fB MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 



4. This equation tvas not integrated ; but by a skilful use of its properties, the 

 problem of attractions was greatly simplified by Laplace. He laid down a theorem, 

 respecting the surfaces of all spheroids of small deviation, that their radii vectores 

 might be developed into series, every term of which would satisfy the above equa- 

 tion ; and he also gave a method of expansion. By means of this theorem, the pro- 

 blem could be solved for spheroidal bodies which differ but little from spheres ; but 

 its generality has been greatly restricted by the researches of subsequent writers*, by 

 whom it has been shown that it is true only for bodies whose radii are expressible in 

 rational and integral functions of (jJ , '/(I — (Jt/^) cos <p', -/(l — jO.'^) sin 9'. Among these 

 are the ellipsoid and elliptical spheroid, and a large class of other spheroids. In these 

 papers I have adopted a different proceeding ; I integrate the equation itself generally, 

 and determine the arbitrary functions contained in the integrals by the circumstances 

 of the problem itself. In consequence of the peculiar form which P^ then takes, V 

 may be found by effecting the operations indicated, which are only explicit integra- 

 tions. 



5. I shall now proceed to integrate this equation. 



Consider /// and <p as functions of two new variables X and Y, to be determined 

 from the equations, 



dY = ^{d^ + J^df.), 



where k and k' are the roots of the equation (1 — /a^^ ^2 _|_ ^ — q These roots 



1 — /* 



are — — —3-, whence we obtain 



1 - jW,2 ' 



1+^ o^^ V _ ^ „ 1 / T 1.^ 1 ± ^ 



X = ^ + iN/-llog^,andY = ^-iy-llogf^. . . (4.) 



— \dff.) dX^'^ ^ d(L dif.dXdY^\dfi.) rfY2 + 



dfu^ — \dfx./ dX^ ^ ^ dfu dii. dXdY ^ XdiiJ dY^ "^ rf^tt^ dX "^ dfu^ dY' 





- 1 ^P 2 ^P„ 1 rf^P 2^i/-l /rfP„ dP\ 



- — (1 _ f^^f dX^ "^ (1 - f^T ^X^Y ~" (1 - /*2)2 ^Y2 + (1 _ fx2)2 \dX "" 7Y/ ' 



dV^_dXdv dYdv^ _ y-::! (dv^ dv\ 



dix, — d[x, dX'^ dfx, dY — 1 -f*2 \dX dY )' 



d^V,_(dXYd^V dXdY d^V, (dYyd^V d^XdV d^Y dV^ 



df ~ \d<p/ dX^ 'T' "^ d<p d(p dXdY"^ \d<^) dY^ "T" df dX "*■ df dY ' 



<72 p /72 p ^2 p 



— f/X2"T-^rfX^Y"T" dY'' 

 Substituting these in (3.), we obtain 



4 d^V 



* See two articles by Mr. Ivory in the Philosophical Transactions, 1812. 



