224 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON 



projected would give a spheroid of which the semiaxes are 

 R, R cos 7, R cos 7, whereas in (3) an ellipsoid has been 

 obtained. It will be observed, however, that there is 

 something arbitrary in the mode of projection. The sole 

 condition to which the projected solid is subjected is 



§Adz = cos 2 yY ; 



but this condition is not sufficient to determine the equa- 

 tion to the projected solid. If L be the length of the 

 primitive axis, then if we draw two parallel planes distant 

 L cos 7, these planes being parallel to the plane of x, y, 

 any solids terminated by these planes and having equal 

 sections made by any plane parallel to them, will fulfil the 

 above condition. Hence we have some choice as to the 

 manner of laying the successive slices of the projected 

 solid on one another; we have just found an ellipsoid as 

 a particular case, but if the elliptical sections parallel to 

 the plane of x } y had been piled on one another, so that 

 all the centres lay in the same vertical line, we should 

 obtain a spheroid of which the semiaxes are R, R cos 7, 

 R cos 7. 



The locus of the centres in (3) may be got as follows. 

 Let a section be made by a plane parallel to plane of x, y 

 and distant f from that plane ; write also X + X t for x, and 

 Y + Yj for y. Let X, and Y, be so determined that the 

 coefficients of X and Y vanish. This leads to the con- 

 dition 



(X,— a) (» 4 + »*/*) +n 7 -lm(J 1 -b)-Z i nl =0, 

 (Y, — *)(»*+ »**»*) +n*lm(X l -a)-t;nm=o. 



Hence, X„ Y„ Z r being the coordinates of the centre of 

 the section, we have 



n(Y l — b) = £m, 



n(K l — a) = £l, 



