154 THE ROYAL SOCIETY OF CANADA 



Covariants. It follows, therefore, that any property of 5 expressible 

 entirely in terms of the Invariants and Covariants of (1) is a purely 

 projedive differential property and inversely.* 



The system (1) can be reduced to the canonical J or m 



. . yuu-\- 2byy -\-fy = 0, 

 (6) 



yyy + 2a' y u -\-gy = 0, 



by a transformation of the type (4). When equations (1) are written 

 in this form, the integrability conditions assume the simpler form 



a'uu + g« + 2ba'v + 4a'è„ =0, 



(7) b^ +/. + 2a'bu + 4:ba'u = 0, 



guu -Jvv - Wu - 2a% + 4gè, + 2bg, = 0. 



The most general transformation of the type (4) that leaves the 

 form of the canonical system unchanged is the following: 



(8) u = a{u), v = ^{v), y = c Jau (3v y, 



where a and ^ are functions of u alone and v alone respectively and c 

 is an arbitrary constant. f 

 The functions 



y, z = yu , p = yv , (T = yuv 



are relatively invariant under transformations of the dependent 

 variable that leave the canonical form unchanged; they are called 

 the fundamental semi-covariants of the canonical system. J If we 

 substitute for 3;^^) in these semi-covariants the four independent 

 solutions 3'^^^ 3'(2\ 3;^^^ y^'^^ of the canonical system, we obtain a 

 set of four values for each of the four functions y, z, p, a; these we 

 shall interpret as the homogeneous coordinates of the four points 

 Py, Pz, Pp, Pa. These four semi-covariants therefore determine 

 a moving tetrahedron which is non-degenerate, since the determinant 

 I 3;zpo- 1 is different from zero. In like manner the function 



xii) = xi y'^^ + Xi z^^ + xz p« + Xi (t(^) (i = 1, 2, 3, 4) 



will determine a point Px ; and the unit point of the system of co- 

 ordinates determined by the tetrahedron PyPzPpPa can be so 

 chosen that {xi, X2, Xs, Xi) will be the coordinates of Px when referred 

 to the moving system. 



*For further information concerning the calculation of Invariants and Covariants 

 and the completeness of the system of Invariants, the reader is referred to Wilczyn- 

 ski's memoirs and a paper by G. M. Green, Trans. Am. Math. Soc (Vol. 16, No. 1). 



tMi, p. 256. 



tMi, p. 247. 



