JoLY — nomographic Divisions of Planes, Spheres, and Space. 521 



invariants of the function (^ of Art. 6. Eemembering that cj>{p-a') 

 = — (p-a), &c., the invariants are 



S{p-a'){p-fi'){p-y') S{p-a'){p-fi'){p-y') 



_ ts(p-a')^(p-fi'mp-y) ^^S(p--'){p-(^)(p-y) 



"^^ S{p- a') {p-(S'){p- Y) S{p- a') {p - 13') (p - y') 



and ^j^ 



_ S^{p-a')^{p-^')4>(p -y') ^7^-^(P-")(P-^)(p-y) 



S{p-a'){p-(3'){p-Y) S{p-a'){p-l3'){p-y') ' 



where g^ - viiff'^ + nizg - m^ = is the cubic determining the roots 

 of <t>. 



10. On reduction this cubic takes the form 



aiclS pV((3y + ya + al3) - Safty'] 

 - g [Sp^a'bcV{(3y + ya! + o!^) - % a'lc8o!^y'\ 

 + / {_Bp%aVG' r(/3V + y'a + a^') - ^al'o'Sa(i'y'^ 

 -g^a'b'c'[SpV((3'y' + Ya' + a'l3') - Sa'^'y'^ = 0. 



This, being linear in p, may be regarded as the equation of a plane 

 involving a variable parameter g rationally in the third degree. 



Now, if two roots of the cubic of a linear vector function are 

 equal, two of the axes of the function will in general coincide.^ So 

 then, if the discriminant of the cubic here written is equated to 

 zero, it will determine by its vanishing those points p through which 

 pass two coincident lines of the congruency. In other words, the 

 developable enveloped by the variable plane is the focal surface of 

 the congruency. 



11. Again, regarding s and t as constant in the equation of a 

 line of the congruency given in Art. 3, it is obvious that the plane 

 determined by varying x, y, and % is homographically divided with the 

 two given planes. Eliminating x, y, and %, the result 



S las{p - ct) + a't{p -o.')Jhs{p -j3)+ l't{p - /5')][cs(p -y) + c't{p -y')] = 0, 



involves the ratio s : ^ in the third degree, and agrees with the 



^ Transactions, E.I. A., vol. sxx., p. 600. 



