35 



13. The function v", the sign of which distinguishes between the 

 two preceding cases of osculation, has this remarkable property, that 

 it is independent of the direction of the coordinate axes ; in such a 

 manner that if a, /3, y, be, as before, the cosines of the angles which 

 the ray makes with three given rectangular axes, and if we denote by 

 a' /3' y the new values which these cosines acquire when we refer the 

 ray to three new rectangular axes, we shall have 



S«» J/3= U«W ■*■ J/3' h" V/33y/ + V J^ ~" \^J - 



^2i_^2y__ f^liV J. !!!L !!!!. _ /ill ^ ' _i, !!i ^*'" M'*" V t* 

 3*'' S;3'^ u«'5^7 "*" 3/3" w WW' "^ sy v^ ~ vjyj;;/ = (• ) 



V being, in the first member, a homogeneous function of a, jS, y, and, 

 in the second member of a', /3', y, of the first dimension. To demon- 

 strate this theorem, let us observe that by the known formulae for the 

 transformation of coordinates, we may put 



...../■ 



«=*'./?+ ;8'B -}- yC' cJ = »A + ^A' + yd", 



/3 = «'^' 4- /3'S' + y'C, /S' = «S + /3i5' + yS", ). (K") 



y = x'A" -f /3'i?" -f- y'C", y' = «C' + ^C" + yC<l ; 



^, 5, C, ^', B', C, yJ", B", C", being constant quantities of which 

 only three are arbitrary, and which satisfy the following conditions : 



A^ + D^ + C = 1, A^ -\- A'^ + A"^ = 1, 



A" + B'» + C" = 1, jB* + B'* + B"^ = 1. 



^«» 4- B"* -I- C"* =1, C^ + C" + C"' = 1, 

 AA' + BB' 4- Ce = 0, AB-\- AB' + ^"B" = 0, '^ 

 ^'4" + B'B" + C'C" = 0, BC + B'C + B"C'i = 0, 

 y3'M+ B"5 + C"C = 0, C/4 + C'^' + C"A" = 0. 



This being laid down, we have, by {K")^ and by the nature of 

 partial differentials, 



F 2 



