

 viz. Conceive a man to be so placed that his head is in the direc-i 

 tion of the translated line {ff) and his feet in the opposite 

 direction, and let him turn round till the direction of translation 

 (a) points to the right; then I shall assume the direction in 

 which he looks to be that of the unit (7) which represents the 

 translation u^. According to this rule, it is easy to see that /3a 

 is represented by —y; and generally, supposing a, /8, 7 to be 

 any three units of length at right angles to each other, we have 

 the following equations, viz. 



a^ = 7, ffy-ct, yoizsff, \ 



/3a=-7, 7/3= -«. «7=-y5.J * • ^ ^^ 



llic equivalence here implied may be called equivalence of 



symbolical definition ; it simply implies that the symbols equated 



define the same thing, and may therefore be substituted for each 



other in any symbolical equations. 



It has been shown that vv=-[ccy^\\\ 6)aj3 ; whence it follows, 



.Vv uv = {a;y sin 0)y (5.) 



Now xy sin 6 is the area of the parallelogram formed upon u 



* The translation of a unit along a perpendicular unit may properly be 

 called a Unit of Translation. 



