lO MeadOWCROFT, Treatment of (Uodesic Torsion. 



On squaring and adding it is obvious by inspection 

 that a large number of terms vanish or coalesce, and we 

 easily find 



-Tj = COS' 6 sin w( — 1 + sin" sin' w( — j + cos'w( 



\ds / \ds / \ds J 



■ •■ 11 4 , sin*w . 2 .> .. ., ... , 



+ sin D cos'w + — ^ + sin w cos'w cot o + cos Q cos w 

 sin'W 



+ 2 sin w cot y — ' + 2 siiroj cos'w 



(tS 



fdi 



( — - + sin M cot ) = I + «' (say). 



.■.-,= I -f //■ (vi). 



Substituting in (iv) we find 



-^- = — or <r = (I +//-)/- (vii.) 



p (T as I as 



This method possesses, I think, some advantages in 

 point of directness over the method of moving axes more 

 usually followed, besides avoiding the use of somewhat 

 confusing figures. It should be remarked that for a 

 sphere the equation (iv') reduces to the form 



, d duv 



This merely expresses that the radius of spherical curva- 

 ture is unity or in other words that the curve lies on a 

 sphere of unit radius. 



V. Two Theorems on Skew Surfaces. 

 The two following theorems do not appear to have 

 been previously enunciated although they are simple 

 generalisations of known theorems and can be deduced in 

 a number of ways. 



