and of bicircular quartics 129 



x^ y^ z" 



+ A-.+ ^^ = h 



a + X b + X c + X 



where a > b > c, using A for ellipsoids, ju. for hyperboloids of one 

 sheet, V for hyperboloids of two sheets. Suppose that the straight 

 portion of the curve touches the confocals for which X = 'p,X=^ q, 

 and denote by dw the Cayley separation of two consecutive points 

 of the curve taken in regard to the confocal of parameter A = ^. 

 Putting 



F {x) = ^{x + a) {x ■i-b){x + c) {x - p) {x - q), 



L^=f{X), 3P=f{iJi), N^=f{v), 



the curve is such that 



(A -p)dX ^ r (/x - p) d^i ^ ( {v - p) dv ^ ^^ 



L ' J M ' ] N 



(A -q)dX [ ifx - q) dix ( {v - q) dv _ 



L +j — li ' J iT^-^' 



while, with S^ = F (6), 



[{X -p){X- q) dX ^ f (/z - ?)) ( ^ - q ) dfx , ^{v -p ){v- q) dv 



J {X-d)L } {yi-6)M ' j {v-e)N 



e 



where tv = jdw. By supposing 6 to increase indefinitely, and re- 

 placing 6^dw by ds, we have the corresponding result when 

 Euclidian distance is used. 



In the notation of hyperelliptic functions (see Multiply -periodic 

 functions, Cambridge, 1907, pp. 35, 36, using the p, q as a^, ag ^re 

 there used), we have 



p {x — p) {x — q) dx 



2{d-p){d-q)]^^ x-e y 



y , e,x,s, x,Xo,y /,,^. =o\ ,, x,xo, 1 1 „ J (w ' + A;) 



where ((f)) denotes the place conjugate to (6), Ic is such that '^ {h) 

 vanishes identically, and when d is large the significant terms of 



the functions l,-^, l,^ ^^^ d~^pq and d~^ {p + q). 



Along a straight portion of the curve, joining, suppose, the 



points (Aq, /Xq, Vq), (A, IX, v), the places (A), (/i), {v) of the hyperelliptic 



construct are coresidual with the places (Aq), {ixq), [vq), and we can 



satisfy the identities 



F {x) - [0 {x)f = 4 (X - A) {X - ix){x- v) ix ~f,) (x -/a), 

 F (x) - [ipo {x)f =4:{x- Ao) {x - (Xq) {x - Vq) (x -/J (x -f^), 



VOL. XX. PART I. 9 



