462 



" Referring to the memoir,* we find, adopting the nota- 

 tion so often described, 



p + p = a/I (" + ^'«i + ^«-^ + ^"3)2 + {l> + a(3i + b(ii + c/Sa)^ "1 

 *^ 1 + {c + ayi + by2 + Cysf. J 



Assuming now 



X = ai2 + (5i' + 7i2, n = a^^+ ^^2 + 72^ V = aa^ + /Sg^ + 73^ 

 ^ = 0203+^2/83 + 7273, X=«3«l+^3^1 + 737l' '/' = CHa2+/3lP2+7l72, 



and developing the square root, we obtain : 

 p + p' = 



(-1+ (aiCOs2a + j83COs2/3 + y3COs2y+Mcos/3cos-y + i;cos-ycosa + u)cosa cos/3) -> 

 n < +1 (Xcos^a + ficos'^jS + vcos^y + 20cos;3cosy + 2xcos7COSa + 2>|;cosacos/3) ] (2) 

 «- + (aicos^a + /32COS2/3 + yacos^y + Mcos/3cosy + rcosycosa + 2i;cosacos/3)2 J 



neglecting terms of a higher order than the second. 



" The function arising from this expansion will be (vid. 

 Transactions of the Academy, vol. xxi. p, 153) : 



2F= 2 (Goi + H[5.i + 773) + Du + Ev + Fw ^ 



+ {GX + Hfi + Iv +D(i> + Ex + Fxf,) \ (3) 



+ 20 J 



where <E> denotes function (1), ' 



2 G = lllFop cos^adu, 2H = fi^oio cos^/SfZw 

 2/=£|'Fop COS27CZW 

 i> = Jjl-Fop cosj3 cos7(i(jj -2J = IIJ-^o/o COS7 cosa(^aj 

 i^= JjjFo|0 cosa cos(5d(t>, 



db) being the element of the volume. 



" This function (3) contains twenty-one coefficients, and is 

 quite distinct from the function which may be derived from <J), 

 by introducing arbitrary coefficients. If the terms G, H, I, D, 

 E, F, be retained, the natural state of the body will not be one 

 of free equilibrium, and the equations of a homogeneous body 



• Transactions of the Royal Irish Academy, vol. xxi. p. 155. 



