jEoIotrojric Potential. 451 



with 2p 2 < 1 the order is unchanged, but the points of division 

 lie between — <r and — b 2 , — b 2 and — « 2 , — a 2 and -co . 

 With 2p 2 >l the discriminant is negative for large positive 

 values of X. and there is a positive root of J'(X)=0, say X . 

 As X falls from + 00 the order is now hyperholoid of one 

 sheet changing at \ = \ to ellipsoid, then to hyperboloid of 

 one sheet, and hyperboloid of two sheets. When pqr all 

 e\i>t the two latter changes take place between — c 2 and — 6 2 , 

 and between — b 2 and —a' 2 ; since for X = — c 2 , — 6 2 , —a 2 , 

 the values of J' are cV(a 2 -c 2 )(6 2 -c 2 ), __^2( a 2__ 5 2) (P-c"), 

 a' 2 p 2 (<r — b' 2 ) (b' 2 — c 2 ), i. e. + , — , +, in succession. 



The change at the critical value X takes place through a 

 disk form, and the normal to the disk has a direction given 

 by (fry' —aay=(y'a' —Pfi , )m=(*'P' —yy')n. Quoting (85) 

 with —qr for//... A' = 0,... this is l/p=7n/q=njr ; or the 

 disk faces the translation. The product of squares of its 

 semi-axes is (a + /3 + y)IA a or J(a + /3 + y)/A at i- e. 



S[W + Xu{r(l -,/) + b 2 (l-r 2 )}+\ 2 (l-q 2 -r 2 )~] 



or S{« 2 + X (l -p 2 )} {b 2 + X (l - q 2 )} - Xo 2 SpV- 



The sum of squares of semi-axes is X(«/3— y' 2 )/A a or 

 (2A + XA n X/>)/A a , or in the notation for principal axes it is 

 %a 2 -\- X (3 — S/> 2 ). Both forms are confirmed by the case 

 p = q = Q } c- 2 + X (l — r 2 ) = 0, the product then being (a 2 + X ) 

 (Zr + Xo), the sum <rr + X +6 2 + X . 



§ 33. In isotropic attraction the law of inverse distance is 

 exact for the sphere, and the integral \pdr/4:7rr is deduced 

 for other forms. The integral makes the first differential 

 coefficients continuous at the frontier and satisfies certain 

 differential equations for external and internal spaces. Objec- 

 tions (1) that the law is not true exactly for any shape of 

 element, and (2) that the volume of a body cannot be made 

 up wholly of spheres, appear to have no force. There is just 

 - _ lod reason for considering the similar integral in aeolo- 

 tropie potential as a correct solution, the simple form of the 

 law applying to a particular shape of ellipsoid ; and it is 

 proposed to give some consequences of equating the direct 

 integral to the solution found above by the method of differ- 

 ential equation-. 



In (96) we found for the ellipsoid which has the status 

 of the sphere in isotropic work, a potential =kpT /4: , n , (u T )$. 

 In the element of a potential integral dr is written for t , 

 and z in ?/,. (which is SP# 2 +2P'y<2) is replaced by x—x', xyz 

 referring to the element, and x't/z' to the place at which the 

 potential is reckoned. Consider first the potential at the 



