PROFESSOR CAYLEY ON POLYZOMAL CURVES. 107 



Annex III. — On the Norm of (b — c) ,JK° + (c — a) ^3° +(« — &) */C , when the Centres 



are in a Line. 



The norm of JTJ+ s/V + JW is 



= (1,1,1,-1, -1,-1)(U, V,W)\ 



whence that of Ju+U' + J'V+V + JW+W is 



= (1,1,1,-1,-1, -i) (£7, v,wy 

 + (i,i,i, -1,-1, -i)(w, v, wy 



+ 2(1,1,1,-1, -1,-1) (U, V,W){U\ V',W), 



where the last term is = 2 into 



U'( U-V-W) 

 + V'(-U+ V-W) 

 + W'(-U- V+W). 



And the norm of Ju+U'+U" + JV+V+ V" + */W+ W'+ W is obviously 

 composed in a similar manner. 



Now, applying the formula to obtain the norm of 



(P ~ c) *Ja?+6+u + (c - a) Jtf + 4 + P + («-*) J # + 6 + ^ , 



the expression contains six terms, two of which are at once seen to vanish ; and 

 writing for shortness ( „ ) in place of (1, 1, 1, — 1, — 1, — 1) the remaining terms 

 are 



(„)((&- c ) 2 a , (c-a) 2 /3, (a-iy 7 y 

 + 2 („)((& -c) 2 « , (c-«) 2 /3, (a-by 7 )((b-c) 2 a 2 , {c-a) 2 b 2 , (a-Z>) 2 c 2 ) 

 + 20(„)((&-c) 2 a , (c-a) 2 /3, {ct-b) 2 7 ){{b-c) 2 , (c - a) 2 ,(a-Z>) 2 ) 

 + 20(„)((&-c) 2 a 2 , (c-a) 2 b 2 , (a - b) 2 e 2 ) {(b - c) 2 , (c-a) 2 ,( a -b) 2 ); 



the first of these terms requires no reduction ; the second, omitting the factor 2, 

 is 



{b-c) 2 a [ {b-c) 2 a 2 - (c-a) 2 b 2 - (a-b) 2 c 2 ] 

 + (c-a,yi3[-(b-cya 2 + {c-ayb 2 - (a-b) 2 c 2 ] 

 + (a-b) 2 7 \_-(b-c) 2 a 2 - (c-a) 2 b 2 + (a-by-c 2 ] ; 



which is 



= 2{a — b)(b— c) (c — a) [&c (b — c) a + ca (c — a) /3 + ab (a — b)y] . 



Similarly the third term, omitting the factor 20, is 



(6 - cfa [ (& - cf - (c - a) 2 - (a - Z>) 2 ] 

 + (c - a) 2 /3 [-(6 - c) 2 + (c - a) 2 - (a - &) 2 ] 

 + •(« r- &) f 7 [-(& - c) 2 - (c - «) 2 + (a - &) 2 ] , 



