518 Proceedings of the Royal Irish Academy, 



Prom these 



-T>S(a-y3)(a'-/3')(a"-/3") = V{a' - fi'){a" - 13") Sap(r 



+ r{a" - (3") (a - (3) Sa'/3'a- + r{a - /?) (a' - ^') Sa" (3"<t. 



Thus T is given as a linear vector function of cr or t = </)cr suppose^ 

 and the condition ^Srcr = 0, constrains o- to be parallel to an edge of 

 the cone >S'p^p = 0. 



5. "When F and F' are corresponding points on homographically 

 divided planes, 



xa (tCT - a) + yh {'^ - (3) + %c (^ - y) = 0, 



and xa'{zi' - a') + yh' (w' - y8') + zc' (^' - y') = 0, 



lead as in the last article to the equivalent forms 



xa (t + Vaa) + yh{T+ V[3(t) + zc (t + Vycr) = 0, 

 and xa'ij + FaV) + yh'ij + Fy^'o-) + zc'(t + Fy'o-) = 0. 



From the second of these, 



X : y : % 

 = h'c' V{t + V{3'(r) (t + r/o") : c^a' F{t + Vy'a) (t + FaV) 



: «'J'F(t+ Va'(T){T+V/3'o-). 



But F(t + FySV) (t + Vy'a) = - Ft F(^' - y) o" + F- F/3'o- Fy'cr 



= -<TfSr{f3'-y')+ Sft'y'a-)', 



so the set of ratios is equivalent to the simpler set 



X : y : z 



= b'c'{ST{/3'-y') + Sft'y'(r) : c' a' {S r {y' - a') + S y' a a) 



: a'b'{ST{a' + (3')- Sa'fi'a-). 

 Substituting in the first equation, 



ab'c'{T + Va(r){ST{(3'-y') + Sf3'y'a-) + AcV(T + F/3o-)(>S'T(/-a') + Sy'a'cr) 

 + ca'h'{T+ Fyo-)(>S'T(a'-/3') + Sa'f3'cr) = 0. 



Again, if the ratios x : y : z had been found from the first and sub- 

 stituted in the second equation, the result would have been 



a'3c(T+ Va'(T){ST{{3-y) + S/3y(r) + b'ca{T + V/3'(T){ST{y-a) + Syacr) 

 + c'ab{T^ Vy'a-){ST{a-fi) + Safta) = 0. 



