64 Transactions of the Royal Canadian Institute 



From equations (A) and (S) , we find at once 



(1) xn = (^e,-^),x.^, = -(^e,^^^, 



_ /ddi ddz\ ( dHs \ _ ( de^ ddz \ 



""''-Vdi dp J '^K^'dpdgJ ~ \dq dp J' 

 by virtue of (A), 

 where 



_ dx _ dx _ d' X 



dp dq dpdq 



The corresponding formulae for y and z can be obtained from these by 

 introducing the proper pair of the three solutions 61, 62, 6^. 



If the fundamental magnitudes of the first and second orders of the 

 surface (S) be denoted by E, F, G and L, M, N respectively, we find 

 immediately from the definitions of these quantities and from equations 



(1) the following relations: 



(2) i^-(^.^)>=-<^.f)(.^).C; = .(.f)' 



L = 0,W=((..^)(.,^))i^^)),Ar=o, 



where 



V^ = E G- F\ 



In order to reduce these expressions to their simplest forms, we make use 

 of two formulae from the theory of determinants, viz. : 



where B', B", etc. are the minors of b' , b" , etc. in the determinant 



D 



t? c 

 ' b' c 

 " b" 



tt 



