58 , Pj^of. Gayley, On a system of two tetrads [Feb. 13. 



and we thence find without difficulty for the axis d^^d^ji^^ the 

 equation 



+ 2/{(^-a)(7S-60 + (S-7)(6^-«/3) + (^-6)(a^-7S)} 



+ ^| _ (ge-70- (^a-e^)- (^7-a8)} = 0, 



and the equation of the axis d^^i^^i^^ is obtained herefrom by the 

 interchange of e and ^. 



10. The points d^^ and d^ will lie upon the first mentioned 

 axis if only d^^ lies upon this axis, viz. if we have 



{ a^(Se-70+ ^^{^oi-e^)+ e^{^^-ah)] 



i^ + E-a-Z) 

 + K/3-a)(7S-6^)+(S-7)(6^-«/3) + (r-e)(ayS-7S)} 



(^E-aZ) 

 + { - (Se-7r)- (^«-6yS)- (/37-«S)} 



(- aZ(^ + E) + /3E(a + Z)) = 0. 



Reducing this equation, the factor a — ^ divides out, and we 

 finally obtain 



(7-a)(^S + ^^)(6-^) 

 + {^-h){ar^+eE){^-Z) 

 + (aS - ^7) (e^ - EZ) 

 + (a^-ry8) (eZ-^E) = 0; 

 say this is 



A + BE+CZ + DEZ=0, 

 where 



B = -(ry-a)m . -(a(3-ry8)^+ (/3-S)e^, 



G=-{^-8)a'y + (a^-y8)€ . + (7 - «) e^, 



i) = -(aS-^7)-(/3-a)6-(7-a)^ 

 viz. this is the condition for the existence of the axis d^„d,„d,^d.^d,„. 



2a Id 12 43 42 



We interchange herein e, ^ and also E, Z, and we thus obtain 

 A' + C'E + B'Z + D'EZ = 0, 



where 



A'= . (/8-S)a7e + (7-a)/3S^ + (aS-y87)6^, 



5'=:-(7-a)/SS-(a/S-7S)6 . + (/3-S)e^, 



0' = _(^-8)a7 . +(a/3-7S)^+ (7-a)er, 



viz. this is the condition for the existence of the axis d^ji,J,J,J,„. 



23 13 12 43 42 



