SURFACES OF THE SECOND DEGREE. 85 



In which equations the letter p, q. A, &c. may be accented throughout 

 singly or doubly, striking off three accents from any A which thus 

 obtains three or more. 



By squaring the equations (15), writing V^ for 7", substituting 

 the values just obtained for p^, qr, &c. and then multiplying the same 

 equations together two and two, and making 



Li = *„Co + b^c,, - 2 IX, Z/2 = in,,n, + m^n,, - aj,, - a„k, 



Ni = «„*„ + «o*// - 2 n„n^, Ni = l,,m, + lotn^^ - c^n„ — c„«o, 



we get 



^_ F,-L,-{F,-aK)A + F,A' r,-M,-{F-br,)A+ F.A^ 



**" V,{A-A'){A-A") ' ^~ r,{A-A'){A-A") 



2 F-N,-{r,-cr:)A+V,A' ,„. 

 ^- K{A-A'){A-A") ^^^^' 



„ _ FiCos^ — Lz — jVi cos^—aF'o)A+ FgCosBA^ 

 ^'y~ F{A-A'){A-A") 



_ Fcos tj — Mi- (Fj cos tj -bFp) A + FqCos t/A^ 

 'y"- F,{A-A'){A-A") 



^ F, cos t-'^2-{F, cos ^-c F„) A + F„ cos ^A' 

 "^~ F^{A-A'){A-A") 



in which the letters may be singly or doubly accented as before, and 

 from which the determination of the position of the principal diameters 

 is made to depend directly upon the solution of (34). 



Let the surface whose equation is (3) be referred to another origin 

 and other axes, and let the quantities corresponding . to those already 

 given or deduced, which belong to the new origin or axes, be denoted 

 by the same letters and accents enclosed in brackets [ ]. Thus the 

 angles made by the new axes are [^1 [>;], and [^] ; the coefficients of 



