SURFACES OF THE SECOND DEGREE. 79 



From (4), we find by inspection that the following six quantities 

 are severally equal: 



pa + qft + ry, pa + p'a' + p" a'\ 



p'a' + g^/3' + r'y', qfi + ^/3' + q"fi" (13), 



p"a" + q"(i" + r"7", ry + r'y' + r"y", 



and moreover, that any symmetrical interchanges of accents in the first 

 three, or of letters in the second, give results severally equal to nothing. 

 Such are joa' + g'iS' + ry, p li + p (i' + p" fi" , &c. Let the common value 

 of the first six be T. We have then 



pa -{■ q& + ry = T, 



pa +ql3'+ry'=0 (14), 



pa" + qli" + ry" = 0. 



From which, by obvious multiplications and additions, looking at equa- 

 tions (2), we have 



p +3' cos (^+r cos ri=Ta, 



pcos ^+q +r COS ^=T(i (15), 



p cos t] +q cos^+r = Ty. 



From either of which sets we deduce 



1^ ■¥<f -Vi^-^^qr cos + 2rp cos n-\-^pq cos^= T^ (16), 



and similar relations may be deduced between jo', g', r', and jt>", ^", r' ; 

 T being the same throughout. 



Again, form the several quantities 



flo, /o, &c. or 1 - cos^ f, cos n cos ^- cos f, &c. 



from the second set of equations in (2), and make the results homo- 

 geneous and symmetrical from the first set; for example, write for 

 Oa and /o 



(7« + V«' + 7"«") («/3 + a')3' + a"/3") - \c?^oi^^cl'''\ (fiy+fi'y'+(i"y"), 



