CHAPTER XV. 



THE PLANE REPRESENTATION OF FREEDOM OF THE THIRD ORDER*. 



198. A Fundamental Consideration. 



Let x, y, z denote the Cartesian co-ordinates of a point in the body 

 referred to axes fixed in space. When the body moves into an adjacent 

 position these co-ordinates become, respectively, x + 8x, y + 8y, z + 82, and 

 we have, by a well-known consequence of the rigidity of the body, 



Bx = a + gz hy, 

 8y = b + hx fz, 

 8z = c+fy- gx, 



where a, b, c, /, g, h may be regarded as expressing the six generalized 

 co-ordinates of the twist which the body has received. 



If the body has only three degrees of freedom, its position must be 

 capable of specification by three independent co-ordinates, which we shall 

 call 6 lt 6.,, 6. A . The six quantities, a, b, c, f, g, h, must each be a function 

 of #!, 2 , 3 , so that when the latter are given the former are determined. 

 As all the movements are infinitely small, it is evident that these equations 

 must in general be linear, and of the type 



a = A 1 1 + A,0, + A 3 0. J , 



in Avhich A lt A 2 , A 3 are constants depending on the character of the 

 constraints. . We should similarly have 



b = B 1 1 + B& + B 3 6 3 , 

 and so on for all the others. 



It is a well-known theorem that the new position of the body defined 

 by 0i, 0-2, 3 may be obtained by a twist about a screw of which the axis 

 is defined by the equations 



a + gz hy _ b + hx fz _ c +fy gx 

 ~7~ 9 & 



* Trans. Roy. Irish Acad., Vol. xxix. p. 247 (1888). 



