364 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
though perhaps a better selection might have been made; there is a slight objection 
to the existence of the relation a 2 = 2b 2 , as in the icy-section it brings a cusp of the 
evolute on the ellipse.) We have therefore 
a = 10, /3 = — 35, 7 =25; 
the ellipses in the principal planes of the centro-surface are 
y 2 £ 2 
(5) 2 + (8-937) 2 = lj 
(2-582) 2 + (3-535) 2 = l * 
x 2 y 2 ' 
(4*950) 2 + (2) 2 ” 
and these determine on each axis the two points which are the cusps of the evolutes. 
We have moreover for the umbilicar centre # = 2*988, y = 0, 0= 1*380, and for the 
outcrop x = 1127, y = 1*947, z = 0. 
88. For the delineation of the nodal curve (crunodal portion) we have first to find 
the values of £, £j; these are given in terms of x, y ante No. 33 [p. 334], where y is 
a given function of x, and x extends between the values {— b 2 and — t (a 2 + b 2 + c 2 )} — 25 
and — 26f. It was thought sufficient to divide the interval into 6 equal parts, that is, 
the values of x were taken to be —25, —25*3, ...—26*6. The values of £, £ being 
found, those of y, y x were obtained from them by means of the original equations 
(a 2 + £) 3 (a? + y) = (a 2 + £ : ) (a 2 -f y) &c. viz. we have thus for the determination of y, y 1 
three simple equations, affording a verification of each other. 
For the performance of these calculations (viz. of the values of y, £, £ x , V> Vi) I 
am indebted to the kindness of Mr J. W. L. Glaisher, of Trinity College. The results 
being obtained it is then easy to calculate as well the coordinates (x, y, z) of the 
point on the nodal curve as also the coordinates (X, Y, Z) and (X 1} F 1} Z x ) of the 
corresponding two points on the ellipsoid (these last are of course not required for 
the delineation of the nodal curve, but it was interesting to obtain them). The 
whole series of the results is given in the annexed Table, and from them the drawing 
was constructed. 
I find also in the neighbourhood of the umbilicar centre (if £= — 25 + q), 
8x = *02868 q\ 
8y=± *02484 q 2 , 
8z = -02191 q\ 
and in the neighbourhood of the outcrop (if £ x = — 38*333 + - 7 tj°- ot), 
8x= 1*127 ot, 
8y = — 1*704 -sx, 
8z = + 4"582 •ST'.