228 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON 



tangent at any point to the fixed curve on the primitive 

 solid, then we have 



, dx dy n dz 



cos <f> = -7- cos a + -f cos 8+ -y cos 7, 

 ^ ds ds ds ' 



also 



ds 2, = dx 7, + dy 2, + dz 7, ; 



hence the right side of (5) may be written in the form 

 cte z (2 + cos 2 </>). Since the curve is fixed on the solid, if 

 we suppose the solid to move in any way, we shall have 

 r<v/2 + cos 2 <£ . ds a constant quantity; if this be denoted 

 by C, then we shall have throughout the motion 



§^dsl + ds z y + dsl=C. 



The lengths of the curves on the derived solids will vary 

 as the primitive solid changes its position, but will vary in 

 such a manner as to satisfy the above equation. 



By reference to the expression for ds z , it will be seen 

 that it does not vanish when cos 7 vanishes ; under this 

 last condition the dimensions of Y z perpendicular to the 

 plane of x y y become indefinitely small ; in other words, 

 the solid degenerates into a plane area, simultaneously the 

 projected curve degenerates into a curve of single curva- 

 ture lying on this plane area. 



The equation to the primitive solid being given in the 

 form 



z=(j>{x,y), 



then the superficial area of this solid will be given by the 

 equation 



-tfV'+@)"+(3)**> 



