v / 



ftp uy -" of the new axe*, \\ln.-li an- limited by tl.-- -.\ inde- 



Thew angles, therefore, 

 be ao ohoecn t ,iml conditions shall be 



fulfilled ; hem*, 80 t | </'. and //' hlmll 



vuni.sli Then tin- new equaii"!. \\illbe 



I >*+&?+ t"j +:/,'/ + 2 Af> + 2.V'i + /r-0. (2) 



Now a second transformation may be made to a parallel 

 syntem of axea tlmm^li a new ; <, A:,/), by equat 



s f.jiution 



//y 4- (Tj -* h -' .V M .v -f 2 N"* -h A"' - 0, (8) 



in u MM functions of tl,,. rn, 



nates A, Ar, and y ; and these coordinates may be chosen so 

 I. , .!T' f and JV" \\ill vanish, giving fr the simplified ! 

 of the f<}uatum of the given quadric. 



A'j* + ffy'+ C'^-H/T-O. . 



ll may hapjM-n. li\\r\cr, th.il tin- rhuica given above for 

 tin* direction angles a p 44, , of the new axes is siu-h that 



tre trim of second degree, as C", \\ill 

 alsovanlsli ; thru iMjuation (I) woifld reduce to 



1 ^^y+/f' = 0, . . . 



and the surface is a r\ lind.- Again, if also 



JIT', IF 1 are not iml.-jM-ndrnt, and the values of A, 4r, j as 

 n above are t lien-fore indi-terminate, th.-u h, (. j may 

 be chosen so that, for example, L\ IT', and K' shall vanish ; 

 and the equation of tli<> <}uadri<- Incomes 



1 a + jy^+i>.V"j-0. ... (6) 



If the coefficient* of two quadratic term* rmnUh, M IT and C, a 

 of origin am, then of direction of axes, may be ehoeen eo that the 

 will reduce to the form (0). 



. OKOM. 94 



